Loading ...
Sorry, an error occurred while loading the content.
 

Re: Levi-Civita's "Proof" that there is a non-tidal 3rd rank tensor gravity field does not exist.

Expand Messages
  • JACK SARFATTI
    ... That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published. ... Actually you are wrong
    Message 1 of 26 , Mar 1, 2009

      On Feb 28, 2009, at 8:51 PM, Paul Zielinski wrote:

      JACK SARFATTI wrote:


      On Feb 28, 2009, at 2:49 PM, Paul Zielinski wrote:
      "...it is not so easy to free oneself from the idea that coordinates must have a direct metric significance."

      - A. Einstein, "Autobiographical Notes"

      Back in San Francisco

      re: 




      Paul, Polemics are the last refuge of the man who cannot make a rational relevant refutation. ;-)
      You've already been definitively refuted Jack. Now I'm just having some fun.:-)

      That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published. 

      I have given you a detailed argument that you have not refuted in a rational relevant way.

      But your argument is based on your own confusion about local coordinates in relation to curved spacetime geometry.



      Curving the 4D spacetime of 1916 GR deforms the *geometry* of the manifold (metric relations between spacetime
      points), NOT the coordinate assignments.

      Actually you are wrong about that. The choice of static local non-inertial frames changes in the SSS case. The covariant acceleration of the static frame is GM/r^2 so when you write

      {LC(M')}^rtt - {LC(M)}^rtt = G(M' - M)/r^2

      this is a conceptual relationship between two physically different frames of reference that actually do not simultaneously exist in a real physical situation. Your thesis is not even wrong simply garbled fragmentary ungrounded thinking in my opinion. I use physical detectors as local frame of reference.

      The meaning of the classical general relativity principle is the computation of local invariants based upon local measurements of two locally coincident detectors in arbitrary relative instantaneous motion measuring the same process up to quantum uncertainty limits. 

      You can leave the local coordinate charts (MAPS from R^4 to the points
      of the spacetime manifold) undisturbed (FIXED, CONSTANT, UNCHANGING, INVARIANT) while you deform
      the manifold in any local spacetime region of interest.

      In other words, you can arrange things such that under the gravitational deformation of the metric around any given spacetime
      point x you end up with the same local chart around x as you started with. The change in the intrinsic geometry doesn't force
      ANY change in the local coordinate assignments around x.

      The very concept of "gravitational deformation" is purely theoretical and cannot actually be measured unless e.g. the Earth loses chunks of matter in a collision with a large asteroid for example. It is not a valid measure of the actual local gravitational field i.e. tetrad 1-form e^I (Rovelli's definition)


      That is something you still don't seem to understand. Hence the Einstein quote. You have not been able to fully separate the
      concept of coordinate intervals from the concept of metric relations between points. I think this is the root of your problem,
      and is the reason why you are having so much trouble understanding Levi-Civita.
      Synopsis, since there are interesting foundational issues here ignored even in some texts, though not in Wheeler's.

      1) Every time you write a particular metric field guv(x), and its associated symmetric torsion-free Levi-Civita connection field for parallel transport of tensors along world lines in 4D spacetime you have chosen a representation in terms of a network of local detectors (AKA local frames of reference).
      Wrong. 

      Irrational response.

      All GCT coordinate representations of a given metric g_uv(x) are representations of a *single* metric field.

      Red Herring. No one says otherwise. 


      You seem to be saying that there is a different "metric field" for every GCT that changes the components of the metric tensor.

      I have said nothing of the kind. You simply do not get the point I am making because your mind is closed on this silly idea you have. It's completely silly - embarrassing in fact to my mind at least.

      The metric fields guv & g'uv induced by distinct sources Tuv & T'uv are not related by a GCT. They are different intrinsic gravity field configurations and they have different sets of static observers (outside event horizons) and, therefore, their formal difference has no direct physical meaning, i.e. no measurement procedures exist.  They cannot simultaneously exist physically.

      If that is what you are saying, then you are missing the whole point of the metric being a GCT tensor.

      Of course that is NOT what I have been saying.
      You are also missing the
      point of defining affine connections on curved manifolds that correct the partial derivatives of tensors for coordinate artifacts.

      Changing the coordinate representation of the metric doesn't affect the intrinsic spacetime geometry.

      You are seriously confused about this. Your Einsteinian optics are preventing you from understanding the actual math.
      The simplest "hydrogen atom" (as it were) example in Einstein's 1915 GR is the SSS vacuum solution, written in the texts as

      g00 = gtt = - 1/g11 = - 1/grr = 1 - rs/r

      rs/r > 1

      rs = 2GM(source)/c^2

      Area of concentric sphere is A = 4pir^2

      usual spherical polar & azimuthal coordinate line element added.

      You can use any local coordinate chart around any given (well-behaved) point of interest, and hold that local chart invariant
      under a gravitational deformation of the manifold.

      You are wrong. Physically you need to specify a set (fiber bundle) of local tangent frames that in principle are implementable by actual detectors. Here the choice of static frames (where they exist) shows my point very clearly. This completely eludes you. It is the fundamental error you have made. You are thinking like a mathematician and completely miss the physics. Different source configurations have physically different bundles of e.g. static local frames that are the basis for the contingent representation chosen for pragmatic convenience only. In any case I stop reading here and end the discussion. Think what you like but

      1) No journal will accept your idea and rightly so.

      2) No one pays any attention to it (except me here) and rightly so.

      That's my opinion.




      You are systematically confusing actual geometric changes in the first partial derivatives g_ij, k with coordinate change.

      The primary purpose of constructing an affine connection is to correct the derivatives of the metric for coordinate artifacts.

      So the premise here is that a distinction must be made between coordinate contributions and actual geometric contributions
      to the partial derivatives of tensors. This also applies to the metric tensor g_uv!

      Now Paul you look at the above as a purely mathematical problem.
      Nonsense. You cannot invalidate a mathematical theorem by attaching a physical interpretation to the math. To suggest that you
      can is completely wrong headed IMO.

      You have to distinguish mathematically between actual geometric changes and mere coordinate changes in the partial derivatives
      of tensors, including those of the metric tensor.

      That is what the affine deformation model I've been trying to explain to you does so nicely  -- for the metric tensor g_uv and its
      associated metric-compatible LC connection.
      That is a profound error of conceptualization that you as The Profundus Philofawzer of Physics Extraordinaire ;-) ought to be well aware of - the "informal language" (David Bohm) the interpretation of the naked mathematique.
      Look Jack, you just got the math wrong, that's all. Do you want to repair the situation, or not?

      As far as I'm concerned both the the mathematical issues *and* the issues of physical interpretation are now completely settled,
      based on

      (1) Einstein's 1917 interpretation of the LC connection field LC^i_jk(x) as representing the non-tidal field strength in GR;

      (2) Levi-Civita's authoritative proof that the difference between two LC connections (his 2 and 2') respectively compatible
      with any two *geometrically* distinct metrics (his 1 and 1') on a single analytical manifold is a non-trivial 3-index GCT tensor
      rho^i_jk.

      At this point any quibbling on your side about how different coordinate representations of the Riemann metric are supposed to constitute
      distinct "metric fields" (which is mathematical nonsense, since g_uv is a tensor object) is just whistling past the graveyard as far as
      I'm concerned.
      The above representation is for static (i.e. constant r, theta, phi) detectors. When M =/= 0 , they are covariantly accelerating in order to stay still in curved spacetime. This is Alice in Wonderland. You are not in Kansas anymore. You are not in Flatland. This is a violation of the Victorian Age engineer's common sense.
      It's also a violation of the mathematics. Levi-Civita's proof shows that the frame acceleration field, the strength of which is represented by
      a non-tensor coordinate correction term X^i_jk, is only accidentally "glued on" to the actual gravitational field, represented to first order
      by the gravitational deformation tensor G^i_jk.

      That is the whole point of the linear LC decomposition

      LC^i_jk = G^i_jk + X^i_jk

      where the gravitational deformation tensor G^i_jk is a particular instance of Levi-Civita's rho^i_jk.

      Your insistence that different coordinate representations of g_uv constitute distinct metric fields is just a fallacy. It
      ignores the tensor character of g_uv.
      Newton's First Law of Motion in Einsteinian 1916 classical geometrodynamical language generalizes to

      All covariantly non-accelerating massive test particles follow  timelike geodesics of the gravitational field.
      Splitting out the gravitational deformation tensor G^i_jk doesn't change any of that Jack. So why do you keep bringing
      this up?
      Newton's Second Law of Motion generalizes to

      D^2x^u/ds^2 = F^u/m

      where m is the invariant rest mass of the test particle (assumed here not to be ejecting mass like a rocket) and ds is the proper time of m along the worldline. F^u is the non-gravity applied force pushing the test particle off-geodesic inducing g-force on the test particle.
      All fully covariant and consistent with tensor G^i_jk. So what's your point?
      More generally, Newton's 2nd Law in curved spacetime for massive test particle motion is

      (D/ds)(mdx^u/ds) = (Dm/ds)dx^u/ds + mD^2x^u/ds^2 = F^u(non-gravity)

      Newton's 3rd law is not valid for global total momenta because translation symmetry is broken - see Noether's theorem.

      Instead we have the local stress-energy current density conservation law

      D^vTuv(non-gravity source fields curving spacetime) = 0

      2) g-force and the Levi-Civita connection field {LC}

      Back to the simplest case

      D^2x^u/ds^2 = F^u/m

      We now do a Godel self-reference and let the test particle detect itself.

      Note, the connection field (LC Christoffel symbol )usually has nothing to do with the test particle, but is a property of the detector, except in the degenerate self-referential case.
      I'm sorry Jack but IMO this is complete and utter nonsense.

      The geodesics are covariantly defined in GR and have nothing to do with "detectors" or "observers" -- except as to mere
      *appearances*. The LC connection field includes linearly independent contributions from the coordinates *and from the
      intrinsic geometry*. Otherwise, how could it be a metric compatible connnection?

      Of course a straight SR free-fall trajectory for example *looks* curved to a uniformly accelerating observer *who ignores
      his own accelerating motion*! So what? That doesn't affect the actual geometry of the geodesic in the slightest. And neither
      does curved geometry force any changes in local coordinate charts. You are completely and hopelessly confused here.
      These are two completely different animals.

      I'm not going to read any further.

      Z.




      The Baron Munchausen is weightless, no inertial g-force, in free-fall parabolic path (timelike geodesic in curved spacetime)
      He does feel g-force initially when cannon fires

      In particular think of the above SSS case in which everything can be concretely computed. Z never bothers to compute an example. This is why his thesis here is not even wrong in my opinion.

      i = 1,2,3 = spacelike components

      i = 0 is the timelike component

      Let's look only at the i = 1 radial component in the static LNIF 

      D^2r/ds^2 = d^2r/ds^2 + {LC}^ruv(dx^u/ds)(dx^v/ds)

      However, in the self-referential "diagonal" case

      dx^i/ds = 0

      dx^o/ds = 1 (c = 1)

      d^2r/ds^2 = 0

      Therefore, the g-force per unit rest mass measured on the test particle itself is

      {LC}^r00 ~ +GM(source)/r^2

      This, however, is not an objective non-tidal non-zero tensor field. It is purely contingent based on an arbitrary, though pragmatically contingent, choice since we here on Earth closely approximate this situation.

      3) Z's Red Herring of the Gravity Deformation

      Start from M = 0 (flat space time) obviously 

      {LC}^r00(M = 0) = 0 

      here the static local non-inertial observer limits to the global inertial observer - a point overlooked by Z because he does not calculate a simple example.

      Now increase to &M

      {LC}^r00(&M) = G&M/r^2

      This is not a tensor obviously.

      The difference is not a tensor, i.e.,

      {LC}^r00(0) - LC^r00(&M) = G&M/r^2

      No tensors here!

      Note the natural assumption here is the adiabatic transformation of the global static inertial frame GIF to the local static non-inertial frame LNIF. The contingent choice of convenience is "static".

      Every time I hear a philofawzer say it's "pure math" or "your argument is mathematical nonsense" I reach for my delete button. ;-)

      "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
      • Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."
      • "Imagination is more important than knowledge."
        • "Reality is merely an illusion, albeit a very persistent one."
        • "The only real valuable thing is intuition."
          • "Anyone who has never made a mistake has never tried anything new."
          • "Great spirits have often encountered violent opposition from weak minds."
          • "God does not care about our mathematical difficulties. He integrates empirically."
          • "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
          • "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
          • http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html

      Jack Sarfatti wrote:
      What's fundamentally wrong in z' s argument is that he does not understand the breakdown of the global frame assumption of special relativity. You simply cannot use purely formal arguments. You must specify what detectors you mean. Physics is about detector responses.

      Sent from my iPhone

      On Feb 28, 2009, at 8:17 AM, Jack Sarfatti <sarfatti@...> wrote:

      Start with flat spacetime. The lc connection field is zero for the global inertial observer.
      Now introduce a small SSS  Tuv matter source which induces an non zero lc field for the static non-inertial observer that adiabatically emerges if you impose a fixed r constraint.
      Since the difference is a tensor
      That difference must be zero otherwise you have a contradiction.

      Sent from my iPhone

      On Feb 28, 2009, at 1:15 AM, Paul Zielinski <iksnileiz@...> wrote:

      Nonsense.

      The common non-tensor coordinate part just cancels out of the difference between the two LC connections,
      which are themselves NOT GCT tensors due to the common coordinate parts that cancel.

      You are just confused.

      Z.

      Jack Sarfatti wrote:
      Problem with the deformation approach is that it does not work. Start with flat spacetime and it implies lc is a tensor.

      Back in Sf 

      Sent from my iPhone

      On Feb 27, 2009, at 1:31 PM, Jack Sarfatti <sarfatti@...> wrote:

      So you are claiming that eq 2 is within a single network of observers e.g static Lnifs in the SSS case. We add a perturbation to tuv source inducing a metric perturbation. The take difference in lc connections and that is the 3rd rank tensor?
      Why did you not say that clearly to begin with?
      In any case it is not conceptually important to the foundations of relativity.

      Only the tetrad and spin connection cartan 1-forms are intrinsic nontidal measures of the gravity field ie scalar zero rank tensors under gct aka local t4. About to take off from jfk.

      Sent from my iPhone

      On Feb 27, 2009, at 4:00 PM, Paul Zielinski <iksnileiz@...> wrote:

      You showed in detail that if you replace Levi-Civita's two different metrics 1 and 1' with the *same*
      metric represented in two different coordinate charts, you get a very different quantity when you take
      the difference of the corresponding compatible connections that obviously is not a tensor..

      Which is just another way of saying that you don't understand -- or don't want to understand -- Levi-
      Civita's proof.

      I told you what Levi-Civita's rho^i_jk means mathematically: it represents the effect of a deformation
      of the geometry of the manifold (from metric 1 to metric 1') on the Levi-Civita connection, free of
      any non-tensor contribution from the coordinates.

      In the context of GR, since in Einstein's model the net observed non-tidal gravitational field strength is
      represented by the LC connection, clearly the tensor rho^i_jk represents the effect on the non-tidal field
      strength of the change in the 4D spacetime geometry when you go from 1 to 1' -- free of any contribution
      from observer frame acceleration.

      If you take metric 1 to be geometrically Minkowski, then the difference G^i_jk between the two LC
      connections obviously represents the effect on the gravity-free LC connection field of the change from
      Minkowski geometry (n_uv) to gravitationally deformed geometry (g_uv).

      Which means that the tensor field G^i_jk(x) represents the *actual* (i.e., objective, frame-independent)
      non-tidal (first-order) gravitational field strength in GR, free of frame acceleration artifacts ("fictitious
      forces").

      Z.

      Jack Sarfatti wrote:
      There is no physics at all in that snippet eq 2.

      Sent from my iPhone at jfk boarding for Sfo 

      On Feb 26, 2009, at 5:04 PM, Paul Zielinski <iksnileiz@...> wrote:

      Actually, we are talking about Levi-Civita's idea here. He's the one with two different metrics on a single
      analytical manifold, with a common local coordinate chart around each point, under general transformations
      of which his rho^i_jk (the difference between two metric-compatible connections) transforms as a 3-index
      tensor.


      I showed in detail it is not a tensor if the usual physics is added to the naked math. It is a contingent relation between two networks of pairs of locally coincident detectors.






    • JACK SARFATTI
      Paul none of this stuff had been published. What has been published has no relationship to your interpretation of what it means. Your basic methodology is
      Message 2 of 26 , Mar 1, 2009
        Paul none of this stuff had been published. What has been published has no relationship to your interpretation of what it means. Your basic methodology is flawed, which is why I say your thesis is not even wrong. You do not understand the subtle role of local frames. When you deform the gravity field by changing the source matter field configuration Tuv(matter) ---> T'uv(matter'), you also adiabatically change the local reference frames! You cannot consistently keep those frames fixed for a given kind of constraint e.g. the static observer constraint. You can imagine it mathematically, but it is not possible physically, therefore, you cannot claim, as you do claim, that your purely formal "deformation" change is the consistent basis for an actual operational definition of an intrinsic objective real locally observable non-tidal gravity field that is a GCT 3rd rank tensor. 

        As an example: deform M to M' in the SSS case for static locally accelerating detectors each at fixed r, theta, phi

        g = GM/r^2

        g' = GM'/r^2

        r is same in both cases, but g =/= g' and these g's are the actual accelerations of the local static frames in each case. They are different frames!
        You cannot subtract them and claim you have a tensor in the same frame! It is physical nonsense operationally even though you can make an abstract formal process inside your demented mind! ;-) It does not exist in the laboratory, which is what physics is all about. 

        On Mar 1, 2009, at 1:16 PM, Paul Zielinski wrote:

        JACK SARFATTI wrote:

        On Feb 28, 2009, at 8:51 PM, Paul Zielinski wrote:

        JACK SARFATTI wrote:


        On Feb 28, 2009, at 2:49 PM, Paul Zielinski wrote:
        "...it is not so easy to free oneself from the idea that coordinates must have a direct metric significance."

        - A. Einstein, "Autobiographical Notes"

        Back in San Francisco

        re: 




        Paul, Polemics are the last refuge of the man who cannot make a rational relevant refutation. ;-)
        You've already been definitively refuted Jack. Now I'm just having some fun.:-)

        That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published.
        Most of this stuff has already been published.

        If you think you can go up against Levi-Civita on the math, then I say it is you who is delusioal.

        I have given you a detailed argument that you have not refuted in a rational relevant way.

        But your argument is based on your own confusion about local coordinates in relation to curved spacetime geometry.



        Curving the 4D spacetime of 1916 GR deforms the *geometry* of the manifold (metric relations between spacetime
        points), NOT the coordinate assignments.

        Actually you are wrong about that.
        Really?
        The choice of static local non-inertial frames changes in the SSS case.
        Because the geometry changes. Of course the change of geometry affects the geodesics. What is an "LIF"
        in one geometry will not be an "LIF" in another. However, this has nothing essentially to do with coordinate
        transformations. The changes in the geodesics are geometrical.

        So you are still seriously confused. If you want to understand this, you should think about coordinate systems,
        not observer frames.
        The covariant acceleration of the static frame is GM/r^2 so when you write

        {LC(M')}^rtt - {LC(M)}^rtt = G(M' - M)/r^2

        this is a conceptual relationship between two physically different frames of reference that actually do not simultaneously exist in a real physical situation.
        The relationship we are really talking about here is between two sets of geodesics covariantly determined by different
        metrics according to the textbook standard geodesic equation.
        Your thesis is not even wrong simply garbled fragmentary ungrounded thinking in my opinion.
        Jack, it's too late for this. You just got the math wrong, that's all.

        The change from n_ij to g_ij does not force *any* changes in local coordinate charts around any given point of
        interest on the manifold. The geodesics will of course change with the geometry, but coordinate transformations
        are simply not relevant here. It is the geometry that changes, not the coordinates.

        Your belief to the contrary is what is truly delusional here. You cannot invalidate mathematical theorems with physical
        interpretations.
        I use physical detectors as local frame of reference.

        The meaning of the classical general relativity principle is the computation of local invariants based upon local measurements of two locally coincident detectors in arbitrary relative instantaneous motion measuring the same process up to quantum uncertainty limits.
        We've already deconstructed this weasel term "local". In this context it means, among other things, that you can't
        talk about the frame-invariant gravitational acceleration of test object geodesics with respect to world lines of sources
        -- which is what I call Einstein's "make-believe physics". This is like putting blinkers on a cart horse.

        The fact is that once you take account of the gravitational acceleration of test objects with respect to sources, Einstein's
        equivalence model simply evaporates. You cannot produce gravitational acceleration by accelerating an observer's frame
        of reference -- not in Newton's theory, and not in GR either. To suppose that you can is a form of insanity IMHO.
        You can leave the local coordinate charts (MAPS from R^4 to the points
        of the spacetime manifold) undisturbed (FIXED, CONSTANT, UNCHANGING, INVARIANT) while you deform
        the manifold in any local spacetime region of interest.

        In other words, you can arrange things such that under the gravitational deformation of the metric around any given spacetime
        point x you end up with the same local chart around x as you started with. The change in the intrinsic geometry doesn't force
        ANY change in the local coordinate assignments around x.

        The very concept of "gravitational deformation" is purely theoretical
        What do you think Einstein's field equations

        G_uv = kT_uv

        are all about Jack? They tell us how matter deforms the geometry of spacetime! This *is* Einstein's theory of gravity!
        and cannot actually be measured unless e.g. the Earth loses chunks of matter in a collision with a large asteroid for example. It is not a valid measure of the actual local gravitational field i.e. tetrad 1-form e^I (Rovelli's definition)
        You have to be kidding, Given a single SSS source, all you have to do is measure g, the acceleration of test particle geodesics in
        relation to the corresponding flat-spacetime geodesics, that is due to gravity. This is all determined in GR by the geodesic equation:

        <moz-screenshot-43.jpg>

        This is a generally covariant equation.

        Are you still trying to say that in GR there is no way of comparing the curved space geodesic trajectories with the corredponding flat
        space trajectories? And that there is no way of observing such changes empirically?

        That would make Einstein's theory completely useless as a theory of gravity. But of course this is nonsense.

        That is something you still don't seem to understand. Hence the Einstein quote. You have not been able to fully separate the
        concept of coordinate intervals from the concept of metric relations between points. I think this is the root of your problem,
        and is the reason why you are having so much trouble understanding Levi-Civita.
        Synopsis, since there are interesting foundational issues here ignored even in some texts, though not in Wheeler's.

        1) Every time you write a particular metric field guv(x), and its associated symmetric torsion-free Levi-Civita connection field for parallel transport of tensors along world lines in 4D spacetime you have chosen a representation in terms of a network of local detectors (AKA local frames of reference).
        Wrong. 

        Irrational response.

        All GCT coordinate representations of a given metric g_uv(x) are representations of a *single* metric field.

        Red Herring. No one says otherwise.
        You just said otherwise. What did you mean then by "a particular metric field guv(x)"?

        You seem to be saying that there is a different "metric field" for every GCT that changes the components of the metric tensor.

        I have said nothing of the kind.
        It sounded like you were saying this.
        You simply do not get the point I am making because your mind is closed on this silly idea you have. It's completely silly - embarrassing in fact to my mind at least.
        That changes in the geometry do not force changes in the local coordinate charts in GR? You think that's a silly idea?

        It is a mathematical fact with the consequence that the Einstein field, represented by LC^i_jk(x), is decomposable into an
        actual gravitational field represented by a tensor G^i_jk, and a non-tensor X^i_jk which accounts for all non-tensor coordinate
        contributions to the LC connection field:

        LC^i_jk = G^i_jk + X^i_jk

        Which means that in 1916 GR, frame acceleration fields are every bit as fictitious in relation to actual gravitational fields as
        they are in Newton's theory in relation to actual Newtonian forces.

        And you think this is a "silly idea"?
        The metric fields guv & g'uv induced by distinct sources Tuv & T'uv are not related by a GCT. They are different intrinsic gravity field configurations and they have different sets of static observers (outside event horizons) and, therefore, their formal difference has no direct physical meaning, i.e. no measurement procedures exist.  They cannot simultaneously exist physically.

        If that is what you are saying, then you are missing the whole point of the metric being a GCT tensor.

        Of course that is NOT what I have been saying.
        OK. Good.

        Then you should have no trouble understanding why the difference between an L:C connection compatible with a curved metric and
        the LC connection compatible with a flat-level Minkowski metric is a tensor.
        You are also missing the
        point of defining affine connections on curved manifolds that correct the partial derivatives of tensors for coordinate artifacts.

        Changing the coordinate representation of the metric doesn't affect the intrinsic spacetime geometry.

        You are seriously confused about this. Your Einsteinian optics are preventing you from understanding the actual math.
        The simplest "hydrogen atom" (as it were) example in Einstein's 1915 GR is the SSS vacuum solution, written in the texts as

        g00 = gtt = - 1/g11 = - 1/grr = 1 - rs/r

        rs/r > 1

        rs = 2GM(source)/c^2

        Area of concentric sphere is A = 4pir^2

        usual spherical polar & azimuthal coordinate line element added.

        You can use any local coordinate chart around any given (well-behaved) point of interest, and hold that local chart invariant
        under a gravitational deformation of the manifold.

        You are wrong.
        That is your delusion Jack.
        Physically you need to specify a set (fiber bundle) of local tangent frames that in principle are implementable by actual detectors.
        In Einstein's coordinate frame model, which is a relic of 1905 SR, yes.
        Here the choice of static frames (where they exist) shows my point very clearly.
        This is where you get really confused. Of course changes in the spacetime geometry will change the classification of trajectories,
        since the geodesics chang -- but this happens regardless of whether you hold the local coordinate charts invariant under the
        deformation of the geometry, or not.

        Of course you can say that the *rest* frames of freely falling test objects will change, and that this is associated with a change in
        the coordinate charts associated with the rest frames, but this is immaterial and irrelevant to the point I'm making. It is
        a trivial observation. There is nothing that compels us to observe a freely falling test object from its own rest frame, regardless
        of changes in the geometry.

        You might as well observe that the rest frame of any object changes when you accelerate the object. So what?
        This completely eludes you.
        What eludes me is why you think this matters. It's a tautology. Trivial.
        It is the fundamental error you have made. You are thinking like a mathematician and completely miss the physics.
        If what you mean here by "physics" is the completely trivial observation that the *rest* frames of freely falling test objects change when
        the geodesics change, then I think I am quite correct in ignoring it. It is of no consequence. It is immaterial to my argument.
        It is trivial and irrelevant. It adds nothing of any substance to the fact that when the spacetime geometry changes, the geodesics change.

        This is why I think it's better to think about mathematical coordinates instead of observer frames -- to avoid these kinds of pitfalls. I think
        this is a good example of the kind of confusion that Einstein's coordinate frame model can generate in the mind of a physicist.
        Different source configurations have physically different bundles of e.g. static local frames that are the basis for the contingent representation chosen for pragmatic convenience only.
        As far as I'm concerned, this is just another way of saying that the geodesics change wen the spacetime geometry changes.
        You are simply substituting pseudo-relativistic talk about "LIFs" and "LNIFs" for objective talk about covariantly determined
        gravitationally deformed geodesics. Your favorite card trick!
        In any case I stop reading here and end the discussion.
        I think it's time to move on. Let's talk about gauge theory.
        Think what you like but

        1) No journal will accept your idea and rightly so.
        Most of this has *already* been published Jack. Some of it -- like Levi-Civita's theorem -- many years ago.

        Poltorak has already published his stuff, working with a metric-affine model.

        A. A. Logunov has published reams on this. Like V. Fock, he says: "No such thing as general relativity".

        No one serious in this field actually believes in classic Einstein equivalence anymore Jack. Just look at the most
        recent text books on GR.

        2) No one pays any attention to it (except me here) and rightly so.

        That's my opinion.
        And you are perfectly entitled to your opinions. You are free to make all your own mistakes.

        Z.




        You are systematically confusing actual geometric changes in the first partial derivatives g_ij, k with coordinate change.

        The primary purpose of constructing an affine connection is to correct the derivatives of the metric for coordinate artifacts.

        So the premise here is that a distinction must be made between coordinate contributions and actual geometric contributions
        to the partial derivatives of tensors. This also applies to the metric tensor g_uv!

        Now Paul you look at the above as a purely mathematical problem.
        Nonsense. You cannot invalidate a mathematical theorem by attaching a physical interpretation to the math. To suggest that you
        can is completely wrong headed IMO.

        You have to distinguish mathematically between actual geometric changes and mere coordinate changes in the partial derivatives
        of tensors, including those of the metric tensor.

        That is what the affine deformation model I've been trying to explain to you does so nicely  -- for the metric tensor g_uv and its
        associated metric-compatible LC connection.
        That is a profound error of conceptualization that you as The Profundus Philofawzer of Physics Extraordinaire ;-) ought to be well aware of - the "informal language" (David Bohm) the interpretation of the naked mathematique.
        Look Jack, you just got the math wrong, that's all. Do you want to repair the situation, or not?

        As far as I'm concerned both the the mathematical issues *and* the issues of physical interpretation are now completely settled,
        based on

        (1) Einstein's 1917 interpretation of the LC connection field LC^i_jk(x) as representing the non-tidal field strength in GR;

        (2) Levi-Civita's authoritative proof that the difference between two LC connections (his 2 and 2') respectively compatible
        with any two *geometrically* distinct metrics (his 1 and 1') on a single analytical manifold is a non-trivial 3-index GCT tensor
        rho^i_jk.

        At this point any quibbling on your side about how different coordinate representations of the Riemann metric are supposed to constitute
        distinct "metric fields" (which is mathematical nonsense, since g_uv is a tensor object) is just whistling past the graveyard as far as
        I'm concerned.
        The above representation is for static (i.e. constant r, theta, phi) detectors. When M =/= 0 , they are covariantly accelerating in order to stay still in curved spacetime. This is Alice in Wonderland. You are not in Kansas anymore. You are not in Flatland. This is a violation of the Victorian Age engineer's common sense.
        It's also a violation of the mathematics. Levi-Civita's proof shows that the frame acceleration field, the strength of which is represented by
        a non-tensor coordinate correction term X^i_jk, is only accidentally "glued on" to the actual gravitational field, represented to first order
        by the gravitational deformation tensor G^i_jk.

        That is the whole point of the linear LC decomposition

        LC^i_jk = G^i_jk + X^i_jk

        where the gravitational deformation tensor G^i_jk is a particular instance of Levi-Civita's rho^i_jk.

        Your insistence that different coordinate representations of g_uv constitute distinct metric fields is just a fallacy. It
        ignores the tensor character of g_uv.
        Newton's First Law of Motion in Einsteinian 1916 classical geometrodynamical language generalizes to

        All covariantly non-accelerating massive test particles follow  timelike geodesics of the gravitational field.
        Splitting out the gravitational deformation tensor G^i_jk doesn't change any of that Jack. So why do you keep bringing
        this up?
        Newton's Second Law of Motion generalizes to

        D^2x^u/ds^2 = F^u/m

        where m is the invariant rest mass of the test particle (assumed here not to be ejecting mass like a rocket) and ds is the proper time of m along the worldline. F^u is the non-gravity applied force pushing the test particle off-geodesic inducing g-force on the test particle.
        All fully covariant and consistent with tensor G^i_jk. So what's your point?
        More generally, Newton's 2nd Law in curved spacetime for massive test particle motion is

        (D/ds)(mdx^u/ds) = (Dm/ds)dx^u/ds + mD^2x^u/ds^2 = F^u(non-gravity)

        Newton's 3rd law is not valid for global total momenta because translation symmetry is broken - see Noether's theorem.

        Instead we have the local stress-energy current density conservation law

        D^vTuv(non-gravity source fields curving spacetime) = 0

        2) g-force and the Levi-Civita connection field {LC}

        Back to the simplest case

        D^2x^u/ds^2 = F^u/m

        We now do a Godel self-reference and let the test particle detect itself.

        Note, the connection field (LC Christoffel symbol )usually has nothing to do with the test particle, but is a property of the detector, except in the degenerate self-referential case.
        I'm sorry Jack but IMO this is complete and utter nonsense.

        The geodesics are covariantly defined in GR and have nothing to do with "detectors" or "observers" -- except as to mere
        *appearances*. The LC connection field includes linearly independent contributions from the coordinates *and from the
        intrinsic geometry*. Otherwise, how could it be a metric compatible connnection?

        Of course a straight SR free-fall trajectory for example *looks* curved to a uniformly accelerating observer *who ignores
        his own accelerating motion*! So what? That doesn't affect the actual geometry of the geodesic in the slightest. And neither
        does curved geometry force any changes in local coordinate charts. You are completely and hopelessly confused here.
        These are two completely different animals.

        I'm not going to read any further.

        Z.




        The Baron Munchausen is weightless, no inertial g-force, in free-fall parabolic path (timelike geodesic in curved spacetime)
        He does feel g-force initially when cannon fires

        In particular think of the above SSS case in which everything can be concretely computed. Z never bothers to compute an example. This is why his thesis here is not even wrong in my opinion.

        i = 1,2,3 = spacelike components

        i = 0 is the timelike component

        Let's look only at the i = 1 radial component in the static LNIF 

        D^2r/ds^2 = d^2r/ds^2 + {LC}^ruv(dx^u/ds)(dx^v/ds)

        However, in the self-referential "diagonal" case

        dx^i/ds = 0

        dx^o/ds = 1 (c = 1)

        d^2r/ds^2 = 0

        Therefore, the g-force per unit rest mass measured on the test particle itself is

        {LC}^r00 ~ +GM(source)/r^2

        This, however, is not an objective non-tidal non-zero tensor field. It is purely contingent based on an arbitrary, though pragmatically contingent, choice since we here on Earth closely approximate this situation.

        3) Z's Red Herring of the Gravity Deformation

        Start from M = 0 (flat space time) obviously 

        {LC}^r00(M = 0) = 0 

        here the static local non-inertial observer limits to the global inertial observer - a point overlooked by Z because he does not calculate a simple example.

        Now increase to &M

        {LC}^r00(&M) = G&M/r^2

        This is not a tensor obviously.

        The difference is not a tensor, i.e.,

        {LC}^r00(0) - LC^r00(&M) = G&M/r^2

        No tensors here!

        Note the natural assumption here is the adiabatic transformation of the global static inertial frame GIF to the local static non-inertial frame LNIF. The contingent choice of convenience is "static".

        Every time I hear a philofawzer say it's "pure math" or "your argument is mathematical nonsense" I reach for my delete button. ;-)

        "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
        • Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."
        • "Imagination is more important than knowledge."
          • "Reality is merely an illusion, albeit a very persistent one."
          • "The only real valuable thing is intuition."
            • "Anyone who has never made a mistake has never tried anything new."
            • "Great spirits have often encountered violent opposition from weak minds."
            • "God does not care about our mathematical difficulties. He integrates empirically."
            • "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
            • "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
            • http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html

        Jack Sarfatti wrote:
        What's fundamentally wrong in z' s argument is that he does not understand the breakdown of the global frame assumption of special relativity. You simply cannot use purely formal arguments. You must specify what detectors you mean. Physics is about detector responses.

        Sent from my iPhone

        On Feb 28, 2009, at 8:17 AM, Jack Sarfatti <sarfatti@...> wrote:

        Start with flat spacetime. The lc connection field is zero for the global inertial observer.
        Now introduce a small SSS  Tuv matter source which induces an non zero lc field for the static non-inertial observer that adiabatically emerges if you impose a fixed r constraint.
        Since the difference is a tensor
        That difference must be zero otherwise you have a contradiction.

        Sent from my iPhone

        On Feb 28, 2009, at 1:15 AM, Paul Zielinski <iksnileiz@...> wrote:

        Nonsense.

        The common non-tensor coordinate part just cancels out of the difference between the two LC connections,
        which are themselves NOT GCT tensors due to the common coordinate parts that cancel.

        You are just confused.

        Z.

        Jack Sarfatti wrote:
        Problem with the deformation approach is that it does not work. Start with flat spacetime and it implies lc is a tensor.

        Back in Sf 

        Sent from my iPhone

        On Feb 27, 2009, at 1:31 PM, Jack Sarfatti <sarfatti@...> wrote:

        So you are claiming that eq 2 is within a single network of observers e.g static Lnifs in the SSS case. We add a perturbation to tuv source inducing a metric perturbation. The take difference in lc connections and that is the 3rd rank tensor?
        Why did you not say that clearly to begin with?
        In any case it is not conceptually important to the foundations of relativity.

        Only the tetrad and spin connection cartan 1-forms are intrinsic nontidal measures of the gravity field ie scalar zero rank tensors under gct aka local t4. About to take off from jfk.

        Sent from my iPhone

        On Feb 27, 2009, at 4:00 PM, Paul Zielinski <iksnileiz@...> wrote:

        You showed in detail that if you replace Levi-Civita's two different metrics 1 and 1' with the *same*
        metric represented in two different coordinate charts, you get a very different quantity when you take
        the difference of the corresponding compatible connections that obviously is not a tensor..

        Which is just another way of saying that you don't understand -- or don't want to understand -- Levi-
        Civita's proof.

        I told you what Levi-Civita's rho^i_jk means mathematically: it represents the effect of a deformation
        of the geometry of the manifold (from metric 1 to metric 1') on the Levi-Civita connection, free of
        any non-tensor contribution from the coordinates.

        In the context of GR, since in Einstein's model the net observed non-tidal gravitational field strength is
        represented by the LC connection, clearly the tensor rho^i_jk represents the effect on the non-tidal field
        strength of the change in the 4D spacetime geometry when you go from 1 to 1' -- free of any contribution
        from observer frame acceleration.

        If you take metric 1 to be geometrically Minkowski, then the difference G^i_jk between the two LC
        connections obviously represents the effect on the gravity-free LC connection field of the change from
        Minkowski geometry (n_uv) to gravitationally deformed geometry (g_uv).

        Which means that the tensor field G^i_jk(x) represents the *actual* (i.e., objective, frame-independent)
        non-tidal (first-order) gravitational field strength in GR, free of frame acceleration artifacts ("fictitious
        forces").

        Z.

        Jack Sarfatti wrote:
        There is no physics at all in that snippet eq 2.

        Sent from my iPhone at jfk boarding for Sfo 

        On Feb 26, 2009, at 5:04 PM, Paul Zielinski <iksnileiz@...> wrote:

        Actually, we are talking about Levi-Civita's idea here. He's the one with two different metrics on a single
        analytical manifold, with a common local coordinate chart around each point, under general transformations
        of which his rho^i_jk (the difference between two metric-compatible connections) transforms as a 3-index
        tensor.


        I showed in detail it is not a tensor if the usual physics is added to the naked math. It is a contingent relation between two networks of pairs of locally coincident detectors.








      • JACK SARFATTI
        PS Hawking s picture Fig 1.11 lower right should make my point obvious. Imagine changing the mass of Earth with a density, but keeping the hovering rockets at
        Message 3 of 26 , Mar 1, 2009
          PS Hawking's picture Fig 1.11 lower right should make my point obvious. Imagine changing the mass of Earth with a density, but keeping the hovering rockets at same distance from center of Earth.

          On Mar 1, 2009, at 2:03 PM, JACK SARFATTI wrote:

          Paul none of this stuff had been published. What has been published has no relationship to your interpretation of what it means. Your basic methodology is flawed, which is why I say your thesis is not even wrong. You do not understand the subtle role of local frames. When you deform the gravity field by changing the source matter field configuration Tuv(matter) ---> T'uv(matter'), you also adiabatically change the local reference frames! You cannot consistently keep those frames fixed for a given kind of constraint e.g. the static observer constraint. You can imagine it mathematically, but it is not possible physically, therefore, you cannot claim, as you do claim, that your purely formal "deformation" change is the consistent basis for an actual operational definition of an intrinsic objective real locally observable non-tidal gravity field that is a GCT 3rd rank tensor. 

          As an example: deform M to M' in the SSS case for static locally accelerating detectors each at fixed r, theta, phi

          g = GM/r^2

          g' = GM'/r^2

          r is same in both cases, but g =/= g' and these g's are the actual accelerations of the local static frames in each case. They are different frames!
          You cannot subtract them and claim you have a tensor in the same frame! It is physical nonsense operationally even though you can make an abstract formal process inside your demented mind! ;-) It does not exist in the laboratory, which is what physics is all about. 

          On Mar 1, 2009, at 1:16 PM, Paul Zielinski wrote:

          JACK SARFATTI wrote:

          On Feb 28, 2009, at 8:51 PM, Paul Zielinski wrote:

          JACK SARFATTI wrote:


          On Feb 28, 2009, at 2:49 PM, Paul Zielinski wrote:
          "...it is not so easy to free oneself from the idea that coordinates must have a direct metric significance."

          - A. Einstein, "Autobiographical Notes"

          Back in San Francisco

          re: 




          Paul, Polemics are the last refuge of the man who cannot make a rational relevant refutation. ;-)
          You've already been definitively refuted Jack. Now I'm just having some fun.:-)

          That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published.
          Most of this stuff has already been published.

          If you think you can go up against Levi-Civita on the math, then I say it is you who is delusioal.

          I have given you a detailed argument that you have not refuted in a rational relevant way.

          But your argument is based on your own confusion about local coordinates in relation to curved spacetime geometry.



          Curving the 4D spacetime of 1916 GR deforms the *geometry* of the manifold (metric relations between spacetime
          points), NOT the coordinate assignments.

          Actually you are wrong about that.
          Really?
          The choice of static local non-inertial frames changes in the SSS case.
          Because the geometry changes. Of course the change of geometry affects the geodesics. What is an "LIF"
          in one geometry will not be an "LIF" in another. However, this has nothing essentially to do with coordinate
          transformations. The changes in the geodesics are geometrical.

          So you are still seriously confused. If you want to understand this, you should think about coordinate systems,
          not observer frames.
          The covariant acceleration of the static frame is GM/r^2 so when you write

          {LC(M')}^rtt - {LC(M)}^rtt = G(M' - M)/r^2

          this is a conceptual relationship between two physically different frames of reference that actually do not simultaneously exist in a real physical situation.
          The relationship we are really talking about here is between two sets of geodesics covariantly determined by different
          metrics according to the textbook standard geodesic equation.
          Your thesis is not even wrong simply garbled fragmentary ungrounded thinking in my opinion.
          Jack, it's too late for this. You just got the math wrong, that's all.

          The change from n_ij to g_ij does not force *any* changes in local coordinate charts around any given point of
          interest on the manifold. The geodesics will of course change with the geometry, but coordinate transformations
          are simply not relevant here. It is the geometry that changes, not the coordinates.

          Your belief to the contrary is what is truly delusional here. You cannot invalidate mathematical theorems with physical
          interpretations.
          I use physical detectors as local frame of reference.

          The meaning of the classical general relativity principle is the computation of local invariants based upon local measurements of two locally coincident detectors in arbitrary relative instantaneous motion measuring the same process up to quantum uncertainty limits.
          We've already deconstructed this weasel term "local". In this context it means, among other things, that you can't
          talk about the frame-invariant gravitational acceleration of test object geodesics with respect to world lines of sources
          -- which is what I call Einstein's "make-believe physics". This is like putting blinkers on a cart horse.

          The fact is that once you take account of the gravitational acceleration of test objects with respect to sources, Einstein's
          equivalence model simply evaporates. You cannot produce gravitational acceleration by accelerating an observer's frame
          of reference -- not in Newton's theory, and not in GR either. To suppose that you can is a form of insanity IMHO.
          You can leave the local coordinate charts (MAPS from R^4 to the points
          of the spacetime manifold) undisturbed (FIXED, CONSTANT, UNCHANGING, INVARIANT) while you deform
          the manifold in any local spacetime region of interest.

          In other words, you can arrange things such that under the gravitational deformation of the metric around any given spacetime
          point x you end up with the same local chart around x as you started with. The change in the intrinsic geometry doesn't force
          ANY change in the local coordinate assignments around x.

          The very concept of "gravitational deformation" is purely theoretical
          What do you think Einstein's field equations

          G_uv = kT_uv

          are all about Jack? They tell us how matter deforms the geometry of spacetime! This *is* Einstein's theory of gravity!
          and cannot actually be measured unless e.g. the Earth loses chunks of matter in a collision with a large asteroid for example. It is not a valid measure of the actual local gravitational field i.e. tetrad 1-form e^I (Rovelli's definition)
          You have to be kidding, Given a single SSS source, all you have to do is measure g, the acceleration of test particle geodesics in
          relation to the corresponding flat-spacetime geodesics, that is due to gravity. This is all determined in GR by the geodesic equation:

          <moz-screenshot-43.jpg>

          This is a generally covariant equation.

          Are you still trying to say that in GR there is no way of comparing the curved space geodesic trajectories with the corredponding flat
          space trajectories? And that there is no way of observing such changes empirically?

          That would make Einstein's theory completely useless as a theory of gravity. But of course this is nonsense.

          That is something you still don't seem to understand. Hence the Einstein quote. You have not been able to fully separate the
          concept of coordinate intervals from the concept of metric relations between points. I think this is the root of your problem,
          and is the reason why you are having so much trouble understanding Levi-Civita.
          Synopsis, since there are interesting foundational issues here ignored even in some texts, though not in Wheeler's.

          1) Every time you write a particular metric field guv(x), and its associated symmetric torsion-free Levi-Civita connection field for parallel transport of tensors along world lines in 4D spacetime you have chosen a representation in terms of a network of local detectors (AKA local frames of reference).
          Wrong. 

          Irrational response.

          All GCT coordinate representations of a given metric g_uv(x) are representations of a *single* metric field.

          Red Herring. No one says otherwise.
          You just said otherwise. What did you mean then by "a particular metric field guv(x)"?

          You seem to be saying that there is a different "metric field" for every GCT that changes the components of the metric tensor.

          I have said nothing of the kind.
          It sounded like you were saying this.
          You simply do not get the point I am making because your mind is closed on this silly idea you have. It's completely silly - embarrassing in fact to my mind at least.
          That changes in the geometry do not force changes in the local coordinate charts in GR? You think that's a silly idea?

          It is a mathematical fact with the consequence that the Einstein field, represented by LC^i_jk(x), is decomposable into an
          actual gravitational field represented by a tensor G^i_jk, and a non-tensor X^i_jk which accounts for all non-tensor coordinate
          contributions to the LC connection field:

          LC^i_jk = G^i_jk + X^i_jk

          Which means that in 1916 GR, frame acceleration fields are every bit as fictitious in relation to actual gravitational fields as
          they are in Newton's theory in relation to actual Newtonian forces.

          And you think this is a "silly idea"?
          The metric fields guv & g'uv induced by distinct sources Tuv & T'uv are not related by a GCT. They are different intrinsic gravity field configurations and they have different sets of static observers (outside event horizons) and, therefore, their formal difference has no direct physical meaning, i.e. no measurement procedures exist.  They cannot simultaneously exist physically.

          If that is what you are saying, then you are missing the whole point of the metric being a GCT tensor.

          Of course that is NOT what I have been saying.
          OK. Good.

          Then you should have no trouble understanding why the difference between an L:C connection compatible with a curved metric and
          the LC connection compatible with a flat-level Minkowski metric is a tensor.
          You are also missing the
          point of defining affine connections on curved manifolds that correct the partial derivatives of tensors for coordinate artifacts.

          Changing the coordinate representation of the metric doesn't affect the intrinsic spacetime geometry.

          You are seriously confused about this. Your Einsteinian optics are preventing you from understanding the actual math.
          The simplest "hydrogen atom" (as it were) example in Einstein's 1915 GR is the SSS vacuum solution, written in the texts as

          g00 = gtt = - 1/g11 = - 1/grr = 1 - rs/r

          rs/r > 1

          rs = 2GM(source)/c^2

          Area of concentric sphere is A = 4pir^2

          usual spherical polar & azimuthal coordinate line element added.

          You can use any local coordinate chart around any given (well-behaved) point of interest, and hold that local chart invariant
          under a gravitational deformation of the manifold.

          You are wrong.
          That is your delusion Jack.
          Physically you need to specify a set (fiber bundle) of local tangent frames that in principle are implementable by actual detectors.
          In Einstein's coordinate frame model, which is a relic of 1905 SR, yes.
          Here the choice of static frames (where they exist) shows my point very clearly.
          This is where you get really confused. Of course changes in the spacetime geometry will change the classification of trajectories,
          since the geodesics chang -- but this happens regardless of whether you hold the local coordinate charts invariant under the
          deformation of the geometry, or not.

          Of course you can say that the *rest* frames of freely falling test objects will change, and that this is associated with a change in
          the coordinate charts associated with the rest frames, but this is immaterial and irrelevant to the point I'm making. It is
          a trivial observation. There is nothing that compels us to observe a freely falling test object from its own rest frame, regardless
          of changes in the geometry.

          You might as well observe that the rest frame of any object changes when you accelerate the object. So what?
          This completely eludes you.
          What eludes me is why you think this matters. It's a tautology. Trivial.
          It is the fundamental error you have made. You are thinking like a mathematician and completely miss the physics.
          If what you mean here by "physics" is the completely trivial observation that the *rest* frames of freely falling test objects change when
          the geodesics change, then I think I am quite correct in ignoring it. It is of no consequence. It is immaterial to my argument.
          It is trivial and irrelevant. It adds nothing of any substance to the fact that when the spacetime geometry changes, the geodesics change.

          This is why I think it's better to think about mathematical coordinates instead of observer frames -- to avoid these kinds of pitfalls. I think
          this is a good example of the kind of confusion that Einstein's coordinate frame model can generate in the mind of a physicist.
          Different source configurations have physically different bundles of e.g. static local frames that are the basis for the contingent representation chosen for pragmatic convenience only.
          As far as I'm concerned, this is just another way of saying that the geodesics change wen the spacetime geometry changes.
          You are simply substituting pseudo-relativistic talk about "LIFs" and "LNIFs" for objective talk about covariantly determined
          gravitationally deformed geodesics. Your favorite card trick!
          In any case I stop reading here and end the discussion.
          I think it's time to move on. Let's talk about gauge theory.
          Think what you like but

          1) No journal will accept your idea and rightly so.
          Most of this has *already* been published Jack. Some of it -- like Levi-Civita's theorem -- many years ago.

          Poltorak has already published his stuff, working with a metric-affine model.

          A. A. Logunov has published reams on this. Like V. Fock, he says: "No such thing as general relativity".

          No one serious in this field actually believes in classic Einstein equivalence anymore Jack. Just look at the most
          recent text books on GR.

          2) No one pays any attention to it (except me here) and rightly so.

          That's my opinion.
          And you are perfectly entitled to your opinions. You are free to make all your own mistakes.

          Z.




          You are systematically confusing actual geometric changes in the first partial derivatives g_ij, k with coordinate change.

          The primary purpose of constructing an affine connection is to correct the derivatives of the metric for coordinate artifacts.

          So the premise here is that a distinction must be made between coordinate contributions and actual geometric contributions
          to the partial derivatives of tensors. This also applies to the metric tensor g_uv!

          Now Paul you look at the above as a purely mathematical problem.
          Nonsense. You cannot invalidate a mathematical theorem by attaching a physical interpretation to the math. To suggest that you
          can is completely wrong headed IMO.

          You have to distinguish mathematically between actual geometric changes and mere coordinate changes in the partial derivatives
          of tensors, including those of the metric tensor.

          That is what the affine deformation model I've been trying to explain to you does so nicely  -- for the metric tensor g_uv and its
          associated metric-compatible LC connection.
          That is a profound error of conceptualization that you as The Profundus Philofawzer of Physics Extraordinaire ;-) ought to be well aware of - the "informal language" (David Bohm) the interpretation of the naked mathematique.
          Look Jack, you just got the math wrong, that's all. Do you want to repair the situation, or not?

          As far as I'm concerned both the the mathematical issues *and* the issues of physical interpretation are now completely settled,
          based on

          (1) Einstein's 1917 interpretation of the LC connection field LC^i_jk(x) as representing the non-tidal field strength in GR;

          (2) Levi-Civita's authoritative proof that the difference between two LC connections (his 2 and 2') respectively compatible
          with any two *geometrically* distinct metrics (his 1 and 1') on a single analytical manifold is a non-trivial 3-index GCT tensor
          rho^i_jk.

          At this point any quibbling on your side about how different coordinate representations of the Riemann metric are supposed to constitute
          distinct "metric fields" (which is mathematical nonsense, since g_uv is a tensor object) is just whistling past the graveyard as far as
          I'm concerned.
          The above representation is for static (i.e. constant r, theta, phi) detectors. When M =/= 0 , they are covariantly accelerating in order to stay still in curved spacetime. This is Alice in Wonderland. You are not in Kansas anymore. You are not in Flatland. This is a violation of the Victorian Age engineer's common sense.
          It's also a violation of the mathematics. Levi-Civita's proof shows that the frame acceleration field, the strength of which is represented by
          a non-tensor coordinate correction term X^i_jk, is only accidentally "glued on" to the actual gravitational field, represented to first order
          by the gravitational deformation tensor G^i_jk.

          That is the whole point of the linear LC decomposition

          LC^i_jk = G^i_jk + X^i_jk

          where the gravitational deformation tensor G^i_jk is a particular instance of Levi-Civita's rho^i_jk.

          Your insistence that different coordinate representations of g_uv constitute distinct metric fields is just a fallacy. It
          ignores the tensor character of g_uv.
          Newton's First Law of Motion in Einsteinian 1916 classical geometrodynamical language generalizes to

          All covariantly non-accelerating massive test particles follow  timelike geodesics of the gravitational field.
          Splitting out the gravitational deformation tensor G^i_jk doesn't change any of that Jack. So why do you keep bringing
          this up?
          Newton's Second Law of Motion generalizes to

          D^2x^u/ds^2 = F^u/m

          where m is the invariant rest mass of the test particle (assumed here not to be ejecting mass like a rocket) and ds is the proper time of m along the worldline. F^u is the non-gravity applied force pushing the test particle off-geodesic inducing g-force on the test particle.
          All fully covariant and consistent with tensor G^i_jk. So what's your point?
          More generally, Newton's 2nd Law in curved spacetime for massive test particle motion is

          (D/ds)(mdx^u/ds) = (Dm/ds)dx^u/ds + mD^2x^u/ds^2 = F^u(non-gravity)

          Newton's 3rd law is not valid for global total momenta because translation symmetry is broken - see Noether's theorem.

          Instead we have the local stress-energy current density conservation law

          D^vTuv(non-gravity source fields curving spacetime) = 0

          2) g-force and the Levi-Civita connection field {LC}

          Back to the simplest case

          D^2x^u/ds^2 = F^u/m

          We now do a Godel self-reference and let the test particle detect itself.

          Note, the connection field (LC Christoffel symbol )usually has nothing to do with the test particle, but is a property of the detector, except in the degenerate self-referential case.
          I'm sorry Jack but IMO this is complete and utter nonsense.

          The geodesics are covariantly defined in GR and have nothing to do with "detectors" or "observers" -- except as to mere
          *appearances*. The LC connection field includes linearly independent contributions from the coordinates *and from the
          intrinsic geometry*. Otherwise, how could it be a metric compatible connnection?

          Of course a straight SR free-fall trajectory for example *looks* curved to a uniformly accelerating observer *who ignores
          his own accelerating motion*! So what? That doesn't affect the actual geometry of the geodesic in the slightest. And neither
          does curved geometry force any changes in local coordinate charts. You are completely and hopelessly confused here.
          These are two completely different animals.

          I'm not going to read any further.

          Z.




          The Baron Munchausen is weightless, no inertial g-force, in free-fall parabolic path (timelike geodesic in curved spacetime)
          He does feel g-force initially when cannon fires

          In particular think of the above SSS case in which everything can be concretely computed. Z never bothers to compute an example. This is why his thesis here is not even wrong in my opinion.

          i = 1,2,3 = spacelike components

          i = 0 is the timelike component

          Let's look only at the i = 1 radial component in the static LNIF 

          D^2r/ds^2 = d^2r/ds^2 + {LC}^ruv(dx^u/ds)(dx^v/ds)

          However, in the self-referential "diagonal" case

          dx^i/ds = 0

          dx^o/ds = 1 (c = 1)

          d^2r/ds^2 = 0

          Therefore, the g-force per unit rest mass measured on the test particle itself is

          {LC}^r00 ~ +GM(source)/r^2

          This, however, is not an objective non-tidal non-zero tensor field. It is purely contingent based on an arbitrary, though pragmatically contingent, choice since we here on Earth closely approximate this situation.

          3) Z's Red Herring of the Gravity Deformation

          Start from M = 0 (flat space time) obviously 

          {LC}^r00(M = 0) = 0 

          here the static local non-inertial observer limits to the global inertial observer - a point overlooked by Z because he does not calculate a simple example.

          Now increase to &M

          {LC}^r00(&M) = G&M/r^2

          This is not a tensor obviously.

          The difference is not a tensor, i.e.,

          {LC}^r00(0) - LC^r00(&M) = G&M/r^2

          No tensors here!

          Note the natural assumption here is the adiabatic transformation of the global static inertial frame GIF to the local static non-inertial frame LNIF. The contingent choice of convenience is "static".

          Every time I hear a philofawzer say it's "pure math" or "your argument is mathematical nonsense" I reach for my delete button. ;-)

          "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
          • Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."
          • "Imagination is more important than knowledge."
            • "Reality is merely an illusion, albeit a very persistent one."
            • "The only real valuable thing is intuition."
              • "Anyone who has never made a mistake has never tried anything new."
              • "Great spirits have often encountered violent opposition from weak minds."
              • "God does not care about our mathematical difficulties. He integrates empirically."
              • "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
              • "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
              • http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html

          Jack Sarfatti wrote:
          What's fundamentally wrong in z' s argument is that he does not understand the breakdown of the global frame assumption of special relativity. You simply cannot use purely formal arguments. You must specify what detectors you mean. Physics is about detector responses.

          Sent from my iPhone

          On Feb 28, 2009, at 8:17 AM, Jack Sarfatti <sarfatti@...> wrote:

          Start with flat spacetime. The lc connection field is zero for the global inertial observer.
          Now introduce a small SSS  Tuv matter source which induces an non zero lc field for the static non-inertial observer that adiabatically emerges if you impose a fixed r constraint.
          Since the difference is a tensor
          That difference must be zero otherwise you have a contradiction.

          Sent from my iPhone

          On Feb 28, 2009, at 1:15 AM, Paul Zielinski <iksnileiz@...> wrote:

          Nonsense.

          The common non-tensor coordinate part just cancels out of the difference between the two LC connections,
          which are themselves NOT GCT tensors due to the common coordinate parts that cancel.

          You are just confused.

          Z.

          Jack Sarfatti wrote:
          Problem with the deformation approach is that it does not work. Start with flat spacetime and it implies lc is a tensor.

          Back in Sf 

          Sent from my iPhone

          On Feb 27, 2009, at 1:31 PM, Jack Sarfatti <sarfatti@...> wrote:

          So you are claiming that eq 2 is within a single network of observers e.g static Lnifs in the SSS case. We add a perturbation to tuv source inducing a metric perturbation. The take difference in lc connections and that is the 3rd rank tensor?
          Why did you not say that clearly to begin with?
          In any case it is not conceptually important to the foundations of relativity.

          Only the tetrad and spin connection cartan 1-forms are intrinsic nontidal measures of the gravity field ie scalar zero rank tensors under gct aka local t4. About to take off from jfk.

          Sent from my iPhone

          On Feb 27, 2009, at 4:00 PM, Paul Zielinski <iksnileiz@...> wrote:

          You showed in detail that if you replace Levi-Civita's two different metrics 1 and 1' with the *same*
          metric represented in two different coordinate charts, you get a very different quantity when you take
          the difference of the corresponding compatible connections that obviously is not a tensor..

          Which is just another way of saying that you don't understand -- or don't want to understand -- Levi-
          Civita's proof.

          I told you what Levi-Civita's rho^i_jk means mathematically: it represents the effect of a deformation
          of the geometry of the manifold (from metric 1 to metric 1') on the Levi-Civita connection, free of
          any non-tensor contribution from the coordinates.

          In the context of GR, since in Einstein's model the net observed non-tidal gravitational field strength is
          represented by the LC connection, clearly the tensor rho^i_jk represents the effect on the non-tidal field
          strength of the change in the 4D spacetime geometry when you go from 1 to 1' -- free of any contribution
          from observer frame acceleration.

          If you take metric 1 to be geometrically Minkowski, then the difference G^i_jk between the two LC
          connections obviously represents the effect on the gravity-free LC connection field of the change from
          Minkowski geometry (n_uv) to gravitationally deformed geometry (g_uv).

          Which means that the tensor field G^i_jk(x) represents the *actual* (i.e., objective, frame-independent)
          non-tidal (first-order) gravitational field strength in GR, free of frame acceleration artifacts ("fictitious
          forces").

          Z.

          Jack Sarfatti wrote:
          There is no physics at all in that snippet eq 2.

          Sent from my iPhone at jfk boarding for Sfo 

          On Feb 26, 2009, at 5:04 PM, Paul Zielinski <iksnileiz@...> wrote:

          Actually, we are talking about Levi-Civita's idea here. He's the one with two different metrics on a single
          analytical manifold, with a common local coordinate chart around each point, under general transformations
          of which his rho^i_jk (the difference between two metric-compatible connections) transforms as a 3-index
          tensor.


          I showed in detail it is not a tensor if the usual physics is added to the naked math. It is a contingent relation between two networks of pairs of locally coincident detectors.









        • JACK SARFATTI
          ... Paul, your argument is so vague and shifting that I don t know what it is except to say that I do not buy your thesis that there is a 3rd rank non-tidal
          Message 4 of 26 , Mar 1, 2009

            On Mar 1, 2009, at 3:46 PM, Paul Zielinski wrote:

            Nice illustration.

            However, how is it relevant to what we have been talking about?

            Paul, your argument is so vague and shifting that I don't know what it is except to say that I do not buy your thesis that there is a 3rd rank non-tidal gravity field tensor in Einstein's 1915 GR or in any sane variation on it. The expression you suggest does not work because it has no local observable meaning in terms of possible operational definitions that an experimental physicist can do. Your formal arguments fall on deaf ears and you are banging your head against a brick wall. There is a non-tidal gravity field however, it is the set of four tetrad Cartan 1-forms e^I = e^Iue^u as explained by Rovelli in Ch 2 of his "Quantum Gravity". The set {e^I} is a set of zero-rank GCT tensors, i.e. invariant local scalar fields under iii local T4 (x) group and it is a Lorentz group first-rank tensor under ii.




            You are systematically confusing the objective geometrical changes in the geodesics with changes in the
            "rest frames" of free test objects.

            No I am not. This shows you have not understood even one word of my argument. You are completely confused. The hovering static LNIF accelerating observers (rockets blasting to stand still in the gravity field)  pictured by Hawking are not the rest frames of free test objects.


            Furthermore the objective changes in the timelike geodesics are seen in the change in the geodesic deviation of neighboring unaccelerating detectors. Gravity deformation is not the proper way to define the gravity field which is there even if there is no such deformation. Your position is clueless to my mind.

            What you write below has nothing to do with what I have been professing here. Nothing at all. You are on a different planet in an alternate universe not even in the same ball park.

            This allows you to pretend that you can transmute statements about
            objective spacetime geometry into statements about observer reference frames.

            You are just playing verbal tricks on yourself. Don't you get it? This is just sophistry.

            Look, everyone agrees that you cannot globally "transform away" a non-homogeneous gravitational field,
            which is the whole point of Hawking's picture.

            So what's *your* point Jack?

            Z.

            JACK SARFATTI wrote:
            PS Hawking's picture Fig 1.11 lower right should make my point obvious. Imagine changing the mass of Earth with a density, but keeping the hovering rockets at same distance from center of Earth.
            <mime-attachment.jpeg>
            On Mar 1, 2009, at 2:03 PM, JACK SARFATTI wrote:

            Paul none of this stuff had been published. What has been published has no relationship to your interpretation of what it means. Your basic methodology is flawed, which is why I say your thesis is not even wrong. You do not understand the subtle role of local frames. When you deform the gravity field by changing the source matter field configuration Tuv(matter) ---> T'uv(matter'), you also adiabatically change the local reference frames! You cannot consistently keep those frames fixed for a given kind of constraint e.g. the static observer constraint. You can imagine it mathematically, but it is not possible physically, therefore, you cannot claim, as you do claim, that your purely formal "deformation" change is the consistent basis for an actual operational definition of an intrinsic objective real locally observable non-tidal gravity field that is a GCT 3rd rank tensor. 

            As an example: deform M to M' in the SSS case for static locally accelerating detectors each at fixed r, theta, phi

            g = GM/r^2

            g' = GM'/r^2

            r is same in both cases, but g =/= g' and these g's are the actual accelerations of the local static frames in each case. They are different frames!
            You cannot subtract them and claim you have a tensor in the same frame! It is physical nonsense operationally even though you can make an abstract formal process inside your demented mind! ;-) It does not exist in the laboratory, which is what physics is all about. 

            On Mar 1, 2009, at 1:16 PM, Paul Zielinski wrote:

            JACK SARFATTI wrote:

            On Feb 28, 2009, at 8:51 PM, Paul Zielinski wrote:

            JACK SARFATTI wrote:


            On Feb 28, 2009, at 2:49 PM, Paul Zielinski wrote:
            "...it is not so easy to free oneself from the idea that coordinates must have a direct metric significance."

            - A. Einstein, "Autobiographical Notes"

            Back in San Francisco

            re: 




            Paul, Polemics are the last refuge of the man who cannot make a rational relevant refutation. ;-)
            You've already been definitively refuted Jack. Now I'm just having some fun.:-)

            That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published.
            Most of this stuff has already been published.

            If you think you can go up against Levi-Civita on the math, then I say it is you who is delusioal.

            I have given you a detailed argument that you have not refuted in a rational relevant way.

            But your argument is based on your own confusion about local coordinates in relation to curved spacetime geometry.



            Curving the 4D spacetime of 1916 GR deforms the *geometry* of the manifold (metric relations between spacetime
            points), NOT the coordinate assignments.

            Actually you are wrong about that.
            Really?
            The choice of static local non-inertial frames changes in the SSS case.
            Because the geometry changes. Of course the change of geometry affects the geodesics. What is an "LIF"
            in one geometry will not be an "LIF" in another. However, this has nothing essentially to do with coordinate
            transformations. The changes in the geodesics are geometrical.

            So you are still seriously confused. If you want to understand this, you should think about coordinate systems,
            not observer frames.
            The covariant acceleration of the static frame is GM/r^2 so when you write

            {LC(M')}^rtt - {LC(M)}^rtt = G(M' - M)/r^2

            this is a conceptual relationship between two physically different frames of reference that actually do not simultaneously exist in a real physical situation.
            The relationship we are really talking about here is between two sets of geodesics covariantly determined by different
            metrics according to the textbook standard geodesic equation.
            Your thesis is not even wrong simply garbled fragmentary ungrounded thinking in my opinion.
            Jack, it's too late for this. You just got the math wrong, that's all.

            The change from n_ij to g_ij does not force *any* changes in local coordinate charts around any given point of
            interest on the manifold. The geodesics will of course change with the geometry, but coordinate transformations
            are simply not relevant here. It is the geometry that changes, not the coordinates.

            Your belief to the contrary is what is truly delusional here. You cannot invalidate mathematical theorems with physical
            interpretations.
            I use physical detectors as local frame of reference.

            The meaning of the classical general relativity principle is the computation of local invariants based upon local measurements of two locally coincident detectors in arbitrary relative instantaneous motion measuring the same process up to quantum uncertainty limits.
            We've already deconstructed this weasel term "local". In this context it means, among other things, that you can't
            talk about the frame-invariant gravitational acceleration of test object geodesics with respect to world lines of sources
            -- which is what I call Einstein's "make-believe physics". This is like putting blinkers on a cart horse.

            The fact is that once you take account of the gravitational acceleration of test objects with respect to sources, Einstein's
            equivalence model simply evaporates. You cannot produce gravitational acceleration by accelerating an observer's frame
            of reference -- not in Newton's theory, and not in GR either. To suppose that you can is a form of insanity IMHO.
            You can leave the local coordinate charts (MAPS from R^4 to the points
            of the spacetime manifold) undisturbed (FIXED, CONSTANT, UNCHANGING, INVARIANT) while you deform
            the manifold in any local spacetime region of interest.

            In other words, you can arrange things such that under the gravitational deformation of the metric around any given spacetime
            point x you end up with the same local chart around x as you started with. The change in the intrinsic geometry doesn't force
            ANY change in the local coordinate assignments around x.

            The very concept of "gravitational deformation" is purely theoretical
            What do you think Einstein's field equations

            G_uv = kT_uv

            are all about Jack? They tell us how matter deforms the geometry of spacetime! This *is* Einstein's theory of gravity!
            and cannot actually be measured unless e.g. the Earth loses chunks of matter in a collision with a large asteroid for example. It is not a valid measure of the actual local gravitational field i.e. tetrad 1-form e^I (Rovelli's definition)
            You have to be kidding, Given a single SSS source, all you have to do is measure g, the acceleration of test particle geodesics in
            relation to the corresponding flat-spacetime geodesics, that is due to gravity. This is all determined in GR by the geodesic equation:

            <moz-screenshot-43.jpg>

            This is a generally covariant equation.

            Are you still trying to say that in GR there is no way of comparing the curved space geodesic trajectories with the corredponding flat
            space trajectories? And that there is no way of observing such changes empirically?

            That would make Einstein's theory completely useless as a theory of gravity. But of course this is nonsense.

            That is something you still don't seem to understand. Hence the Einstein quote. You have not been able to fully separate the
            concept of coordinate intervals from the concept of metric relations between points. I think this is the root of your problem,
            and is the reason why you are having so much trouble understanding Levi-Civita.
            Synopsis, since there are interesting foundational issues here ignored even in some texts, though not in Wheeler's.

            1) Every time you write a particular metric field guv(x), and its associated symmetric torsion-free Levi-Civita connection field for parallel transport of tensors along world lines in 4D spacetime you have chosen a representation in terms of a network of local detectors (AKA local frames of reference).
            Wrong. 

            Irrational response.

            All GCT coordinate representations of a given metric g_uv(x) are representations of a *single* metric field.

            Red Herring. No one says otherwise.
            You just said otherwise. What did you mean then by "a particular metric field guv(x)"?

            You seem to be saying that there is a different "metric field" for every GCT that changes the components of the metric tensor.

            I have said nothing of the kind.
            It sounded like you were saying this.
            You simply do not get the point I am making because your mind is closed on this silly idea you have. It's completely silly - embarrassing in fact to my mind at least.
            That changes in the geometry do not force changes in the local coordinate charts in GR? You think that's a silly idea?

            It is a mathematical fact with the consequence that the Einstein field, represented by LC^i_jk(x), is decomposable into an
            actual gravitational field represented by a tensor G^i_jk, and a non-tensor X^i_jk which accounts for all non-tensor coordinate
            contributions to the LC connection field:

            LC^i_jk = G^i_jk + X^i_jk

            Which means that in 1916 GR, frame acceleration fields are every bit as fictitious in relation to actual gravitational fields as
            they are in Newton's theory in relation to actual Newtonian forces.

            And you think this is a "silly idea"?
            The metric fields guv & g'uv induced by distinct sources Tuv & T'uv are not related by a GCT. They are different intrinsic gravity field configurations and they have different sets of static observers (outside event horizons) and, therefore, their formal difference has no direct physical meaning, i.e. no measurement procedures exist.  They cannot simultaneously exist physically.

            If that is what you are saying, then you are missing the whole point of the metric being a GCT tensor.

            Of course that is NOT what I have been saying.
            OK. Good.

            Then you should have no trouble understanding why the difference between an L:C connection compatible with a curved metric and
            the LC connection compatible with a flat-level Minkowski metric is a tensor.
            You are also missing the
            point of defining affine connections on curved manifolds that correct the partial derivatives of tensors for coordinate artifacts.

            Changing the coordinate representation of the metric doesn't affect the intrinsic spacetime geometry.

            You are seriously confused about this. Your Einsteinian optics are preventing you from understanding the actual math.
            The simplest "hydrogen atom" (as it were) example in Einstein's 1915 GR is the SSS vacuum solution, written in the texts as

            g00 = gtt = - 1/g11 = - 1/grr = 1 - rs/r

            rs/r > 1

            rs = 2GM(source)/c^2

            Area of concentric sphere is A = 4pir^2

            usual spherical polar & azimuthal coordinate line element added.

            You can use any local coordinate chart around any given (well-behaved) point of interest, and hold that local chart invariant
            under a gravitational deformation of the manifold.

            You are wrong.
            That is your delusion Jack.
            Physically you need to specify a set (fiber bundle) of local tangent frames that in principle are implementable by actual detectors.
            In Einstein's coordinate frame model, which is a relic of 1905 SR, yes.
            Here the choice of static frames (where they exist) shows my point very clearly.
            This is where you get really confused. Of course changes in the spacetime geometry will change the classification of trajectories,
            since the geodesics chang -- but this happens regardless of whether you hold the local coordinate charts invariant under the
            deformation of the geometry, or not.

            Of course you can say that the *rest* frames of freely falling test objects will change, and that this is associated with a change in
            the coordinate charts associated with the rest frames, but this is immaterial and irrelevant to the point I'm making. It is
            a trivial observation. There is nothing that compels us to observe a freely falling test object from its own rest frame, regardless
            of changes in the geometry.

            You might as well observe that the rest frame of any object changes when you accelerate the object. So what?
            This completely eludes you.
            What eludes me is why you think this matters. It's a tautology. Trivial.
            It is the fundamental error you have made. You are thinking like a mathematician and completely miss the physics.
            If what you mean here by "physics" is the completely trivial observation that the *rest* frames of freely falling test objects change when
            the geodesics change, then I think I am quite correct in ignoring it. It is of no consequence. It is immaterial to my argument.
            It is trivial and irrelevant. It adds nothing of any substance to the fact that when the spacetime geometry changes, the geodesics change.

            This is why I think it's better to think about mathematical coordinates instead of observer frames -- to avoid these kinds of pitfalls. I think
            this is a good example of the kind of confusion that Einstein's coordinate frame model can generate in the mind of a physicist.
            Different source configurations have physically different bundles of e.g. static local frames that are the basis for the contingent representation chosen for pragmatic convenience only.
            As far as I'm concerned, this is just another way of saying that the geodesics change wen the spacetime geometry changes.
            You are simply substituting pseudo-relativistic talk about "LIFs" and "LNIFs" for objective talk about covariantly determined
            gravitationally deformed geodesics. Your favorite card trick!
            In any case I stop reading here and end the discussion.
            I think it's time to move on. Let's talk about gauge theory.
            Think what you like but

            1) No journal will accept your idea and rightly so.
            Most of this has *already* been published Jack. Some of it -- like Levi-Civita's theorem -- many years ago.

            Poltorak has already published his stuff, working with a metric-affine model.

            A. A. Logunov has published reams on this. Like V. Fock, he says: "No such thing as general relativity".

            No one serious in this field actually believes in classic Einstein equivalence anymore Jack. Just look at the most
            recent text books on GR.

            2) No one pays any attention to it (except me here) and rightly so.

            That's my opinion.
            And you are perfectly entitled to your opinions. You are free to make all your own mistakes.

            Z.




            You are systematically confusing actual geometric changes in the first partial derivatives g_ij, k with coordinate change.

            The primary purpose of constructing an affine connection is to correct the derivatives of the metric for coordinate artifacts.

            So the premise here is that a distinction must be made between coordinate contributions and actual geometric contributions
            to the partial derivatives of tensors. This also applies to the metric tensor g_uv!

            Now Paul you look at the above as a purely mathematical problem.
            Nonsense. You cannot invalidate a mathematical theorem by attaching a physical interpretation to the math. To suggest that you
            can is completely wrong headed IMO.

            You have to distinguish mathematically between actual geometric changes and mere coordinate changes in the partial derivatives
            of tensors, including those of the metric tensor.

            That is what the affine deformation model I've been trying to explain to you does so nicely  -- for the metric tensor g_uv and its
            associated metric-compatible LC connection.
            That is a profound error of conceptualization that you as The Profundus Philofawzer of Physics Extraordinaire ;-) ought to be well aware of - the "informal language" (David Bohm) the interpretation of the naked mathematique.
            Look Jack, you just got the math wrong, that's all. Do you want to repair the situation, or not?

            As far as I'm concerned both the the mathematical issues *and* the issues of physical interpretation are now completely settled,
            based on

            (1) Einstein's 1917 interpretation of the LC connection field LC^i_jk(x) as representing the non-tidal field strength in GR;

            (2) Levi-Civita's authoritative proof that the difference between two LC connections (his 2 and 2') respectively compatible
            with any two *geometrically* distinct metrics (his 1 and 1') on a single analytical manifold is a non-trivial 3-index GCT tensor
            rho^i_jk.

            At this point any quibbling on your side about how different coordinate representations of the Riemann metric are supposed to constitute
            distinct "metric fields" (which is mathematical nonsense, since g_uv is a tensor object) is just whistling past the graveyard as far as
            I'm concerned.
            The above representation is for static (i.e. constant r, theta, phi) detectors. When M =/= 0 , they are covariantly accelerating in order to stay still in curved spacetime. This is Alice in Wonderland. You are not in Kansas anymore. You are not in Flatland. This is a violation of the Victorian Age engineer's common sense.
            It's also a violation of the mathematics. Levi-Civita's proof shows that the frame acceleration field, the strength of which is represented by
            a non-tensor coordinate correction term X^i_jk, is only accidentally "glued on" to the actual gravitational field, represented to first order
            by the gravitational deformation tensor G^i_jk.

            That is the whole point of the linear LC decomposition

            LC^i_jk = G^i_jk + X^i_jk

            where the gravitational deformation tensor G^i_jk is a particular instance of Levi-Civita's rho^i_jk.

            Your insistence that different coordinate representations of g_uv constitute distinct metric fields is just a fallacy. It
            ignores the tensor character of g_uv.
            Newton's First Law of Motion in Einsteinian 1916 classical geometrodynamical language generalizes to

            All covariantly non-accelerating massive test particles follow  timelike geodesics of the gravitational field.
            Splitting out the gravitational deformation tensor G^i_jk doesn't change any of that Jack. So why do you keep bringing
            this up?
            Newton's Second Law of Motion generalizes to

            D^2x^u/ds^2 = F^u/m

            where m is the invariant rest mass of the test particle (assumed here not to be ejecting mass like a rocket) and ds is the proper time of m along the worldline. F^u is the non-gravity applied force pushing the test particle off-geodesic inducing g-force on the test particle.
            All fully covariant and consistent with tensor G^i_jk. So what's your point?
            More generally, Newton's 2nd Law in curved spacetime for massive test particle motion is

            (D/ds)(mdx^u/ds) = (Dm/ds)dx^u/ds + mD^2x^u/ds^2 = F^u(non-gravity)

            Newton's 3rd law is not valid for global total momenta because translation symmetry is broken - see Noether's theorem.

            Instead we have the local stress-energy current density conservation law

            D^vTuv(non-gravity source fields curving spacetime) = 0

            2) g-force and the Levi-Civita connection field {LC}

            Back to the simplest case

            D^2x^u/ds^2 = F^u/m

            We now do a Godel self-reference and let the test particle detect itself.

            Note, the connection field (LC Christoffel symbol )usually has nothing to do with the test particle, but is a property of the detector, except in the degenerate self-referential case.
            I'm sorry Jack but IMO this is complete and utter nonsense.

            The geodesics are covariantly defined in GR and have nothing to do with "detectors" or "observers" -- except as to mere
            *appearances*. The LC connection field includes linearly independent contributions from the coordinates *and from the
            intrinsic geometry*. Otherwise, how could it be a metric compatible connnection?

            Of course a straight SR free-fall trajectory for example *looks* curved to a uniformly accelerating observer *who ignores
            his own accelerating motion*! So what? That doesn't affect the actual geometry of the geodesic in the slightest. And neither
            does curved geometry force any changes in local coordinate charts. You are completely and hopelessly confused here.
            These are two completely different animals.

            I'm not going to read any further.

            Z.




            The Baron Munchausen is weightless, no inertial g-force, in free-fall parabolic path (timelike geodesic in curved spacetime)
            He does feel g-force initially when cannon fires

            In particular think of the above SSS case in which everything can be concretely computed. Z never bothers to compute an example. This is why his thesis here is not even wrong in my opinion.

            i = 1,2,3 = spacelike components

            i = 0 is the timelike component

            Let's look only at the i = 1 radial component in the static LNIF 

            D^2r/ds^2 = d^2r/ds^2 + {LC}^ruv(dx^u/ds)(dx^v/ds)

            However, in the self-referential "diagonal" case

            dx^i/ds = 0

            dx^o/ds = 1 (c = 1)

            d^2r/ds^2 = 0

            Therefore, the g-force per unit rest mass measured on the test particle itself is

            {LC}^r00 ~ +GM(source)/r^2

            This, however, is not an objective non-tidal non-zero tensor field. It is purely contingent based on an arbitrary, though pragmatically contingent, choice since we here on Earth closely approximate this situation.

            3) Z's Red Herring of the Gravity Deformation

            Start from M = 0 (flat space time) obviously 

            {LC}^r00(M = 0) = 0 

            here the static local non-inertial observer limits to the global inertial observer - a point overlooked by Z because he does not calculate a simple example.

            Now increase to &M

            {LC}^r00(&M) = G&M/r^2

            This is not a tensor obviously.

            The difference is not a tensor, i.e.,

            {LC}^r00(0) - LC^r00(&M) = G&M/r^2

            No tensors here!

            Note the natural assumption here is the adiabatic transformation of the global static inertial frame GIF to the local static non-inertial frame LNIF. The contingent choice of convenience is "static".

            Every time I hear a philofawzer say it's "pure math" or "your argument is mathematical nonsense" I reach for my delete button. ;-)

            "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
            • Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."
            • "Imagination is more important than knowledge."
              • "Reality is merely an illusion, albeit a very persistent one."
              • "The only real valuable thing is intuition."
                • "Anyone who has never made a mistake has never tried anything new."
                • "Great spirits have often encountered violent opposition from weak minds."
                • "God does not care about our mathematical difficulties. He integrates empirically."
                • "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
                • "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
                • http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html

            Jack Sarfatti wrote:
            What's fundamentally wrong in z' s argument is that he does not understand the breakdown of the global frame assumption of special relativity. You simply cannot use purely formal arguments. You must specify what detectors you mean. Physics is about detector responses.

            Sent from my iPhone

            On Feb 28, 2009, at 8:17 AM, Jack Sarfatti <sarfatti@...> wrote:

            Start with flat spacetime. The lc connection field is zero for the global inertial observer.
            Now introduce a small SSS  Tuv matter source which induces an non zero lc field for the static non-inertial observer that adiabatically emerges if you impose a fixed r constraint.
            Since the difference is a tensor
            That difference must be zero otherwise you have a contradiction.

            Sent from my iPhone

            On Feb 28, 2009, at 1:15 AM, Paul Zielinski <iksnileiz@...> wrote:

            Nonsense.

            The common non-tensor coordinate part just cancels out of the difference between the two LC connections,
            which are themselves NOT GCT tensors due to the common coordinate parts that cancel.

            You are just confused.

            Z.

            Jack Sarfatti wrote:
            Problem with the deformation approach is that it does not work. Start with flat spacetime and it implies lc is a tensor.

            Back in Sf 

            Sent from my iPhone

            On Feb 27, 2009, at 1:31 PM, Jack Sarfatti <sarfatti@...> wrote:

            So you are claiming that eq 2 is within a single network of observers e.g static Lnifs in the SSS case. We add a perturbation to tuv source inducing a metric perturbation. The take difference in lc connections and that is the 3rd rank tensor?
            Why did you not say that clearly to begin with?
            In any case it is not conceptually important to the foundations of relativity.

            Only the tetrad and spin connection cartan 1-forms are intrinsic nontidal measures of the gravity field ie scalar zero rank tensors under gct aka local t4. About to take off from jfk.

            Sent from my iPhone

            On Feb 27, 2009, at 4:00 PM, Paul Zielinski <iksnileiz@...> wrote:

            You showed in detail that if you replace Levi-Civita's two different metrics 1 and 1' with the *same*
            metric represented in two different coordinate charts, you get a very different quantity when you take
            the difference of the corresponding compatible connections that obviously is not a tensor..

            Which is just another way of saying that you don't understand -- or don't want to understand -- Levi-
            Civita's proof.

            I told you what Levi-Civita's rho^i_jk means mathematically: it represents the effect of a deformation
            of the geometry of the manifold (from metric 1 to metric 1') on the Levi-Civita connection, free of
            any non-tensor contribution from the coordinates.

            In the context of GR, since in Einstein's model the net observed non-tidal gravitational field strength is
            represented by the LC connection, clearly the tensor rho^i_jk represents the effect on the non-tidal field
            strength of the change in the 4D spacetime geometry when you go from 1 to 1' -- free of any contribution
            from observer frame acceleration.

            If you take metric 1 to be geometrically Minkowski, then the difference G^i_jk between the two LC
            connections obviously represents the effect on the gravity-free LC connection field of the change from
            Minkowski geometry (n_uv) to gravitationally deformed geometry (g_uv).

            Which means that the tensor field G^i_jk(x) represents the *actual* (i.e., objective, frame-independent)
            non-tidal (first-order) gravitational field strength in GR, free of frame acceleration artifacts ("fictitious
            forces").

            Z.

            Jack Sarfatti wrote:
            There is no physics at all in that snippet eq 2.

            Sent from my iPhone at jfk boarding for Sfo 

            On Feb 26, 2009, at 5:04 PM, Paul Zielinski <iksnileiz@...> wrote:

            Actually, we are talking about Levi-Civita's idea here. He's the one with two different metrics on a single
            analytical manifold, with a common local coordinate chart around each point, under general transformations
            of which his rho^i_jk (the difference between two metric-compatible connections) transforms as a 3-index
            tensor.


            I showed in detail it is not a tensor if the usual physics is added to the naked math. It is a contingent relation between two networks of pairs of locally coincident detectors.











          • JACK SARFATTI
            this is worthless polemical philofawzy - bad and bogus pseudo-physics arguing from authority with misrepresentations of the sources cited to my way of
            Message 5 of 26 , Mar 1, 2009
              this is worthless polemical philofawzy - bad and bogus pseudo-physics arguing from authority with misrepresentations of the sources cited to my way of thinking. Any theory that is not grounded in operational definitions at some point is completely worthless as theoretical physics in my opinion.
              On Mar 1, 2009, at 3:33 PM, Paul Zielinski wrote:

              JACK SARFATTI wrote:
              Paul none of this stuff had been published. What has been published has no relationship to your interpretation of what it means.
              The math has been published. It's textbook differential geometry. Levi-Civita's book is just one example.

              Many, many critiques of the equivalence principle have been published. Fock wrote about this in the 1950, saying that there is no
              "general relativity" in Einstein's theory of gravity.

              Utter nonsense. 

              The principle of general relativity is a procedure for locally coincident detectors in arbitrary relative motion to compute the same frame-invariant local observables from their frame-dependent data. End of story. All these guys are simply wrong - period.

              Synge's critique is very famous. Ohanian and Ruffini for example explicitly disavowed
              Einstein equivalence in their textbook published in the 1990s. And so on and so forth.

              Studies of quasi-global homogeneous solutions of the GR field equations (with R^i_jkl = 0 almost everywhere) have been published
              (e.g., Vilenkin's papers) which conclude that there is objective gravitational acceleration of free test objects with respect to the source
              in a flat field.

              Since you cannot produce such acceleration by any frame transformation, that means that Einstein equivalence is already a DEAD
              DUCK The only object available for accounting for such acceleration of geodesics in Einstein's theory is the tensor G^i_jk extraced
              from the LC connection field, as I have now explained in considerable detail..

              You are completely deluded on this. You have done nothing of the kind. It is your fantasy that you think you did. You cannot even compute an example of what you mean in the simple SSS case.


              The insistence that you cannot empirically observe such objective gravitational acceleration and distinguish it from frame acceleration
              -- advanced under the false rubric of "locality" -- is a form of insanity IMO. It is a nutty idea that is completely detached from objective
              physical reality.

              Prove what you say with a concrete calculation. I did. You are spouting word salad above. Polemics no physics.

              Your basic methodology is flawed, which is why I say your thesis is not even wrong.

              Yes, you claim that it is not possible to empirically measure the acceleration due to gravity at the surface of the earth

              I never said anything like that. That is a complete misrepresentation of your confused irrational mind grasping at straws to maintain your severe obsessional delusion. I am saying what Hawking says here in this picture:


              -- something that
              school children do quite routinely when learning about basic physics. You claim that objective gravitational acceleration of test objects
              has no operational meaning.

              What did Freud say about those who are in the grip of a delusion?
              You do not understand the subtle role of local frames.
              I understand that they are represented mathematically in GR by local coordinate charts, and that such charts are completely independent
              of the geometry abd can be held completely imvariant under deformations of the spacetime geometry.

              Now if you want to play pointless sophistical word games with phrases like "rest frame", go right ahead. Different folks, different strokes!
              When you deform the gravity field by changing the source matter field configuration Tuv(matter) ---> T'uv(matter'), you also adiabatically change the local reference frames!
              Nonsense. This is a mirage Jack. A MIRAGE.
              You cannot consistently keep those frames fixed for a given kind of constraint e.g. the static observer constraint.
              So you can't maintain a "hovering frame" with non-gravitational forces while you switch on an SSS field?

              If you stay at fixed r it is not the same frame. If you shift to a new r with same acceleration as before it is still not the same frame. Your LC difference is therefore not a local quantity that can be measured in a single local frame. I do not have the patience to read the rest of your screed below.


              Of course you can. You can keep a gravitationally attracted test object on exactly the same trajectory as a free test object in the absence
              of the field, simply by supporting the test object in the field. Then the trajectory of the gravity-free free test object and that of the supported
              gravitationally attracted test object can be made identical. Then the Einstein coordinate chart remains fixed when you switch on the field and
              the compensating non-gravity support.  

              This is complete nonsense. Even if they take the same world line, in the first case it is a timelike geodesic in the flat spacetime. In the second case it is off timelike geodesic in the now deformed curved spacetime. Apples & Oranges never the twain shall meet. What you suggest here is comedic Laputan sillyness to my mind -- ridiculous Rube Goldberg balderdash.
                                                                          
              You can imagine it mathematically, but it is not possible physically,
              I just showed above that it is physically possible. You can maintain a hovering trajectory when the field is switched on.

              Oh yeah, show us how to switch on a large gravity field. Good trick if you can do it. You can do it on a small scale locally with very sensitive instruments, but even if you do, it does not give a 3rd rank GCT tensor non-tidal gravity field.

              g = - GM/r^2 

              is not the acceleration of a real non-tidal gravity field. It is the covariant acceleration of the static detector hovering in the real tidal gravity field.

              That's the point you do not understand.

              When you write for a test particle moving along x^u(s)

              D^2x^u/ds^2 = d^2x^u/ds^2 + {LC}^uvw(dx^v/ds)(dx^w/ds)

              The quantity {LC}^uvw is not a GCT 3rd rank tensor nor does it contain one, it is a description of the detector near the test particle.

              You cannot invoke deformation. There is only one {LC} here - not two!



              No need to change
              the local Einstein coordinate chart representing the observer's reference frame.
              therefore, you cannot claim, as you do claim, that your purely formal "deformation" change is the consistent basis for an actual operational definition of an intrinsic objective real locally observable non-tidal gravity field that is a GCT 3rd rank tensor.
              I can certainly do it mathematically, and that is all that is required to define and extract the deformation tensor G^i_jk.

              But in fact you can also implement this physically with *Einstein* coordinate charts that track the observer's spatial position,
              as I showed above.
              As an example: deform M to M' in the SSS case for static locally accelerating detectors each at fixed r, theta, phi

              g = GM/r^2

              g' = GM'/r^2

              r is same in both cases, but g =/= g' and these g's are the actual accelerations of the local static frames in each case. They are different frames!
              But you do not have to use those frames. You can arrange things such that the Einstein chart is held fixed under the deformation. Then the coordinate
              intervals do not change -- only the metric relations between spacetime points.

              This is your problem Jack -- right here. This is the mote in your eye.

              All you are really saying here is that the objective spacetime geometry of the geodesics changes when you change the SSS field
              -- which of course no one denies.
              You cannot subtract them and claim you have a tensor in the same frame! It is physical nonsense operationally even though you can make an abstract formal process inside your demented mind! ;-) It does not exist in the laboratory, which is what physics is all about.
              I can and have.

              Nothing forces any particular choice of observer frame when the SSS field is switched on Jack. There is no law that says
              we must do our physics in the rest frame of a freely falling test object.

              After all, isn't this supposed to be a generally covariant theory? What does that mean, if you can't use any Einstein frame
              and associated local coordinate chart that you find to be convenient for analyzing your problem?

              Z.

              On Mar 1, 2009, at 1:16 PM, Paul Zielinski wrote:

              JACK SARFATTI wrote:

              On Feb 28, 2009, at 8:51 PM, Paul Zielinski wrote:

              JACK SARFATTI wrote:


              On Feb 28, 2009, at 2:49 PM, Paul Zielinski wrote:
              "...it is not so easy to free oneself from the idea that coordinates must have a direct metric significance."

              - A. Einstein, "Autobiographical Notes"

              Back in San Francisco

              re: 




              Paul, Polemics are the last refuge of the man who cannot make a rational relevant refutation. ;-)
              You've already been definitively refuted Jack. Now I'm just having some fun.:-)

              That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published.
              Most of this stuff has already been published.

              If you think you can go up against Levi-Civita on the math, then I say it is you who is delusioal.

              I have given you a detailed argument that you have not refuted in a rational relevant way.

              But your argument is based on your own confusion about local coordinates in relation to curved spacetime geometry.



              Curving the 4D spacetime of 1916 GR deforms the *geometry* of the manifold (metric relations between spacetime
              points), NOT the coordinate assignments.

              Actually you are wrong about that.
              Really?
              The choice of static local non-inertial frames changes in the SSS case.
              Because the geometry changes. Of course the change of geometry affects the geodesics. What is an "LIF"
              in one geometry will not be an "LIF" in another. However, this has nothing essentially to do with coordinate
              transformations. The changes in the geodesics are geometrical.

              So you are still seriously confused. If you want to understand this, you should think about coordinate systems,
              not observer frames.
              The covariant acceleration of the static frame is GM/r^2 so when you write

              {LC(M')}^rtt - {LC(M)}^rtt = G(M' - M)/r^2

              this is a conceptual relationship between two physically different frames of reference that actually do not simultaneously exist in a real physical situation.
              The relationship we are really talking about here is between two sets of geodesics covariantly determined by different
              metrics according to the textbook standard geodesic equation.
              Your thesis is not even wrong simply garbled fragmentary ungrounded thinking in my opinion.
              Jack, it's too late for this. You just got the math wrong, that's all.

              The change from n_ij to g_ij does not force *any* changes in local coordinate charts around any given point of
              interest on the manifold. The geodesics will of course change with the geometry, but coordinate transformations
              are simply not relevant here. It is the geometry that changes, not the coordinates.

              Your belief to the contrary is what is truly delusional here. You cannot invalidate mathematical theorems with physical
              interpretations.
              I use physical detectors as local frame of reference.

              The meaning of the classical general relativity principle is the computation of local invariants based upon local measurements of two locally coincident detectors in arbitrary relative instantaneous motion measuring the same process up to quantum uncertainty limits.
              We've already deconstructed this weasel term "local". In this context it means, among other things, that you can't
              talk about the frame-invariant gravitational acceleration of test object geodesics with respect to world lines of sources
              -- which is what I call Einstein's "make-believe physics". This is like putting blinkers on a cart horse.

              The fact is that once you take account of the gravitational acceleration of test objects with respect to sources, Einstein's
              equivalence model simply evaporates. You cannot produce gravitational acceleration by accelerating an observer's frame
              of reference -- not in Newton's theory, and not in GR either. To suppose that you can is a form of insanity IMHO.
              You can leave the local coordinate charts (MAPS from R^4 to the points
              of the spacetime manifold) undisturbed (FIXED, CONSTANT, UNCHANGING, INVARIANT) while you deform
              the manifold in any local spacetime region of interest.

              In other words, you can arrange things such that under the gravitational deformation of the metric around any given spacetime
              point x you end up with the same local chart around x as you started with. The change in the intrinsic geometry doesn't force
              ANY change in the local coordinate assignments around x.

              The very concept of "gravitational deformation" is purely theoretical
              What do you think Einstein's field equations

              G_uv = kT_uv

              are all about Jack? They tell us how matter deforms the geometry of spacetime! This *is* Einstein's theory of gravity!
              and cannot actually be measured unless e.g. the Earth loses chunks of matter in a collision with a large asteroid for example. It is not a valid measure of the actual local gravitational field i.e. tetrad 1-form e^I (Rovelli's definition)
              You have to be kidding, Given a single SSS source, all you have to do is measure g, the acceleration of test particle geodesics in
              relation to the corresponding flat-spacetime geodesics, that is due to gravity. This is all determined in GR by the geodesic equation:

              <moz-screenshot-43.jpg>

              This is a generally covariant equation.

              Are you still trying to say that in GR there is no way of comparing the curved space geodesic trajectories with the corredponding flat
              space trajectories? And that there is no way of observing such changes empirically?

              That would make Einstein's theory completely useless as a theory of gravity. But of course this is nonsense.

              That is something you still don't seem to understand. Hence the Einstein quote. You have not been able to fully separate the
              concept of coordinate intervals from the concept of metric relations between points. I think this is the root of your problem,
              and is the reason why you are having so much trouble understanding Levi-Civita.
              Synopsis, since there are interesting foundational issues here ignored even in some texts, though not in Wheeler's.

              1) Every time you write a particular metric field guv(x), and its associated symmetric torsion-free Levi-Civita connection field for parallel transport of tensors along world lines in 4D spacetime you have chosen a representation in terms of a network of local detectors (AKA local frames of reference).
              Wrong. 

              Irrational response.

              All GCT coordinate representations of a given metric g_uv(x) are representations of a *single* metric field.

              Red Herring. No one says otherwise.
              You just said otherwise. What did you mean then by "a particular metric field guv(x)"?

              You seem to be saying that there is a different "metric field" for every GCT that changes the components of the metric tensor.

              I have said nothing of the kind.
              It sounded like you were saying this.
              You simply do not get the point I am making because your mind is closed on this silly idea you have. It's completely silly - embarrassing in fact to my mind at least.
              That changes in the geometry do not force changes in the local coordinate charts in GR? You think that's a silly idea?

              It is a mathematical fact with the consequence that the Einstein field, represented by LC^i_jk(x), is decomposable into an
              actual gravitational field represented by a tensor G^i_jk, and a non-tensor X^i_jk which accounts for all non-tensor coordinate
              contributions to the LC connection field:

              LC^i_jk = G^i_jk + X^i_jk

              Which means that in 1916 GR, frame acceleration fields are every bit as fictitious in relation to actual gravitational fields as
              they are in Newton's theory in relation to actual Newtonian forces.

              And you think this is a "silly idea"?
              The metric fields guv & g'uv induced by distinct sources Tuv & T'uv are not related by a GCT. They are different intrinsic gravity field configurations and they have different sets of static observers (outside event horizons) and, therefore, their formal difference has no direct physical meaning, i.e. no measurement procedures exist.  They cannot simultaneously exist physically.

              If that is what you are saying, then you are missing the whole point of the metric being a GCT tensor.

              Of course that is NOT what I have been saying.
              OK. Good.

              Then you should have no trouble understanding why the difference between an L:C connection compatible with a curved metric and
              the LC connection compatible with a flat-level Minkowski metric is a tensor.
              You are also missing the
              point of defining affine connections on curved manifolds that correct the partial derivatives of tensors for coordinate artifacts.

              Changing the coordinate representation of the metric doesn't affect the intrinsic spacetime geometry.

              You are seriously confused about this. Your Einsteinian optics are preventing you from understanding the actual math.
              The simplest "hydrogen atom" (as it were) example in Einstein's 1915 GR is the SSS vacuum solution, written in the texts as

              g00 = gtt = - 1/g11 = - 1/grr = 1 - rs/r

              rs/r > 1

              rs = 2GM(source)/c^2

              Area of concentric sphere is A = 4pir^2

              usual spherical polar & azimuthal coordinate line element added.

              You can use any local coordinate chart around any given (well-behaved) point of interest, and hold that local chart invariant
              under a gravitational deformation of the manifold.

              You are wrong.
              That is your delusion Jack.
              Physically you need to specify a set (fiber bundle) of local tangent frames that in principle are implementable by actual detectors.
              In Einstein's coordinate frame model, which is a relic of 1905 SR, yes.
              Here the choice of static frames (where they exist) shows my point very clearly.
              This is where you get really confused. Of course changes in the spacetime geometry will change the classification of trajectories,
              since the geodesics chang -- but this happens regardless of whether you hold the local coordinate charts invariant under the
              deformation of the geometry, or not.

              Of course you can say that the *rest* frames of freely falling test objects will change, and that this is associated with a change in
              the coordinate charts associated with the rest frames, but this is immaterial and irrelevant to the point I'm making. It is
              a trivial observation. There is nothing that compels us to observe a freely falling test object from its own rest frame, regardless
              of changes in the geometry.

              You might as well observe that the rest frame of any object changes when you accelerate the object. So what?
              This completely eludes you.
              What eludes me is why you think this matters. It's a tautology. Trivial.
              It is the fundamental error you have made. You are thinking like a mathematician and completely miss the physics.
              If what you mean here by "physics" is the completely trivial observation that the *rest* frames of freely falling test objects change when
              the geodesics change, then I think I am quite correct in ignoring it. It is of no consequence. It is immaterial to my argument.
              It is trivial and irrelevant. It adds nothing of any substance to the fact that when the spacetime geometry changes, the geodesics change.

              This is why I think it's better to think about mathematical coordinates instead of observer frames -- to avoid these kinds of pitfalls. I think
              this is a good example of the kind of confusion that Einstein's coordinate frame model can generate in the mind of a physicist.
              Different source configurations have physically different bundles of e.g. static local frames that are the basis for the contingent representation chosen for pragmatic convenience only.
              As far as I'm concerned, this is just another way of saying that the geodesics change wen the spacetime geometry changes.
              You are simply substituting pseudo-relativistic talk about "LIFs" and "LNIFs" for objective talk about covariantly determined
              gravitationally deformed geodesics. Your favorite card trick!
              In any case I stop reading here and end the discussion.
              I think it's time to move on. Let's talk about gauge theory.
              Think what you like but

              1) No journal will accept your idea and rightly so.
              Most of this has *already* been published Jack. Some of it -- like Levi-Civita's theorem -- many years ago.

              Poltorak has already published his stuff, working with a metric-affine model.

              A. A. Logunov has published reams on this. Like V. Fock, he says: "No such thing as general relativity".

              No one serious in this field actually believes in classic Einstein equivalence anymore Jack. Just look at the most
              recent text books on GR.

              2) No one pays any attention to it (except me here) and rightly so.

              That's my opinion.
              And you are perfectly entitled to your opinions. You are free to make all your own mistakes.

              Z.




              You are systematically confusing actual geometric changes in the first partial derivatives g_ij, k with coordinate change.

              The primary purpose of constructing an affine connection is to correct the derivatives of the metric for coordinate artifacts.

              So the premise here is that a distinction must be made between coordinate contributions and actual geometric contributions
              to the partial derivatives of tensors. This also applies to the metric tensor g_uv!

              Now Paul you look at the above as a purely mathematical problem.
              Nonsense. You cannot invalidate a mathematical theorem by attaching a physical interpretation to the math. To suggest that you
              can is completely wrong headed IMO.

              You have to distinguish mathematically between actual geometric changes and mere coordinate changes in the partial derivatives
              of tensors, including those of the metric tensor.

              That is what the affine deformation model I've been trying to explain to you does so nicely  -- for the metric tensor g_uv and its
              associated metric-compatible LC connection.
              That is a profound error of conceptualization that you as The Profundus Philofawzer of Physics Extraordinaire ;-) ought to be well aware of - the "informal language" (David Bohm) the interpretation of the naked mathematique.
              Look Jack, you just got the math wrong, that's all. Do you want to repair the situation, or not?

              As far as I'm concerned both the the mathematical issues *and* the issues of physical interpretation are now completely settled,
              based on

              (1) Einstein's 1917 interpretation of the LC connection field LC^i_jk(x) as representing the non-tidal field strength in GR;

              (2) Levi-Civita's authoritative proof that the difference between two LC connections (his 2 and 2') respectively compatible
              with any two *geometrically* distinct metrics (his 1 and 1') on a single analytical manifold is a non-trivial 3-index GCT tensor
              rho^i_jk.

              At this point any quibbling on your side about how different coordinate representations of the Riemann metric are supposed to constitute
              distinct "metric fields" (which is mathematical nonsense, since g_uv is a tensor object) is just whistling past the graveyard as far as
              I'm concerned.
              The above representation is for static (i.e. constant r, theta, phi) detectors. When M =/= 0 , they are covariantly accelerating in order to stay still in curved spacetime. This is Alice in Wonderland. You are not in Kansas anymore. You are not in Flatland. This is a violation of the Victorian Age engineer's common sense.
              It's also a violation of the mathematics. Levi-Civita's proof shows that the frame acceleration field, the strength of which is represented by
              a non-tensor coordinate correction term X^i_jk, is only accidentally "glued on" to the actual gravitational field, represented to first order
              by the gravitational deformation tensor G^i_jk.

              That is the whole point of the linear LC decomposition

              LC^i_jk = G^i_jk + X^i_jk

              where the gravitational deformation tensor G^i_jk is a particular instance of Levi-Civita's rho^i_jk.

              Your insistence that different coordinate representations of g_uv constitute distinct metric fields is just a fallacy. It
              ignores the tensor character of g_uv.
              Newton's First Law of Motion in Einsteinian 1916 classical geometrodynamical language generalizes to

              All covariantly non-accelerating massive test particles follow  timelike geodesics of the gravitational field.
              Splitting out the gravitational deformation tensor G^i_jk doesn't change any of that Jack. So why do you keep bringing
              this up?
              Newton's Second Law of Motion generalizes to

              D^2x^u/ds^2 = F^u/m

              where m is the invariant rest mass of the test particle (assumed here not to be ejecting mass like a rocket) and ds is the proper time of m along the worldline. F^u is the non-gravity applied force pushing the test particle off-geodesic inducing g-force on the test particle.
              All fully covariant and consistent with tensor G^i_jk. So what's your point?
              More generally, Newton's 2nd Law in curved spacetime for massive test particle motion is

              (D/ds)(mdx^u/ds) = (Dm/ds)dx^u/ds + mD^2x^u/ds^2 = F^u(non-gravity)

              Newton's 3rd law is not valid for global total momenta because translation symmetry is broken - see Noether's theorem.

              Instead we have the local stress-energy current density conservation law

              D^vTuv(non-gravity source fields curving spacetime) = 0

              2) g-force and the Levi-Civita connection field {LC}

              Back to the simplest case

              D^2x^u/ds^2 = F^u/m

              We now do a Godel self-reference and let the test particle detect itself.

              Note, the connection field (LC Christoffel symbol )usually has nothing to do with the test particle, but is a property of the detector, except in the degenerate self-referential case.
              I'm sorry Jack but IMO this is complete and utter nonsense.

              The geodesics are covariantly defined in GR and have nothing to do with "detectors" or "observers" -- except as to mere
              *appearances*. The LC connection field includes linearly independent contributions from the coordinates *and from the
              intrinsic geometry*. Otherwise, how could it be a metric compatible connnection?

              Of course a straight SR free-fall trajectory for example *looks* curved to a uniformly accelerating observer *who ignores
              his own accelerating motion*! So what? That doesn't affect the actual geometry of the geodesic in the slightest. And neither
              does curved geometry force any changes in local coordinate charts. You are completely and hopelessly confused here.
              These are two completely different animals.

              I'm not going to read any further.

              Z.




              The Baron Munchausen is weightless, no inertial g-force, in free-fall parabolic path (timelike geodesic in curved spacetime)
              He does feel g-force initially when cannon fires

              In particular think of the above SSS case in which everything can be concretely computed. Z never bothers to compute an example. This is why his thesis here is not even wrong in my opinion.

              i = 1,2,3 = spacelike components

              i = 0 is the timelike component

              Let's look only at the i = 1 radial component in the static LNIF 

              D^2r/ds^2 = d^2r/ds^2 + {LC}^ruv(dx^u/ds)(dx^v/ds)

              However, in the self-referential "diagonal" case

              dx^i/ds = 0

              dx^o/ds = 1 (c = 1)

              d^2r/ds^2 = 0

              Therefore, the g-force per unit rest mass measured on the test particle itself is

              {LC}^r00 ~ +GM(source)/r^2

              This, however, is not an objective non-tidal non-zero tensor field. It is purely contingent based on an arbitrary, though pragmatically contingent, choice since we here on Earth closely approximate this situation.

              3) Z's Red Herring of the Gravity Deformation

              Start from M = 0 (flat space time) obviously 

              {LC}^r00(M = 0) = 0 

              here the static local non-inertial observer limits to the global inertial observer - a point overlooked by Z because he does not calculate a simple example.

              Now increase to &M

              {LC}^r00(&M) = G&M/r^2

              This is not a tensor obviously.

              The difference is not a tensor, i.e.,

              {LC}^r00(0) - LC^r00(&M) = G&M/r^2

              No tensors here!

              Note the natural assumption here is the adiabatic transformation of the global static inertial frame GIF to the local static non-inertial frame LNIF. The contingent choice of convenience is "static".

              Every time I hear a philofawzer say it's "pure math" or "your argument is mathematical nonsense" I reach for my delete button. ;-)

              "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
              • Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."
              • "Imagination is more important than knowledge."
                • "Reality is merely an illusion, albeit a very persistent one."
                • "The only real valuable thing is intuition."
                  • "Anyone who has never made a mistake has never tried anything new."
                  • "Great spirits have often encountered violent opposition from weak minds."
                  • "God does not care about our mathematical difficulties. He integrates empirically."
                  • "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
                  • "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
                  • http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html

              Jack Sarfatti wrote:
              What's fundamentally wrong in z' s argument is that he does not understand the breakdown of the global frame assumption of special relativity. You simply cannot use purely formal arguments. You must specify what detectors you mean. Physics is about detector responses.

              Sent from my iPhone

              On Feb 28, 2009, at 8:17 AM, Jack Sarfatti <sarfatti@...> wrote:

              Start with flat spacetime. The lc connection field is zero for the global inertial observer.
              Now introduce a small SSS  Tuv matter source which induces an non zero lc field for the static non-inertial observer that adiabatically emerges if you impose a fixed r constraint.
              Since the difference is a tensor
              That difference must be zero otherwise you have a contradiction.

              Sent from my iPhone

              On Feb 28, 2009, at 1:15 AM, Paul Zielinski <iksnileiz@...> wrote:

              Nonsense.

              The common non-tensor coordinate part just cancels out of the difference between the two LC connections,
              which are themselves NOT GCT tensors due to the common coordinate parts that cancel.

              You are just confused.

              Z.

              Jack Sarfatti wrote:
              Problem with the deformation approach is that it does not work. Start with flat spacetime and it implies lc is a tensor.

              Back in Sf 

              Sent from my iPhone

              On Feb 27, 2009, at 1:31 PM, Jack Sarfatti <sarfatti@...> wrote:

              So you are claiming that eq 2 is within a single network of observers e.g static Lnifs in the SSS case. We add a perturbation to tuv source inducing a metric perturbation. The take difference in lc connections and that is the 3rd rank tensor?
              Why did you not say that clearly to begin with?
              In any case it is not conceptually important to the foundations of relativity.

              Only the tetrad and spin connection cartan 1-forms are intrinsic nontidal measures of the gravity field ie scalar zero rank tensors under gct aka local t4. About to take off from jfk.

              Sent from my iPhone

              On Feb 27, 2009, at 4:00 PM, Paul Zielinski <iksnileiz@...> wrote:

              You showed in detail that if you replace Levi-Civita's two different metrics 1 and 1' with the *same*
              metric represented in two different coordinate charts, you get a very different quantity when you take
              the difference of the corresponding compatible connections that obviously is not a tensor..

              Which is just another way of saying that you don't understand -- or don't want to understand -- Levi-
              Civita's proof.

              I told you what Levi-Civita's rho^i_jk means mathematically: it represents the effect of a deformation
              of the geometry of the manifold (from metric 1 to metric 1') on the Levi-Civita connection, free of
              any non-tensor contribution from the coordinates.

              In the context of GR, since in Einstein's model the net observed non-tidal gravitational field strength is
              represented by the LC connection, clearly the tensor rho^i_jk represents the effect on the non-tidal field
              strength of the change in the 4D spacetime geometry when you go from 1 to 1' -- free of any contribution
              from observer frame acceleration.

              If you take metric 1 to be geometrically Minkowski, then the difference G^i_jk between the two LC
              connections obviously represents the effect on the gravity-free LC connection field of the change from
              Minkowski geometry (n_uv) to gravitationally deformed geometry (g_uv).

              Which means that the tensor field G^i_jk(x) represents the *actual* (i.e., objective, frame-independent)
              non-tidal (first-order) gravitational field strength in GR, free of frame acceleration artifacts ("fictitious
              forces").

              Z.

              Jack Sarfatti wrote:
              There is no physics at all in that snippet eq 2.

              Sent from my iPhone at jfk boarding for Sfo 

              On Feb 26, 2009, at 5:04 PM, Paul Zielinski <iksnileiz@...> wrote:

              Actually, we are talking about Levi-Civita's idea here. He's the one with two different metrics on a single
              analytical manifold, with a common local coordinate chart around each point, under general transformations
              of which his rho^i_jk (the difference between two metric-compatible connections) transforms as a 3-index
              tensor.


              I showed in detail it is not a tensor if the usual physics is added to the naked math. It is a contingent relation between two networks of pairs of locally coincident detectors.










            • JACK SARFATTI
              ... To me it shows exactly the opposite of what you say. You do not understand the meaning of the word gravitational field in Einstein s 1915 General
              Message 6 of 26 , Mar 1, 2009

                On Mar 1, 2009, at 6:12 PM, Paul Zielinski wrote:

                What this picture illustrates is exactly what I said it does: there is no actual physical equivalence of gravitation and frame acceleration
                in a non-homogeneous field (which every physical gravitational field is, except locally).

                To me it shows exactly the opposite of what you say. You do not understand the meaning of the word "gravitational field" in Einstein's 1915 General Relativity.

                It shows that you cannot replicate such a field
                globally with frame acceleration; and neither can you globally transform such a field away with frame acceleration.

                Red Herring. Einstein never said you could. You seem confused on every major idea distinguishing Einstein's explanation of gravity from Newton's. In Einstein's theory there is no such thing as an objective local non-tidal gravity force field. There is that, of course, in Newton's theory of gravity.


                Then the Einsteinian die hards retreat to their next trench and say, "well, not globally, but *locally* you can do this. Therefore, a gravitational
                field is *locally* equivalent to frame acceleration."

                This shows how crazy you are. You are the die hard here. You are the throw back. In any case, the "Einstein die hards" as you call them control physics as a profession. You don't have a chance with your crank thesis - and rightly so in my opinion.


                But this doesn't work either, because

                (1) Local tidal effects cannot be replicated and cannot be transformed away by frame acceleration, either globally or locally, with
                the sole exception of a globally perfectly homogeneous matter-induced gravitational field -- which doesn't exist;

                Another Red Herring, Straw Man.


                (2) Measurements of non-tidal gravitational acceleration d^2r/dt^2 of test objects with respect to sources can be made completely
                "local", and since such gravitational acceleration cannot be produced or eliminated by frame acceleration, we can use such measurements
                to *locally* distinguish between a frame acceleration field and a real gravitational field.

                So the "locality" condition fails to do the job, Einstein's equivalence idea thus goes up in smoke. RED HERRING.

                A heuristically fruitful red herring, but a red herring none the less. Not to be taken seriously -- in the context of justification.

                Z.



                JACK SARFATTI wrote:

                On Mar 1, 2009, at 3:46 PM, Paul Zielinski wrote:

                Nice illustration.

                However, how is it relevant to what we have been talking about?

                Paul, your argument is so vague and shifting that I don't know what it is except to say that I do not buy your thesis that there is a 3rd rank non-tidal gravity field tensor in Einstein's 1915 GR or in any sane variation on it. The expression you suggest does not work because it has no local observable meaning in terms of possible operational definitions that an experimental physicist can do. Your formal arguments fall on deaf ears and you are banging your head against a brick wall. There is a non-tidal gravity field however, it is the set of four tetrad Cartan 1-forms e^I = e^Iue^u as explained by Rovelli in Ch 2 of his "Quantum Gravity". The set {e^I} is a set of zero-rank GCT tensors, i.e. invariant local scalar fields under iii local T4 (x) group and it is a Lorentz group first-rank tensor under ii.

                <mime-attachment.jpeg>


                You are systematically confusing the objective geometrical changes in the geodesics with changes in the
                "rest frames" of free test objects.

                No I am not. This shows you have not understood even one word of my argument. You are completely confused. The hovering static LNIF accelerating observers (rockets blasting to stand still in the gravity field)  pictured by Hawking are not the rest frames of free test objects.

                <mime-attachment.jpeg>

                Furthermore the objective changes in the timelike geodesics are seen in the change in the geodesic deviation of neighboring unaccelerating detectors. Gravity deformation is not the proper way to define the gravity field which is there even if there is no such deformation. Your position is clueless to my mind.

                What you write below has nothing to do with what I have been professing here. Nothing at all. You are on a different planet in an alternate universe not even in the same ball park.

                This allows you to pretend that you can transmute statements about
                objective spacetime geometry into statements about observer reference frames.

                You are just playing verbal tricks on yourself. Don't you get it? This is just sophistry.

                Look, everyone agrees that you cannot globally "transform away" a non-homogeneous gravitational field,
                which is the whole point of Hawking's picture.

                So what's *your* point Jack?

                Z.

                JACK SARFATTI wrote:
                PS Hawking's picture Fig 1.11 lower right should make my point obvious. Imagine changing the mass of Earth with a density, but keeping the hovering rockets at same distance from center of Earth.
                <mime-attachment.jpeg>
                On Mar 1, 2009, at 2:03 PM, JACK SARFATTI wrote:

                Paul none of this stuff had been published. What has been published has no relationship to your interpretation of what it means. Your basic methodology is flawed, which is why I say your thesis is not even wrong. You do not understand the subtle role of local frames. When you deform the gravity field by changing the source matter field configuration Tuv(matter) ---> T'uv(matter'), you also adiabatically change the local reference frames! You cannot consistently keep those frames fixed for a given kind of constraint e.g. the static observer constraint. You can imagine it mathematically, but it is not possible physically, therefore, you cannot claim, as you do claim, that your purely formal "deformation" change is the consistent basis for an actual operational definition of an intrinsic objective real locally observable non-tidal gravity field that is a GCT 3rd rank tensor. 

                As an example: deform M to M' in the SSS case for static locally accelerating detectors each at fixed r, theta, phi

                g = GM/r^2

                g' = GM'/r^2

                r is same in both cases, but g =/= g' and these g's are the actual accelerations of the local static frames in each case. They are different frames!
                You cannot subtract them and claim you have a tensor in the same frame! It is physical nonsense operationally even though you can make an abstract formal process inside your demented mind! ;-) It does not exist in the laboratory, which is what physics is all about. 

                On Mar 1, 2009, at 1:16 PM, Paul Zielinski wrote:

                JACK SARFATTI wrote:

                On Feb 28, 2009, at 8:51 PM, Paul Zielinski wrote:

                JACK SARFATTI wrote:


                On Feb 28, 2009, at 2:49 PM, Paul Zielinski wrote:
                "...it is not so easy to free oneself from the idea that coordinates must have a direct metric significance."

                - A. Einstein, "Autobiographical Notes"

                Back in San Francisco

                re: 




                Paul, Polemics are the last refuge of the man who cannot make a rational relevant refutation. ;-)
                You've already been definitively refuted Jack. Now I'm just having some fun.:-)

                That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published.
                Most of this stuff has already been published.

                If you think you can go up against Levi-Civita on the math, then I say it is you who is delusioal.

                I have given you a detailed argument that you have not refuted in a rational relevant way.

                But your argument is based on your own confusion about local coordinates in relation to curved spacetime geometry.



                Curving the 4D spacetime of 1916 GR deforms the *geometry* of the manifold (metric relations between spacetime
                points), NOT the coordinate assignments.

                Actually you are wrong about that.
                Really?
                The choice of static local non-inertial frames changes in the SSS case.
                Because the geometry changes. Of course the change of geometry affects the geodesics. What is an "LIF"
                in one geometry will not be an "LIF" in another. However, this has nothing essentially to do with coordinate
                transformations. The changes in the geodesics are geometrical.

                So you are still seriously confused. If you want to understand this, you should think about coordinate systems,
                not observer frames.
                The covariant acceleration of the static frame is GM/r^2 so when you write

                {LC(M')}^rtt - {LC(M)}^rtt = G(M' - M)/r^2

                this is a conceptual relationship between two physically different frames of reference that actually do not simultaneously exist in a real physical situation.
                The relationship we are really talking about here is between two sets of geodesics covariantly determined by different
                metrics according to the textbook standard geodesic equation.
                Your thesis is not even wrong simply garbled fragmentary ungrounded thinking in my opinion.
                Jack, it's too late for this. You just got the math wrong, that's all.

                The change from n_ij to g_ij does not force *any* changes in local coordinate charts around any given point of
                interest on the manifold. The geodesics will of course change with the geometry, but coordinate transformations
                are simply not relevant here. It is the geometry that changes, not the coordinates.

                Your belief to the contrary is what is truly delusional here. You cannot invalidate mathematical theorems with physical
                interpretations.
                I use physical detectors as local frame of reference.

                The meaning of the classical general relativity principle is the computation of local invariants based upon local measurements of two locally coincident detectors in arbitrary relative instantaneous motion measuring the same process up to quantum uncertainty limits.
                We've already deconstructed this weasel term "local". In this context it means, among other things, that you can't
                talk about the frame-invariant gravitational acceleration of test object geodesics with respect to world lines of sources
                -- which is what I call Einstein's "make-believe physics". This is like putting blinkers on a cart horse.

                The fact is that once you take account of the gravitational acceleration of test objects with respect to sources, Einstein's
                equivalence model simply evaporates. You cannot produce gravitational acceleration by accelerating an observer's frame
                of reference -- not in Newton's theory, and not in GR either. To suppose that you can is a form of insanity IMHO.
                You can leave the local coordinate charts (MAPS from R^4 to the points
                of the spacetime manifold) undisturbed (FIXED, CONSTANT, UNCHANGING, INVARIANT) while you deform
                the manifold in any local spacetime region of interest.

                In other words, you can arrange things such that under the gravitational deformation of the metric around any given spacetime
                point x you end up with the same local chart around x as you started with. The change in the intrinsic geometry doesn't force
                ANY change in the local coordinate assignments around x.

                The very concept of "gravitational deformation" is purely theoretical
                What do you think Einstein's field equations

                G_uv = kT_uv

                are all about Jack? They tell us how matter deforms the geometry of spacetime! This *is* Einstein's theory of gravity!
                and cannot actually be measured unless e.g. the Earth loses chunks of matter in a collision with a large asteroid for example. It is not a valid measure of the actual local gravitational field i.e. tetrad 1-form e^I (Rovelli's definition)
                You have to be kidding, Given a single SSS source, all you have to do is measure g, the acceleration of test particle geodesics in
                relation to the corresponding flat-spacetime geodesics, that is due to gravity. This is all determined in GR by the geodesic equation:

                <moz-screenshot-43.jpg>

                This is a generally covariant equation.

                Are you still trying to say that in GR there is no way of comparing the curved space geodesic trajectories with the corredponding flat
                space trajectories? And that there is no way of observing such changes empirically?

                That would make Einstein's theory completely useless as a theory of gravity. But of course this is nonsense.

                That is something you still don't seem to understand. Hence the Einstein quote. You have not been able to fully separate the
                concept of coordinate intervals from the concept of metric relations between points. I think this is the root of your problem,
                and is the reason why you are having so much trouble understanding Levi-Civita.
                Synopsis, since there are interesting foundational issues here ignored even in some texts, though not in Wheeler's.

                1) Every time you write a particular metric field guv(x), and its associated symmetric torsion-free Levi-Civita connection field for parallel transport of tensors along world lines in 4D spacetime you have chosen a representation in terms of a network of local detectors (AKA local frames of reference).
                Wrong. 

                Irrational response.

                All GCT coordinate representations of a given metric g_uv(x) are representations of a *single* metric field.

                Red Herring. No one says otherwise.
                You just said otherwise. What did you mean then by "a particular metric field guv(x)"?

                You seem to be saying that there is a different "metric field" for every GCT that changes the components of the metric tensor.

                I have said nothing of the kind.
                It sounded like you were saying this.
                You simply do not get the point I am making because your mind is closed on this silly idea you have. It's completely silly - embarrassing in fact to my mind at least.
                That changes in the geometry do not force changes in the local coordinate charts in GR? You think that's a silly idea?

                It is a mathematical fact with the consequence that the Einstein field, represented by LC^i_jk(x), is decomposable into an
                actual gravitational field represented by a tensor G^i_jk, and a non-tensor X^i_jk which accounts for all non-tensor coordinate
                contributions to the LC connection field:

                LC^i_jk = G^i_jk + X^i_jk

                Which means that in 1916 GR, frame acceleration fields are every bit as fictitious in relation to actual gravitational fields as
                they are in Newton's theory in relation to actual Newtonian forces.

                And you think this is a "silly idea"?
                The metric fields guv & g'uv induced by distinct sources Tuv & T'uv are not related by a GCT. They are different intrinsic gravity field configurations and they have different sets of static observers (outside event horizons) and, therefore, their formal difference has no direct physical meaning, i.e. no measurement procedures exist.  They cannot simultaneously exist physically.

                If that is what you are saying, then you are missing the whole point of the metric being a GCT tensor.

                Of course that is NOT what I have been saying.
                OK. Good.

                Then you should have no trouble understanding why the difference between an L:C connection compatible with a curved metric and
                the LC connection compatible with a flat-level Minkowski metric is a tensor.
                You are also missing the
                point of defining affine connections on curved manifolds that correct the partial derivatives of tensors for coordinate artifacts.

                Changing the coordinate representation of the metric doesn't affect the intrinsic spacetime geometry.

                You are seriously confused about this. Your Einsteinian optics are preventing you from understanding the actual math.
                The simplest "hydrogen atom" (as it were) example in Einstein's 1915 GR is the SSS vacuum solution, written in the texts as

                g00 = gtt = - 1/g11 = - 1/grr = 1 - rs/r

                rs/r > 1

                rs = 2GM(source)/c^2

                Area of concentric sphere is A = 4pir^2

                usual spherical polar & azimuthal coordinate line element added.

                You can use any local coordinate chart around any given (well-behaved) point of interest, and hold that local chart invariant
                under a gravitational deformation of the manifold.

                You are wrong.
                That is your delusion Jack.
                Physically you need to specify a set (fiber bundle) of local tangent frames that in principle are implementable by actual detectors.
                In Einstein's coordinate frame model, which is a relic of 1905 SR, yes.
                Here the choice of static frames (where they exist) shows my point very clearly.
                This is where you get really confused. Of course changes in the spacetime geometry will change the classification of trajectories,
                since the geodesics chang -- but this happens regardless of whether you hold the local coordinate charts invariant under the
                deformation of the geometry, or not.

                Of course you can say that the *rest* frames of freely falling test objects will change, and that this is associated with a change in
                the coordinate charts associated with the rest frames, but this is immaterial and irrelevant to the point I'm making. It is
                a trivial observation. There is nothing that compels us to observe a freely falling test object from its own rest frame, regardless
                of changes in the geometry.

                You might as well observe that the rest frame of any object changes when you accelerate the object. So what?
                This completely eludes you.
                What eludes me is why you think this matters. It's a tautology. Trivial.
                It is the fundamental error you have made. You are thinking like a mathematician and completely miss the physics.
                If what you mean here by "physics" is the completely trivial observation that the *rest* frames of freely falling test objects change when
                the geodesics change, then I think I am quite correct in ignoring it. It is of no consequence. It is immaterial to my argument.
                It is trivial and irrelevant. It adds nothing of any substance to the fact that when the spacetime geometry changes, the geodesics change.

                This is why I think it's better to think about mathematical coordinates instead of observer frames -- to avoid these kinds of pitfalls. I think
                this is a good example of the kind of confusion that Einstein's coordinate frame model can generate in the mind of a physicist.
                Different source configurations have physically different bundles of e.g. static local frames that are the basis for the contingent representation chosen for pragmatic convenience only.
                As far as I'm concerned, this is just another way of saying that the geodesics change wen the spacetime geometry changes.
                You are simply substituting pseudo-relativistic talk about "LIFs" and "LNIFs" for objective talk about covariantly determined
                gravitationally deformed geodesics. Your favorite card trick!
                In any case I stop reading here and end the discussion.
                I think it's time to move on. Let's talk about gauge theory.
                Think what you like but

                1) No journal will accept your idea and rightly so.
                Most of this has *already* been published Jack. Some of it -- like Levi-Civita's theorem -- many years ago.

                Poltorak has already published his stuff, working with a metric-affine model.

                A. A. Logunov has published reams on this. Like V. Fock, he says: "No such thing as general relativity".

                No one serious in this field actually believes in classic Einstein equivalence anymore Jack. Just look at the most
                recent text books on GR.

                2) No one pays any attention to it (except me here) and rightly so.

                That's my opinion.
                And you are perfectly entitled to your opinions. You are free to make all your own mistakes.

                Z.




                You are systematically confusing actual geometric changes in the first partial derivatives g_ij, k with coordinate change.

                The primary purpose of constructing an affine connection is to correct the derivatives of the metric for coordinate artifacts.

                So the premise here is that a distinction must be made between coordinate contributions and actual geometric contributions
                to the partial derivatives of tensors. This also applies to the metric tensor g_uv!

                Now Paul you look at the above as a purely mathematical problem.
                Nonsense. You cannot invalidate a mathematical theorem by attaching a physical interpretation to the math. To suggest that you
                can is completely wrong headed IMO.

                You have to distinguish mathematically between actual geometric changes and mere coordinate changes in the partial derivatives
                of tensors, including those of the metric tensor.

                That is what the affine deformation model I've been trying to explain to you does so nicely  -- for the metric tensor g_uv and its
                associated metric-compatible LC connection.
                That is a profound error of conceptualization that you as The Profundus Philofawzer of Physics Extraordinaire ;-) ought to be well aware of - the "informal language" (David Bohm) the interpretation of the naked mathematique.
                Look Jack, you just got the math wrong, that's all. Do you want to repair the situation, or not?

                As far as I'm concerned both the the mathematical issues *and* the issues of physical interpretation are now completely settled,
                based on

                (1) Einstein's 1917 interpretation of the LC connection field LC^i_jk(x) as representing the non-tidal field strength in GR;

                (2) Levi-Civita's authoritative proof that the difference between two LC connections (his 2 and 2') respectively compatible
                with any two *geometrically* distinct metrics (his 1 and 1') on a single analytical manifold is a non-trivial 3-index GCT tensor
                rho^i_jk.

                At this point any quibbling on your side about how different coordinate representations of the Riemann metric are supposed to constitute
                distinct "metric fields" (which is mathematical nonsense, since g_uv is a tensor object) is just whistling past the graveyard as far as
                I'm concerned.
                The above representation is for static (i.e. constant r, theta, phi) detectors. When M =/= 0 , they are covariantly accelerating in order to stay still in curved spacetime. This is Alice in Wonderland. You are not in Kansas anymore. You are not in Flatland. This is a violation of the Victorian Age engineer's common sense.
                It's also a violation of the mathematics. Levi-Civita's proof shows that the frame acceleration field, the strength of which is represented by
                a non-tensor coordinate correction term X^i_jk, is only accidentally "glued on" to the actual gravitational field, represented to first order
                by the gravitational deformation tensor G^i_jk.

                That is the whole point of the linear LC decomposition

                LC^i_jk = G^i_jk + X^i_jk

                where the gravitational deformation tensor G^i_jk is a particular instance of Levi-Civita's rho^i_jk.

                Your insistence that different coordinate representations of g_uv constitute distinct metric fields is just a fallacy. It
                ignores the tensor character of g_uv.
                Newton's First Law of Motion in Einsteinian 1916 classical geometrodynamical language generalizes to

                All covariantly non-accelerating massive test particles follow  timelike geodesics of the gravitational field.
                Splitting out the gravitational deformation tensor G^i_jk doesn't change any of that Jack. So why do you keep bringing
                this up?
                Newton's Second Law of Motion generalizes to

                D^2x^u/ds^2 = F^u/m

                where m is the invariant rest mass of the test particle (assumed here not to be ejecting mass like a rocket) and ds is the proper time of m along the worldline. F^u is the non-gravity applied force pushing the test particle off-geodesic inducing g-force on the test particle.
                All fully covariant and consistent with tensor G^i_jk. So what's your point?
                More generally, Newton's 2nd Law in curved spacetime for massive test particle motion is

                (D/ds)(mdx^u/ds) = (Dm/ds)dx^u/ds + mD^2x^u/ds^2 = F^u(non-gravity)

                Newton's 3rd law is not valid for global total momenta because translation symmetry is broken - see Noether's theorem.

                Instead we have the local stress-energy current density conservation law

                D^vTuv(non-gravity source fields curving spacetime) = 0

                2) g-force and the Levi-Civita connection field {LC}

                Back to the simplest case

                D^2x^u/ds^2 = F^u/m

                We now do a Godel self-reference and let the test particle detect itself.

                Note, the connection field (LC Christoffel symbol )usually has nothing to do with the test particle, but is a property of the detector, except in the degenerate self-referential case.
                I'm sorry Jack but IMO this is complete and utter nonsense.

                The geodesics are covariantly defined in GR and have nothing to do with "detectors" or "observers" -- except as to mere
                *appearances*. The LC connection field includes linearly independent contributions from the coordinates *and from the
                intrinsic geometry*. Otherwise, how could it be a metric compatible connnection?

                Of course a straight SR free-fall trajectory for example *looks* curved to a uniformly accelerating observer *who ignores
                his own accelerating motion*! So what? That doesn't affect the actual geometry of the geodesic in the slightest. And neither
                does curved geometry force any changes in local coordinate charts. You are completely and hopelessly confused here.
                These are two completely different animals.

                I'm not going to read any further.

                Z.




                The Baron Munchausen is weightless, no inertial g-force, in free-fall parabolic path (timelike geodesic in curved spacetime)
                He does feel g-force initially when cannon fires

                In particular think of the above SSS case in which everything can be concretely computed. Z never bothers to compute an example. This is why his thesis here is not even wrong in my opinion.

                i = 1,2,3 = spacelike components

                i = 0 is the timelike component

                Let's look only at the i = 1 radial component in the static LNIF 

                D^2r/ds^2 = d^2r/ds^2 + {LC}^ruv(dx^u/ds)(dx^v/ds)

                However, in the self-referential "diagonal" case

                dx^i/ds = 0

                dx^o/ds = 1 (c = 1)

                d^2r/ds^2 = 0

                Therefore, the g-force per unit rest mass measured on the test particle itself is

                {LC}^r00 ~ +GM(source)/r^2

                This, however, is not an objective non-tidal non-zero tensor field. It is purely contingent based on an arbitrary, though pragmatically contingent, choice since we here on Earth closely approximate this situation.

                3) Z's Red Herring of the Gravity Deformation

                Start from M = 0 (flat space time) obviously 

                {LC}^r00(M = 0) = 0 

                here the static local non-inertial observer limits to the global inertial observer - a point overlooked by Z because he does not calculate a simple example.

                Now increase to &M

                {LC}^r00(&M) = G&M/r^2

                This is not a tensor obviously.

                The difference is not a tensor, i.e.,

                {LC}^r00(0) - LC^r00(&M) = G&M/r^2

                No tensors here!

                Note the natural assumption here is the adiabatic transformation of the global static inertial frame GIF to the local static non-inertial frame LNIF. The contingent choice of convenience is "static".

                Every time I hear a philofawzer say it's "pure math" or "your argument is mathematical nonsense" I reach for my delete button. ;-)

                "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
                • Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."
                • "Imagination is more important than knowledge."
                  • "Reality is merely an illusion, albeit a very persistent one."
                  • "The only real valuable thing is intuition."
                    • "Anyone who has never made a mistake has never tried anything new."
                    • "Great spirits have often encountered violent opposition from weak minds."
                    • "God does not care about our mathematical difficulties. He integrates empirically."
                    • "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
                    • "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
                    • http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html

                Jack Sarfatti wrote:
                What's fundamentally wrong in z' s argument is that he does not understand the breakdown of the global frame assumption of special relativity. You simply cannot use purely formal arguments. You must specify what detectors you mean. Physics is about detector responses.

                Sent from my iPhone

                On Feb 28, 2009, at 8:17 AM, Jack Sarfatti <sarfatti@...> wrote:

                Start with flat spacetime. The lc connection field is zero for the global inertial observer.
                Now introduce a small SSS  Tuv matter source which induces an non zero lc field for the static non-inertial observer that adiabatically emerges if you impose a fixed r constraint.
                Since the difference is a tensor
                That difference must be zero otherwise you have a contradiction.

                Sent from my iPhone

                On Feb 28, 2009, at 1:15 AM, Paul Zielinski <iksnileiz@...> wrote:

                Nonsense.

                The common non-tensor coordinate part just cancels out of the difference between the two LC connections,
                which are themselves NOT GCT tensors due to the common coordinate parts that cancel.

                You are just confused.

                Z.

                Jack Sarfatti wrote:
                Problem with the deformation approach is that it does not work. Start with flat spacetime and it implies lc is a tensor.

                Back in Sf 

                Sent from my iPhone

                On Feb 27, 2009, at 1:31 PM, Jack Sarfatti <sarfatti@...> wrote:

                So you are claiming that eq 2 is within a single network of observers e.g static Lnifs in the SSS case. We add a perturbation to tuv source inducing a metric perturbation. The take difference in lc connections and that is the 3rd rank tensor?
                Why did you not say that clearly to begin with?
                In any case it is not conceptually important to the foundations of relativity.

                Only the tetrad and spin connection cartan 1-forms are intrinsic nontidal measures of the gravity field ie scalar zero rank tensors under gct aka local t4. About to take off from jfk.

                Sent from my iPhone

                On Feb 27, 2009, at 4:00 PM, Paul Zielinski <iksnileiz@...> wrote:

                You showed in detail that if you replace Levi-Civita's two different metrics 1 and 1' with the *same*
                metric represented in two different coordinate charts, you get a very different quantity when you take
                the difference of the corresponding compatible connections that obviously is not a tensor..

                Which is just another way of saying that you don't understand -- or don't want to understand -- Levi-
                Civita's proof.

                I told you what Levi-Civita's rho^i_jk means mathematically: it represents the effect of a deformation
                of the geometry of the manifold (from metric 1 to metric 1') on the Levi-Civita connection, free of
                any non-tensor contribution from the coordinates.

                In the context of GR, since in Einstein's model the net observed non-tidal gravitational field strength is
                represented by the LC connection, clearly the tensor rho^i_jk represents the effect on the non-tidal field
                strength of the change in the 4D spacetime geometry when you go from 1 to 1' -- free of any contribution
                from observer frame acceleration.

                If you take metric 1 to be geometrically Minkowski, then the difference G^i_jk between the two LC
                connections obviously represents the effect on the gravity-free LC connection field of the change from
                Minkowski geometry (n_uv) to gravitationally deformed geometry (g_uv).

                Which means that the tensor field G^i_jk(x) represents the *actual* (i.e., objective, frame-independent)
                non-tidal (first-order) gravitational field strength in GR, free of frame acceleration artifacts ("fictitious
                forces").

                Z.

                Jack Sarfatti wrote:
                There is no physics at all in that snippet eq 2.

                Sent from my iPhone at jfk boarding for Sfo 

                On Feb 26, 2009, at 5:04 PM, Paul Zielinski <iksnileiz@...> wrote:

                Actually, we are talking about Levi-Civita's idea here. He's the one with two different metrics on a single
                analytical manifold, with a common local coordinate chart around each point, under general transformations
                of which his rho^i_jk (the difference between two metric-compatible connections) transforms as a 3-index
                tensor.


                I showed in detail it is not a tensor if the usual physics is added to the naked math. It is a contingent relation between two networks of pairs of locally coincident detectors.













              • JACK SARFATTI
                you are chasing your tail again this is not interesting
                Message 7 of 26 , Mar 1, 2009
                  you are chasing your tail again
                  this is not interesting 
                  On Mar 1, 2009, at 8:10 PM, Paul Zielinski wrote:

                  JACK SARFATTI wrote:

                  On Mar 1, 2009, at 6:12 PM, Paul Zielinski wrote:

                  What this picture illustrates is exactly what I said it does: there is no actual physical equivalence of gravitation and frame acceleration
                  in a non-homogeneous field (which every physical gravitational field is, except locally).

                  To me it shows exactly the opposite of what you say. You do not understand the meaning of the word "gravitational field" in Einstein's 1915 General Relativity.

                  It shows that you cannot replicate such a field
                  globally with frame acceleration; and neither can you globally transform such a field away with frame acceleration.

                  Red Herring. Einstein never said you could.
                  I didn't say that Einstein ever said you could do this. After all, Einstein was no dummy.

                  What he did do is consider the case of an imaginary  perfectly homogeneous gravitational field, comparing it to a uniformly
                  accelerating frame of reference.

                  Now, there is in fact no such thing as a matter-induced *globally* homogeneous gravitational field.

                  So according to you, I suppose, his famous thought experiment "has no operation meaning"? :-)
                  You seem confused on every major idea distinguishing Einstein's explanation of gravity from Newton's. In Einstein's theory there is no such thing as an objective local non-tidal gravity force field.
                  That's right, but the only reason for this is Einstein's version of the equivalence principle -- which as it turns out is
                  empirically false, and not only that, it has no actual support in the mathematical framework of GR!

                  Dump Einstein's equivalence idea (I mean the one that literally identifies the intrinsic physical nature of gravitation
                  and frame acceleration) and the whole mess clears up Everything is then fully covariant (described by the tensors G^i_jk
                  and R^i_jkl) and we recover the traditional fundamental distinction between fictitious frame acceleration fields, on the
                  one hand, and actual non-tidal gravitational fields, on the other.

                  Two very different animals. Even in GR.
                  There is that, of course, in Newton's theory of gravity.
                  Same in GR, once you put Einstein's "white rabbit" back into the magician's hat.

                  Then the Einsteinian die hards retreat to their next trench and say, "well, not globally, but *locally* you can do this. Therefore, a gravitational
                  field is *locally* equivalent to frame acceleration."

                  This shows how crazy you are. You are the die hard here. You are the throw back. In any case, the "Einstein die hards" as you call them control physics as a profession.
                  I think this is going to change.
                  You don't have a chance with your crank thesis - and rightly so in my opinion.
                  The equivalence principle is empirically false -- you can locally distinguish non-tidal gravitational attraction from frame acceleration by
                  empirical means. This is because you cannot produce non-tidal gravitational attraction of free test particle geodesics with respect to sources
                  with frame acceleration! You can't do it Jack. Apples and oranges! And this is fully local, empirical, and operational.

                  As the relativist J. L. Synge famously wrote, to the extent that Einstein's equivalence principle (even in its "local" version) is non-trivial,
                  it is empirically false!
                  But this doesn't work either, because

                  (1) Local tidal effects cannot be replicated and cannot be transformed away by frame acceleration, either globally or locally, with
                  the sole exception of a globally perfectly homogeneous matter-induced gravitational field -- which doesn't exist;

                  Another Red Herring, Straw Man.


                  (2) Measurements of non-tidal gravitational acceleration d^2r/dt^2 of test objects with respect to sources can be made completely
                  "local", and since such gravitational acceleration cannot be produced or eliminated by frame acceleration, we can use such measurements
                  to *locally* distinguish between a frame acceleration field and a real gravitational field.

                  So the "locality" condition fails to do the job, Einstein's equivalence idea thus goes up in smoke. RED HERRING.

                  A heuristically fruitful red herring, but a red herring none the less. Not to be taken seriously -- in the context of justification.

                  Z.
                  No response to (2) from Jack. Probably because he doesn't have one?

                  It should be obvious that you can *locally* measure the acceleration of free test objects with respect to a typical gravitational source as
                  they approach the source boundary -- and this is a frame-invariant empirically measurable quantity that cannot be produced or eliminated,
                  or augmented or diminshed, by accelerating any observer's frame of reference!

                  So, Einstein equivalence as written is *empirically* false! There is in fact no "local equivalence" of frame acceleration and actual gravitation.

                  The Emperor, as they say, has no clothes.

                  Z



                  JACK SARFATTI wrote:

                  On Mar 1, 2009, at 3:46 PM, Paul Zielinski wrote:

                  Nice illustration.

                  However, how is it relevant to what we have been talking about?

                  Paul, your argument is so vague and shifting that I don't know what it is except to say that I do not buy your thesis that there is a 3rd rank non-tidal gravity field tensor in Einstein's 1915 GR or in any sane variation on it. The expression you suggest does not work because it has no local observable meaning in terms of possible operational definitions that an experimental physicist can do. Your formal arguments fall on deaf ears and you are banging your head against a brick wall. There is a non-tidal gravity field however, it is the set of four tetrad Cartan 1-forms e^I = e^Iue^u as explained by Rovelli in Ch 2 of his "Quantum Gravity". The set {e^I} is a set of zero-rank GCT tensors, i.e. invariant local scalar fields under iii local T4 (x) group and it is a Lorentz group first-rank tensor under ii.

                  <mime-attachment.jpeg>


                  You are systematically confusing the objective geometrical changes in the geodesics with changes in the
                  "rest frames" of free test objects.

                  No I am not. This shows you have not understood even one word of my argument. You are completely confused. The hovering static LNIF accelerating observers (rockets blasting to stand still in the gravity field)  pictured by Hawking are not the rest frames of free test objects.

                  <mime-attachment.jpeg>

                  Furthermore the objective changes in the timelike geodesics are seen in the change in the geodesic deviation of neighboring unaccelerating detectors. Gravity deformation is not the proper way to define the gravity field which is there even if there is no such deformation. Your position is clueless to my mind.

                  What you write below has nothing to do with what I have been professing here. Nothing at all. You are on a different planet in an alternate universe not even in the same ball park.

                  This allows you to pretend that you can transmute statements about
                  objective spacetime geometry into statements about observer reference frames.

                  You are just playing verbal tricks on yourself. Don't you get it? This is just sophistry.

                  Look, everyone agrees that you cannot globally "transform away" a non-homogeneous gravitational field,
                  which is the whole point of Hawking's picture.

                  So what's *your* point Jack?

                  Z.

                  JACK SARFATTI wrote:
                  PS Hawking's picture Fig 1.11 lower right should make my point obvious. Imagine changing the mass of Earth with a density, but keeping the hovering rockets at same distance from center of Earth.
                  <mime-attachment.jpeg>
                  On Mar 1, 2009, at 2:03 PM, JACK SARFATTI wrote:

                  Paul none of this stuff had been published. What has been published has no relationship to your interpretation of what it means. Your basic methodology is flawed, which is why I say your thesis is not even wrong. You do not understand the subtle role of local frames. When you deform the gravity field by changing the source matter field configuration Tuv(matter) ---> T'uv(matter'), you also adiabatically change the local reference frames! You cannot consistently keep those frames fixed for a given kind of constraint e.g. the static observer constraint. You can imagine it mathematically, but it is not possible physically, therefore, you cannot claim, as you do claim, that your purely formal "deformation" change is the consistent basis for an actual operational definition of an intrinsic objective real locally observable non-tidal gravity field that is a GCT 3rd rank tensor. 

                  As an example: deform M to M' in the SSS case for static locally accelerating detectors each at fixed r, theta, phi

                  g = GM/r^2

                  g' = GM'/r^2

                  r is same in both cases, but g =/= g' and these g's are the actual accelerations of the local static frames in each case. They are different frames!
                  You cannot subtract them and claim you have a tensor in the same frame! It is physical nonsense operationally even though you can make an abstract formal process inside your demented mind! ;-) It does not exist in the laboratory, which is what physics is all about. 

                  On Mar 1, 2009, at 1:16 PM, Paul Zielinski wrote:

                  JACK SARFATTI wrote:

                  On Feb 28, 2009, at 8:51 PM, Paul Zielinski wrote:

                  JACK SARFATTI wrote:


                  On Feb 28, 2009, at 2:49 PM, Paul Zielinski wrote:
                  "...it is not so easy to free oneself from the idea that coordinates must have a direct metric significance."

                  - A. Einstein, "Autobiographical Notes"

                  Back in San Francisco

                  re: 




                  Paul, Polemics are the last refuge of the man who cannot make a rational relevant refutation. ;-)
                  You've already been definitively refuted Jack. Now I'm just having some fun.:-)

                  That is delusional Paul. You have not thought deeply about what you are saying. In any case try to get your idea published.
                  Most of this stuff has already been published.

                  If you think you can go up against Levi-Civita on the math, then I say it is you who is delusioal.

                  I have given you a detailed argument that you have not refuted in a rational relevant way.

                  But your argument is based on your own confusion about local coordinates in relation to curved spacetime geometry.



                  Curving the 4D spacetime of 1916 GR deforms the *geometry* of the manifold (metric relations between spacetime
                  points), NOT the coordinate assignments.

                  Actually you are wrong about that.
                  Really?
                  The choice of static local non-inertial frames changes in the SSS case.
                  Because the geometry changes. Of course the change of geometry affects the geodesics. What is an "LIF"
                  in one geometry will not be an "LIF" in another. However, this has nothing essentially to do with coordinate
                  transformations. The changes in the geodesics are geometrical.

                  So you are still seriously confused. If you want to understand this, you should think about coordinate systems,
                  not observer frames.
                  The covariant acceleration of the static frame is GM/r^2 so when you write

                  {LC(M')}^rtt - {LC(M)}^rtt = G(M' - M)/r^2

                  this is a conceptual relationship between two physically different frames of reference that actually do not simultaneously exist in a real physical situation.
                  The relationship we are really talking about here is between two sets of geodesics covariantly determined by different
                  metrics according to the textbook standard geodesic equation.
                  Your thesis is not even wrong simply garbled fragmentary ungrounded thinking in my opinion.
                  Jack, it's too late for this. You just got the math wrong, that's all.

                  The change from n_ij to g_ij does not force *any* changes in local coordinate charts around any given point of
                  interest on the manifold. The geodesics will of course change with the geometry, but coordinate transformations
                  are simply not relevant here. It is the geometry that changes, not the coordinates.

                  Your belief to the contrary is what is truly delusional here. You cannot invalidate mathematical theorems with physical
                  interpretations.
                  I use physical detectors as local frame of reference.

                  The meaning of the classical general relativity principle is the computation of local invariants based upon local measurements of two locally coincident detectors in arbitrary relative instantaneous motion measuring the same process up to quantum uncertainty limits.
                  We've already deconstructed this weasel term "local". In this context it means, among other things, that you can't
                  talk about the frame-invariant gravitational acceleration of test object geodesics with respect to world lines of sources
                  -- which is what I call Einstein's "make-believe physics". This is like putting blinkers on a cart horse.

                  The fact is that once you take account of the gravitational acceleration of test objects with respect to sources, Einstein's
                  equivalence model simply evaporates. You cannot produce gravitational acceleration by accelerating an observer's frame
                  of reference -- not in Newton's theory, and not in GR either. To suppose that you can is a form of insanity IMHO.
                  You can leave the local coordinate charts (MAPS from R^4 to the points
                  of the spacetime manifold) undisturbed (FIXED, CONSTANT, UNCHANGING, INVARIANT) while you deform
                  the manifold in any local spacetime region of interest.

                  In other words, you can arrange things such that under the gravitational deformation of the metric around any given spacetime
                  point x you end up with the same local chart around x as you started with. The change in the intrinsic geometry doesn't force
                  ANY change in the local coordinate assignments around x.

                  The very concept of "gravitational deformation" is purely theoretical
                  What do you think Einstein's field equations

                  G_uv = kT_uv

                  are all about Jack? They tell us how matter deforms the geometry of spacetime! This *is* Einstein's theory of gravity!
                  and cannot actually be measured unless e.g. the Earth loses chunks of matter in a collision with a large asteroid for example. It is not a valid measure of the actual local gravitational field i.e. tetrad 1-form e^I (Rovelli's definition)
                  You have to be kidding, Given a single SSS source, all you have to do is measure g, the acceleration of test particle geodesics in
                  relation to the corresponding flat-spacetime geodesics, that is due to gravity. This is all determined in GR by the geodesic equation:

                  <moz-screenshot-43.jpg>

                  This is a generally covariant equation.

                  Are you still trying to say that in GR there is no way of comparing the curved space geodesic trajectories with the corredponding flat
                  space trajectories? And that there is no way of observing such changes empirically?

                  That would make Einstein's theory completely useless as a theory of gravity. But of course this is nonsense.

                  That is something you still don't seem to understand. Hence the Einstein quote. You have not been able to fully separate the
                  concept of coordinate intervals from the concept of metric relations between points. I think this is the root of your problem,
                  and is the reason why you are having so much trouble understanding Levi-Civita.
                  Synopsis, since there are interesting foundational issues here ignored even in some texts, though not in Wheeler's.

                  1) Every time you write a particular metric field guv(x), and its associated symmetric torsion-free Levi-Civita connection field for parallel transport of tensors along world lines in 4D spacetime you have chosen a representation in terms of a network of local detectors (AKA local frames of reference).
                  Wrong. 

                  Irrational response.

                  All GCT coordinate representations of a given metric g_uv(x) are representations of a *single* metric field.

                  Red Herring. No one says otherwise.
                  You just said otherwise. What did you mean then by "a particular metric field guv(x)"?

                  You seem to be saying that there is a different "metric field" for every GCT that changes the components of the metric tensor.

                  I have said nothing of the kind.
                  It sounded like you were saying this.
                  You simply do not get the point I am making because your mind is closed on this silly idea you have. It's completely silly - embarrassing in fact to my mind at least.
                  That changes in the geometry do not force changes in the local coordinate charts in GR? You think that's a silly idea?

                  It is a mathematical fact with the consequence that the Einstein field, represented by LC^i_jk(x), is decomposable into an
                  actual gravitational field represented by a tensor G^i_jk, and a non-tensor X^i_jk which accounts for all non-tensor coordinate
                  contributions to the LC connection field:

                  LC^i_jk = G^i_jk + X^i_jk

                  Which means that in 1916 GR, frame acceleration fields are every bit as fictitious in relation to actual gravitational fields as
                  they are in Newton's theory in relation to actual Newtonian forces.

                  And you think this is a "silly idea"?
                  The metric fields guv & g'uv induced by distinct sources Tuv & T'uv are not related by a GCT. They are different intrinsic gravity field configurations and they have different sets of static observers (outside event horizons) and, therefore, their formal difference has no direct physical meaning, i.e. no measurement procedures exist.  They cannot simultaneously exist physically.

                  If that is what you are saying, then you are missing the whole point of the metric being a GCT tensor.

                  Of course that is NOT what I have been saying.
                  OK. Good.

                  Then you should have no trouble understanding why the difference between an L:C connection compatible with a curved metric and
                  the LC connection compatible with a flat-level Minkowski metric is a tensor.
                  You are also missing the
                  point of defining affine connections on curved manifolds that correct the partial derivatives of tensors for coordinate artifacts.

                  Changing the coordinate representation of the metric doesn't affect the intrinsic spacetime geometry.

                  You are seriously confused about this. Your Einsteinian optics are preventing you from understanding the actual math.
                  The simplest "hydrogen atom" (as it were) example in Einstein's 1915 GR is the SSS vacuum solution, written in the texts as

                  g00 = gtt = - 1/g11 = - 1/grr = 1 - rs/r

                  rs/r > 1

                  rs = 2GM(source)/c^2

                  Area of concentric sphere is A = 4pir^2

                  usual spherical polar & azimuthal coordinate line element added.

                  You can use any local coordinate chart around any given (well-behaved) point of interest, and hold that local chart invariant
                  under a gravitational deformation of the manifold.

                  You are wrong.
                  That is your delusion Jack.
                  Physically you need to specify a set (fiber bundle) of local tangent frames that in principle are implementable by actual detectors.
                  In Einstein's coordinate frame model, which is a relic of 1905 SR, yes.
                  Here the choice of static frames (where they exist) shows my point very clearly.
                  This is where you get really confused. Of course changes in the spacetime geometry will change the classification of trajectories,
                  since the geodesics chang -- but this happens regardless of whether you hold the local coordinate charts invariant under the
                  deformation of the geometry, or not.

                  Of course you can say that the *rest* frames of freely falling test objects will change, and that this is associated with a change in
                  the coordinate charts associated with the rest frames, but this is immaterial and irrelevant to the point I'm making. It is
                  a trivial observation. There is nothing that compels us to observe a freely falling test object from its own rest frame, regardless
                  of changes in the geometry.

                  You might as well observe that the rest frame of any object changes when you accelerate the object. So what?
                  This completely eludes you.
                  What eludes me is why you think this matters. It's a tautology. Trivial.
                  It is the fundamental error you have made. You are thinking like a mathematician and completely miss the physics.
                  If what you mean here by "physics" is the completely trivial observation that the *rest* frames of freely falling test objects change when
                  the geodesics change, then I think I am quite correct in ignoring it. It is of no consequence. It is immaterial to my argument.
                  It is trivial and irrelevant. It adds nothing of any substance to the fact that when the spacetime geometry changes, the geodesics change.

                  This is why I think it's better to think about mathematical coordinates instead of observer frames -- to avoid these kinds of pitfalls. I think
                  this is a good example of the kind of confusion that Einstein's coordinate frame model can generate in the mind of a physicist.
                  Different source configurations have physically different bundles of e.g. static local frames that are the basis for the contingent representation chosen for pragmatic convenience only.
                  As far as I'm concerned, this is just another way of saying that the geodesics change wen the spacetime geometry changes.
                  You are simply substituting pseudo-relativistic talk about "LIFs" and "LNIFs" for objective talk about covariantly determined
                  gravitationally deformed geodesics. Your favorite card trick!
                  In any case I stop reading here and end the discussion.
                  I think it's time to move on. Let's talk about gauge theory.
                  Think what you like but

                  1) No journal will accept your idea and rightly so.
                  Most of this has *already* been published Jack. Some of it -- like Levi-Civita's theorem -- many years ago.

                  Poltorak has already published his stuff, working with a metric-affine model.

                  A. A. Logunov has published reams on this. Like V. Fock, he says: "No such thing as general relativity".

                  No one serious in this field actually believes in classic Einstein equivalence anymore Jack. Just look at the most
                  recent text books on GR.

                  2) No one pays any attention to it (except me here) and rightly so.

                  That's my opinion.
                  And you are perfectly entitled to your opinions. You are free to make all your own mistakes.

                  Z.




                  You are systematically confusing actual geometric changes in the first partial derivatives g_ij, k with coordinate change.

                  The primary purpose of constructing an affine connection is to correct the derivatives of the metric for coordinate artifacts.

                  So the premise here is that a distinction must be made between coordinate contributions and actual geometric contributions
                  to the partial derivatives of tensors. This also applies to the metric tensor g_uv!

                  Now Paul you look at the above as a purely mathematical problem.
                  Nonsense. You cannot invalidate a mathematical theorem by attaching a physical interpretation to the math. To suggest that you
                  can is completely wrong headed IMO.

                  You have to distinguish mathematically between actual geometric changes and mere coordinate changes in the partial derivatives
                  of tensors, including those of the metric tensor.

                  That is what the affine deformation model I've been trying to explain to you does so nicely  -- for the metric tensor g_uv and its
                  associated metric-compatible LC connection.
                  That is a profound error of conceptualization that you as The Profundus Philofawzer of Physics Extraordinaire ;-) ought to be well aware of - the "informal language" (David Bohm) the interpretation of the naked mathematique.
                  Look Jack, you just got the math wrong, that's all. Do you want to repair the situation, or not?

                  As far as I'm concerned both the the mathematical issues *and* the issues of physical interpretation are now completely settled,
                  based on

                  (1) Einstein's 1917 interpretation of the LC connection field LC^i_jk(x) as representing the non-tidal field strength in GR;

                  (2) Levi-Civita's authoritative proof that the difference between two LC connections (his 2 and 2') respectively compatible
                  with any two *geometrically* distinct metrics (his 1 and 1') on a single analytical manifold is a non-trivial 3-index GCT tensor
                  rho^i_jk.

                  At this point any quibbling on your side about how different coordinate representations of the Riemann metric are supposed to constitute
                  distinct "metric fields" (which is mathematical nonsense, since g_uv is a tensor object) is just whistling past the graveyard as far as
                  I'm concerned.
                  The above representation is for static (i.e. constant r, theta, phi) detectors. When M =/= 0 , they are covariantly accelerating in order to stay still in curved spacetime. This is Alice in Wonderland. You are not in Kansas anymore. You are not in Flatland. This is a violation of the Victorian Age engineer's common sense.
                  It's also a violation of the mathematics. Levi-Civita's proof shows that the frame acceleration field, the strength of which is represented by
                  a non-tensor coordinate correction term X^i_jk, is only accidentally "glued on" to the actual gravitational field, represented to first order
                  by the gravitational deformation tensor G^i_jk.

                  That is the whole point of the linear LC decomposition

                  LC^i_jk = G^i_jk + X^i_jk

                  where the gravitational deformation tensor G^i_jk is a particular instance of Levi-Civita's rho^i_jk.

                  Your insistence that different coordinate representations of g_uv constitute distinct metric fields is just a fallacy. It
                  ignores the tensor character of g_uv.
                  Newton's First Law of Motion in Einsteinian 1916 classical geometrodynamical language generalizes to

                  All covariantly non-accelerating massive test particles follow  timelike geodesics of the gravitational field.
                  Splitting out the gravitational deformation tensor G^i_jk doesn't change any of that Jack. So why do you keep bringing
                  this up?
                  Newton's Second Law of Motion generalizes to

                  D^2x^u/ds^2 = F^u/m

                  where m is the invariant rest mass of the test particle (assumed here not to be ejecting mass like a rocket) and ds is the proper time of m along the worldline. F^u is the non-gravity applied force pushing the test particle off-geodesic inducing g-force on the test particle.
                  All fully covariant and consistent with tensor G^i_jk. So what's your point?
                  More generally, Newton's 2nd Law in curved spacetime for massive test particle motion is

                  (D/ds)(mdx^u/ds) = (Dm/ds)dx^u/ds + mD^2x^u/ds^2 = F^u(non-gravity)

                  Newton's 3rd law is not valid for global total momenta because translation symmetry is broken - see Noether's theorem.

                  Instead we have the local stress-energy current density conservation law

                  D^vTuv(non-gravity source fields curving spacetime) = 0

                  2) g-force and the Levi-Civita connection field {LC}

                  Back to the simplest case

                  D^2x^u/ds^2 = F^u/m

                  We now do a Godel self-reference and let the test particle detect itself.

                  Note, the connection field (LC Christoffel symbol )usually has nothing to do with the test particle, but is a property of the detector, except in the degenerate self-referential case.
                  I'm sorry Jack but IMO this is complete and utter nonsense.

                  The geodesics are covariantly defined in GR and have nothing to do with "detectors" or "observers" -- except as to mere
                  *appearances*. The LC connection field includes linearly independent contributions from the coordinates *and from the
                  intrinsic geometry*. Otherwise, how could it be a metric compatible connnection?

                  Of course a straight SR free-fall trajectory for example *looks* curved to a uniformly accelerating observer *who ignores
                  his own accelerating motion*! So what? That doesn't affect the actual geometry of the geodesic in the slightest. And neither
                  does curved geometry force any changes in local coordinate charts. You are completely and hopelessly confused here.
                  These are two completely different animals.

                  I'm not going to read any further.

                  Z.




                  The Baron Munchausen is weightless, no inertial g-force, in free-fall parabolic path (timelike geodesic in curved spacetime)
                  He does feel g-force initially when cannon fires

                  In particular think of the above SSS case in which everything can be concretely computed. Z never bothers to compute an example. This is why his thesis here is not even wrong in my opinion.

                  i = 1,2,3 = spacelike components

                  i = 0 is the timelike component

                  Let's look only at the i = 1 radial component in the static LNIF 

                  D^2r/ds^2 = d^2r/ds^2 + {LC}^ruv(dx^u/ds)(dx^v/ds)

                  However, in the self-referential "diagonal" case

                  dx^i/ds = 0

                  dx^o/ds = 1 (c = 1)

                  d^2r/ds^2 = 0

                  Therefore, the g-force per unit rest mass measured on the test particle itself is

                  {LC}^r00 ~ +GM(source)/r^2

                  This, however, is not an objective non-tidal non-zero tensor field. It is purely contingent based on an arbitrary, though pragmatically contingent, choice since we here on Earth closely approximate this situation.

                  3) Z's Red Herring of the Gravity Deformation

                  Start from M = 0 (flat space time) obviously 

                  {LC}^r00(M = 0) = 0 

                  here the static local non-inertial observer limits to the global inertial observer - a point overlooked by Z because he does not calculate a simple example.

                  Now increase to &M

                  {LC}^r00(&M) = G&M/r^2

                  This is not a tensor obviously.

                  The difference is not a tensor, i.e.,

                  {LC}^r00(0) - LC^r00(&M) = G&M/r^2

                  No tensors here!

                  Note the natural assumption here is the adiabatic transformation of the global static inertial frame GIF to the local static non-inertial frame LNIF. The contingent choice of convenience is "static".

                  Every time I hear a philofawzer say it's "pure math" or "your argument is mathematical nonsense" I reach for my delete button. ;-)

                  "As far as the laws of mathematics refer to reality, they are not certain, as far as they are certain, they do not refer to reality."
                  • Any intelligent fool can make things bigger, more complex, and more violent. It takes a touch of genius -- and a lot of courage -- to move in the opposite direction."
                  • "Imagination is more important than knowledge."
                    • "Reality is merely an illusion, albeit a very persistent one."
                    • "The only real valuable thing is intuition."
                      • "Anyone who has never made a mistake has never tried anything new."
                      • "Great spirits have often encountered violent opposition from weak minds."
                      • "God does not care about our mathematical difficulties. He integrates empirically."
                      • "Do not worry about your difficulties in Mathematics. I can assure you mine are still greater."
                      • "Not everything that counts can be counted, and not everything that can be counted counts." (Sign hanging in Einstein's office at Princeton)
                      • http://rescomp.stanford.edu/~cheshire/EinsteinQuotes.html

                  Jack Sarfatti wrote:
                  What's fundamentally wrong in z' s argument is that he does not understand the breakdown of the global frame assumption of special relativity. You simply cannot use purely formal arguments. You must specify what detectors you mean. Physics is about detector responses.

                  Sent from my iPhone

                  On Feb 28, 2009, at 8:17 AM, Jack Sarfatti <sarfatti@...> wrote:

                  Start with flat spacetime. The lc connection field is zero for the global inertial observer.
                  Now introduce a small SSS  Tuv matter source which induces an non zero lc field for the static non-inertial observer that adiabatically emerges if you impose a fixed r constraint.
                  Since the difference is a tensor
                  That difference must be zero otherwise you have a contradiction.

                  Sent from my iPhone

                  On Feb 28, 2009, at 1:15 AM, Paul Zielinski <iksnileiz@...> wrote:

                  Nonsense.

                  The common non-tensor coordinate part just cancels out of the difference between the two LC connections,
                  which are themselves NOT GCT tensors due to the common coordinate parts that cancel.

                  You are just confused.

                  Z.

                  Jack Sarfatti wrote:
                  Problem with the deformation approach is that it does not work. Start with flat spacetime and it implies lc is a tensor.

                  Back in Sf 

                  Sent from my iPhone

                  On Feb 27, 2009, at 1:31 PM, Jack Sarfatti <sarfatti@...> wrote:

                  So you are claiming that eq 2 is within a single network of observers e.g static Lnifs in the SSS case. We add a perturbation to tuv source inducing a metric perturbation. The take difference in lc connections and that is the 3rd rank tensor?
                  Why did you not say that clearly to begin with?
                  In any case it is not conceptually important to the foundations of relativity.

                  Only the tetrad and spin connection cartan 1-forms are intrinsic nontidal measures of the gravity field ie scalar zero rank tensors under gct aka local t4. About to take off from jfk.

                  Sent from my iPhone

                  On Feb 27, 2009, at 4:00 PM, Paul Zielinski <iksnileiz@...> wrote:

                  You showed in detail that if you replace Levi-Civita's two different metrics 1 and 1' with the *same*
                  metric represented in two different coordinate charts, you get a very different quantity when you take
                  the difference of the corresponding compatible connections that obviously is not a tensor..

                  Which is just another way of saying that you don't understand -- or don't want to understand -- Levi-
                  Civita's proof.

                  I told you what Levi-Civita's rho^i_jk means mathematically: it represents the effect of a deformation
                  of the geometry of the manifold (from metric 1 to metric 1') on the Levi-Civita connection, free of
                  any non-tensor contribution from the coordinates.

                  In the context of GR, since in Einstein's model the net observed non-tidal gravitational field strength is
                  represented by the LC connection, clearly the tensor rho^i_jk represents the effect on the non-tidal field
                  strength of the change in the 4D spacetime geometry when you go from 1 to 1' -- free of any contribution
                  from observer frame acceleration.

                  If you take metric 1 to be geometrically Minkowski, then the difference G^i_jk between the two LC
                  connections obviously represents the effect on the gravity-free LC connection field of the change from
                  Minkowski geometry (n_uv) to gravitationally deformed geometry (g_uv).

                  Which means that the tensor field G^i_jk(x) represents the *actual* (i.e., objective, frame-independent)
                  non-tidal (first-order) gravitational field strength in GR, free of frame acceleration artifacts ("fictitious
                  forces").

                  Z.

                  Jack Sarfatti wrote:
                  There is no physics at all in that snippet eq 2.

                  Sent from my iPhone at jfk boarding for Sfo 

                  On Feb 26, 2009, at 5:04 PM, Paul Zielinski <iksnileiz@...> wrote:

                  Actually, we are talking about Levi-Civita's idea here. He's the one with two different metrics on a single
                  analytical manifold, with a common local coordinate chart around each point, under general transformations
                  of which his rho^i_jk (the difference between two metric-compatible connections) transforms as a 3-index
                  tensor.


                  I showed in detail it is not a tensor if the usual physics is added to the naked math. It is a contingent relation between two networks of pairs of locally coincident detectors.















                Your message has been successfully submitted and would be delivered to recipients shortly.