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Re: Z rocking on his hobby horse rolls off his meds again! ;-)

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  • JACK SARFATTI
    This only proves that you cannot read objectively read as I said all along. You distort the text to conform to your delusionary fixation. You actually say
    Message 1 of 17 , Dec 1, 2009
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      This only proves that you cannot read objectively read as I said all along.
      You distort the text to conform to your delusionary fixation.
      You actually say nothing substantial below. You only pontificate. No details.
      Your remarks on "acceleration" are gibberish. You confound two different uses of the same term in Einstein's GR as I explained in detail yesterday. Your entire confused not-even-wrong critique of the equivalence principle is based on your garbled fusion of these two meanings of the same term. You are not able to properly parse context.

      On Nov 30, 2009, at 9:49 PM, Paul Zielinski wrote:

      I just re-read Feynman's "Cargo Cult" speech and I have to say he gives a stunningly accurate
      description of your bone-headed dogmatic attitude to criticism of Einstein's equivalence principle,
      and your parrot-like replication of orthodox textbook positions.

      Feynman's point is that a true scientist must encourage efforts to expose problems with even
      his most cherished theories and hypotheses, and to welcome hard-hitting criticism.

      With regard to orthodox GR -- the formal theorems of which you slavishly recite in true "Cargo
      Cult style" -- I'm doing exactly what Feynman is asking for.

      So what's your point?

      Also, I see no mention of "cocktail party philosophers" or "philofawzical" thinking in this talk.

      So I still say that you are seriously misrepresenting Feynman's attitude to philosophers in general, and
      are completely reversing his position in my case. Feynman's "cocktail party philosophers" were the positivists
      who took the position that Eimstein's theories of "relativity" had eliminated the ether and rendered all motion
      purely relative -- which according to Feynman is not true, but is rather a myth that was founded on ignorance
      of the details of the actual theories in question. Which is almost exactly what I've been saying here -- there is
      no such thing as "general relativity".

      So in my case -- since I'm taking a diametrically opposed position to the positivist whereby acceleration in GR
      is not relative but absolute -- you have precisely inverted the meaning of Feynman's remarks!

      Z.

      Appendix - the two meanings of "acceleration" in Einstein's GR that Z confounds.

      1. relative acceleration between two locally coincident observers Alice and Bob

      2. Absolute acceleration of the centers of mass of each of them individually - detected as g-force on the observers,

      Alice and Bob are measuring the same set of events {C} up to any limits set by the quantum principle.




      My refutation of Z's crackpot theory is his nonsense that

      1) the equivalence principle is wrong

      2) there is a physically meaningful non-zero 3rd rank "nonmetricity" "tensor" inside the Levi-Civita connection - utter hogwash based on a nonsensical Rube Goldberg fantasy of a second connection. Sure once you make up your own rules and cheat you can say any stupid thing you like and Z does.


      Ironically, among such "cocktail party philosophers" was the younger Einstein himself, before he himself 
      recanted in 1918!

      It was precisely such "philofawzical" reasoning that motivated Einstein's attempts to generalize the 1905
      relativity principle to accelerating motion, leading directly to Einstein's famous equivalence principle that 
      literally identified fictitious and actual matter-produced gravitational fields.

      Z simply does not understand the different meanings of "acceleration" in Einstein's theory. He garbles them and falls into confusion.

      Meaning 1: The field equations retain their form (covariance) for locally coincident observers in arbitrary relative motion.

      Meaning 2: Acceleration is absolute - physically measured by g-force. Any point test particle-observer Alice on a timelike Levi-Civita connection geodesic world line has zero absolute (tensor covariant) 4-acceleration - and is weightless. Similarly, any such test particle Bob-observer pushed off that geodesic by a non-gravity force has absolute acceleration and feels g-force (aka weight).

      However, both observers see the same field equations! The latter is the principle of general relativity.

      The unaccelerated weightless Alice sees

      GIJ + /\zpfnIJ + kTIJ = 0

      I,J are LIF indices

      nIJ = flat Minkowski metric of Einstein's 1905 SR

      GIJ is the Einstein curved space-time tensor

      TIJ is the non-gravity field energy-momentum-stress tensor

      The locally coincident heavy Bob sees

      Guv + /\zpfguv + kTuv = 0

      where u,v are the LNIF indices

      e.g. the metric tensor field tetrad e^Iu transformation is

      guv(LNIF) = nIJ(LIF)e^Iue^Jv

      similarly for G & T

      very simple.

      Z muddies clear waters, makes simple things hard.

      In reality Feynman's actual position was the exact opposite -- that attempts to generalize the 1905 principle to 
      "general relativity" were misguided, and that Einstein's theories do not in fact support a pure relational theory of 
      spacetime as various pseudo-positivists once believed, and that spacetime is consequently absolute, as opposed
      to relational.

      Feynman cites Ehrenfest's two-clock problem as proof that Einstein's 1905 theory, as reformulated using 
      Minkowski's geometric spacetime model, does not in fact support a relational view of space and time. Because
      in Einstein's theories acceleration is in fact absolute, and not relative.

      In other words, Feynman is arguing against Einstein's classic concept of "general relativity" -- a generalization of
      the 1905 relativity principle to accelerating and rotating motion -- and not for it. So Feynman actually agrees with 
      my position on GR, and rejects Jack's.

      Z.

      On Nov 30, 2009, at 2:13 PM, Paul Zielinski wrote:

      What drugs are you pushing here Jack? Ones that would make me as disconnected from
      reality as you are?

      You clearly have no idea of what Feynman was actually referring to when he ridiculed the
      "philofawzical" reasoning of what he called "cocktail party philosophers".

      You are completely misrepresenting Feynman's position on Einstein's theories, which actually
      agree quite closely with mine, and not with yours. Feynman was ridiculing the pseudo-positivist
      philosophers who once argued that Einstein's theories of so-called "relativity" support a purely
      relational view of space and time -- which according to Feynman (and myself, among many
      others) they do not

      If acceleration is absolute, then there is no such thing as "general relativity" as Einstein originally
      defined it. That's all there is to it. Feynman's position was that according to Einstein's actual 1915
      theory of gravity, acceleration  is absolute.

      The sin of the "cocktail party philosophers" was to ignore this fact about Einstein's actual theories
      of space, time, and gravitation.

      You do not appear to have the slightest understanding of the logical relationship between the
      question of the existence of the ether, on the one hand, and Einstein's classic concept of a
      generalized relativity principle ("general relativity"), on the other.

      More on this later.

      Z.

      JACK SARFATTI wrote:
      Z you are obviously off your meds - or need some.










    • JACK SARFATTI
      New film Tristram Shandy ... RIght - and pseudo-scientists like you have closed minds, twist every rational comment on their crackpot theories into their
      Message 2 of 17 , Dec 1, 2009
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        New film Tristram Shandy
        On Dec 1, 2009, at 11:31 AM, Paul Zielinski wrote:



        On Tue, Dec 1, 2009 at 10:36 AM, JACK SARFATTI <sarfatti@...> wrote:
        This only proves that you cannot read objectively read as I said all along.
         
        Actually I think it is just the latest example of your own severe dyslexia.
         
        Feynman's talk was about intelectual honesty in science, and how it separates science from
        pseudo-science.

        RIght - and pseudo-scientists like you have closed minds, twist every rational comment on their crackpot theories into their opposites - adjust their perceptions to fit their delusional fixations. More relevant to your delusional garbled "work" on relativity in my opinion is Feynman's 1961 letter to his wife

        Feynman told me to always try to prove myself wrong - and I do. You have no conception of that.
        I welcome intelligent relevant refutations of my own "crazy" ideas.
         
        Your parrot-like recitations from aging textbooks on GR are an excellent illustration of the
        kind of thing that Feynman is referring to in his speech. 

        classic crackpot comment - as if "aging" makes the textbooks wrong.

        For the record Zielinski puts Misner, Thorne & Wheeler's Gravitation at the top of his silly list.
         
        You distort the text to conform to your delusionary fixation.
         
        Not at all. I actually understand Feynman -- whereas evidently you don't.
         
        For example, you copied the formula
         
        e^a_u
         
        for the tetrad coefficients from Rovelli's book without understanding its mathematical meaning.

        How? Prove your phony allegation.

        e^a_u maps locally zero g-force LIF "alice" to non-zero g-force LNIF "umberto"

        alice and umberto are locally coincident (solves hole problem of 1918)

        that is the physical meaning of the symbol

        e^a(lice) = e^a_ue^u(mberto)

        e^a is LIF basis

        e^u is LNIF basis

        guv(LNIF) = (Minkowski LIF)abe^aue^bv

        that's almost all a physicist need know from the standard text book stuff


        In general relativity, a frame field (also called a tetrad or vierbein) is an orthonormal set of four vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by \vec{e}_0 and the three spacelike unit vector fields by \vec{e}_1, \vec{e}_2, \, \vec{e}_3. All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.
        Frames were introduced into general relativity by Hermann Weyl in 1929[1].
        ...

        Nonspinning and inertial frames

        Some frames are nicer than others. Particularly in vacuum or electrovacuum solutions, the physical experience of inertial observers (who feel no forces) may be of particular interest. The mathematical characterization of an inertial frame is very simple: the integral curves of the timelike unit vector field must define a geodesic congruence, or in other words, its acceleration vector must vanish:
         \nabla_{\vec{e}_{0}} \, \vec{e}_0 = 0
        It is also often desirable to ensure that the spatial triad carried by each observer does not rotate. In this case, the triad can be viewed as being gyrostabilized. The criterion for a nonspinning inertial (NSI) frame is again very simple:
         \nabla_{\vec{e}_0} \, \vec{e}_j = 0, \; \; j = 0 \dots 3
        This says that as we move along the worldline of each observer, their spatial triad is parallel-transported. Nonspinning inertial frames hold a special place in general relativity, because they are as close as we can get in a curved Lorentzian manifold to the Lorentz frames used in special relativity (these are special nonspinning inertial frames in the Minkowski vacuum).
        More generally, if the acceleration of our observers is nonzero\nabla_{\vec{e}_0}\,\vec{e}_0 \neq 0, we can replace the covariant derivatives
         \nabla_{\vec{e}_0} \, \vec{e}_j, \; j = 1 \dots 3
        with the (spatially projected) Fermi-Walker derivatives to define a nonspinning frame.
        Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion. However, a given frame field might very well be defined on only part of the manifold.

        [edit]Example: Static observers in Schwarzschild vacuum

        It will be instructive to consider in some detail a few simple examples. Consider the famous Schwarzschild vacuum that models spacetime outside an isolated nonspinning spherically symmetric massive object, such as a star. In most textbooks one finds the metric tensor written in terms of a static polar spherical chart, as follows:
        ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)
         -\infty < t < \infty, \; 2 m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
        More formally, the metric tensor can be expanded with respect to the coordinate cobasis as
        g = -(1-2m/r) \, dt \otimes dt + \frac{1}{1-2m/r} \, dr \otimes dr + r^2 \, d\theta \otimes d\theta + r^2 \sin(\theta)^2 \, d\phi \otimes d\phi
        A coframe can be read off from this expression:
         \sigma^0 = -\sqrt{1-2m/r} \, dt, \; \sigma^1 = \frac{dr}{\sqrt{1-2m/r}}, \; \sigma^2 = r d\theta, \; \sigma^3 = r \sin(\theta) d\phi
        To see that this coframe really does correspond to the Schwarzschild metric tensor, just plug this coframe into
        g = -\sigma^0 \otimes \sigma^0 + \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3
        The frame dual to the coframe is
         \vec{e}_0 = \frac{1}{\sqrt{1-2m/r}} \partial_t, \; \vec{e}_1 = \sqrt{1-2m/r} \partial_r, \; \vec{e}_2 = \frac{1}{r} \partial_\theta, \; \vec{e}_3 = \frac{1}{r \sin(\theta)} \partial_\phi
        (The minus sign on σ0 ensures that \vec{e}_0 is future pointing.) This is the frame that models the experience of static observers who use rocket engines to "hover" over the massive object. The thrust they require to maintain their position is given by the magnitude of the acceleration vector
         \nabla_{\vec{e}_0} \vec{e}_0 = \frac{m/r^2}{\sqrt{1-2m/r}} \, \vec{e}_1
        This is radially outward pointing, since the observers need to accelerate away from the object to avoid falling toward it. On the other hand, the spatially projected Fermi derivatives of the spatial basis vectors (with respect to \vec{e}_0) vanish, so this is a nonspinning frame.
        The components of various tensorial quantities with respect to our frame and its dual coframe can now be computed.
        For example, the tidal tensor for our static observers is defined using tensor notation (for a coordinate basis) as
         E[X]_{ab} = R_{ambn} \, X^m \, X^n
        where we write \vec{X} = \vec{e}_0  to avoid cluttering the notation. Its only non-zero components with respect to our coframe turn out to be
         E[X]_{11} = -2m/r^3, \; E[X]_{22} = E[X]_{33} = m/r^3
        The corresponding coordinate basis components are
         E[X]_{rr} = -2m/r^3/(1-2m/r), \; E[X]_{\theta \theta} = m/r, \; E[X]_{\phi \phi} = m \sin(\theta)^2/r
        (A quick note concerning notation: many authors put carets over abstract indices referring to a frame. When writing down specific components, it is convenient to denote frame components by 0,1,2,3 and coordinate components by t,r,θ,φ. Since an expression like Sab = 36m / r doesn't make sense as a tensor equation, there should be no possibility of confusion.)
        Compare the tidal tensor Φ of Newtonian gravity, which is the traceless part of the Hessian of the gravitational potential U. Using tensor notation for a tensor field defined on three-dimensional euclidean space, this can be written
        \Phi_{ij} = U_{,i j} - \frac{1}{3} {U^{,k}}_{,k} \, \eta_{ij}
        The reader may wish to crank this through (notice that the trace term actually vanishes identically when U is harmonic) and compare results with the following elementary approach: we can compare the gravitational forces on two nearby observers lying on the same radial line:
         m/(r+h)^2 - m/r^2 = -2m/r^3 \, h + 3m/r^4 \, h^2 + O(h^3)
        Because in discussing tensors we are dealing with multilinear algebra, we retain only first order terms, so Φ11 = − 2m / r3. Similarly, we can compare the gravitational force on two nearby observers lying on the same sphere r = r0. Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to the sphere which has magnitude
         \frac{m}{r_0^2} \, \sin(\theta) \approx \frac{m}{r_0^2} \, \frac{h}{r_0} = \frac{m}{r_0^3} \, h
        By using the small angle approximation, we have ignored all terms of order O(h2), so the tangential components are Φ22 = Φ33 = m / r3. Here, we are referring to the obvious frame obtained from the polar spherical chart for our three-dimensional euclidean space:
         \vec{\epsilon}_1 = \partial_r, \; \vec{\epsilon}_2 = \frac{1}{r} \, \partial_\theta, \; \vec{\epsilon}_3 = \frac{1}{r \sin \theta} \, \partial_\phi
        Plainly, the coordinate components E[X]_{\theta \theta}, \, E[X]_{\phi \phi} computed above don't even scale the right way, so they clearly cannot correspond to what an observer will measure even approximately. (By coincidence, the Newtonian tidal tensor components agree exactly with the relativistic tidal tensor components we wrote out above.)

        [edit]Example: Lemaître observers in the Schwarzschild vacuum

        To find an inertial frame, we can boost our static frame in the \vec{e}_1 direction by an undetermined boost parameter (depending on the radial coordinate), compute the acceleration vector of the new undetermined frame, set this equal to zero, and solve for the unknown boost parameter. The result will be a frame which we can use to study the physical experience of observers who fall freely and radially toward the massive object. By appropriately choosing an integration constant, we obtain the frame of Lemaître observers, who fall in from rest at spatial infinity. (This phrase doesn't make sense, but the reader will no doubt have no difficulty in understanding our meaning.) In the static polar spherical chart, this frame can be written
        \vec{f}_0 = \frac{1}{1-2m/r} \, \partial_t - \sqrt{2m/r} \, \partial_r
        \vec{f}_1 = \partial_r - \frac{\sqrt{2m/r}}{1-2m/r} \, \partial_t
        \vec{f}_3 = \frac{1}{r} \, \partial_\theta
        \vec{f}_3 = \frac{1}{r \sin(\theta)} \, \partial_\phi
        Note that \vec{e}_0 \neq \vec{f}_0, \; \vec{e}_1 \neq \vec{f}_1, and that \vec{e}_0 "leans inwards", as it should, since its integral curves are timelike geodesics representing the world lines of infalling observers. Indeed, since the covariant derivatives of all four basis vectors (taken with respect to \vec{e}_0) vanish identically, our new frame is a nonspinning inertial frame.
        If our massive object is in fact a (nonrotating) black hole, we probably wish to follow the experience of the Lemaître observers as they fall through the event horizon at r = 2m. Since the static polar spherical coordinates have a coordinate singularity at the horizon, we'll need to switch to a more appropriate coordinate chart. The simplest possible choice is to define a new time coordinate by
         T(t,r) = t - \int \frac{\sqrt{2m/r}}{1-2m/r} \, dr = t + 2 \sqrt{2mr} + 2m \log \left( \frac{\sqrt{r}-\sqrt{2m}}{\sqrt{r}+\sqrt{2m}} \right)
        This gives the Painlevé chart. The new line element is
         ds^2 = -dT^2 + \left( dr + \sqrt{2m/r} \, dT \right)^2 + r^2 \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)
         -\infty < T < \infty, \; 0 < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
        With respect to the Painlevé chart, the Lemaître frame is
        \vec{f}_0 = \partial_T - \sqrt{2m/r} \, \partial_r
        \vec{f}_1 = \partial_r
        \vec{f}_2 = \frac{1}{r} \, \partial_\theta
        \vec{f}_3 = \frac{1}{r \sin(\theta)} \, \partial_\phi
        Notice that their spatial triad looks exactly like the frame for three-dimensional euclidean space which we mentioned above (when we computed the Newtonian tidal tensor). Indeed, thespatial hyperslices T = T0 turn out to be locally isometric to flat three-dimensional euclidean space! (This is a remarkable and rather special property of the Schwarzschild vacuum; most spacetimes do not admit a slicing into flat spatial sections.)
        The tidal tensor taken with respect to the Lemaître observers is
         E[Y]_{ab} = R_{ambn} \, Y^m \, Y^n
        where we write Y = \vec{f}_0 to avoid cluttering the notation. This is a different tensor from the one we obtained above, because it is defined using a different family of observers. Nonetheless, its nonvanishing components look familiar: E[Y]_{11} = -2m/r^3, \, E[Y]_{22} = E[Y]_{33} = m/r^3. (This is again a rather special property of the Schwarzschild vacuum.)
        Notice that there is simply no way of defining static observers on or inside the event horizon. On the other hand, the Lemaître observers are not defined on the entire exterior regioncovered by the static polar spherical chart either, so in these examples, neither the Lemaître frame nor the static frame are defined on the entire manifold.

        [edit]Example: Hagihara observers in the Schwarzschild vacuum

        In the same way that we found the Lemaître observers, we can boost our static frame in the \vec{e}_3 direction by an undetermined parameter (depending on the radial coordinate), compute the acceleration vector, and require that this vanish in the equatorial plane θ = π / 2. The new Hagihara frame describes the physical experience of observers in stable circular orbits around our massive object. It was apparently first discussed by the distinguished (and mathematically gifted) astronomer Yusuke Hagihara.
        In the static polar spherical chart, the Hagihara frame is
        \vec{h}_0 = \frac{1}{\sqrt{1-3m/r}} \, \partial_t + \frac{\sqrt{m/r^3}}{\sqrt{1-3m/r} \, \sin(\theta)} \, \partial_\phi
        \vec{h}_1 = \sqrt{1-2m/r} \, \partial_r
        \vec{h}_2 = \frac{1}{r} \, \partial_\theta
        \vec{h}_3 = \frac{\sqrt{1-2m/r}}{\sqrt{1-3m/r} \,\sin(\theta)} \, \partial_\phi - \frac{\sqrt{m/r^3}}{\sqrt{1-2m/r} \, \sqrt{1-3m/r}} \, \partial_t
        which in the equatorial plane becomes
      • JACK SARFATTI
        ... should obviously be e^u(LNIF) = (Minkowski)^u + A^u(LNIF) example with the dual vector field basis for the static rocket observers in Hawking s Fig 1.11
        Message 3 of 17 , Dec 1, 2009
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          On Dec 1, 2009, at 12:49 PM, JACK SARFATTI wrote:

          e^a(LIF) = (Minkowski)^a + A^a(LNIF)

          should obviously be

          e^u(LNIF) = (Minkowski)^u + A^u(LNIF)

          example with the dual vector field basis for the static rocket observers in Hawking's Fig 1.11 below


          eab433eaa735908da0d931f48916402f.png


          for example, Taylor series expand

          1/(1 - 2m/r)^1/2 ~ 1 + m/r + .....

          A0 = m/r + ....

          note also the acceleration of these static LNIFs is

          16fbe4981a993a850e29f0d57d8f4880.png





        • JACK SARFATTI
          OK enough Your remarks below about relativity theory are a classic crackpot delusional rant - unintelligible to everyone but yourself. There is no actual
          Message 4 of 17 , Dec 1, 2009
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            OK enough
            Your remarks below about relativity theory are a classic crackpot delusional rant - unintelligible to everyone but yourself.
            There is no actual scientific content in your polemic. You remind me of Roy who screams in the street outside Caffe Trieste.

            On Dec 1, 2009, at 1:23 PM, Paul Zielinski wrote:

            Your "rules of GR" are just like the airplane mockups of Feynman''s cargo cult. You
            dogmatically recite the "rules of GR" without having any in-depth understanding of where
            they came from or why they are there.
             
            The reason Einstein supposed that the gravitational field literally vanishes in free fall is that
            he believed this would allow him to generalize his 1905 relativity principle to apply to accelerating
            frames of reference. If on the other hand it turns out not to be possible to generalize the 1905
            principle in this manner as Einstein once hoped (which is now the mainstream view) , then the
            theoretical motivation for Einstein's rather naive supposition collapses.

            Completely confused nonsense for the reasons I gave. You garble at least two different meanings of "acceleration".
            Newton's gravitation field does completely vanish in free fall. That is a correct idea.
             
            While I understand this very well, since I'm thoroughly familiar with the historical genesis and
            original theoretical motivation for the actual Einstein equivalence principle, you on the other hand
            are completely oblivious, and content with the mere husk of Einstein's principle in the form of
            the so-called "EEP".
             
            Thus in his "Cargo Cult" speech Feynman is not talking about people like me, who are interested
            in the whys and the wherefors -- why the rules of GR are what they are, and where they came
            from -- he's talking about people like YOU.

            To your crackpot way of thinking, that is quite understandable.
             

             
            On Tue, Dec 1, 2009 at 12:49 PM, JACK SARFATTI <sarfatti@...> wrote:
            New film Tristram Shandy
            On Dec 1, 2009, at 11:31 AM, Paul Zielinski wrote:



            On Tue, Dec 1, 2009 at 10:36 AM, JACK SARFATTI <sarfatti@...> wrote:
            This only proves that you cannot read objectively read as I said all along.
             
            Actually I think it is just the latest example of your own severe dyslexia.
             
            Feynman's talk was about intelectual honesty in science, and how it separates science from
            pseudo-science.

            RIght - and pseudo-scientists like you have closed minds, twist every rational comment on their crackpot theories into their opposites - adjust their perceptions to fit their delusional fixations. More relevant to your delusional garbled "work" on relativity in my opinion is Feynman's 1961 letter to his wife
            <Feynmanwarsaw.jpg>

            Feynman told me to always try to prove myself wrong - and I do. You have no conception of that.
            I welcome intelligent relevant refutations of my own "crazy" ideas.
             
            Your parrot-like recitations from aging textbooks on GR are an excellent illustration of the
            kind of thing that Feynman is referring to in his speech. 

            classic crackpot comment - as if "aging" makes the textbooks wrong.

            For the record Zielinski puts Misner, Thorne & Wheeler's Gravitation at the top of his silly list.

             
            You distort the text to conform to your delusionary fixation.
             
            Not at all. I actually understand Feynman -- whereas evidently you don't.
             
            For example, you copied the formula
             
            e^a_u
             
            for the tetrad coefficients from Rovelli's book without understanding its mathematical meaning.

            How? Prove your phony allegation.

            e^a_u maps locally zero g-force LIF "alice" to non-zero g-force LNIF "umberto"

            alice and umberto are locally coincident (solves hole problem of 1918)

            that is the physical meaning of the symbol

            e^a(lice) = e^a_ue^u(mberto)

            e^a is LIF basis

            e^u is LNIF basis

            guv(LNIF) = (Minkowski LIF)abe^aue^bv

            that's almost all a physicist need know from the standard text book stuff


            In general relativity, a frame field (also called a tetrad or vierbein) is an orthonormal set of four vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by <6732d76ad8298e5ba3ffe100f0e991ff.png> and the three spacelike unit vector fields by <bc55961067897955a649b1d824643abc.png>. All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.
            Frames were introduced into general relativity by Hermann Weyl in 1929[1].
            ...

            Nonspinning and inertial frames

            Some frames are nicer than others. Particularly in vacuum or electrovacuum solutions, the physical experience of inertial observers (who feel no forces) may be of particular interest. The mathematical characterization of an inertial frame is very simple: the integral curves of the timelike unit vector field must define a geodesic congruence, or in other words, its acceleration vector must vanish:
            <217ae0c116127009fe1e5210c0ef09bc.png>
            It is also often desirable to ensure that the spatial triad carried by each observer does not rotate. In this case, the triad can be viewed as being gyrostabilized. The criterion for a nonspinning inertial (NSI) frame is again very simple:
            <f4aed4fe6d5e72697945f3712da267b6.png>
            This says that as we move along the worldline of each observer, their spatial triad is parallel-transported. Nonspinning inertial frames hold a special place in general relativity, because they are as close as we can get in a curved Lorentzian manifold to the Lorentz frames used in special relativity (these are special nonspinning inertial frames in the Minkowski vacuum).
            More generally, if the acceleration of our observers is nonzero<1d2fb17e857b3c93c84570b3a757a2fc.png>, we can replace the covariant derivatives
            <e07092eb9452024b2afea537f6770632.png>
            with the (spatially projected) Fermi-Walker derivatives to define a nonspinning frame.
            Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion. However, a given frame field might very well be defined on only part of the manifold.

            [edit]Example: Static observers in Schwarzschild vacuum

            It will be instructive to consider in some detail a few simple examples. Consider the famous Schwarzschild vacuum that models spacetime outside an isolated nonspinning spherically symmetric massive object, such as a star. In most textbooks one finds the metric tensor written in terms of a static polar spherical chart, as follows:
            <db9fb6841c8b970adc13eee622d0a12e.png>
            <edaaaae4151e0b88ab2e66e1e447b345.png>
            More formally, the metric tensor can be expanded with respect to the coordinate cobasis as
            <0e6e16d3f67ae54ac8f53dc4c14900bc.png>
            A coframe can be read off from this expression:
            <8e96a6a7e96eb32fa8d0327e7b64f215.png>
            To see that this coframe really does correspond to the Schwarzschild metric tensor, just plug this coframe into
            <0b4eb1063c4983662ff3fb066d579363.png>
            The frame dual to the coframe is
            <eab433eaa735908da0d931f48916402f.png>
            (The minus sign on σ0 ensures that <6732d76ad8298e5ba3ffe100f0e991ff.png> is future pointing.) This is the frame that models the experience of static observers who use rocket engines to "hover" over the massive object. The thrust they require to maintain their position is given by the magnitude of the acceleration vector
            <16fbe4981a993a850e29f0d57d8f4880.png>
            This is radially outward pointing, since the observers need to accelerate away from the object to avoid falling toward it. On the other hand, the spatially projected Fermi derivatives of the spatial basis vectors (with respect to <6732d76ad8298e5ba3ffe100f0e991ff.png>) vanish, so this is a nonspinning frame.
            The components of various tensorial quantities with respect to our frame and its dual coframe can now be computed.
            For example, the tidal tensor for our static observers is defined using tensor notation (for a coordinate basis) as
            <332e989add658a0cccd5fd3c584984bc.png>
            where we write <031e8f2cc6e2a0feec77d487ea4ba735.png> to avoid cluttering the notation. Its only non-zero components with respect to our coframe turn out to be
            <7d9bc473c4089dd8bb2fce43ba25e626.png>
            The corresponding coordinate basis components are
            <bfa6af33fc6587abb806cdece7f3f934.png>
            (A quick note concerning notation: many authors put carets over abstract indices referring to a frame. When writing down specific components, it is convenient to denote frame components by 0,1,2,3 and coordinate components by t,r,θ,φ. Since an expression like Sab = 36m / r doesn't make sense as a tensor equation, there should be no possibility of confusion.)
            Compare the tidal tensor Φ of Newtonian gravity, which is the traceless part of the Hessian of the gravitational potential U. Using tensor notation for a tensor field defined on three-dimensional euclidean space, this can be written
            <048208b27c7e1ee53231bbf9035ebd49.png>
            The reader may wish to crank this through (notice that the trace term actually vanishes identically when U is harmonic) and compare results with the following elementary approach: we can compare the gravitational forces on two nearby observers lying on the same radial line:
            <5cf988827ed65cc0b0c27e749a85bf7a.png>
            Because in discussing tensors we are dealing with multilinear algebra, we retain only first order terms, so Φ11 = − 2m / r3. Similarly, we can compare the gravitational force on two nearby observers lying on the same sphere r = r0. Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to the sphere which has magnitude
            <e4ea57662cabf6913e852fa40b1a40f3.png>
            By using the small angle approximation, we have ignored all terms of order O(h2), so the tangential components are Φ22 = Φ33 = m / r3. Here, we are referring to the obvious frame obtained from the polar spherical chart for our three-dimensional euclidean space:
            <5ccb9b42534824ed1899c755f1217713.png>
            Plainly, the coordinate components <1bb08ecb0754bb0a397136d39b5f206a.png> computed above don't even scale the right way, so they clearly cannot correspond to what an observer will measure even approximately. (By coincidence, the Newtonian tidal tensor components agree exactly with the relativistic tidal tensor components we wrote out above.)

            [edit]Example: Lemaître observers in the Schwarzschild vacuum

            To find an inertial frame, we can boost our static frame in the <d6d1d332cb6ebaab8bcf05cab4f5f423.png> direction by an undetermined boost parameter (depending on the radial coordinate), compute the acceleration vector of the new undetermined frame, set this equal to zero, and solve for the unknown boost parameter. The result will be a frame which we can use to study the physical experience of observers who fall freely and radially toward the massive object. By appropriately choosing an integration constant, we obtain the frame of Lemaître observers, who fall in from rest at spatial infinity. (This phrase doesn't make sense, but the reader will no doubt have no difficulty in understanding our meaning.) In the static polar spherical chart, this frame can be written
            <caae3636baf2c8c120135e47af7160e8.png>
            <66b77d5ddfc2ed58ebf2b77946304946.png>
            <f11b476fd4cf147a90209ff9d7bc32b9.png>
            <18fdc6478f93aae07585ec1d2d83b16d.png>
            Note that <33b63d6e91dc3d1451125d3ed6649394.png>, and that <6732d76ad8298e5ba3ffe100f0e991ff.png> "leans inwards", as it should, since its integral curves are timelike geodesics representing the world lines of infalling observers. Indeed, since the covariant derivatives of all four basis vectors (taken with respect to <6732d76ad8298e5ba3ffe100f0e991ff.png>) vanish identically, our new frame is a nonspinning inertial frame.
            If our massive object is in fact a (nonrotating) black hole, we probably wish to follow the experience of the Lemaître observers as they fall through the event horizon at r = 2m. Since the static polar spherical coordinates have a coordinate singularity at the horizon, we'll need to switch to a more appropriate coordinate chart. The simplest possible choice is to define a new time coordinate by
            <d14571bb800582180b1675fff607be5b.png>
            This gives the Painlevé chart. The new line element is
            <6b267b6e13d0cd9fbc905d5d5bc0aab7.png>
            <077e4c7a3953ca4913714764cd7cd2cc.png>
            With respect to the Painlevé chart, the Lemaître frame is
            <ecf418663b6403a4d19ff095d2540c6d.png>
            <af408f83eebf27b195bf675b00a5a9fa.png>
            <799d97077013b4f9e841e91b73f1273b.png>
            <18fdc6478f93aae07585ec1d2d83b16d.png>
            Notice that their spatial triad looks exactly like the frame for three-dimensional euclidean space which we mentioned above (when we computed the Newtonian tidal tensor). Indeed, thespatial hyperslices T = T0 turn out to be locally isometric to flat three-dimensional euclidean space! (This is a remarkable and rather special property of the Schwarzschild vacuum; most spacetimes do not admit a slicing into flat spatial sections.)
            The tidal tensor taken with respect to the Lemaître observers is
            <ae27b80b790915b7bc75215f66aac36b.png>
            where we write <39ea879bc0b495ad7e08ff5020d7c492.png> to avoid cluttering the notation. This is a different tensor from the one we obtained above, because it is defined using a different family of observers. Nonetheless, its nonvanishing components look familiar: <00860a33e76826ab9493cc4c52bbf50e.png>. (This is again a rather special property of the Schwarzschild vacuum.)
            Notice that there is simply no way of defining static observers on or inside the event horizon. On the other hand, the Lemaître observers are not defined on the entire exterior regioncovered by the static polar spherical chart either, so in these examples, neither the Lemaître frame nor the static frame are defined on the entire manifold.

            [edit]Example: Hagihara observers in the Schwarzschild vacuum

            In the same way that we found the Lemaître observers, we can boost our static frame in the <37a4859ff1e0b7df66bcd7beae9d7f73.png> direction by an undetermined parameter (depending on the radial coordinate), compute the acceleration vector, and require that this vanish in the equatorial plane θ = π / 2. The new Hagihara frame describes the physical experience of observers in stable circular orbits around our massive object. It was apparently first discussed by the distinguished (and mathematically gifted) astronomer Yusuke Hagihara.
            In the static polar spherical chart, the Hagihara frame is
            <8a8dbe1577940bdca4e01855a144c17d.png>
            <c5402f484daf0ed998f34718471c5204.png>
            <66c0ae419e866e3126abc6feb844bc27.png>
            <5a1a6b0e43bc04c94960ec9e98461c85.png>
            which in the equatorial plane becomes
            <0f7011280e14d82f2f14a16afcc87ab9.png>
            <c5402f484daf0ed998f34718471c5204.png>
            <66c0ae419e866e3126abc6feb844bc27.png>
            <b8166b64ddf851545c6dc981b22317fc.png>
            The tidal tensor E[Z]ab where <d2f7d68bab7ad9ad56c8295a17a88027.png> turns out to be given (in the equatorial plane) by
            <e45706e0b4b4f14c1fb6c903a3c10ce0.png>
            <d840b376f54f00a27bc902f3ed952c13.png>
            <9b8a42cf2c4795313d7c1d35daa7f63d.png>
            Thus, compared to a static observer hovering at a given coordinate radius, a Hagihara observer in a stable circular orbit with the same coordinate radius will measure radial tidal forces which are slightly larger in magnitude, and transverse tidal forces which are no longer isotropic (but slightly larger orthogonal to the direction of motion).
            Note that the Hagihara frame is only defined on the region r > 3m. Indeed, stable circular orbits only exist on r > 6m, so the frame should not be used inside this locus.
            Computing Fermi derivatives shows that the frame field just given is in fact spinning with respect to a gyrostabilized frame. The principal reason why is easy to spot: in this frame, each Hagihara observer keeps his spatial vectors radially aligned, so <fc227fe555c967cd14f562ce71d2df46.png> rotate about <71353eb1b47a69c3e1456ef37264b62b.png> as the observer orbits around the central massive object. However, after correcting for this observation, a small precession of the spin axis of a gyroscope carried by a Hagihara observer still remains; this is the de Sitter precession effect (also called the geodetic precessioneffect).
            Any other pseudo-mathematical GIGO you throw in is part of your delusionary model and is completely irrelevant to real physics.

            There is also my own new physics theory of emergent gravity from the partial cohering of the false vacuum at the moment of inflation to our observable universe bounded by our past particle horizon and our future event horizon retrocausal hologram.

            <DavisFig1_1conformal.jpg>

            e^a(LIF) = (Minkowski)^a + A^a(LNIF)

            A^a(LNIF) are the "Acceleration" field Cartan 1-forms

            formally they come from localizing rigid T4 to locally variable T4(x)

            in addition they are, in my new theory, derivable from eight SU3 QCD gluon strong force vacuum condensate post-inflation singular multi-valued  (Dirac string --> H. Kleinert) Goldstone phases - an idea still under construction.

             
            Another example: you confuse the infinitesimal correspondence principle "EEP" with Einstein's
            generalized relativity principle. You haven't the slightest clue as to Einstein's actual motivations
            for formulating his equivalence principle. Now that's what I would call "cargo cult" -- and I'm sure
            Feynman would agree, based on his other remarks about GR.

            Gibberish.

             
            To top it all off, Feynman makes it very clear that when he talks about "cocktail party philosophers"
            he is targeting the positivists who once argued that Einstein's theories of relativity render all
            motion relative -- which Feynman says ignores the technical details of thew actual theories. He
            actually says in so many words that the 1905 relativity principle of Einstein is not generalized
            to apply to accelerating motion in Einstein's theory of gravitation, against the "cocktail party
            philosophers" who mistakenly believe differently.

            Hogwash. What part of Feynman's text do you mean? You always make wild inferences not in the actual text.

             
            That is exactly my position, and was also Vladimir Fock's 1950s position, on 1915 GR.
             
            So you have the whole thing ass-backwards Jack. Feynman agrees with me -- and you are the
            real "cargo cultist" here. Because you imitate the *forms* of Einstein's theories but are oblivious
            to the actual substance of Einstein's original thinking regarding his relativity principles,
             
            You actually say nothing substantial below.
             
            If so, then neither does Feynman. However, I don't agree with you. Feynman is saying that a
            real scientist must "bend over backwards" to find problems with his theories -- not use every
            rhetorical trick in the book to suppress counterarguments and contrary data in order to prop up
            dogma, as the AGW fraudsters at East Anglia have done in climatology, and as you have been
            doing right here in the context of GR.
             
            You are in fact one of the worst offenders, with your knee-jerk dogmatism and abysnmal historical
            ignorance regarding Einstein's equivalence concept.
             
            You only pontificate. No details.
             
            Actually it is Feynman who is pontificating in his "Cargo Cult" talk.
             
            It's your citation Jack.
             
            Your remarks on "acceleration" are gibberish. You confound two different uses of the same term in Einstein's GR as I explained in detail yesterday.
             
            Look STUPID -- if there is an absolute standard of acceleration, then ALL forms of acceleration are absolute and not relative.
             
            OF COURSE I distingish between different different kinds of acceleration in GR -- but Feynman's position is that there is an
            absolute reference for acceleration in GR, and the 1905 relativity principle does not generalize to accelerating motion
            in GR.
             
            That was exactly Fock's position regarding GR. And it is also the same as mine.
             
            None of this is consistent with Einstein's own version of the equivalence principle.
             
            Your entire confused not-even-wrong critique of the equivalence principle is based on your garbled fusion of these two meanings of the same term.
             
            Actually it is based on carefully distingishing between the various meanings of "acceleration".
             
            An absolute standard of acceleration applies to *all* kinds of accleration. 
             
            So this is typical spaghetti code Jack. Your "logic" here is hopeless.
             
            Time to put you out to pasture...
             
            You are not able to properly parse context.
             
            No, you are simply dyslexic, that's all. You have not only misunderstood Feynman, you have actually
            reversed the intended meaning of his comments!
             
            His actual position on GR and Einstein equivalence is essentially the same as mine, for much the
            same reasons. 
             
            And your entire attitude to GR and Einstein equivalence clearly qualifies as "cargo cult" in Feynman's
            definition of the term.
             
            So as usual, you have turned the whole thing upside-down and inside-out. 
             
            Z.
             

            On Nov 30, 2009, at 9:49 PM, Paul Zielinski wrote:

            I just re-read Feynman's "Cargo Cult" speech and I have to say he gives a stunningly accurate
            description of your bone-headed dogmatic attitude to criticism of Einstein's equivalence principle,
            and your parrot-like replication of orthodox textbook positions.

            Feynman's point is that a true scientist must encourage efforts to expose problems with even
            his most cherished theories and hypotheses, and to welcome hard-hitting criticism.

            With regard to orthodox GR -- the formal theorems of which you slavishly recite in true "Cargo
            Cult style" -- I'm doing exactly what Feynman is asking for.

            So what's your point?

            Also, I see no mention of "cocktail party philosophers" or "philofawzical" thinking in this talk.

            So I still say that you are seriously misrepresenting Feynman's attitude to philosophers in general, and
            are completely reversing his position in my case. Feynman's "cocktail party philosophers" were the positivists
            who took the position that Eimstein's theories of "relativity" had eliminated the ether and rendered all motion
            purely relative -- which according to Feynman is not true, but is rather a myth that was founded on ignorance
            of the details of the actual theories in question. Which is almost exactly what I've been saying here -- there is
            no such thing as "general relativity".

            So in my case -- since I'm taking a diametrically opposed position to the positivist whereby acceleration in GR
            is not relative but absolute -- you have precisely inverted the meaning of Feynman's remarks!

            Z.

            Appendix - the two meanings of "acceleration" in Einstein's GR that Z confounds.

            1. relative acceleration between two locally coincident observers Alice and Bob

            2. Absolute acceleration of the centers of mass of each of them individually - detected as g-force on the observers,

            Alice and Bob are measuring the same set of events {C} up to any limits set by the quantum principle.





            My refutation of Z's crackpot theory is his nonsense that

            1) the equivalence principle is wrong

            2) there is a physically meaningful non-zero 3rd rank "nonmetricity" "tensor" inside the Levi-Civita connection - utter hogwash based on a nonsensical Rube Goldberg fantasy of a second connection. Sure once you make up your own rules and cheat you can say any stupid thing you like and Z does.


            Ironically, among such "cocktail party philosophers" was the younger Einstein himself, before he himself 
            recanted in 1918!

            It was precisely such "philofawzical" reasoning that motivated Einstein's attempts to generalize the 1905
            relativity principle to accelerating motion, leading directly to Einstein's famous equivalence principle that 
            literally identified fictitious and actual matter-produced gravitational fields.

            Z simply does not understand the different meanings of "acceleration" in Einstein's theory. He garbles them and falls into confusion.

            Meaning 1: The field equations retain their form (covariance) for locally coincident observers in arbitrary relative motion.

            Meaning 2: Acceleration is absolute - physically measured by g-force. Any point test particle-observer Alice on a timelike Levi-Civita connection geodesic world line has zero absolute (tensor covariant) 4-acceleration - and is weightless. Similarly, any such test particle Bob-observer pushed off that geodesic by a non-gravity force has absolute acceleration and feels g-force (aka weight).

            However, both observers see the same field equations! The latter is the principle of general relativity.

            The unaccelerated weightless Alice sees

            GIJ + /\zpfnIJ + kTIJ = 0

            I,J are LIF indices

            nIJ = flat Minkowski metric of Einstein's 1905 SR

            GIJ is the Einstein curved space-time tensor

            TIJ is the non-gravity field energy-momentum-stress tensor

            The locally coincident heavy Bob sees

            Guv + /\zpfguv + kTuv = 0

            where u,v are the LNIF indices

            e.g. the metric tensor field tetrad e^Iu transformation is

            guv(LNIF) = nIJ(LIF)e^Iue^Jv

            similarly for G & T

            very simple.

            Z muddies clear waters, makes simple things hard.

            In reality Feynman's actual position was the exact opposite -- that attempts to generalize the 1905 principle to 
            "general relativity" were misguided, and that Einstein's theories do not in fact support a pure relational theory of 
            spacetime as various pseudo-positivists once believed, and that spacetime is consequently absolute, as opposed
            to relational.

            Feynman cites Ehrenfest's two-clock problem as proof that Einstein's 1905 theory, as reformulated using 
            Minkowski's geometric spacetime model, does not in fact support a relational view of space and time. Because
            in Einstein's theories acceleration is in fact absolute, and not relative.

            In other words, Feynman is arguing against Einstein's classic concept of "general relativity" -- a generalization of
            the 1905 relativity principle to accelerating and rotating motion -- and not for it. So Feynman actually agrees with 
            my position on GR, and rejects Jack's.

            Z.

            On Nov 30, 2009, at 2:13 PM, Paul Zielinski wrote:

            What drugs are you pushing here Jack? Ones that would make me as disconnected from
            reality as you are?

            You clearly have no idea of what Feynman was actually referring to when he ridiculed the
            "philofawzical" reasoning of what he called "cocktail party philosophers".

            You are completely misrepresenting Feynman's position on Einstein's theories, which actually
            agree quite closely with mine, and not with yours. Feynman was ridiculing the pseudo-positivist
            philosophers who once argued that Einstein's theories of so-called "relativity" support a purely
            relational view of space and time -- which according to Feynman (and myself, among many
            others) they do not

            If acceleration is absolute, then there is no such thing as "general relativity" as Einstein originally
            defined it. That's all there is to it. Feynman's position was that according to Einstein's actual 1915
            theory of gravity, acceleration  is absolute.

            The sin of the "cocktail party philosophers" was to ignore this fact about Einstein's actual theories
            of space, time, and gravitation.

            You do not appear to have the slightest understanding of the logical relationship between the
            question of the existence of the ether, on the one hand, and Einstein's classic concept of a
            generalized relativity principle ("general relativity"), on the other.

            More on this later.

            Z.

            JACK SARFATTI wrote:
            Z you are obviously off your meds - or need some.














          • JACK SARFATTI
            ... JS: It does literally vanish. That you think it does not shows you are a crackpot in my opinion. ... JS: In fact it does do exactly that. That you think it
            Message 5 of 17 , Dec 1, 2009
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              On Dec 1, 2009, at 1:30 PM, JACK SARFATTI wrote:

               


              On Dec 1, 2009, at 1:23 PM, Paul Zielinski wrote:

               
              The reason Einstein supposed that the gravitational field literally vanishes in free fall

              JS: It does literally vanish. That you think it does not shows you are a crackpot in my opinion.


              PZ: is that he believed this would allow him to generalize his 1905 relativity principle to apply to accelerating
              frames of reference. 

              JS: In fact it does do exactly that. That you think it does not shows once again that you are a crackpot in my opinion.

              PZ: If on the other hand it turns out not to be possible to generalize the 1905
              principle in this manner as Einstein once hoped

              JS: False premise.

              PZ: (which is now the mainstream view)

              JS: Completely false. Any philofawzers who allegedly say that are cranks like you. They are not by any stretch of imagination "mainstream".

              PZ: then the
              theoretical motivation for Einstein's rather naive supposition collapses.

              JS: Your delusional fantasy.


            • JACK SARFATTI
              I have explained them. However since you completely lack the ability to correctly connect the formal symbols to physical operations what I have said is over
              Message 6 of 17 , Dec 1, 2009
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                I have explained them. However since you completely lack the ability to correctly connect the formal symbols to physical operations what I have said is over your head under your radar.

                OK genius, you explain them.

                On Dec 1, 2009, at 4:22 PM, Paul Zielinski wrote:

                Jack, anyone can copy and paste a bunch of formulas into an e-mail message. The hard part is figuring
                out what they mean.

                As far as I'm aware you still haven't even understood the true meaning of the coefficients e^a_u in the
                standard tetrad model. Why not start there?

                Z.

                JACK SARFATTI wrote:
                Paul it's obvious you cannot apply math to your pseudo-physics rants.

                On Dec 1, 2009, at 1:31 PM, Paul Zielinski wrote:

                Shouldn't you try to figure out what the tetrad coefficients e^a_u actually mean before reciting all these
                formulas?
                 
                What is the point of covering the board with a bunch of formulas whose meaning eludes you?
                 
                How is this different from Feynman's cargo cult mockups? Unless you can explain the actual meaning
                of the expressions you are reciting?

                Polemics, sophistry.

                On Tue, Dec 1, 2009 at 1:10 PM, JACK SARFATTI <sarfatti@...> wrote:

                On Dec 1, 2009, at 12:49 PM, JACK SARFATTI wrote:

                e^a(LIF) = (Minkowski)^a + A^a(LNIF)

                should obviously be

                e^u(LNIF) = (Minkowski)^u + A^u(LNIF)

                example with the dual vector field basis for the static rocket observers in Hawking's Fig 1.11 below


                <eab433eaa735908da0d931f48916402f.png>


                for example, Taylor series expand

                1/(1 - 2m/r)^1/2 ~ 1 + m/r + .....

                A0 = m/r + ....

                note also the acceleration of these static LNIFs is

                <16fbe4981a993a850e29f0d57d8f4880.png>

                <equivalence2.jpg>








              • JACK SARFATTI
                ... Yes, you are completely confused. ... You garble the tensor curvature with the g-force. Of course there is a 4th rank curvature tensor Ruvwl that does not
                Message 7 of 17 , Dec 1, 2009
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                  On Dec 1, 2009, at 4:16 PM, Paul Zielinski wrote:

                  JACK SARFATTI wrote:

                  On Dec 1, 2009, at 1:30 PM, JACK SARFATTI wrote:

                   


                  On Dec 1, 2009, at 1:23 PM, Paul Zielinski wrote:

                   
                  The reason Einstein supposed that the gravitational field literally vanishes in free fall

                  JS: It does literally vanish. That you think it does not shows you are a crackpot in my opinion.

                  That *I* think it does not?

                  Yes, you are completely confused.

                  The contemporary textbook view is that in GR there either is or is not a gravitational field at any given spacetime point,
                  regardless of any observer's frame of reference. According to the contemporary mainstream view the gravitational field
                  of GR does not generally vanish in any LIF, unless it vanishes in every other frame of reference.

                  You garble the tensor curvature with the g-force. Of course there is a 4th rank curvature tensor Ruvwl that does not vanish in every frame if it does not vanish in one frame. That is not true for the g-force that vanishes in the LIFs. The g-force is not a tensor. Newton's gravity force = g-force.

                  Aren't you getting a little confused?

                  No.

                  PZ: is that he believed this would allow him to generalize his 1905 relativity principle to apply to accelerating
                  frames of reference. 

                  JS: In fact it does do exactly that. That you think it does not shows once again that you are a crackpot in my opinion.

                  That you hold this opinion shows that you are the worst example of the kind of thing Feynman warned about in his 1974 "Cargo
                  Cult" speech.

                  This is not the contemporary mainstream view Jack. You are disconnected from reality. You are living in your private fantasy
                  bubble.

                  Cheap false polemic.

                  PZ: If on the other hand it turns out not to be possible to generalize the 1905
                  principle in this manner as Einstein once hoped

                  JS: False premise.
                  That's not the contemporary view of the matter Jack. You are out of the mainstream here.
                  PZ: (which is now the mainstream view)

                  JS: Completely false. Any philofawzers who allegedly say that are cranks like you. They are not by any stretch of imagination "mainstream".

                  For example, Michael Friedman, "Foundations of Space-Time Theories".

                  Show the relevant text. I have never not even once see you correctly interpret a text in this field.

                  Or John Norton, "What was Einstein's principle of equivalence?".

                  Show the relevant text. I have never not even once see you correctly interpret a text in this field.

                  Or just about any recent textbook on GR.

                  Show the relevant text. I have never not even once see you correctly interpret a text in this field.

                  You simply don't know what you are talking about. This is knee-jerk dogmatics of the worst sort.

                  Also very ill informed.

                  As you admitted, your relativity clock stopped in 1915 -- at the latest.

                  PZ: then the
                  theoretical motivation for Einstein's rather naive supposition collapses.

                  JS: Your delusional fantasy.

                  That's the whole point of trying to fully implement Mach's principle within the theoretical framework of GR --
                  to get complete Machian relativity of all motion.

                  This is irrelevant to the above point.

                  Einstein made several attempts at this, but ultimately gave
                  up on the whole idea. That was his last desperate attempt to implement "general relativity" inside the framework
                  of GR.

                  Gibberish.

                  FOR THE RECORD.

                  Z.


                • JACK SARFATTI
                  Now you must show what precise text in that pdf you think in your deranged mind justifies your delusion and refutes my stubbornly persistent illusion!
                  Message 8 of 17 , Dec 1, 2009
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                    Now you must show what precise text in that pdf you think in your deranged mind justifies your delusion and refutes my stubbornly persistent illusion!
                    We had a crank like you from Budapest at ISSO in 1999 that Joe Firmage funded who, like you, threw stacks of good papers at us, that had nothing to do with his idiotic ideas.

                    On Dec 1, 2009, at 7:06 PM, Paul Zielinski wrote:

                    JACK SARFATTI wrote:

                    On Dec 1, 2009, at 4:16 PM, Paul Zielinski wrote:

                    JACK SARFATTI wrote:

                    On Dec 1, 2009, at 1:30 PM, JACK SARFATTI wrote:

                     


                    On Dec 1, 2009, at 1:23 PM, Paul Zielinski wrote:

                     
                    The reason Einstein supposed that the gravitational field literally vanishes in free fall

                    JS: It does literally vanish. That you think it does not shows you are a crackpot in my opinion.

                    That *I* think it does not?

                    Yes, you are completely confused.
                    Projection.

                    The contemporary textbook view is that in GR there either is or is not a gravitational field at any given spacetime point,
                    regardless of any observer's frame of reference. According to the contemporary mainstream view the gravitational field
                    of GR does not generally vanish in any LIF, unless it vanishes in every other frame of reference.

                    You garble the tensor curvature with the g-force. Of course there is a 4th rank curvature tensor Ruvwl that does not vanish in every frame if it does not vanish in one frame. That is not true for the g-force that vanishes in the LIFs. The g-force is not a tensor. Newton's gravity force = g-force.
                    I think I'll take that as a "yes".
                    Aren't you getting a little confused?

                    No.

                    I think you are very confused Jack. Einstein's view of GR wherein an arbitary gravitational field literally vanishes in the LIF at any spacetime
                    point is not consistent with the contemporary mainstream view.

                    You can't have it both ways. You can't have both Einstein's interpretation *and* the contemporary understanding of GR. They are not mutually
                    consistent.

                    You want to keep your cake, and you want to eat it too.
                    PZ: is that he believed this would allow him to generalize his 1905 relativity principle to apply to accelerating
                    frames of reference. 

                    JS: In fact it does do exactly that. That you think it does not shows once again that you are a crackpot in my opinion.

                    That you hold this opinion shows that you are the worst example of the kind of thing Feynman warned about in his 1974 "Cargo
                    Cult" speech.

                    This is not the contemporary mainstream view Jack. You are disconnected from reality. You are living in your private fantasy
                    bubble.

                    Cheap false polemic.
                    No Jack. According to the modern view of GR, the gravitational field does not vanish in any LIF at a spacetime point x unless
                    it vanishes in *every other reference frame* at x.

                    PZ: If on the other hand it turns out not to be possible to generalize the 1905
                    principle in this manner as Einstein once hoped

                    JS: False premise.
                    That's not the contemporary view of the matter Jack. You are out of the mainstream here.
                    PZ: (which is now the mainstream view)

                    JS: Completely false. Any philofawzers who allegedly say that are cranks like you. They are not by any stretch of imagination "mainstream".

                    For example, Michael Friedman, "Foundations of Space-Time Theories".

                    Show the relevant text. I have never not even once see you correctly interpret a text in this field.
                    Jack, this is well known. Read Norton. Read Friedman. This is practically a cliche in contemporary gravitational
                    physics.

                    I've already given you a whole bibliography on this -- for your edification.

                    The problem is that you don't read anything -- or if you do, it just bounces off.

                    Or John Norton, "What was Einstein's principle of equivalence?".

                    Show the relevant text. I have never not even once see you correctly interpret a text in this field.
                    Google it.

                    Or just about any recent textbook on GR.

                    Show the relevant text. I have never not even once see you correctly interpret a text in this field.
                    J. B. Hartle's, for example.

                    You simply don't know what you are talking about. This is knee-jerk dogmatics of the worst sort.

                    Also very ill informed.

                    As you admitted, your relativity clock stopped in 1915 -- at the latest.

                    PZ: then the
                    theoretical motivation for Einstein's rather naive supposition collapses.

                    JS: Your delusional fantasy.

                    That's the whole point of trying to fully implement Mach's principle within the theoretical framework of GR --
                    to get complete Machian relativity of all motion.

                    This is irrelevant to the above point.
                    No it is not.

                    You don't have the slightest in depth understand of anything! You are just out of it.
                    Einstein made several attempts at this, but ultimately gave
                    up on the whole idea. That was his last desperate attempt to implement "general relativity" inside the framework
                    of GR.

                    Gibberish.
                    No, it's *a historical fact*.

                    Evidently you are still functionally illiterate in this subject area. Sad situation.

                    I've attached an easy paper by Peter Brown explaining Einstein's view, in comparison to what he calls the "modern" view, of GR.
                    Take it from there.

                    Z.

                    FOR THE RECORD.

                    Z.



                    <Einstein's Field_0204044.pdf>

                  • JACK SARFATTI
                    ... http://plato.stanford.edu/entries/spacetime-holearg/ ... this last equation formally expresses the Einstein Equivalence Principle EEP that Newton s gravity
                    Message 9 of 17 , Dec 1, 2009
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                      On Dec 1, 2009, at 12:49 PM, JACK SARFATTI wrote:


                      The 16 tetrad coefficients e^a_u maps locally zero g-force LIF "alice" to non-zero g-force LNIF "umberto"

                      alice and umberto are locally coincident (solves hole problem of 1918)

                      http://plato.stanford.edu/entries/spacetime-holearg/

                      that is the physical meaning of the symbol e^a_u and its dual e^u_a

                      e^a(lice) = e^a_ue^u(mberto)

                      e^a is LIF basis

                      e^u is LNIF basis

                      guv(LNIF) = (Minkowski LIF)abe^aue^bv

                      this last equation formally expresses the Einstein Equivalence Principle EEP that Newton's gravity force vanishes in a LIF (Local Inertial Frame).

                      EEP implies covariance of Einstein's gravity field equations, i.e., their form is invariant under all locally coincident frame transformations.

                      There are several distinct groups of locally coincident frame transformations.

                      I. Rovelli's "iii" = T4(x) = GCT = LNIF(non-zero g-force) ---> LNIF'(non-zero g'-force)

                      Guv + kTuv = 0 --->  Gu'v' + kTu'v' = 0 

                      where

                      Gu'v'(LNIF') = x^u',u x^v',vGuv(LNIF)

                      etc.

                      II. Rovelli's "ii" SO(1,3) Lorentz group LIF(zero g-force) ---> LIF'(zero g-force)

                      where

                      Gab(LIF) + kTab(LIF) = 0 ---> Ga'b'(LIF') + kTa'b'(LIF') = 0

                      where

                      Ga'b'(LIF') = SO(1,3)a'^aSO(1.3)b'^bGab(LIF)

                      III. Tetrad map  LIF <---> LNIF

                      where

                      Gab(LIF) = ea^ueb^vGuv(LNIF)

                      Guv(LNIF) = eu^aev^bGab(LIF)

                      The above equations show that Einstein's original conception is completely correct. The principle of general relativity is that the field equations of all the matter fields including gravity are covariant (i.e. form-invariant) under all physically meaningful frame transformations. In particular the two locally coincident observers looking at the same events can be in arbitrary individual absolute acceleration - unlike special relativity where they both must not accelerate. In addition to that absolute covariance we have the EEP that Newton's gravity force disappears at the centers of mass of LIFs. Newton's gravity force is 100% inertial g-force, i.e. artifact of an accelerating frame. 

                      That's almost all a physicist need know from the standard text book stuff.

                      There is also my own new physics theory of emergent gravity from the partial cohering of the false vacuum at the moment of inflation to our observable universe bounded by our past particle horizon and our future event horizon retrocausal hologram.




                      e^u(LNIF) = (Minkowski)^u + A^u(LNIF)

                      A^u(LNIF) are the "Acceleration" field Cartan 1-forms

                      formally they come from localizing rigid T4 to locally variable T4(x)

                      in addition they are, in my new theory, derivable from eight SU3 QCD gluon strong force vacuum condensate post-inflation singular multi-valued  (Dirac string --> H. Kleinert) Goldstone phases - an idea still under construction.

                       
                      In general relativity, a frame field (also called a tetrad or vierbein) is an orthonormal set of four vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. The timelike unit vector field is often denoted by \vec{e}_0 and the three spacelike unit vector fields by \vec{e}_1, \vec{e}_2, \, \vec{e}_3. All tensorial quantities defined on the manifold can be expressed using the frame field and its dual coframe field.
                      Frames were introduced into general relativity by Hermann Weyl in 1929[1].

                      Physical interpretation

                      Frame fields always correspond to a family of ideal observers immersed in the given spacetime; the integral curves of the timelike unit vector field are the worldlines of these observers, and at each event along a given worldline, the three spacelike unit vector fields specify the spatial triad carried by the observer. The triad may be thought of as defining the spatial coordinate axes of a local laboratory frame, which is valid very near the observer's worldline.
                      In general, the worldlines of these observers need not be timelike geodesics. If any of the worldlines bends away from a geodesic path in some region, we can think of the observers astest particles that accelerate by using ideal rocket engines with a thrust equal to the magnitude of their acceleration vector. Alternatively, if our observer is attached to a bit of matter in a ball of fluid in hydrostatic equilibrium, this bit of matter will in general be accelerated outward by the net effect of pressure holding up the fluid ball against the attraction of its own gravity. Other possibilities include an observer attached to a free charged test particle in an electrovacuum solution, which will of course be accelerated by the Lorentz force, or an observer attached to a spinning test particle, which may be accelerated by a spin-spin force.
                      It is important to recognize that frames are geometric objects. That is, vector fields make sense (in a smooth manifold) independently of choice of a coordinate chart, and (in a Lorentzian manifold), so do the notions of orthogonality and length. Thus, just like vector fields and other geometric quantities, frame fields can be represented in various coordinate charts. But computations of the components of tensorial quantities, with respect to a given frame, will always yield the same result, whichever coordinate chart is used to represent the frame.
                      These fields are required to write the Dirac equation in curved spacetime.

                      [edit]Specifying a frame

                      To write down a frame, a coordinate chart on the Lorentzian manifold needs to be chosen. Then, every vector field on the manifold can be written down as a linear combination of the fourcoordinate basis vector fields:
                       \vec{X} = X^j \, \partial_{x^j}
                      (Here, the Einstein summation convention is used, and the vector fields are thought of as first order linear differential operators, and the components Xj are often called contravariant components.) In particular, the vector fields in the frame can be expressed this way:
                       \vec{e}_j = {e_j}^k \, \partial_{x^k}
                      In "designing" a frame, one naturally needs to ensure, using the given metric, that the four vector fields are everywhere orthonormal.
                      Once a signature is adopted (in the case of a four-dimensional Lorentzian manifold, the signature is -1 + 3), by duality every vector has a dual covector and conversely. Thus, every frame field is associated with a unique coframe field, and vice versa.

                      [edit]Specifying the metric using a coframe

                      Alternatively, the metric tensor can be specified by writing down a coframe in terms of a coordinate basis and stipulating that the metric tensor is given by
                      g = -\sigma^0 \otimes \sigma^0 + \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3
                      This is just a fancy way of saying that the coframe is orthonormal. Whether this is used to obtain the metric tensor after writing down the frame (and passing to the dual coframe), or starting with the metric tensor and using it to verify that a frame has been obtained by other means, it must always hold true.

                      [edit]Relationship with metric tensor, in a coordinate basis

                      The vierbein field, e^\mu_a, has two indices: \mu \, labels the general spacetime coordinate and a \, labels the local lorentz spacetime or local laboratory coordinates.
                      The vierbein field or frame fields can be regarded as the square root of the metric tensorg^{\mu \nu} \,, since in a coordinate basis,
                      g^{\mu \nu}= e^{\mu}_{\ a} e^{\nu}_{\ b} \eta^{ab} \,
                      where \eta^{ab} \, is the Lorentz metric.
                      Local lorentz indices are raised and lowered with the lorentz metric in the same way as general spacetime coordinates are raised and lowered with the metric tensor. For example:
                      T^a = \eta^{ab} T_b \,
                      The vierbein fields enables conversion between spacetime and local lorentz indices. For example:
                      T_a = e^\mu_a T_\mu \,
                      The vierbein field itself can be manipulated in the same fashion:
                      e^\nu_a = e^\mu_a e^\nu_\mu \,, since  e^\nu_\mu = \delta^\nu_\mu \,
                      And these can combine.
                      T^a = e_\mu^a T^\mu \,
                      A few more examples: Spacetime and local lorentz coordinates can be mixed together:
                      T^{\mu a}=e_\nu^a T^{\mu \nu}
                      The local lorentz coordinates transform differently from the general spacetime coordinates. Under a general coordinate transformation we have:
                      T'^{\mu a} = \frac{\partial x'^\mu}{\partial x^\nu}T^{\nu a}
                      whilst under a local lorentz transformation we have:
                      T'^{\mu a} = \Lambda(x)^a_{\ b} T^{\mu b}.

                      [edit]Comparison with coordinate basis

                      Coordinate basis vectors have the special property that their Lie brackets pairwise vanish. Except in locally flat regions, at least some Lie brackets of vector fields from a frame will notvanish. The resulting baggage needed to compute with them is acceptable, as components of tensorial objects with respect to a frame (but not with respect to a coordinate basis) have a direct interpretation in terms of measurements made by the family of ideal observers corresponding the frame.
                      Coordinate basis vectors can very well be null, which, by definition, cannot happen for frame vectors.

                      [edit]Nonspinning and inertial frames

                      Some frames are nicer than others. Particularly in vacuum or electrovacuum solutions, the physical experience of inertial observers (who feel no forces) may be of particular interest. The mathematical characterization of an inertial frame is very simple: the integral curves of the timelike unit vector field must define a geodesic congruence, or in other words, its acceleration vector must vanish:
                       \nabla_{\vec{e}_{0}} \, \vec{e}_0 = 0
                      It is also often desirable to ensure that the spatial triad carried by each observer does not rotate. In this case, the triad can be viewed as being gyrostabilized. The criterion for anonspinning inertial (NSI) frame is again very simple:
                       \nabla_{\vec{e}_0} \, \vec{e}_j = 0, \; \; j = 0 \dots 3
                      This says that as we move along the worldline of each observer, their spatial triad is parallel-transported. Nonspinning inertial frames hold a special place in general relativity, because they are as close as we can get in a curved Lorentzian manifold to the Lorentz frames used in special relativity (these are special nonspinning inertial frames in the Minkowski vacuum).
                      More generally, if the acceleration of our observers is nonzero, \nabla_{\vec{e}_0}\,\vec{e}_0 \neq 0, we can replace the covariant derivatives
                       \nabla_{\vec{e}_0} \, \vec{e}_j, \; j = 1 \dots 3
                      with the (spatially projected) Fermi-Walker derivatives to define a nonspinning frame.
                      Given a Lorentzian manifold, we can find infinitely many frame fields, even if we require additional properties such as inertial motion. However, a given frame field might very well be defined on only part of the manifold.

                      [edit]Example: Static observers in Schwarzschild vacuum

                      It will be instructive to consider in some detail a few simple examples. Consider the famous Schwarzschild vacuum that models spacetime outside an isolated nonspinning spherically symmetric massive object, such as a star. In most textbooks one finds the metric tensor written in terms of a static polar spherical chart, as follows:
                      ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)
                       -\infty < t < \infty, \; 2 m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
                      More formally, the metric tensor can be expanded with respect to the coordinate cobasis as
                      g = -(1-2m/r) \, dt \otimes dt + \frac{1}{1-2m/r} \, dr \otimes dr + r^2 \, d\theta \otimes d\theta + r^2 \sin(\theta)^2 \, d\phi \otimes d\phi
                      A coframe can be read off from this expression:
                       \sigma^0 = -\sqrt{1-2m/r} \, dt, \; \sigma^1 = \frac{dr}{\sqrt{1-2m/r}}, \; \sigma^2 = r d\theta, \; \sigma^3 = r \sin(\theta) d\phi
                      To see that this coframe really does correspond to the Schwarzschild metric tensor, just plug this coframe into
                      g = -\sigma^0 \otimes \sigma^0 + \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3
                      The frame dual to the coframe is
                       \vec{e}_0 = \frac{1}{\sqrt{1-2m/r}} \partial_t, \; \vec{e}_1 = \sqrt{1-2m/r} \partial_r, \; \vec{e}_2 = \frac{1}{r} \partial_\theta, \; \vec{e}_3 = \frac{1}{r \sin(\theta)} \partial_\phi
                      (The minus sign on σ0 ensures that \vec{e}_0 is future pointing.) This is the frame that models the experience of static observers who use rocket engines to "hover" over the massive object. The thrust they require to maintain their position is given by the magnitude of the acceleration vector
                       \nabla_{\vec{e}_0} \vec{e}_0 = \frac{m/r^2}{\sqrt{1-2m/r}} \, \vec{e}_1
                      This is radially outward pointing, since the observers need to accelerate away from the object to avoid falling toward it. On the other hand, the spatially projected Fermi derivatives of the spatial basis vectors (with respect to \vec{e}_0) vanish, so this is a nonspinning frame.
                      The components of various tensorial quantities with respect to our frame and its dual coframe can now be computed.
                      For example, the tidal tensor for our static observers is defined using tensor notation (for a coordinate basis) as
                    • JACK SARFATTI
                      Example: Static observers in Schwarzschild vacuum For example ( c = G = 1) e^0(LNIF) = - (1 - 2m/r)^1/2dt = -[1 + (1/2)(2m/r) + (1x1/2x4)(2m/r)^2 +
                      Message 10 of 17 , Dec 1, 2009
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                        Example: Static observers in Schwarzschild vacuum

                        For example ( c = G = 1)

                        e^0(LNIF) = - (1 - 2m/r)^1/2dt = -[1 + (1/2)(2m/r) + (1x1/2x4)(2m/r)^2 + (1x1x3/2x4x6)(2m/r)^3 + (1x1x3x5/2x4x6x8)(2m/r)^4 + ...]dt

                        e^0(LIF) = -dt

                        e^0(LNIF) = e^0(LIF) + A^0(LNIF)

                        A^0(LNIF) = A^0tdt

                        We have the static LNIF "acceleration field" infinite Taylor series expansion for 2m/r < 1

                        A^0t = - [1 + (1/2)(2m/r) + (1x1/2x4)(2m/r)^2 + (1x1x3/2x4x6)(2m/r)^3 + (1x1x3x5/2x4x6x8)(2m/r)^4 + ...]

                        The A's vanish when r ---> infinity and m ---> 0

                        Do not confuse this with the actual universal inertial g-force per unit test particle

                        g = (m/r^2)(1 - 2m/r)^-1/2

                        "It will be instructive to consider in some detail a few simple examples. Consider the famous Schwarzschild vacuum that models spacetime outside an isolated nonspinning spherically symmetric massive object, such as a star. In most textbooks one finds the metric tensor written in terms of a static polar spherical chart, as follows:
                        ds^2 = -(1-2m/r) \, dt^2 + \frac{dr^2}{1-2m/r} + r^2 \, \left( d\theta^2 + \sin(\theta)^2 \, d\phi^2 \right)
                         -\infty < t < \infty, \; 2 m < r < \infty, \; 0 < \theta < \pi, \; -\pi < \phi < \pi
                        More formally, the metric tensor can be expanded with respect to the coordinate cobasis as
                        g = -(1-2m/r) \, dt \otimes dt + \frac{1}{1-2m/r} \, dr \otimes dr + r^2 \, d\theta \otimes d\theta + r^2 \sin(\theta)^2 \, d\phi \otimes d\phi
                        A coframe can be read off from this expression:
                         \sigma^0 = -\sqrt{1-2m/r} \, dt, \; \sigma^1 = \frac{dr}{\sqrt{1-2m/r}}, \; \sigma^2 = r d\theta, \; \sigma^3 = r \sin(\theta) d\phi
                        To see that this coframe really does correspond to the Schwarzschild metric tensor, just plug this coframe into
                        g = -\sigma^0 \otimes \sigma^0 + \sigma^1 \otimes \sigma^1 + \sigma^2 \otimes \sigma^2 + \sigma^3 \otimes \sigma^3
                        The frame dual to the coframe is
                         \vec{e}_0 = \frac{1}{\sqrt{1-2m/r}} \partial_t, \; \vec{e}_1 = \sqrt{1-2m/r} \partial_r, \; \vec{e}_2 = \frac{1}{r} \partial_\theta, \; \vec{e}_3 = \frac{1}{r \sin(\theta)} \partial_\phi
                        (The minus sign on σ0 ensures that \vec{e}_0 is future pointing.) This is the frame that models the experience of static observers who use rocket engines to "hover" over the massive object. The thrust they require to maintain their position is given by the magnitude of the acceleration vector
                         \nabla_{\vec{e}_0} \vec{e}_0 = \frac{m/r^2}{\sqrt{1-2m/r}} \, \vec{e}_1
                        This is radially outward pointing, since the observers need to accelerate away from the object to avoid falling toward it. On the other hand, the spatially projected Fermi derivatives of the spatial basis vectors (with respect to \vec{e}_0) vanish, so this is a nonspinning frame.
                        The components of various tensorial quantities with respect to our frame and its dual coframe can now be computed.
                        For example, the tidal tensor for our static observers is defined using tensor notation (for a coordinate basis) as
                         E[X]_{ab} = R_{ambn} \, X^m \, X^n
                        where we write \vec{X} = \vec{e}_0  to avoid cluttering the notation. Its only non-zero components with respect to our coframe turn out to be
                         E[X]_{11} = -2m/r^3, \; E[X]_{22} = E[X]_{33} = m/r^3
                        The corresponding coordinate basis components are
                         E[X]_{rr} = -2m/r^3/(1-2m/r), \; E[X]_{\theta \theta} = m/r, \; E[X]_{\phi \phi} = m \sin(\theta)^2/r
                        (A quick note concerning notation: many authors put carets over abstract indices referring to a frame. When writing down specific components, it is convenient to denote frame components by 0,1,2,3 and coordinate components by t,r,θ,φ. Since an expression like Sab = 36m / r doesn't make sense as a tensor equation, there should be no possibility of confusion.)
                        Compare the tidal tensor Φ of Newtonian gravity, which is the traceless part of the Hessian of the gravitational potential U. Using tensor notation for a tensor field defined on three-dimensional euclidean space, this can be written
                        \Phi_{ij} = U_{,i j} - \frac{1}{3} {U^{,k}}_{,k} \, \eta_{ij}
                        The reader may wish to crank this through (notice that the trace term actually vanishes identically when U is harmonic) and compare results with the following elementary approach: we can compare the gravitational forces on two nearby observers lying on the same radial line:
                         m/(r+h)^2 - m/r^2 = -2m/r^3 \, h + 3m/r^4 \, h^2 + O(h^3)
                        Because in discussing tensors we are dealing with multilinear algebra, we retain only first order terms, so Φ11 = − 2m / r3. Similarly, we can compare the gravitational force on two nearby observers lying on the same sphere r = r0. Using some elementary trigonometry and the small angle approximation, we find that the force vectors differ by a vector tangent to the sphere which has magnitude
                         \frac{m}{r_0^2} \, \sin(\theta) \approx \frac{m}{r_0^2} \, \frac{h}{r_0} = \frac{m}{r_0^3} \, h
                        By using the small angle approximation, we have ignored all terms of order O(h2), so the tangential components are Φ22 = Φ33 = m / r3. Here, we are referring to the obvious frame obtained from the polar spherical chart for our three-dimensional euclidean space:
                         \vec{\epsilon}_1 = \partial_r, \; \vec{\epsilon}_2 = \frac{1}{r} \, \partial_\theta, \; \vec{\epsilon}_3 = \frac{1}{r \sin \theta} \, \partial_\phi
                        Plainly, the coordinate components E[X]_{\theta \theta}, \, E[X]_{\phi \phi} computed above don't even scale the right way, so they clearly cannot correspond to what an observer will measure even approximately. (By coincidence, the Newtonian tidal tensor components agree exactly with the relativistic tidal tensor components we wrote out above.)"
                        Wiki article excerpt
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