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Re: Accelerations in the Lorentz group

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  • Jack Sarfatti
    Thanks, good points. ... -- CREATE, COMMUNICATE, COLLABORATE http://stardrive.org
    Message 1 of 2 , Feb 1, 2001
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      Thanks, good points.

      Bill Page wrote:

      > On Wednesday, January 31, 2001 3:15 AM
      > <RKiehn2352@...> wrote:
      >
      > > I am now trying to absorb Kleinert's point of view. But I believe
      > > that the difference between a fluid and a solid has to do with the
      > > non-singular symmetric metric constraint imposed upon a solid,
      > > and on the associated stress and strain. I realize that I am not very
      > > clear on this point.
      >
      > I am not sure that this makes sense. It seems to me that we might
      > (in special circumstances) want to use a singular metric in both
      > solids and fluids for the topological reasons that you have discussed
      > elsewhere.
      >
      > I think the constraint of symmetry of the metric is a separate issue
      > that has more to do with the formalism then with the physical
      > interpretation. The tetrad formalism allows us to absorb what
      > would otherwise required a non-symmetric metric into objects
      > such as the contorsion part of the connection.
      >
      > It seems to me that the distinction between fluid and solid has
      > more to do with the difference between "plasticity" versus
      > "elasticity". I think this distinction has to do with the difference
      > between a U_n geometry with both curvature and torsion versus
      > an A_n geometry (or better: teleparallel geometry) where the
      > total curvature is zero, but torsion is allowed. The plasticity that
      > is observed in solids is greatly constrained dynamically compared
      > to the plasticity in fluids. Between plastic deformation "events"
      > in crystals, we have only elastic deformations. In solid state, these
      > elastic deformations give rise to a range of effects like phonon
      > propagation etc.
      >
      > I realize that I am also not very clear on this distinction. <grin>
      >
      > > In a fluid there can be rigid body rotation - like in a solid- and
      > > for which there is a curl or vorticity associated with the velocity
      > > field, but it seems to me that the concept of circulation without
      > > vorticity does not appear in a solid, but can appear in a fluid.
      >
      > Of course there is also a sense in which fluids are "4-dimensional"
      > dynamic "objects" while solids are "3-dimensional" by definition.
      > I am not sure how to define "circulation" vs. vorticity in a solid
      > unless we are specifically discussing plastic deformations of
      > solids.
      >
      > > I conjecture that it is the imposed symmetries, and the simple
      > > connectedness, or single component constraints that yield these
      > > differences. I will try to make these thoughts more precise a bit
      > > later on.
      > > **
      >
      > I also look forward to discussing this further.
      >
      > > I have recently thinking about accelerations in inertial frames. I have
      > > found an interesting class of conformal Lorentz transformations
      > > which can be put into correspondence with the generators of the
      > > spiral wakes known as the Rayliegh-Taylor instability and the
      > > Kelvin-Helmholtz instability.
      > > http://www22.pair.com/csdc/pdf/poitier.pdf
      >
      > The above is a very pretty paper. I think it is clearly written and
      > presented some very useful new work. How well was it received
      > by the fluid dynamics people?
      >
      > > The title of the work will be
      > > **
      > > Torsion and Curvature of Lorentz transformations
      > >
      > > ABSTRACT: Counter to popular opinion, the equivalence class
      > > of Lorentz transformations need not be free from torsion and
      > > curvature. If the generators of the Lorentz transformations are
      > > constants, then all subspaces are torsion and curvature free.
      > > However, when the generators of the Lorentz transformations are
      > > not constants, such that rotational and translational accelerations
      > > are admitted, the larger equivalence class of inertial frames
      > > based on such Lorentz transformations includes subspaces with
      > > torsion and curvature.
      >
      > Yes! Of course you are talking about Lorentz transformations
      > in a particular context, say from an observer's frame of reference
      > to the comoving frame of reference with respect to an object.
      > The changes in the such a transformation correspond to acceleration
      > and rotation of the object.
      >
      > Such transformations also occur as the "(pseudo-)orthogonal factor"
      > of a general frame matrix, tetrad or vierbien field.
      >
      > > In all cases the null line element with Minkowski signature is
      > > preserved. Hence an electromagnetic signal, seen as a propagating
      > > E field discontinuity, is the same for all observers, even in the
      > > accelerated systems. The accelerations produce a basis set of
      > > non-closed vierbein 1-forms, but the Pfaff dimension of all such
      > > 1-forms for the primitive Lorentz generators is less than 4.
      >
      > That is an interesting observation. This would correspond to
      > the set of Ricci rotations or symmetric contorsions, right?
      >
      > > If the Lorentz group is enlarged to the conformal (Lorentz) group,
      > > thereby permitting expansions or contractions of finite line
      > > elements, but still preserving the null line element, then the
      > > vierbien constructions with both accelerations and non-constant
      > > expansion factors, lambda, are of Pfaff dimension 4 (conjecture:
      > > and are irreversible).
      >
      > If your conjecture is true, then I think that would answer in the
      > positive a question you asked Gennady Shipov some months
      > back about possibility of irreversibility in his theory of the
      > physical vacuum.
      >
      > > Similar remarks can be made if the Lorentz group is enlarged
      > > to admit Light speeds, C(x,y,z,t), that are not constants, but vary
      > > with position and time. The line element is still preserved, but
      > > the 1-form basis elements are not closed, hence can support
      > > torsion and curvature.
      >
      > Yes. In these generalization you are moving to more general
      > frame matrices. But you can do so and still remain within
      > what Shipov calls A4.
      >
      > > It is shown that there exist a class of conformal Lorentz
      > > transformations that are related to the hydrodynamic spiral
      > > instabilities known as the Kelvin-Helmholtz instability and
      > > the Rayleigh-Taylor instability. It is known that the limit sets
      > > of such instabilities form long lived coherent structures or
      > > wakes in a fluid.
      > > ****
      > > I think this is what Gennady is implying in his spaces of absolute
      > > parallelism.
      >
      > YES! I agree completely.
      >
      > > Jack will like this for the accelerations are related to non-holonomic
      > > constraints.
      > > **
      >
      > Yes again. I want to understand better the relationship between
      > such non-holonomic constraints expressed, for example, in the
      > formalism of Schouten's A_n versus the notion of torsion and
      > the Riemann tensor in teleparallel geometries. This is a subject
      > that David Cyganski and I have been working on (on and off)
      > for some time.
      >
      > Cheers,
      > Bill Page.

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