Re: Accelerations in the Lorentz group
- 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
> 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
> > 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.
> Bill Page.
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