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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