elastic forces and relativity

[mvonkann2000@yahoo.com]

As I understand history, before general relativity Einstein tried
and failed to find a Lorenz invariant description of gravitational
forces that would reduce suitably to Newtonian gravity in appropriate
cases (in retrospect it seems obvious why such a description cannot
exist). How do elastic forces work relativistically? I am quite sure
that there does not exist a Lorenz invariant force that reduces to F =
-kx to a first approximation. This question is not particularly
important since elastic forces are not fundamental, but it seems to me
it would be a cute pure math exercise to find a reasonably elegent
tensor equation that would reduce in the limit to F = -kx for small k,
x, and m. So what would the analogue of the Einstein tensor be if a
fundamental force had a Newtonian limit of F = -kx and that had been
Einstein's pressing concern rather than gravity? Perhaps this question
is more fiction than physics, but I think there might exist an elegant
mathematical answer. Any thoughts?

[tessel@um.bot]

On Mon, 19 Jun 2006, mvonkann2000@yahoo.com wrote:

> How do elastic forces work relativistically? I am quite sure that there
> does not exist a Lorenz invariant force that reduces to F = -kx to a
> first approximation.

Oyvind Gron,
"Covariant formulation of Hooke's Law",
Am. J. Phys. 49 (1981)

> This question is not particularly important since elastic forces are not
> fundamental

Actually, relativistic elasticity is interesting to researchers working on
highly compact objects like neutron stars! You can search the arXiv for a
recent Ph.D. thesis on a formulation of the theory of elasticity suitable
for curved spacetimes; Brandon Carter and a bunch of others have written a
bunch of papers on this topic.

As you probably know, there is a large literature on alleged "relativistic
paradoxes". Unfortunately most of these papers seem to be written by
authors who neglected to familiarize themselves with earlier work, or to
think things through, or even to try to write clearly. The result tends
to challenge the view that science improves monotonically on an even
front, as it were, since in this area one finds considerable
"backtracking" (rediscoveries of -flawed- arguments long since debunked),
especially since the advent of the arXiv!

Be this as it may, the point here is that some of these authors try to
appeal (often without realizing what they are doing) to some kind of
material model, but they rarely seem to have even thought about the
limitations of Hooke's law. Gron is exceptional in that he knows the
literature and took enough care to notice that Hooke's law is not
relativistic! See his review in

http://digilander.libero.it/solciclos/

Thought experiments involving springs are also useful in thinking about
curvature, e.g. there is some indication that null curvature singularities
occuring in certain exact gravitational plane wave solutions might be
survivable by objects exhibiting appropriate motion, because the blowup
occurs to rapidly to stretch a spring very much.

> So what would the analogue of the Einstein tensor be if a fundamental
> force had a Newtonian limit of F = -kx and that had been Einstein's
> pressing concern rather than gravity?

Not sure I understand the question. Maybe you are asking what the
stress-energy tensor looks like inside an idealized relativistic model of
an elastic rod under tension?

"T. Essel"

[Bossavit]

T. Essel:

>Gron is exceptional in that he knows the
>literature and took enough care to notice
>that Hooke's law is not relativistic!

There is also

B. Rothenstein: "A simple way to the relativistic
Hooke's law distances", Am. J. Phys., 53,
1 (1985), pp. 87-8.

which I remember as slightly simpler than
Gron's paper, "Covariant formulation of
Hooke's law", Am. J. Phys., 49, 1 (1981),
pp. 28-30.

[Mike]

Thanks to you and T. Essel for the references.

Mike

Bossavit wrote:
> T. Essel:
>
> >Gron is exceptional in that he knows the
> >literature and took enough care to notice
> >that Hooke's law is not relativistic!
>
> There is also
>
> B. Rothenstein: "A simple way to the relativistic
> Hooke's law distances", Am. J. Phys., 53,
> 1 (1985), pp. 87-8.
>
> which I remember as slightly simpler than
> Gron's paper, "Covariant formulation of
> Hooke's law", Am. J. Phys., 49, 1 (1981),
> pp. 28-30.