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Elastic forces and relativity

  1. Jun 21, 2006 #1
    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?
     
  2. jcsd
  3. Jun 22, 2006 #2
    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"
     
  4. Jun 24, 2006 #3
    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.
     
  5. Jun 25, 2006 #4
    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.
     
  6. Sep 22, 2006 #5
    'Would T. Essel provide a rigorous proof that Einstein\'s General Relativity requires of necessity that a singularity must only occur where the Riemann tensor scalar curvature invariant is unbounded. No relativist in the history of the subject has ever proved this tacit assumption, upon which the black hole and the big bang rely. \r\n\r\nIn the alternative, which is equivalent for the purpose, would T. Essel provide a rigorous proof that a geometry is not entirely determined by the form of its line element. \r\n\r\nThe requested proofs will actually require some original thought, instead of regurgitation of the claims of other relativists. Anything less than the requested proofs is only hot air.'
     
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