Is There a Relativistic Version of Hooke's Law for Materials?

[Dale]

After a few recent discussions I have realized that I don’t know anything about a relativistic theory of materials. Does anyone have a good link for a reference about Hooke’s law in relativity, or something similar. Homogeneous isotropic linear media is fine.

 

[PeterDonis]

I've always found Greg Egan's page on Relativistic Elasticity to be a good starting point:

http://www.gregegan.net/SCIENCE/Rindler/SimpleElasticity.html

He also gives a reference at the end that treats the subject in much more detail.

 

[pervect]

My thinking is that one starts with a congruence of worldlines that represent the motion and deformation of the body in question. If one imagine the body is made up of small particles, then this congruence is a timelike unit vector field that is the four-velocity of each particle that defines the body.

Then one can compute the expansion scalar, shear tensor, and the vorticity tensor from the congruence. The computation process revolves around the projection operator that projects space-time into a spatial part and a time part, it's known as the kinematic decomposition of the time-like congruence. https://en.wikipedia.org/w/index.php?title=Congruence_(general_relativity)&oldid=737290097.

Some of the detailed work here involves showing that the expansion, shear, and vorticity tensors are in fact tensors. Most of the textbooks I have skip over congruences completely, and if they do consider them at all, they only consider geodesic congruences and not more general congruences. But we really need to be able to handle non-geodesic congruences for this type of problem. Wiki is one of the few sources that I've seen that talk about non-geodesic congruences. There's not a huge difference, the basic difference is that the four-acceleration of a point on a geodesic congruence is always zero, but on a general congruence it's not necessarily zero. The focus on geodesic congruences I've noted probably represents what I read more than what exists in the literature, my suspicion is basically that physicists mostly leave non-geodesic congurences to the mathematicians.

The material model in tensor form, I believe, says that when you know the expansion, shear, and the vorticity tensors of the congurence, the material model allows you to compute the associated stress-energy tensor. (Though I don't think Egan's model involves the vorticity tensor at all, from what I can recall. I'm leaving it in for completeness, it seems like it might be needed in general, though possibly one could argue somehow that the vorticity doesn't contribute to the model).

Then the equations to be satisfied are that the stress-energy tensor is divergence free, i.e. $\nabla_a T^{ab} = 0$. This apporoach needs to be expanded to take into account how one deals with constraints and interactions with other bodies, it's oriented towards the analysis of force-free bodies.

I don't have any references for any of this alas. It's really just my thoughts, based on trying to give a high-level and very abstract description of Egan's approach as I recall it from a very long time ago when I spent more time with it.

 

[PeterDonis]

My thinking is that one starts with a congruence of worldlines that represent the motion and deformation of the body in question.

Just to be clear, in a problem of this sort, one usually doesn't know in advance exactly which congruence of worldlines represents the motion of the body. However, as Egan describes, one approach (the second of two that he describes) is to think of the body as described by some (as yet unknown) congruence of worldlines, and write down a set of differential equations that describe the congruence, and look for solutions to those equations.

 

[Grinkle]

relativistic theory of materials

What does this mean?

SR -> how different reference frames calculate classically useful material properties with respect to each other?

GR -> how classically useful material properties change with the curvature of space time?

Something else?

 

[Dale]

I've always found Greg Egan's page on Relativistic Elasticity to be a good starting point:

http://www.gregegan.net/SCIENCE/Rindler/SimpleElasticity.html

He also gives a reference at the end that treats the subject in much more detail.

This is very helpful. I will need to go through it a second time. I think I will see if I can use it to construct a simple harmonic oscillator solution.

 

[Dale]

Then the equations to be satisfied are that the stress-energy tensor is divergence free, i.e. ∇aTab=0∇aTab=0\nabla_a T^{ab} = 0.

Surely you need more than that. That would lead to the same stress energy tensor regardless of the stiffness of the material itself. I think that there needs to be some mapping from the shear and expansion tensor to the stress energy tensor.

 

[Dale]

What does this mean?

I think what I really mean is a manifestly covariant generalization of Hooke’s law

 

[PeterDonis]

SR -> how different reference frames calculate classically useful material properties with respect to each other?

GR -> how classically useful material properties change with the curvature of space time?

Something else?

All of the above.

The main additional element that I see, over and above what you put under "SR" and "GR", both of which are important, is how the material properties determine what worldlines the individual pieces of the material follow.