[Probably a very niche question, but I hope someone here has useful information.]
All treatments I've found so far of elastic materials in general relativity use a notion variously called "material projection", or "matter space", or "body manifold", or other similar names.
The idea is to map every event (t,x) in spacetime M to a point (X) of a 3-dimensional manifold B:
M→Bby(t,x)↦X.
The manifold
B represents a material body (say a rubber ball), seen as abstracted from its evolution in time – and therefore only considered in its topological properties. The counter-image of a point
X in the body manifold is the world-line in spacetime of that particular point of the body.
Then, by considering a reference metric in the body manifold B, and comparing it (in a manner of speaking) with the metric of spacetime M, one manages to introduce the notion of strain of the material body, and so on.
One among the many reviews of this approach is for example that by Carter & al..
In the past, at least one way to frame elasticity in general relativity was proposed, by Synge in 1959, which avoided the use of a "body manifold", and instead managed to define a notion of strain from spacetime fields such as the matter-number current.
Do you know of any current works on elasticity in general relativity which avoid using the "body manifold" framework?
Cheers!