I come from a continuum mechanics background, and I make numerical simulations of fluids/solids using the Finite Element Method. The basic equation we solve then is Newton's law of motion, written in terms of relevant vectors and tensors. Using a Lagrangian description we have the equilibrium equation:
$\rho \underline{\ddot u} = div(\underline{\underline{\sigma}}) + \underline{f}_v$
where $\underline{\ddot u}$ is the acceleration 3-vector (second time derivative of the displacement vector), $\underline{\underline{\sigma}}$ is the second-order Cauchy stress tensor and $\underline{f}_v$ is the 3-vector of external forces.
The system of equations is completed by a constitutive equation, linking the stress tensor to a measure of strain (typical solids) or a measure of strain rate (viscoelastic solids and fluids). For the sake of the example let us consider a usual linear elastic relation:
$\underline{\underline{\sigma}} = \mathbb{C} : \underline{\underline{\varepsilon}}$
with $\mathbb{C}$ being the 4th-order elasticity tensor and $\underline{\underline{\varepsilon}}$ the linear strain tensor, classically defined in the compatibility equation as the symmetric part of the displacement gradient:
$\underline{\underline{\varepsilon}} = \frac{1}{2}(\underline{\underline{\nabla}} u + \underline{\underline{\nabla}}^T u)$.
Now, the equations I stated are all I need in order to discretize the system and simulate small deformations of an arbitrary geometry under a system of forces in the context of Newtonian mechanics, using the Finite Element Method.
What I want to know is what is the way to adapt these equations so that they satisfy General Relativity. That is, I want to simulate the deformation of a mechanical structure when the velocities involved are close to the speed of light and/or when a very massive object is near.
I am familiar with nonlinear elasticity if needed, but as far as tensors go I'm unfamiliar with co-variant/contra-variant notation and I prefer intrinsic notation, even though I'll take answers expressed in any way.
What form do the equilibrium, constitutive and compatibility equations take? Is the simulation of deformation of bodies in a relativistic context something that was properly done already? Does the elasticity tensor need to be redefined in terms of the metric tensor maybe?
I couldn't find any good reference that addresses this issue, even though I feel like this is possible to achieve. I would be very thankful for any help on this matter.
Best Answer
The answer you're looking for seems to be contained in
Rezzolla & Zanotti: Relativistic Hydrodynamics (Oxford U.P. 2013)
https://books.google.com/books/?id=KU2oAAAAQBAJ
but it is not a trivial generalization. Quoting Disconzi's On the well-posedness of relativistic viscous fluids (Nonlinearity 27 (2014) 1915, arXiv:1310.1954):
Similar problems exist for elastic materials.
You may also take a look at Chapter 15 ("Relativistic continuum mechanics") of Maugin's Continuum Mechanics Through the Twentieth Century (Springer 2013), and its references:
https://books.google.com/books?id=-QhAAAAAQBAJ
and at Bressan's Relativistic Theories of Materials (Springer 1978):
https://books.google.com/books?id=kMTuCAAAQBAJ
General-relativistic continuum mechanics unfortunately has not been given a clear mathematical and conceptual framework yet. Newtonian continuum mechanics is easy to summarize:
And at this point we have a well-defined set of partial differential equations in a number of unknown fields, for which we can set up well-defined initial- & boundary-value problems to be solved analytically or numerically. (An expanded but analogous framework accommodates electromagnetism and continua with internal structure.)
This framework and steps are very neat – we clearly know what the fields are, which of them are dependent and which independent; what are the equations valid for all materials, and what are the equations constitutive to each material. I've never seen a clearly defined procedure like the one above for general relativity, although I believe it could be extracted from Rezzolla & Zanotti's or Bressan's books. Moreover, the core of general-relativistic community uses a different jargon and way of thinking.
Most general-relativity books tell you that the Einstein equations determine everything, but they are not so clear about which fields in them are independent and which dependent; even Misner et al.'s Gravitation (ch. 21) has a long discussion and explanation about this point. It was only with 3+1 formulations and the work of Arnowitt, Deser, Misner, York, and others around the 1970s that this point got clarified. Then they tell you that we need "special" additional equations for the stress tensor – that is, constitutive equations. Sometimes other conservation equations, like baryonic number (basically rest-mass), are added with no real explanation. This is a sample of books where "constitutive equations" are mentioned explicitly (only once or twice in most of them):
but they constitute a very small minority in the huge relativistic literature.
Yet, the general-relativistic community cannot be criticized for the confused conceptual state and somehow confused language of the subject. Newtonian continuum mechanics can be neatly formulated today because it has been refined over several centuries. General relativity is still very young instead, and its conceptual refinement still in progress. Some of the steps in the Newtonian framework become extremely complicated in general relativity. For example: step 1. (choose an inertial frame) cannot be done so simply. The Einstein equations, evolved from initial conditions, construct a reference frame "along the way", while they determine the dynamics. This gives rise to peculiar fields like "lapse" and "shift", which aren't really physical, and all sorts of redundancy (gauge freedom) in the equations.
Another example: the metric becomes a dynamical field variable, and you suddenly realize that it is hidden almost everywhere in the Newtonian framework – divergences, curls, vectors/covectors... So its evolution can't be easily divided among some new balance and constitutive equations (like we can do with electromagnetism instead). Are all of its appearances in the Newtonian framework dynamically significant? or can the metric be eliminated from some places? There's some research today on this "de-metrization" of Newton's equations; see for example Segev's Metric-independent analysis of the stress-energy tensor, J. Math. Phys. 43 (2002) 3220. This line of research has shown that some Newtonian physical objects actually don't need a metric: they be expressed via differential forms and other metric-free differential-geometrical objects (e.g., van Dantzig's On the geometrical representation of elementary physical objects and the relations between geometry and physics, Nieuw Archief voor Wiskunde II (1954) 73; there is a vast literature on this, let me know if you want more references). This is still work in progress – which means that it's obviously not completely clear yet how mass-energy-momentum-stress and metric are coupled.
To conclude, I think another good starting point to understand how things work in general-relativistic continuum mechanics is to look in books on numerical formulations of general relativity and matter dynamics. The conceptual framework in them is a bit confused, but you can see how they actually do it. If from the practice of these books you manage to reverse-engineer a framework like the Newtonian one above, please write a pedagogical paper about it!
Here are some books and reviews on numerical relativity with continua:
I'm happy to provide or look for additional references.