Here is Wikipedia's diagram of the stress-energy tensor in general relativity:
I notice that all of its elements are what would be termed "macroscopic" quantities in thermodynamics. That is, in statistical mechanics we would usuallt define these quantities in terms of an ensemble of systems, rather than in terms of the microscopic state of a single system. (This doesn't make much difference for large systems, but for small ones it does.) This observation leads me to a number of questions – I hope it's ok to post them all as a single Question, since they're so closely related:
- Am I correct in inferring that general relativity is actually a macroscopic, or phenomenological theory, rather than a theory about the microscopic level?
- Was Einstein explicit about this in deriving it? Or did he simply start by assuming that matter can be modelled as a continuously subdivisible fluid and take it from there?
- If general relativity is a macroscopic theory, what does the microscopic picture look like? (I'd expect that this is actually unknown, hence all the excitement about holography and whatnot, but perhaps I'm being naïve in thinking that.)
- Are there cases in which this continuous fluid approximation breaks down? For example, what if there are two weakly interacting fluids with different pressures?
- If general relativity is a macroscopic theory, does it imply that space and time themselves are macroscopic concepts?
- Is this related to the whole "is gravity an entropic force?" debate from a few years ago?
Best Answer
Before answering, I would like to say that the difference between macroscopic and microscopic is not made in terms of ensembles of systems; in fact, quantum mechanics has an ensemble interpretation. About your questions, my answers are the following:
Yes. General relativity is a pre-quantum theory, which means that does not account for the discrete particle-like structure of matter. Particularly, I never use the term "phenomenological theory", which I consider a misnomer.
Yes, Einstein, Grossmann, and Hilbert explicitly ignored the structure of matter when developed general relativity.
There is not microscopic picture of general relativity, because this is a (geo)metric theory. Somehow as there is not a microscopic picture of geometric optics. Of course there is a microscopic picture of physical optics which we call quantum optics. A quantum gravity is currently under active research. A first step is the quantum field theory of gravitons whose "microscopic picture" is close to that of quantum electrodynamics.
There are many cases where the continuous fluid approximation used in general relativity breaks down. E.g. if there are shock waves in your interacting fluids, then they cannot be described by a continuous fluid model. The best that you can do is to describe matter at the mesoscopic level and gravity at the macroscopic level. An example is the Einstein/Vlasov approach. Matter (e.g. a collision-less plasma) is described by the Vlasov kinetic equation, but $g_{\mu\nu}$ is obtained from an approximated energy-momentum tensor $T_{\mu\nu}$ which is computed from averaging over matter with the help of the kinetic $f(x,p,t)$ (see eq. 32 in above link). Both mesoscopic and microscopic descriptions of gravity are entirely outside the scope of GR.
No. Because the (geo)metric model of general relativity is not fundamental, as Feynman already noted [1]:
The underlying quantum theory of gravity uses, essentially, the same space and time as quantum mechanics.
No. There are lots of flawed thermodynamic analogies found in the general relativity literature (black hole thermodynamics being the more popular of them).
[1] Feynman Lectures on Gravitation 1995: Addison-Wesley Publishing Company; Massachusetts; John Preskill; Kip S. Thorne (foreword); Brian Hatfield (Editor). Feynman. Richard P.; Morinigo, B. Fernando; Wagner, William G.