Here is Wikipedia's diagram of the stress-energy tensor in general relativity:

enter image description here

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?
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    $\begingroup$ While some quantum/stat mech types might want you to believe anything macroscopic is phenomenological curve-fitting and somehow less pure, that's no more true than saying all non-relativistic physics is phenomenological and thus without a theoretical basis. $\endgroup$ – user10851 Nov 23 '12 at 8:39
  • $\begingroup$ @ChrisWhite pheonomenological $\ne$ without a theoretical basis. $\endgroup$ – Nathaniel Nov 23 '12 at 8:40
  • $\begingroup$ ..and certainly $\ne$ curve fitting! $\endgroup$ – Nathaniel Nov 23 '12 at 8:40
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    $\begingroup$ @ChrisWhite in other words, I didn't mean in the slightest to imply anything negative about general relativity by suggesting that it's a macroscopic theory. As a thermodynamicist, I consider macroscopic theories to be a very pure form of physics indeed. $\endgroup$ – Nathaniel Nov 23 '12 at 8:42
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    $\begingroup$ Very well - I guess I've just been jaded by seeing too many physicists deride other branches of physics for not using primitive enough principles. $\endgroup$ – user10851 Nov 23 '12 at 8:45

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]:

    It is one of the peculiar aspects of the theory of gravitation, that is has both a field interpretation and a geometrical interpretation. [...] The geometrical interpretation is not really necessary or essential to physics.

    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.

  • $\begingroup$ What would you say does define the microscopic/macroscopic distinction, if not the use of ensembles? In my view, the difference is that macroscopic quantities require the entropy to be defined in order for them to make sense (e.g. pressure can be defined as $-\partial U/\partial V |_{S,{N_i}}$, in which entropy must be held constant when calculating the derivative), and you can't have entropy without an ensemble. $\endgroup$ – Nathaniel Nov 25 '12 at 8:56
  • $\begingroup$ As stated above QM is a theory of ensembles (Rev. Mod. Phys. 1970: 42, 358-381). The amount of atomic-molecular details are what differentiates each level in the hierarchy of ensembles: thermodynamic level <---> hydrodynamic level <---> Boltzmann level <---> ··· QM level. The two first levels are collectively labelled as macroscopic description, QM is microscopic description and the Boltzmann level is often considered mesoscopic description. $\endgroup$ – juanrga Nov 26 '12 at 21:03
  • $\begingroup$ Sure, you can interpret QM as an ensemble theory. I'm in favour of this, though the vast majority of physicists aren't. But I would say that this interpretation makes QM a macroscopic theory, precisely because it supposes a level below the QM level, which describes the individual members of the ensemble. (By the way, if you write "@Nathaniel" in your replies I'll be notified of them. There are some fairly complicated rules about when you need to do this and when you don't.) $\endgroup$ – Nathaniel Nov 30 '12 at 2:14
  • $\begingroup$ More constructively, you say that the fluid model of GR breaks down in the case of shock waves, which makes sense - but how does one do general relativity in such cases? You can't write down Einstein's equations if there isn't a (single, unique) stress-energy tensor, so what takes its place? $\endgroup$ – Nathaniel Nov 30 '12 at 3:27
  • $\begingroup$ @Nathaniel: The ensemble interpretation of QM is rather agnostic regarding the existence of a sub-quantum level. If this level exists I would say that both the quantum and the sub-quantum are microscopic levels, somehow as both the thermodynamic and the hydrodynamic are macroscopic levels. I am glad to see that you favour the ensemble interpretation. $\endgroup$ – juanrga Nov 30 '12 at 19:17

To quote the Wikipedia article

$T^{ik}$ represent flux of $i^{th}$ component of linear momentum across the $x^k$ surface

so the definition is actually microscopic, in the sense that you can in principle calculate the momentum for every particle in an ensemble. However the momentum flux corresponds to what we mean by shear stress and pressure so this is what we'd use in practice. As we reduce the size of the system our approximation of the momentum fluxes by macroscopic concepts becomes poor and we just put in the momenta explicitly.

There is no sense in which GR is a macroscopic theory, well, not until we get to quantum gravity but this has a much smaller length scale than what we normally mean by microscopic. It's just that we may wish to use macroscopic approximations when we construct the stress-energy tensor.

  • $\begingroup$ What do you mean by "we just put in the momenta explicitly"? Does this mean it's possible to do GR with ideal point masses instead of a continuous fluid model? $\endgroup$ – Nathaniel Nov 23 '12 at 7:21
  • $\begingroup$ Also, I really don't think a "momentum flux" is a microscopic concept at all. You either have to average over an ensemble or over a nonzero time period in order for it to be meaningful - otherwise it will just be zero (if no particle crossed the surface) or infinite. $\endgroup$ – Nathaniel Nov 23 '12 at 7:22
  • $\begingroup$ @Nathaniel: if you treat your particle as a point mass you will indeed get an infinite momentum flux, but then the particle would be a black hole so it would have infinite density. I'm not sure how you include a black hole in the stress-energy tensor: presumably you'd treat it as being the size of it's event horizon. That's finite so you'd get a finite momentum flux. Or maybe you'd need to use a proper (currently non-existant) quantum gravity approach. $\endgroup$ – John Rennie Nov 23 '12 at 8:53

General relativity is a classical theory, so it makes sense at all levels, though that's different from being correct at all levels (it shouldn't be). The energy-momentum tensor doesn't intrinsically have anything to do with statistical mechanics or fluids at all. Its size just reflects that gravity is a spin-2 field.

For a particle with charge $q$ in its rest frame with worldline $\xi^\mu(\tau)$ with four-velocity $u$ as a function of its proper time $\tau$ is, the appropriate source density for the field of spins 0,1,2 (respectively) would be: $$\rho(x^\sigma) = q\int\delta^4(x^\sigma-\xi^\sigma(\tau))\,\mathrm{d}\tau$$ $$J^\mu(x^\sigma) = q\int u^\mu\delta^4(x^\sigma-\xi^\sigma(\tau))\,\mathrm{d}\tau$$ $$T^{\mu\nu}(x^\sigma) = q\int u^\mu u^\nu\delta^4(x^\sigma-\xi^\sigma(\tau))\,\mathrm{d}\tau$$ The electromagnetic field is spin-1, and the electromagnetic charge density is actually a four-current $J^\mu$, and there is nothing conceptually strange about a lone point-like charge. It just means the four-current is described using an appropriate Dirac delta function over its worldline, in a slightly more complicated way than having a Dirac delta function for a charge density $\rho$.

In general, a four-current $J^\mu$ means that an observer with four-velocity $v$ measures a charge density $J^\mu v_\mu.$ Similarly, a 2-tensor $T^{\mu\nu}$ means that such an observer measures a four-current density $T^{\mu\nu}v_\mu$ and charge density $T^{\mu\nu}v_\mu v_\nu$. For GTR, the charge is mass-energy and the four-current is the four-momentum.

Thus, microscopically, it's exactly the same theory.

The problem isn't that we can't get a sensible $T^{\mu\nu}$ for ideal point-masses. In flat spacetime, it's easy, and indeed $T^{\mu\nu}$ is sensible and useful even in STR. The problem is peculiar to GTR rather than the conceptual nature of $T^{\mu\nu}$: the theory says spacetime won't be flat and that you'll get a black holes with a singularity. To try to fix this, Einstein invented the wormhole ("Einstein-Rosen bridge") and attempted to replace point-particles with them. The proposal doesn't actually work for that purpose, though.


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?

Judge by yourself with this excerpt from the Princeton lectures (1921), published in english as "The Principle of Relativity". When departing from Poisson's equation in his heuristic search for the field equations of GR, he states:

We have seen, indeed, that in a more complete analysis the energy tensor can be regarded only as a provisional means of representing matter. In reality, matter consists of electrically charged particles, and is to be regarded itself as a part, in fact, the principal part, of the electromagnetic field. It is only the circumstance that we have not sufficient knowledge of the electromagnetic field of concentrated charges that compels us, provisionally, to leave undetermined in presenting the theory, the true form of this tensor. From this point of view our problem now is to introduce a tensor, $T_{\mu\nu}$, of the second rank, whose structure we do not know provisionally, and which includes in itself the energy density of the electromagnetic field and of ponderable matter; we shall denote this in the following as the ``energy tensor of matter.''


I don't want to un-accept @juranga's perfectly good answer, but for future visitors it's worth recording that the macroscopic nature of general relativity is made very clear in this 1995 paper, in which Ted Jacobson derives Einstein's field equations from $dS = \delta Q/T$, together with the Bekenstein bound. (Plus a few other assumptions to do with special relativity and the Unruh effect.)

In the last few paragraphs of the paper, Jacobson spells out some circumstances in which the thermodynamic assumptions he makes might break down. In particular, he points out that in his argument the time-reversible nature of space-time evolution arises from a near-equilibrium assumption. This assumption would not hold close to the big bang and black hole singularities, and this might lead to space-time behaving in thermodynamically irreversible ways. It's interesting stuff.


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