I've recently been reading some notes (following a paper by J.D. Bekenstein, titled "The Relation between Physical and Gravitational Geometry": http://arxiv.org/abs/gr-qc/9211017) on alternative theories of gravity in which the author considers two metrics:

"One defining the geometry on which matter plays out its dynamics $\bar{g}_{\mu\nu}$, and one which describes gravitation $g_{\mu\nu}$". (In second paragraph of introduction in paper linked to above).

What exactly is meant by this statement? Is it simply that $g_{\mu\nu}$ is a solution to Einstein's equation $G_{\mu\nu}=8\pi T_{\mu\nu}$ such that it defines the geometry of spacetime which manifests itself as gravity? And then the physical metric $\bar{g}_{\mu\nu}$ is the one that matter is coupled to and is related to the gravitational metric $g_{\mu\nu}$ through some transformation (e.g. conformal, disformal, etc.)?

  • 1
    $\begingroup$ You'll have to give the reference for this to be answerable - we can't guess what precise modification the author has in mind. $\endgroup$
    – ACuriousMind
    Aug 22 '15 at 18:31
  • $\begingroup$ @ACuriousMind Sorry, I've updated my question with a link to the original paper I read on the subject. $\endgroup$
    – Will
    Aug 22 '15 at 18:35
  • $\begingroup$ I think you got it precisely the wrong way around, i.e. $\bar g$ should describe the dynamics. $\endgroup$
    – Danu
    Aug 22 '15 at 19:07
  • $\begingroup$ @Danu What do you mean by that? Is it that $\bar{g}$ is a solution to Einstein's equation? If this is the case, then what does Bekenstein mean when he says that $g$ is the metric that defines gravitation? $\endgroup$
    – Will
    Aug 22 '15 at 19:44


The short answer is you have one action you extremize to get Einstein's Field Equation $G_{\alpha\beta}=kT_{\alpha\beta}.$ Which you can think of as equations of motion for the gravitational metric $g_{\alpha\beta}.$ (They determine the second derivatives of the metric in terms of the matter fields and metric and the first derivatives of the metric.) And you have a different action you extremize to get the equations of motion for the matter (instead of them moving on geodesics in the gravitational metric). So it's like there is a different geometry you use for finding out how the matter moves.

To compare the two geometry approach to GR I'll first go into some details about how GR is usually used (in more detail than you might want to see). This is for contrast and comparison purposes. Just skip the next paragraph if if how GR is usually practiced is too bothersome and you need to move on. I personally don't know why the two geometry approach would be called for or even desirable, but if it can agree with observations made so far I won't be judgemental in advance.

In GR fiduciary particles (or test particles) are particles that have vanishing mass and vanishing spin and vanishing charge and so on and that take their limits to zero in some nice way (so they have nice ratios, etc.), so that when they are subject to no interactions besides gravity, they travel on geodesics in the background spacetime whose curvature they contribute to. At least that is the story people like to say. It's hard to even make it precise enough to be right or wrong and there are no test particles in nature so the details will forever be untestable by experiment. It is most applicable if you do GR in a fluid dynamics limit for your matter sources. Then you have fluid element with bulk properties and the components inside an element can be test particles as a continuous density of matter.

So that's the story at least, things move on geodesics when it's just gravity. But you could look at Einstein's Field Equation and decide that you want the spacetime geometry to evolve in way consistent with that so that $G_{\alpha\beta}=kT_{\alpha\beta}.$ But then you could say that you don't want these test particles to move on geodesics. That's your right if you can make it agree with observations so far.

Thus, the two-geometries approach to the formulation of gravitational theory is an important paradigm. Whenever it becomes necessary to formulate a new theory of gravity, a conservative way to proceed in order to avoid immediate conflict with the tests of GR is to invoke a Riemannian metric $g_{\alpha\beta}$, build the Einstein-Hilbert action for the geometry’s dynamics out of it, and effect the departure from standard GR by prescribing the relation between $g_{\alpha\beta}$ and the physical geometry on which matter propagates. Most known theories assume the relation is a simple conformal transformation.

Then on the bottom of page 4 the author states that particle dynamics are determined by extremizing $$S=\frac{1}{2}\int g_{\alpha\beta}\dot x^\alpha \dot x^\beta F(I,H,\Psi)d\lambda$$

So you still have a metric that makes $G_{\alpha\beta}=kT_{\alpha\beta}$ and you can think of that as equations of motion for the gravitational metric (the one that makes $G_{\alpha\beta}$) when given $T_{\alpha\beta}.$ But then there can be another geometry for which $S=\frac{1}{2}\int g_{\alpha\beta}\dot x^\alpha \dot x^\beta F(I,H,\Psi)d\lambda$ is extremized and that geometry tells matter how to move.

This is the essential aspect of a two geometry approach. And you can do it so that free massless particles (actual massless ones that were massless to begin with, not the fiduciary test particles that moved on timelike "curves" even as you pretended to take the limit of no mass), when subjected to the second geometry to get dynamics will still move on curves with null tangents that are null in the first geometry. But in general the paths of particles in a two geometry theory are not geodesics. Again, from the same paper.

We do not require that the trajectories which extremize S in the Finsler geometry coincide with the geodesics of $g_{\alpha\beta}.$ There is no physical basis for such an assumption in our context: the metric $g_{\alpha\beta}$ is for gravitational phenomena, whereas the Finsler geometry is for matter dynamics.

So one metric for matter telling spacetime how to curve and a different one for telling matter how to move. And then you can go into details about how they are related to each other.

  • $\begingroup$ So is that what he means then, that $g_{\mu\nu}$ (the "gravitational metric") describes the curvature of spacetime in the presence of matter. As gravitation is the manifestation of this curvature, is this why he says it describes gravitation (and is referred to as the gravitational metric)? $\endgroup$
    – Will
    Aug 22 '15 at 20:52

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