Justification for excluding gravitational energy from the stress-energy tensor

Question

I did general relativity years ago at Uni and was just revising with the aid of Dirac''s brilliant book; the beauty of this book is that it is so thin and concise. On reading this book I find that I have a few questions regarding energy.

One thing I had not appreciated before was that the energy in the energy tensor only includes all energy excluding gravitational energy. Is this true? What is the evidence for this position? How could we know that this energy term actually excludes gravitational energy?

The only argument that I can see is that the energy in the energy tensor is not fully conserved, so you could infer that there is a missing energy term and that that energy is gravitational energy. But if you take the missing quantity and call it the gravitational energy, this quantity turns out not to be a tensor.

Hence, its form will in general always look different in at least some different coordinate systems regardless of whichever quantities you use to write it out in. This latter point might only have mathematical consequences, but does it have physical consequences?

Answer 1

I came across this passage in Misner, Thorne & Wheeler (20.4) where they first talk about the stress-energy pseudotensor of the gravitational field and how one might calculate a contribution to the local momentum vector... and then memorably state:

Right? No, the question is wrong. The motivation is wrong. The result is wrong. The idea is wrong.

To ask for the amount of electromagnetic energy and momentum in an element of 3-volume makes sense. First, there is one and only one formula for this quantity. Second, and more important, this energy-momentum in principle "has weight." It curves space. It serves as a source term on the righthand side of Einstein's field equations. It produces a relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is observable. Not one of these properties does "local gravitational energy-momentum" possess. There is no unique formula for it, but a multitude of quite distinct formulas. ... Moreover, "local gravitational energy-momentum" has no weight. It does not curve space. It does not serve as a source term on the righthand side of Einstein's field equations. It does not produce any relative geodesic deviation of two nearby world lines that pass through the region of space in question. It is not observable.

As Ben said, one cannot speak of gravitational energy being specified at one point: at the very least it requires integrating over a 4-volume, but even that turns out to be fraught and one gets several competing versions.

Answer 2

Well, first of all you should ask yourself what do you expect from the concept of energy and momentum. Or in other words, what is energy and momentum, really? You have a set of intuitions in mind, but the minimal requirement is that these are quantities that are 1) conserved, and 2) reduce to our "usual" definitions of energy and momentum in suitable limits.

Let us take a look at the Einstein equations $$G_{\mu\nu} = 8\pi T_{\mu\nu}$$ The Einstein tensor $G_{\mu\nu} = R_{\mu\nu} - R g_{\mu\nu}/2$ does have the property of being conserved $G^{\mu\nu}_{\;\;\;;\nu} = 0$, same as the stress-energy tensor. Furthermore, and this is a somewhat nuanced point, certain parts of it can be interpreted as gravitational energy in the weak-field limit.

For instance, in the post-Minkowski limit you have $g_{\mu\nu} = \eta_{\mu\nu} + \epsilon h^{(1)}_{\mu\nu} + \epsilon^2 h^{(2)}_{\mu\nu}+...$. At first order in $\epsilon$ the left-hand side of the Einstein equations just correspond to a linear operator acting on $h^{(1)}_{\mu\nu}$. At second order, however, you have the same operator acting on $h^{(2)}_{\mu\nu}$ and terms quadratic in $h^{(1)}_{\mu\nu}$ that can be interpreted as (minus) the stress-energy tensor of the field $h^{(1)}_{\mu\nu}$ (sourcing the gravitational field correction $h^{(2)}_{\mu\nu}$). So $G_{\mu\nu}$ (or $-G_{\mu\nu}/8\pi$) seems to naturally offer itself as a gravitational stress-energy tensor.

However, as you may have already noticed, when $-G_{\mu\nu}/8\pi$ is moved to the right-hand side of the Einstein equations as a part of stress-energy, then the interpretation of the equations ends up being that the total stress-energy content of any point of space-time is exactly zero. Every order of the post-Minkowski expansion is really just enforcing that statement, at higher and higher order in $\epsilon$.

This may be elegant, but underwhelming, especially in vacuum space-times. In vacuum space-times, such as space-times of inspiraling and merging black holes, a lot can be obviously going on, while this interpretation is just telling us there is no stress-energy ever flowing, present, or exchanged. This is the reason why other notions of energy and momentum of the gravitational field have been introduced. But I am going to be honest, their real value is in keeping track of how the gravitating mass (and/or momentum) of a system evolve in time. After this violent process with outgoing gravitational radiation, how much will this system still attract me gravitationaly? These "nonlocal" formulas mentioned in the other posts here provide the answer. But beyond this pragmatic meaning, I would defer from interpreting them too deeply.

Answer 3

The simplest way to see this is that by the equivalence principle, we can always let the gravitational field at any point have any value we like, including zero. Therefore there is no possibility of defining an energy density of the gravitational field at one point.

Answer 4

There are various definitions of the total global mass-energy contained in a spacetime: the ADM mass, Bondi mass, etc. In present understanding, these require specific conditions such as symmetry in time, or asymptotic flatness.

Then there are an awful lot of quasi-local proposals. The idea is that, since you can't define the gravitational energy at a point, draw a small box around a region of spacetime, and define the gravitational energy inside it. For instance, Hawking's proposal examines how light rays exiting the box diverge; the motivation is mass-energy curves spacetime which affects the light. The proposal by Epp (& Mann & McGrath), studies the acceleration of the walls of the box. Another proposal by Bartnik is more mathematically motivated, I'm told. Most definitions agree for simple cases, apparently.