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In Newtonian mechanics, assuming a spherical uniform mass distribution, the total gravitational potential energy (gravitational self-energy) inside the sphere is

$$U_{gs}=-\frac35\frac{GM^2}R.$$

  1. In general relativity, assuming a spherical uniform mass distribution, what is the total energy value of the gravitational field inside the sphere?

  2. In general relativity, assuming a spherical uniform mass distribution, does "total energy value of the gravitational field inside the sphere" equal the "gravitational self-energy"?

*This question was added after 1 answer. In general relativity, I have seen many articles about the difficulty of defining the energy of the gravitational field.

The point of my question is,

  1. In a weak gravitational field or in near-flat spacetime, can the energy density of the gravitational field be taken from the total gravitational potential energy obtained from Newtonian mechanics, or from the gravitational self-energy, as a (good?) approximation?
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  • $\begingroup$ This is a matter of much less concern because your question has a "GR" tag and did not include a "cosmology" tag, but, as this site caters to all education levels, "inside the sphere" might be misinterpreted as "within the ball" of which the sphere is a surface: In at least one cosmological model whose Arxiv preprints can be found by their author's name (Nikodem Poplawski), the local universes of an inflationary multiverse are analogized as resembling "the skin of a basketball", which might be as thin as the fermions of the Einstein-Cartan Theory that he uses (which have a spatial extent). $\endgroup$
    – Edouard
    May 2, 2022 at 2:12
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    $\begingroup$ This question ultimately hinges on exactly what you mean by a good approximation. GR reduces to Newtonian gravity in an appropriate limit, so anything you can do in Newtonian gravity, you can also do in GR, in that limit. If that is a good enough approximation for you, then there's your answer. However, things become much more complicated if you want the approximation to be good enough that it includes relativistic corrections to the Newtonian limit (en.wikipedia.org/wiki/Gravitational_energy#General_relativity). $\endgroup$
    – Andrew
    May 2, 2022 at 2:41
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    $\begingroup$ It helped a lot. Since there is no generalized application method, I am going to define and use ρ_gs=U/V. By the way, would this cause serious problems in near-flat spacetime? This is what I'm curious about. $\endgroup$
    – D will
    May 2, 2022 at 2:44
  • $\begingroup$ Uniform wrt. what radial coordinate? $\endgroup$
    – Qmechanic ♦
    May 2, 2022 at 7:10

1 Answer 1

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In general relativity, there is no gauge invariant local definition of the energy of a gravitational field. In simpler terms, there is not a well-defined way to define the total energy contained in a finite region of space. So, while you can take a non-relativistic limit and recover Newtonian gravity, it is not possible to unambiguously generalize the idea of the non-relativistic gravitational energy contained in a region, to GR.

For an asymptotically flat spacetime (such as a lump of matter sitting in isolation), you can define a mass or energy associated with the space. But, you cannot localize this energy to a region.

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  • $\begingroup$ Thank you. But~In cosmology, the energy density ρ is used, and the concept of internal energy U/V is also used. So, U_gs=ρ_gsV, ρ_gs=U_gs/V; That is, why not define and use the equivalent energy density corresponding to the gravitational potential energy? $\endgroup$
    – D will
    May 2, 2022 at 1:56
  • $\begingroup$ In an almost flat space-time, why not define and use the energy density for the total gravitational potential energy? Isn't this a good approximation? $\endgroup$
    – D will
    May 2, 2022 at 2:00
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    $\begingroup$ @Dwill $\rho$ refers to the $00$ component of the stress-energy tensor $T_{\mu\nu}$, which gives us a way of defining energy density for matter. What I'm saying is that there is not an unambiguous way to define the stress energy tensor for the gravitational field. (There are some cases where it is useful to do so, like in studying weak gravitational waves, but not in general). $\endgroup$
    – Andrew
    May 2, 2022 at 2:01
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    $\begingroup$ In the weak field limit, you do recover the Newtonian potential, and you can define the energy density of the gravitational field to some approximation as if you were in Newtonian gravity. The problem is that to answer your question, you need to generalize the definition so it applies beyond the Newtonian limit. There's no good way to do that. You might be able to make it work in some special cases in an approximate sense, but ultimately the gravitational energy just isn't a very useful concept in GR in most cases (gravitational waves are one exception where it can be useful). $\endgroup$
    – Andrew
    May 2, 2022 at 2:02
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    $\begingroup$ To be clear, it's known that you can't define a stress-energy tensor for gravity, so the best case is that your definition provides an approximate notion of the gravitational energy density that is good enough for you application. It's very possible that your definition matches this set of criteria -- I can't say without looking in more detail, which realistically I am not going to do. The burden is on you to supply a convincing argument that your definition of gravitational energy density has useful properties when applied to your problem. Since you wrote a paper, we should be asking you. $\endgroup$
    – Andrew
    May 2, 2022 at 3:51

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