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

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?
• 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). May 2, 2022 at 2:12
• 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). May 2, 2022 at 2:41
• 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. May 2, 2022 at 2:44
• Uniform wrt. what radial coordinate? May 2, 2022 at 7:10

• @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). May 2, 2022 at 2:01