# Can the Einstein tensor be written as an integral over real spacetime?

## Background

We know the Stress-energy tensor can be written as:

$$T^{\mu \nu} = \int \mathcal{N}(x,p,t) p^\mu \otimes p^\nu \frac{d V_p}{E}$$

where $$\mathcal{N}(x,p,t)$$ is the distribution function, $$p^\mu$$ is the $$4$$ momentum, $$p^0 = E$$ is the energy and $$d V_p$$ is the momentum phase space.

## Question

Can the Einstein Tensor also be written as an integral over real (not phase) space? If so, what? I think this would be really interesting since that would mean General relativity is telling us something about how real space relates to momentum space.

• Your stress energy tensor can only be written that way because of certain assumptions on the composition of it (it being composed of classical particles). It's not true in general and certainly doesn't seem to apply to the Einstein tensor part. Commented Nov 13, 2022 at 6:54
• @Raskolnikov I was thinking of classical general relativity without electromagnetism Commented Nov 13, 2022 at 7:39
• @KP99 I wasn't thinking of doing anything quantum. In classical mechanics they are independent variables: physics.stackexchange.com/a/318964/150174 Commented Nov 13, 2022 at 7:57
• What do you mean? $G_{\mu\nu}(x)$ and $T_{\mu\nu}(x)$ refer to a single point $x$ in spacetime; there is no spacetime integration. Commented Nov 13, 2022 at 8:43
• It is unclear what the question is about. Are you looking for an integral representation of Einstein equations? Is this a question about Einstein-Vlasov (or similar) system, where your expression for SE tensor applies? Or are interested in averaging problem in GR and in coarse-grained Einstein tensor? Commented Nov 13, 2022 at 9:10