Einstein Field Eqs: Stress Energy Tensor Explained

[Silviu]

Hello! I have just started the Einstein field equations in my readings on GR and I want to make sure I understand the stress energy tensor. If we have a spherical, non-moving, non-spinning source, let's say a neutron star (I don't know much about neutron stars, so I apologize if the non-moving and spinning are realistic). Please tell me if the following are correct. As nothing moves, $T^{0i} = 0$. $T^{ij}$, should be equal to the pressure inside the star (as particles don't move across boundaries to carry momentum). Now for $T^{00}$, this contains $\rho$ the energy density. Now here is where I am a bit confused, does $T^{00}$ also contains the gravitational potential energy at a given point, or the potential energy doesn't exist in GR (i.e. mass curves spacetime and particles move on geodesics there, without needing the notion of potential energy). So let's say just outside the star, where $\rho = 0$ the $T^{00}$ is 0 or is the potential energy created by the star at that point? Thank you!

 

[andrewkirk]

does $T^{00}$ also contains the gravitational potential energy at a given point, or the potential energy doesn't exist in GR (i.e. mass curves spacetime and particles move on geodesics there, without needing the notion of potential energy). So let's say just outside the star, where $\rho = 0$ the $T^{00}$ is 0 or is the potential energy created by the star at that point? Thank you!

You are correct that it does not contain gravitational potential energy, and it is zero outside the star. The energy focused on for most problems in GR is mass-energy and kinetic energy, which merge into one under the GR framework.

 

[PeterDonis]

does $T^{00}$ also contains the gravitational potential energy at a given point

No.

or the potential energy doesn't exist in GR

There is a valid concept of gravitational potential energy in GR. It only applies to a certain class of spacetimes, but the one you are considering is in this class.

(i.e. mass curves spacetime and particles move on geodesics there, without needing the notion of potential energy)

This is correct; although there is a valid concept of gravitational potential energy in GR that applies to certain spacetimes (including the one you are considering), you don't need to use it to make any predictions about spacetime curvature and particle motions.