I'm trying to understand the Einstein Field equation equipped only with training in Riemannian geometry. My question is very simple although I cant extract the answer from the wikipedia page:
Is the "stress-energy" something that can be derived from the pseudo-Riemannian metric (like the Ricci tensor, scalar curvature, and obviously the metric coefficients that appear in the equation) or is it some empirical physics thing like the "constants of nature" that appear in the equation? Or do you need some extra mathematical gadget to specify it? Thanks and apologies in advance if this is utterly nonsensical. Also, as a non-physicists I'm not sure how to tag this either so sorry for that as well.
Best Answer
Good question! From a physical perspective, the stress-energy tensor is the source term for Einstein's equation, kind of like the electric charge and current is the source term for Maxwell's equations. It represents the amounts of energy, momentum, pressure, and stress in the space. Roughly:
$$T = \begin{pmatrix}u & p_x & p_y & p_z \\ p_x & P_{xx} & \sigma_{xy} & \sigma_{xz} \\ p_y & \sigma_{yx} & P_{yy} & \sigma_{yz} \\ p_z & \sigma_{zx} & \sigma_{zy} & P_{zz}\end{pmatrix}$$
Here $u$ is the energy density, the $p$'s are momentum densities, $P$'s are pressures, and $\sigma$'s are shear stresses.
In its most "natural" physical intepretation, Einstein's equation $G^{\mu\nu} = 8\pi T^{\mu\nu}$ (in appropriate units) represents the fact that the curvature of space is determined by the stuff in it. To put that into practice, you measure the amount of stuff in your space, which tells you the components of the stress-energy tensor. Then you try to find a solution for the metric $g_{\mu\nu}$ that gives the proper $G^{\mu\nu}$ such that the equation is satisfied. (The Einstein tensor $G$ is a function of the metric.) In other words, you're measuring $T$ and trying to solve the resulting equation for $G$.
But you can also in principle measure the curvature of space, which tells you $G$ (or you could pick some metric and get $G$ from that), and use that to determine $T$, which tells you how much stuff is in the space. This is what cosmologists do when they try to figure out how the density of the universe compares to the critical density, for example.
It's worth noting that $T$ is a dynamical variable (like electric charge), not a constant (like the speed of light).