# [Physics] How does one actually use the Einstein field equations

general-relativity

I'm experimenting with the Einstein field equations (EFE) and I'm wondering how to actually use them. It seems to me that the simplest are those evolving a vacuum, which imply that the stress-energy-momentum tensor $T_{mn}$ equals zero, because there is no matter, nor energy to curve space-time. How does one solve for a metric for that given situation? If $T_{mn}$ the the entire Einstein tensor $G_{mn}$ equals zero, right? Does flat space-time imply that the Riemann curvature tensor $R^t_{mnk}$ also equal zero? This would imply that its contraction, the Ricci curvature tensor $R_{mn}$, would equal zero, which would also imply that its trace (or its contraction, the Ricci scalar $R$) would also equal zero. Or does the metric equal zero and $R^t_{mnk}$ has a value?

Also, given an a metric (which I invent using basis vectors of my choice), is it possible to find the the energy distribution $T_{mn}$ for that metric?

In practice, given a stress-energy tensor $T_{\mu\nu}$, we may attempt to find solutions to the Einstein field equations using perturbation theory. The basic idea is to expand around a known solution $g_{\mu\nu}$ by a perturbation $h_{\mu\nu}$. In the case of a flat background,

$$\delta G_{\mu\nu} = 8\pi G \delta T_{\mu\nu} = \partial_\mu \partial_\nu h - \partial_\mu \partial_\alpha h^\alpha_\nu -\partial_\nu \partial_\alpha h^\alpha_\mu + \partial_\alpha \partial^\alpha h_{\mu\nu} - \eta_{\mu\nu} \partial_\alpha \partial^\alpha h + \eta_{\mu\nu} \partial_\alpha \partial_\beta h^{\alpha \beta}.$$

In some cases, one may solve the equations exactly, or more typically, employ numerical methods. Another alternative approach when faced with a stress-energy tensor is to try to determine the symmetries the metric may have, and then plug in an ansatz for the metric to yield a set of differential equations which may be more tractable, analytically and numerically.

Yet another alternative is to generate new solutions from old ones through the introduction of pseudopotentials, a method due to Harrison, Eastbrook and Wahlquist, called the method of prolongation structures.

There are several other methods such as those that rely on Lie point symmetries of differential equations. There is also a Backlund transformation, which relies on identifying a simpler differential equation whose solution satisfies a condition involving the solution to the harder problem.

These methods are too involved to present here and require a significant background. They are explained in Exact Solutions to the Einstein Field Equations by H. Stephani et al.

Addressing your other question, if given a metric $g_{\mu\nu}$, of course one can compute $T_{\mu\nu}$ through the Einstein field equations, you just plug it in and tediously compute all the curvature tensors. There are strictly speaking probably some requirements on the functions in $g_{\mu\nu}$, but you can get away with most things, even distributions, such as a delta function, which may lead to a stress-energy tensor describing a brane.