Background Material on General Relativity

Numerical Relativity is the use of a computer to simulate the Einstein-Hilbert field equations. These relate the curvature of space-time, denoted by the trace-free Ricci curvature tensor, and the cosmological constant to the stress-energy-momentum tensor:

It also appears with the Einstein tensor (which is the same thing as the trace-free Ricci curvature tensor)

These equations can be categorized as nonlinear wave equations in 4-dimensional space-time. 

Unlike the space-time of Newton, or even of special relativity, the space-time of general relativity can be curved. This curved space-time has the general form of a pseudo-Riemman manifold, that has the general metric:

Almost immediately after the discovery of the field equations of gravity, Schwarzchild published his solution to a radially symmetric space-time, the Schwarzchild metric (in units in which the speed of light, c, is equal to unity):

Several other analytical solutions soon followed, including Einstien’s own steady-state cosmological model, de Sitter’s exponentially growing vacuum solution, and Friedmann’s more general cosmological model (also called the Friedmann–Lemaître–Robertson–Walker cosmological model.) Except for Einstein’s steady-state universe model, versions of these models are still used as approximations to the cosmology of our universe today.

Unfortunately, to go beyond symmetric geometries to more realistic systems, the Einstein-Hilbert field equations need to be solved numerically.

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