Solutions of the Einstein field equations

"Where appropriate, this article will use the abstract index notation."

Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity. Solving the field equations actually gives Lorentz metrics. Solutions are broadly classed as "exact" or "non-exact".

The Einstein field equations are

$G\_\{ab\}\; ,\; =\; kappa\; T\_\{ab\}$

or more generally

$G\_\{ab\}\; +\; Lambda\; g\_\{ab\}\; ,\; =\; kappa\; T\_\{ab\}$

where $kappa$ is a constant, and the Einstein tensor on the left side of the equation is equated to the stress-energy tensor representing the energy and momentum present in the spacetime. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, the EFE are a system of ten partial differential equations to be solved for the metric.

olving the equations

It is important to realize that the Einstein field equations alone are not enough to determine the evolution of a gravitational system in many cases. They depend on the stress-energy tensor, which in turn depends on the (unknown) metric. If only interested in the weak field limit of the theory, the dynamics of matter can be computed using special relativity methods and/or Newtonian laws of gravity and then placing the resulting stress-energy tensor into the Einstein field equations. But if the exact solution is required or a solution describing strong fields, the evolution of the metric and the stress-energy tensor must be solved for together.

To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine evolution of the stress-energy tensor):

:$T^\{ab\}\{\}\_\{;b\}\; ,\; =\; 0$

This is clearly not enough, as there are only 14 equations (10 from the field equations and 4 from the continuity equation) for 20 unknowns (10 metric components and 10 stress-energy tensor components). Equations of state are missing. In the most general case, it's easy to see that at least 6 more equations are required, possibly more if there are internal degrees of freedom (such as temperature) which may vary throughout space-time.

In practice, it is usually possible to simplify the problem by replacing the full set of equations of state with a simple approximation. Some common approximations are:

* Vacuum::$T\_\{ab\}\; ,\; =\; 0$

* Perfect fluid::$T\_\{ab\}\; ,\; =\; (\; ho\; +\; p)u\_a\; u\_b\; +\; p\; g\_\{ab\}$ where $u^au\_a\; =\; -1!$

Here $ho$ is the mass-energy density measured in a momentary co-moving frame, $u\_a$ is the fluid's 4-velocity vector field, and $p$ is the pressure.

* Non-interacting dust ( a special case of perfect fluid )::$T\_\{ab\}\; ,\; =\; ho\; u\_a\; u\_b$

For a perfect fluid, another equation of state relating density $ho$ and pressure $p$ must be added. This equation will often depend on temperature, so a heat transfer equation is required or the postulate that heat transfer can be neglected.

Next, notice that only 10 of the original 14 equations are independent, because the continuity equation $T^\{ab\}\{\}\_\{;b\}\; =\; 0$ is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant and a "gauge fixing" is needed, i.e. impose 4 constraints on the system, in order to obtain unequivocal results.

A popular choice of gauge is the so-called "De Donder gauge", also known as harmonic gauge or Lorentz gauge.

$g^\{mu\; u\}\; Gamma^\{sigma\}\_\{mu\; u\}\; =\; 0$

In numerical relativity, the preferred gauge is the so-called "3+1 decomposition", based on the ADM formalism. In this decomposition, metric is written in the form

$ds^2\; ,\; =\; (-N\; +\; N^i\; N^j\; gamma\_\{ij\})\; dt^2\; +\; 2N^i\; gamma\_\{ij\}\; dt\; dx^j\; +\; gamma\_\{ij\}\; dx^i\; dx^j$, where $i,j\; =\; 1dots\; 3$

$N$ and $N^i$ can be chosen arbitrarily. The remaining physical degrees of freedom are contained in $gamma\_\{ij\}$, which represents the Riemannian metric on 3-hypersurfaces $t=const$.

Once equations of state are chosen and the gauge is fixed, the complete set of equations can be solved for. Unfortunately, even in the simplest case of gravitational field in the vacuum ( vanishing stress-energy tensor ), the problem turns out too complex to be exactly solvable. To get physical results, we can either turn to numerical methods; try to find exact solutions by imposing symmetries; or try middle-ground approaches such as perturbation methods or linear approximations of the Einstein tensor.

Exact solutions

Exact solutions are Lorentz metrics that are conformable to a physically realistic stress-energy tensor and which are obtained by solving the EFE exactly in closed form.

Non-exact solutions

Those solutions that are not exact are called "non-exact solutions". Such solutions mainly arise due to the difficulty of solving the EFE in closed form and often take the form of approximations to ideal systems. Many non-exact solutions may be devoid of physical content, but serve as useful counterexamples to theoretical conjectures.

Applications

There are practical as well as theoretical reasons for studying solutions of the Einstein field equations.

From a purely mathematical viewpoint, it is interesting to know the set of solutions of the Einstein field equations. Some of these solutions are parametrised by one or more parameters.