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

The Einstein field equations are

${\displaystyle G_{ab}\,=\kappa T_{ab}}$

or more generally

${\displaystyle G_{ab}+\Lambda g_{ab}\,=\kappa T_{ab}}$

where ${\displaystyle \kappa }$ 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.

Solving 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 we're only interested in the weak field limit of the theory, we can compute the dynamics of matter using special relativity methods and/or Newtonian laws of gravity and then plug in the resulting stress-energy tensor into the Einstein field equations. But if we want to obtain the exact solution or to deal with strong fields, we need to solve for the evolution of the metric and the stress-energy tensor at the same time.

We start with two sets of equations:

The Einstein field equations (to determine evolution of the metric):

${\displaystyle G_{\alpha \beta }+\Lambda g_{\alpha \beta }\,=\kappa T_{\alpha \beta }}$

The continuity equation (to determine evolution of the stress-energy tensor):

${\displaystyle T_{;\beta }^{\alpha \beta }=0}$

This is clearly not enough, because we only have 14 equations (10 from the Einstein's equations and 4 from the continuity equation) for 20 unknowns (10 components of the metric and 10 components of the stress-energy tensor). We're missing equations of state. It's easy to see that, in the most general case, we need at least 6 more equations, 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:

${\displaystyle T_{\alpha \beta }\,=0}$

• Perfect fluid:

${\displaystyle T_{\alpha \beta }\,=(\rho +p)u_{\alpha }u_{\beta }+pg_{\alpha \beta }}$ where ${\displaystyle u^{\alpha }u_{\alpha }=-1\!}$

Here ${\displaystyle \rho }$ is the mass-energy density measured in a momentary co-moving frame, ${\displaystyle u_{\alpha }}$ is the fluid's 4-velocity vector field, and ${\displaystyle p}$ is the pressure.

• Non-interacting dust ( a special case of perfect fluid ):

${\displaystyle T_{\alpha \beta }\,=\rho u_{\alpha }u_{\beta }}$

For a perfect fluid, we need to add one more equation of state that relates density ${\displaystyle \rho }$ and pressure ${\displaystyle p}$. This equation will often depend on temperature, so we also need to include a heat transfer equation or to postulate that heat transfer can be neglected.

Next, notice that only 10 of our original 14 equations are independent, because the continuity equation ${\displaystyle T_{;\beta }^{\alpha \beta }=0}$ is a consequence of the Einstein's equations. This reflects the fact that our system is gauge invariant and we need to perform "gauge fixing", 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.

${\displaystyle g^{\mu \nu }\Gamma _{\mu \nu }^{\sigma }=0}$

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

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

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

Once equations of state are chosen and the gauge is fixed, we can try to solve the complete set of equations. 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

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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

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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

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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.

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