Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations of general relativity. Solving the field equations gives a Lorentz manifold. Solutions are broadly classed as exact or non-exact. The Einstein field equations are where is the Einstein tensor, is the cosmological constant, is the metric tensor, is a constant, and is the stress–energy tensor. The Einstein field equations relate the Einstein tensor to the stress–energy tensor, which represents the distribution of energy, momentum and stress in the spacetime manifold. The Einstein tensor is built up from the metric tensor and its partial derivatives; thus, given the stress–energy tensor, the Einstein field equations are a system of ten partial differential equations in which the metric tensor can be solved for.
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 depends on the dynamics of matter and energy, which in turn depends on the gravitational field. If one is 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 the resulting stress–energy tensor can be plugged 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 plus the continuity equation : This is clearly not enough, as there are only 14 equations for 20 unknowns. 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 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:
Perfect fluid:
Here is the mass-energy density measured in a momentary co-moving frame, is the fluid's 4-velocity vector field, and is the pressure.
For a perfect fluid, another equation of state relating density and pressure 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 is a consequence of Einstein's equations. This reflects the fact that the system is gauge invariant A "gauge fixing" is needed, i.e. we need to impose 4 constraints on the coordinate system in order to obtain unequivocal results. These constraints are known as coordinate conditions. A popular choice of gauge is the so-called "De Donder gauge", also known as the harmoniccondition or harmonic gauge 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 and are functions of spacetime coordinates and can be chosen arbitrarily in each point. The remaining physical degrees of freedom are contained in, which represents the Riemannian metric on 3-hypersurfaces. For example, a naive choice of,, would correspond to a so-called synchronous coordinate system: one where t-coordinate coincides with proper time for any comoving observer 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, 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 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.
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. Al Momin argues that Kurt Gödel's solution to these equations do not describe our universe and are therefore approximations.
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.
