Solutions of the Einstein field equations are metrics of spacetimes that result from solving the Einstein field equations (EFE) of general relativity.
Solving the field equations gives a Lorentz manifold.
Solutions are broadly classed as exact or non-exact.
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.
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.
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 the resulting stress–energy tensor can then be plugged into the Einstein field equations.
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 spacetime.
In practice, it is usually possible to simplify the problem by replacing the full set of equations of state with a simple approximation.
For a perfect fluid, another equation of state relating density
This reflects the fact that the system is gauge invariant (in general, absent some symmetry, any choice of a curvilinear coordinate net on the same system would correspond to a numerically different solution.)
A "gauge fixing" is needed, i.e. we need to impose 4 (arbitrary) constraints on the coordinate system in order to obtain unequivocal results.
are functions of spacetime coordinates and can be chosen arbitrarily in each point.
, would correspond to a so-called synchronous coordinate system: one where t-coordinate coincides with proper time for any comoving observer (particle that moves along a fixed
Unfortunately, even in the simplest case of gravitational field in the vacuum (vanishing stress–energy tensor), the problem is 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.
Many non-exact solutions may be devoid of physical content, but serve as useful counterexamples to theoretical conjectures.
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.
From a physical standpoint, knowing the solutions of the Einstein Field Equations allows highly-precise modelling of astrophysical phenomena, including black holes, neutron stars, and stellar systems.
Predictions can be made analytically about the system analyzed; such predictions include the perihelion precession of Mercury, the existence of a co-rotating region inside spinning black holes, and the orbits of objects around massive bodies.