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 are
G μ ν + Λ g μ ν = κ T μ ν , {\displaystyle G_{\mu \nu }+\Lambda g_{\mu \nu }\,=\kappa T_{\mu \nu },}
where G μ ν {\displaystyle G_{\mu \nu }} is the Einstein tensor, Λ {\displaystyle \Lambda } is the cosmological constant (sometimes taken to be zero for simplicity), g μ ν {\displaystyle g_{\mu \nu }} is the metric tensor, κ {\displaystyle \kappa } is a constant, and T μ ν {\displaystyle T_{\mu \nu }} 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 (such as trajectories of moving particles), 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 the resulting stress–energy tensor can then be plugged into the Einstein field equations. But if one requires an exact solution or a solution describing strong fields, the evolution of both the metric and the stress–energy tensor must be solved for at once. To obtain solutions, the relevant equations are the above quoted EFE (in either form) plus the continuity equation (to determine the evolution of the stress–energy tensor):
T a b
; b = 0 . {\displaystyle T^{ab}{}_{;b}\,=0\,.}
These amount to only 14 equations (10 from the field equations and 4 from the continuity equation) and are by themselves insufficient for determining the 20 unknowns (10 metric components and 10 stress–energy tensor components). The 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 spacetime. 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 a b = 0 {\displaystyle T_{ab}\,=0}
Perfect fluid:
T a b = ( ρ + p ) u a u b + p g a b {\displaystyle T_{ab}\,=(\rho +p)u_{a}u_{b}+pg_{ab}} where u a u a = − 1 {\displaystyle u^{a}u_{a}=-1}
Here ρ {\displaystyle \rho } is the mass–energy density measured in a momentary co-moving frame, u a {\displaystyle u_{a}} is the fluid's 4-velocity vector field, and p {\displaystyle p} is the pressure.
Non-interacting dust ( a special case of perfect fluid ):
T a b = ρ u a u b {\displaystyle T_{ab}\,=\rho u_{a}u_{b}}
For a perfect fluid, another equation of state relating density ρ {\displaystyle \rho } and pressure p {\displaystyle 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 a b
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