Solutions of the Einstein field equations
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 a Lorentz manifold. Solutions are broadly classed as exact or non-exact.
The Einstein field equations are
or more generally, if one allows a nonzero cosmological constant,
where 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.
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 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.
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