The Einstein Condition and Einstein's Equation
In local coordinates the condition that (M, g) be an Einstein manifold is simply
Taking the trace of both sides reveals that the constant of proportionality k for Einstein manifolds is related to the scalar curvature R by
where n is the dimension of M.
In general relativity, Einstein's equation with a cosmological constant Λ is
written in geometrized units with G = c = 1. The stress-energy tensor Tab gives the matter and energy content of the underlying spacetime. In a vacuum (a region of spacetime with no matter) Tab = 0, and one can rewrite Einstein's equation in the form (assuming n > 2):
Therefore, vacuum solutions of Einstein's equation are (Lorentzian) Einstein manifolds with k proportional to the cosmological constant.
Read more about this topic: Einstein Manifold
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