Assumptions and Notation
Working in a coordinate chart with coordinates labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is an arbitrary function of 4 variables). The solution is assumed to be spherically symmetric, static and vacuum. For the purposes of this article, these assumptions may be stated as follows (see the relevant links for precise definitions):
(1) A spherically symmetric spacetime is one in which all metric components are unchanged under any rotation-reversal or .
(2) A static spacetime is one in which all metric components are independent of the time coordinate (so that ) and the geometry of the spacetime is unchanged under a time-reversal .
(3) A vacuum solution is one which satisfies the equation . From the Einstein field equations (with zero cosmological constant), this implies that (after contracting and putting ).
Read more about this topic: Deriving The Schwarzschild Solution
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