Continue reading...This will be the first of several articles which will provide, for reference, useful equations for static, spherically symmeetric spacetimes. This is a common special case that is studied in General Relativity, and it has the advantage of having a general solution for the Einstein Field equation that can be expressed in closed form equations. This first article will focus on those closed form equations for that solution.
First, we should be clear about the set of spacetimes we are talking about. The two key properties of this set of spacetimes can be defined in coordinate-independent terms as follows:
(1) A static spacetime has a timelike Killing vector field which is hypersurface orthogonal.
(2) A spherically symmetric spacetime has a 3-parameter group of spacelike Killing vector fields that satisfy the properties of the Lie group SO(3), which describes rotations in 3-dimensional space.
As noted, these...