Schwarzschild Coordinates - Killing Vector Fields

Killing Vector Fields

With respect to the Schwarzschild chart, the Lie algebra of Killing vector fields is generated by the timelike irrotational Killing vector field

and three spacelike Killing vector fields

Here, saying that is irrotational means that the vorticity tensor of the corresponding timelike congruence vanishes; thus, this Killing vector field is hypersurface orthogonal. The fact that our spacetime admits an irrotational timelike Killing vector field is in fact the defining characteristic of a static spacetime. One immediate consequence is that the constant time coordinate surfaces form a family of (isometric) spatial hyperslices. (This is not true for example in the Boyer-Lindquist chart for the exterior region of the Kerr vacuum, where the timelike coordinate vector is not hypersurface orthogonal.)