Isotropic Coordinates - Killing Vector Fields

Killing Vector Fields

The Lie algebra of Killing vector fields of a spherically symmetric static spacetime takes the same form in the isotropic chart as in the Schwarzschild chart. Namely, this algebra 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 the 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 (spacelike hypersurfaces).

Unlike the Schwarzschild chart, the isotropic chart is not well suited for constructing embedding diagrams of these hyperslices.