Schwarzschild Black Hole

Schwarzschild coordinates

The spacetime metric is where is the metric for the unit two-sphere. The horizon is and the singularity is . The lapse function, shift vector, and extrinsic curvature defined by the slices and time flow vector field are:

(1) The geodesic equations are:

Isotropic coordinates

Starting with Schwarzschild coordinates, the transformation gives the line element The horizon is . The lapse function, shift vector, and extrinsic curvature defined by the slices and the time flow vector field are:

(2) The geodesic equations are:

Kerr-Schild coordinates

Starting with Schwarzschild coordinates, the transformation gives the line element The lapse function, shift vector, and extrinsic curvature defined by the slices and the time flow vector field are:

(3) Changing from spherical coordinates , , to Cartesian coordinates gives This can also be written as where is the Minkowski metric, is a covector, and . Also let . Then the inverse metric is . Note that the vector is null in both the Minkowski and physical metrics.

The geodesic equations:

Kruskal coordinates

Starting with Schwarzschild coordinates, let

(4) The line element becomes The lapse function, shift vector, and extrinsic curvature defined by the slices and the time flow vector field are:

(5) Geodesic equations: