Kruskal Dynamics for Radial Geodesics

Abstract

The total spacetime manifold for a Schwarzschild black hole (BH) is believed to be described by the Kruskal coordi-nates and , where r and t are the conventional Schwarzschild radial and time coordinates re-spectively. The relationship between r and t for a test particle moving along a radial or non-radial geodesic is well known. Similarly, the expression for the vacuum Schwarzschild derivative for a geodesic, in terms of the constants of motion, is well known. However, the same is not true for the Kruskal coordinates; and, we derive here the expression for the Kruskal derivative for a radial geodesic in terms of the constants of motion. In particular, it is seen that the value of ) is regular on the Event Horizon of the Black Hole. The regular nature of the Kruskal derivative is in sharp contrast with the Schwarzschild derivative, , at the Event Horizon. We also explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion for a test particle on a radial geodesic. The physical implications of this result will be discussed elsewhere.