Kruskal Dynamics For Radial Geodesics. I

TL;DR

This paper derives explicit expressions for Kruskal coordinates and their derivatives on the event horizon of a Schwarzschild black hole, revealing regularity properties that contrast with Schwarzschild coordinates.

Contribution

It provides the first explicit calculation of Kruskal coordinates at the horizon as functions of constants of motion for radial geodesics.

Findings

Kruskal coordinates u_H and v_H are finite for E<1 at the horizon.

The Kruskal derivative |du/dv| equals 1 and is regular at the horizon.

The Kruskal derivative remains finite, unlike the Schwarzschild derivative |dt/dr| which diverges.

Abstract

The total spacetime manifold for a Schwarzschild black hole (BH) is described by the Kruskal coordinates u=u(r,t) and v=v(r,t), where r and t are the conventional Schwarzschild radial and time coordinates respectively. The relationship between r and t for a test particle moving on 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. Here, we, first, explicitly obtain the value of the Kruskal coordinates on the Event Horizon as a function of the constant of motion (E=energy per unit rest mass) for a test particle on a radial geodesic by directly using the r-t relationship as obtained by Chandrasekhar and also by Misner, Thorne and Wheeler. It is found that u_H and v_H are finite for E <1. And then, 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 |du/dv| (= 1) 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, |dt/dr| =infty, at the Event Horizon.