Novikov Coordinates and the Physical Description of Gravitational Collapse

TL;DR

The paper demonstrates how Novikov coordinates, derived from radial geodesics, offer a physically transparent description of gravitational collapse, clarifying the difference between free-fall and static observer perspectives.

Contribution

It provides a direct derivation of Novikov coordinates from geodesics, offering a clear dynamical interpretation of black-hole and white-hole regions in gravitational collapse.

Findings

Novikov coordinates are obtained from radial geodesics with negative energy.

In Novikov coordinates, collapse completes in finite proper time for free-fall observers.

Schwarzschild coordinates depict an infinite time collapse, which is a coordinate artifact.

Abstract

We show that the Novikov coordinates can be obtained in a direct and physically transparent way from the radial geodesics of massive particles with negative energy in the Schwarzschild spacetime. These geodesics form a complete congruence that covers the entire spacetime. By rectifying this family of trajectories using the proper time as the time coordinate, the Novikov variables naturally emerge, providing a clear dynamical interpretation of the different regions usually identified as black-hole and white-hole sectors. In Novikov coordinates, observers at fixed spatial position follow free-fall trajectories. From their perspective, the gravitational collapse of a dust star is completed in a finite proper time, independently of their initial distance from the star. In contrast, observers described by Schwarzschild-Droste coordinates perceive the boundary of the collapsing star as taking an infinite coordinate time to reach the horizon. We emphasize that Schwarzschild-Droste observers are static with respect to the center of mass of the star and therefore cannot be in free fall. The use of these coordinates implicitly requires the presence of a force that compensates the gravitational attraction. From this viewpoint, the apparent infinite-time collapse is not a physical effect but a coordinate artifact associated with non-inertial observers.