Novikov Coordinates and the Physical Description of Gravitational Collapse

TL;DR

The paper develops Novikov coordinates by constructing them from radial freely falling geodesics in Schwarzschild spacetime, tying the time coordinate to the proper time of fallers and yielding a physically transparent view of horizon crossing. By comparing with Schwarzschild–Droste and Kruskal–Szekeres representations, it shows that the maximal analytic extension can introduce time-reversed sectors (white holes) that are not demanded by realistic collapse. For a spherically symmetric dust cloud, the boundary crosses the horizon in finite proper time $T_H$, resolving the classic 'frozen star' paradox that arises in static Schwarzschild coordinates. The work emphasizes that free-fall motion provides the natural physical extension of the Schwarzschild geometry and clarifies the limits of Novikov coordinates as a pedagogical tool for understanding horizon dynamics in idealized collapse scenarios.

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.

Figures (5)

• Figure 1: Representation of the Schwarzschild spacetime in Lemaître coordinates $(T,R)$. Freely falling worldline is represented by the dashed red line. The green straight-line corresponds to the horizon at $r=2MG$.
• Figure 2: Representation of the Schwarzschild spacetime in Novikov coordinates $(T,\bar{R})$. Freely falling worldline is represented by the dashed red line.
• Figure 3: Schematic representation of the geodesic appearing in Figure \ref{['figure2']} in the Schwarzschild-Droste coordenates.
• Figure 4: Representation of gravitational collapse in Novikov coordinates $(T,\bar{R})$. The star, whose evolution—restricted to those dust particles that at $T=0$ lie outside the Schwarzschild radius—is depicted by the gray region, exhausts its internal energy at $T=0$ and subsequently begins to collapse from an initial radius $R_B$, which corresponds to the Novikov coordinate $\bar{R}_B=\sqrt{2MG,(R_B-2MG)}$. $T_H$ is the time when the boundary crosses the Schwarzschild radius and the horizon is formed. $\bar{R}_O$ is the coordinate of an external observer following a geodesic with $-1/2<E<0$.
• Figure 5: Schematic representation of a physical geodesic with $-1/2<E<0$ in Kruskal spacetime.