Physical Interpretation of Coordinates for the Schwarzschild Metric

TL;DR

This paper explores various coordinate systems for the Schwarzschild metric, aiming to find physically meaningful interpretations and discussing how different coordinates relate to the presence of black holes and the nature of the horizon.

Contribution

It analyzes well-known and lesser-known coordinate systems to understand their physical significance and how they relate to the black hole interior and horizon.

Findings

Coordinate systems differ in their origin placement and physical interpretation.

Some coordinates exclude the black hole interior, potentially offering more physically acceptable models.

Mathematical signature distinctions help differentiate time and space coordinates across horizons.

Abstract

Since its first introduction, the Schwarzschild metric has been written in various coordinate systems. This has been done primarily to understand the nature of the coordinate singularity at the event horizon. However, very often, the mathematics of a coordinate system does not provide a clear physical interpretation. In Schwarzschild's original work, the origin of the radial coordinate was at the event horizon. Hence, there was no black hole. The generally accepted current definition of the radial coordinate has an origin beyond the horizon. This necessitates the discussion of black holes. Here, some well-known and some not-so-well-known coordinate systems will be visited in search of a physical interpretation. It will be noted that they all agree at large radial distances. However, the location of the origin of the radial coordinate can be different for different systems. Some "natural" coordinate systems can exclude the entirety of the interior of a black hole from the physical manifold. Such coordinate systems maybe physically more acceptable as they avoid the issue of the metric signature change across the horizon. Mathematically, the metric signature alone can distinguish time from space coordinates. Hence, coordinate systems that include the interior of the black hole need to switch the physical meanings of time and one of the space coordinates.