Coordinate families for the Schwarzschild geometry based on radial timelike geodesics

TL;DR

This paper investigates various coordinate systems in Schwarzschild spacetime linked to radial geodesics, introducing new families of coordinates and clarifying their relationships to existing ones like Painlevé-Gullstrand and Lemaître coordinates.

Contribution

It constructs new coordinate families based on radial geodesics and elucidates their connections to known coordinate systems in Schwarzschild geometry.

Findings

Introduces LMP and proper-time coordinate families related to radial geodesics.

Shows these coordinate families are distinct but related generalizations of PG time.

Establishes links between these families and Lemaître coordinates.

Abstract

We explore the connections between various coordinate systems associated with observers moving inwardly along radial geodesics in the Schwarzschild geometry. Painlev\'e-Gullstrand (PG) time is adapted to freely falling observers dropped from rest from infinity; Lake-Martel-Poisson (LMP) time coordinates are adapted to observers who start at infinity with non-zero initial inward velocity; Gautreau-Hoffmann (GH) time coordinates are adapted to observers dropped from rest from a finite distance from the black hole horizon. We construct from these an LMP family and a proper-time family of time coordinates, the intersection of which is PG time. We demonstrate that these coordinate families are distinct, but related, one-parameter generalizations of PG time, and show linkage to Lema\^itre coordinates as well.