Near-horizon geodesics for astrophysical and idealised black holes: Coordinate velocity and coordinate acceleration

TL;DR

This paper examines how coordinate choices affect the interpretation of geodesic motion near black hole horizons, emphasizing the importance of horizon-penetrating coordinates for accurate descriptions.

Contribution

It provides a detailed analysis of coordinate velocity and acceleration near black hole horizons, advocating for horizon-penetrating coordinates to avoid misinterpretation.

Findings

Coordinate acceleration can be misleading near horizons.

Horizon-penetrating coordinates offer clearer physical insights.

Coordinate velocity and acceleration behaviors depend on coordinate choice.

Abstract

Geodesics (by definition) have an intrinsic 4-acceleration zero. However, when expressed in terms of coordinates, the coordinate acceleration $d^2 x^i/d t^2$ can very easily be non-zero, and the coordinate velocity $d x^i/d t$ can behave unexpectedly. The situation becomes extremely delicate in the near-horizon limit---for both astrophysical and idealised black holes---where an inappropriate choice of coordinates can quite easily lead to significant confusion. We shall carefully explore the relative merits of horizon-penetrating versus horizon-non-penetrating coordinates, arguing that in the near-horizon limit the coordinate acceleration $d^2 x^i/d t^2$ is best interpreted in terms of horizon-penetrating coordinates.