Near-Horizon Geodesics for Astrophysical and Idealised Black Holes: Coordinate Velocity and Coordinate Acceleration

Geodesics (by definition) have an intrinsic 4-acceleration zero. However, when expressed in terms of coordinates, the coordinate acceleration d(2)x(i) / dt(2) can very easily be non-zero, and the coordinate velocity dx(i) / dt 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) / dt(2) is best interpreted in terms of horizon-penetrating coordinates.