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 $d^2 x^i/d t^2$ 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.