**Difficulty level:**★ ★ ★

Last time we derived the 4-velocity **u** of a small test body moving radially in the Schwarzschild geometry, in terms of *e*, the “energy per unit rest mass”. Another parametrisation is in terms of the 3-speed *V* relative to stationary observers. This turns out to be, in Schwarzschild coordinate expression,

To derive this, first consider the 4-velocity of stationary observers:

We know the “moving” body has 4-velocity **u** of form since the motion is radial. The Lorentz factor for the relative speed is

Evaluating and rearranging yields . Normalisation leads to , after some algebra including use of the identity . We allow also, and define this as inward motion. Carefully considering the sign, this results in the top equation. (An alternate derivation is to perform a local Lorentz boost. *Later articles will discuss this*… The Special Relativity formulae cannot be applied directly to Schwarzschild coordinates.)

Some special cases are noteworthy. For *V*=0, γ=1, and **u** reduces to **u**_{Schw}. This corresponds to . Also we can relate the parametrisation by *V* (and γ) to the parametrisation by *e* via

where the leftmost equation follows from the definition , and subsequently the rightmost equation from γ=γ(*V*). For raindrops with *e*=1, the relative speed reduces to .

We would expect the construction to fail for , as stationary timelike observers cannot exist there, and so the relative speed to them would become meaningless. But curiously, it can actually work for a faster-than-light *V*>1 “Lorentz” boost, as even the authorities MTW (§31.2, explicit acknowledgement) and Taylor & Wheeler (§B.4, implicitly *v*_{rel}>1 for *r*<2*M*) attest. Sometime, I will investigate this further…