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 uSchw. 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 vrel>1 for r<2M) attest. Sometime, I will investigate this further…