Radial motion in the Schwarzschild metric, relative to stationary observers

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,

    \[u^\mu=\left(\gamma\Schw^{-1/2},V\gamma\Schw^{1/2},0,0\right)\]

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

    \[u_{\rm{Schw}}^\mu=\left(\Schw^{-1/2},0,0,0\right)\]

We know the “moving” body has 4-velocity u of form u^\mu=(u^t,u^r,0,0) since the motion is radial. The Lorentz factor \gamma\equiv(1-V^2)^{-1/2} for the relative speed is

    \[\gamma=-\fvec u\cdot\fvec u_{\rm{Schw}}\]

Evaluating and rearranging yields u^t=\gamma\Schw^{-1/2}. Normalisation \fvec u\cdot\fvec u=-1 leads to u^r=\pm V\gamma\Schw^{1/2}, after some algebra including use of the identity \gamma^2-1=V^2\gamma^2. We allow V<0 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 e=-\fvec\xi\cdot\fveclabel{u}{Schw}=\Schw^{1/2}. Also we can relate the parametrisation by V (and γ) to the parametrisation by e via

    \[\gamma=e\Schw^{-1/2}\qquad V=-\frac{1}{e}\sqrt{e^2-1+\frac{2M}{r}}\]

where the leftmost equation follows from the definition e\equiv-\fvec\xi\cdot\fvec u, and subsequently the rightmost equation from γ=γ(V). For raindrops with e=1, the relative speed reduces to V=-\Schwroot.

We would expect the construction to fail for r\le 2M, 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…

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