Radial motion in the Schwarzschild metric, relative to stationary observers

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 u_Schw. 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 textbook (§31.2, explicit acknowledgement) and Taylor & Wheeler (§B.4, implicitly v_rel>1 for r<2M) attest. Sometime, I will investigate this further…

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