Radial motion in the Schwarzschild metric, in terms of e

A nice way to parametrise the 4-velocity u of a small test body moving radially within Schwarzschild spacetime is by the “energy per unit rest mass” e:

${u^\mu=\left(e\Schw^{-1},\pm\eroot,0,0\right)$

For the “±” term, choose the sign based on whether the motion is inwards or outwards. All components are given in Schwarzschild coordinates $(t,r,\theta,\phi)$ . The result was derived as follows. In geometric units, the metric is:

$\Schwmetric$

By definition $e\equiv-\fvec\xi\cdot\fvec u$ , where $\fvec\xi\equiv\partial_t$ is the Killing vector corresponding to the independence of the metric from t, and has components $\xi^\mu=(1,0,0,0)$ (Hartle textbook §9.3). For geodesic (freefalling) motion e is invariant, however even for accelerated motion e is well-defined instantaneously and makes a useful parametrisation.

We want to find $u^\mu=(u^t,u^r,u^\theta,u^\phi)$ say. Rearranging the defining equation for e gives $u^t=e\Schw^{-1}$ . Radial motion means $u^\theta=u^\phi=0$ , so the normalised condition $\fvec u\cdot\fvec u=-1$ yields the remaining component $\abs{u^r}$ . The resulting formula is valid for all $0<r\ne 2M$ , and for e=1 the 4-velocity describes “raindrops” as expected.

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