**Difficulty level:**★ ★ ★

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*:

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

By definition , where is the Killing vector corresponding to the independence of the metric from *t*, and has components (Hartle §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 say. Rearranging the defining equation for *e* gives . Radial motion means , so the normalised condition yields the remaining component . The resulting formula is valid for all , and for *e*=1 the 4-velocity describes “raindrops” as expected.