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.