I thought it might be of interest to derive the first order differential equation for time travel for geodesic motion through a charged nonrotating black hole. Though physically realistic black holes in nature will be of little if any charge, but of extreme spin, the result and math is far more messy but gives similar results to that of a charged nonrotating hole, and its worked out most simply the same way, so this will be the case of a charged but nonrotating hole. No black hole in nature will be Schwarzschild. They will be Kerr like and any radiative aspect will be Vaidya like. As such dump anything you ever worried about from Susskind and Hawking. Neither ever once ever mention Vaidya, because they didn't understand what he did, and because he utterly undermines their publications. Anyway, the nonrotating but charged hole, Reissner-Nordström, solution is
Now the metric elements read off the elements of the coordinate differentials here are all independent of time. Now I know no one here, and hardly any physicists, knows Noether's first name for the same reason that they don't know Vaidya's but do know Einstein and Hawking's first names, so look it up if you feel the need, but anyway in accordance with her theorem that means there is a timelike Killing vector
for that isometry in the metric yielding as a constant of the geodesic motion of
Here
is merely what I name that constant where this
is often called the conserved energy parameter of geodesic motion. In the case that it is far enough from the hole to neglect the mass and charge terms, it is equal to the
of special relativity. Since the metric is diagonal there is only one element corresponding to the nonzero element of that Killing vector that contributes to the sum
Which results in
simplified
QED