Assume a Schwarzschild metric (one spherically symmetric massive body). Given an initial position $r_0$, an initial velocity $v_0$, a Schwarzschild radius $r_s$ and a time coordinate $t$ (not $\tau$), is there an equation that can tell me the current position $r$ of my test particle?
Note: I'm aware that for a given coordinate time $t$ there may be 2 valid positions $r$ (one on each side of the event horizon). My simulation can figure out which way to evaluate a $+/-$ sign if it has to.
Research I've already done:
I dove into the excellent Reflections on Relativity online book, which has a chapter about Radial Paths in a Spherically Symmetrical Field. In that book, the author posits the following set of equations:
before saying "These equations can be integrated in closed form (see below), but they can also be directly integrated numerically using small incremental steps of dτ. For any given initial position and trajectory we can generate the subsequent geodesic path in terms of $r$ as a function of $t$." However, his later equation involves calculating $t$ in terms of $r$, where as I'm looking for the other way around (maybe I missed something).