In a static and spherically symmetric space-time, using the standard coordinates $\{t,r,\theta,\phi\}$ the geodesic equations imply that $$ g_{rr}\biggl(\frac{dr}{d\tau}\biggr)^2+\frac{J^2}{r^2}-\frac{E^2}{g_{tt}}=-1 $$ where $J$ and $E$ are the conserved quantities and $\tau$ is the proper time along the curve. Dividing both sides by $g_{rr}$, one can make the equation have the usual kinetic+potential form $$ \biggl(\frac{dr}{d\tau}\biggr)^2+V_{eff}(r)=E^2 $$ but the potential is in general $E$-dependent (the only case in which it's not is when $g_{tt}g_{rr}=-1$), so that different bodies experience different accelerations if they have different $E$. Does this dependence violates the equivalence principle? Or better, does it violate any of the equivalence principles?

  • $\begingroup$ I like your question! I feel that since it is very simply derived from the metric, which clearly is compatible with the EP, it cannot be in violation. However I haven't found the flaw in your argument yet ;) I have used the equations myself to plot orbits, but I always use M = mu = 1. $\endgroup$ – m4r35n357 Jun 7 at 18:04

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