# Effective potential in static spherically symmetric spacetime and equivalence principle

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?

• 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. – m4r35n357 Jun 7 at 18:04