General Relativity Effective Potentials

[Auburnman]

Homework Statement

Calculate and sketch the effective potentials V (r) for the radial motion of particles,
for the cases of massive and massless particles.
Interpret this effective potential: are there stable orbits? Are particles attracted
to r = 0? Can particles reach r = 0?

Homework Equations

the effective potentials

The Attempt at a Solution

I have gotten the potentials for the massive and massless particles
The first potential is for massive particles V = (L^2)/(2r^2) - (r^2)/(2R^2) - (L^2)/(R^2)
The second potential is for massless particles V = (-1/2) + (L^2)/(2r^2) -(L^2)/(R^2)

 

[NateD]

The interpretation of these potentials is that there are stable orbits around r = 0, since the potential is negative and decreases as r increases. Particles are also attracted to r = 0 since the potential is negative, however they cannot reach r = 0 due to the nature of the potential.