Einstein's Field Equations: Effective Potential Functions

[novice_hack]

I have seen written out in various places (including this forum) the effective potential function that comes from the solutions to the Schwarszschild Geodesic. But I haven't been able to find the effective potential functions for other solutions to Einstein's field equations. Are there effective potential functions for the other solutions? And if so, is there a resource where some or all of them are written out.

 

[bcrowell]

Why "effective?"

Most spacetimes aren't static, and therefore can't be described by a potential.

 

[stevendaryl]

Why "effective?"

Most spacetimes aren't static, and therefore can't be described by a potential.

I can explain why the use of the word "effective". You can treat the quantity $\mathcal{L} = \frac{1}{2} g_{\mu \nu} \frac{dx^\mu}{ds} \frac{dx^\nu}{ds}$ as if it were a Lagrangian in 4-D Newtonian mechanics. Then there are terms that look like "potential energy" and "kinetic energy" terms. They aren't really.

 

[WannabeNewton]

Are there effective potential functions for the other solutions?

Yes. There is an effective potential for the general family of Kerr-Newman metrics. As long as you have the necessary Killing fields you can get enough first integrals of motion to write down an effective 1D potential for the dynamics of a particle whose worldline respects the symmetries of the Killing fields. It is straightforward but tedious to calculate this effective potential. C.f. section 33.5 of MTW. See also http://arxiv.org/pdf/1103.1807.pdf, http://arxiv.org/pdf/1304.2142v1.pdf, and http://www.roma1.infn.it/teongrav/leonardo/bh/bhcap4.pdf

I do not know of an enumeration of effective potentials for every known solution to the EFEs. Note there are also effective potential methods in PPN.