Can Kerr-Newman Geodesics Be Separated in Hamilton-Jacobi Equation with Charge?

[michael879]

I've found the equations of motion for a charged test particle in the Kerr-Newman geometry from a number of sources. However, they aren't very reliable and disagree on small details, so I'm trying to derive it myself. I'm completely stuck at the last step though, where you "use" the separability of the Hamilton-Jacobi equation to get the EOM of [itex]\dot{r}[/itex] and [itex]\dot{\theta}[/itex] without their second derivatives. Here is what I've got so far:

Metric + Potential (only part not derived/checked by me):
[tex]
ds^2\equiv-\dfrac{\rho}{\Delta}dr^2-\rho^2d\theta^2+\dfrac{\Delta}{\rho^2}\left(dt-asin^2\theta d\phi\right)^2-\dfrac{sin^2\theta}{\rho^2}\left((r^2+a^2)d\phi-adt\right)^2 \\
A_\mu\equiv \dfrac{Qr}{\rho^2}\left(\delta^t_\mu-asin^2\theta\delta^\phi_\mu\right) \\
\Delta\equiv r^2+a^2-2Mr+Q^2 \\
\rho^2\equiv r^2+a^2cos\theta
[/tex]
for a black hole with parameters M,a,Q

Geodesic Equation+Hamiltonian:
[tex]
\dfrac{d}{ds}\left(g_{\mu\nu}\dot{x}^\nu -\dfrac{q}{m}A_\mu\right) = \dfrac{1}{2}\partial_\mu g_{\sigma\rho}\dot{x}^\sigma\dot{x}^\rho - \dfrac{q}{m}\dot{x}^\sigma \partial_\mu A_\sigma \\
H=\dfrac{1}{2}mg_{\mu\nu}\dot{x}^\mu\dot{x}^\nu=\dfrac{\kappa}{2}m=(2m)^{-1}g^{\mu\nu}\left(p_\mu+qA_\mu\right)\left(p_\nu+qA_\nu\right)
[/tex]
for a test particle with mass m and charge q, with [itex]\kappa\equiv sgn(m^2)[/itex]. I've also scaled everything by m just to keep it consistent with usual notation in the non-gravitational limit.

The time and rotational symmetries immediately give the first two equations of motion:
[tex]
E\equiv mg_{tt}\dot{t}+mg_{\phi t}\dot{\phi}-qA_t \\
L\equiv qA_\phi - mg_{\phi\phi}\dot{\phi}-mg_{\phi t}\dot{t}
[/tex]

The 2 other solutions to the EL equations give second derivatives of r and θ, so I'm following the steps of the sources I found (all originally from Carter,68). They all claim they are "using" the Hamilton-Jacobi equation to derive the next set of equations. However, as far as I can tell all they really do is assume that [itex]p_r[/itex] is a function of only r and [itex]p_\theta[/itex] is a function of only theta, and then set the two forms of H equal. The Hamilton-Jacobi equations and the S function never even need to be mentioned, but for some reason every one of them does?

Anyway, when I do this step I find:
[tex]
2mH = |m|^2 = g^{tt}\left(E+qA_t\right)^2 + g^{\phi\phi}\left(L-qA_\phi\right)^2-2g^{t\phi}\left(E+qA_t\right)\left(L-qA_\phi\right) - \dfrac{\Delta}{\rho^2} p_r^2-\dfrac{1}{\rho^2}p_\theta^2
[/tex]

I calculated the inverse metric elements to be:
[tex]
g^{\phi\phi}=\dfrac{1}{\rho^2}\left(\dfrac{a^2}{\Delta}-\dfrac{1}{sin^2\theta}\right) \\
g^{t\phi}=\dfrac{a}{\Delta\rho^2}\left(2Mr-Q^2\right) \\
g^{tt}=\dfrac{1}{\rho^2}\left(\dfrac{(r^2+a^2)^2}{\Delta}-a^2sin^2\theta\right)
[/tex]
which can be plugged into the Hamiltonian expression. Multiplying both sides by [itex]\rho^2[/itex] gives you an easily separable equation of r and θ IF q=0. In this case, you can continue by setting the separated equations equal to a constant K and finding the 2 remaining EOM. However, if q≠0, the A terms add a very non-trivial dependence of both r and θ. As far as I can tell this general formula is NOT separable. This doesn't really make sense though because even Carter's original derivation made the general assumption of q≠0! Anybody know what I'm doing wrong?

*edit*
The paper I mentioned can be found at http://journals.aps.org/pr/abstract/10.1103/PhysRev.174.1559

Also, I just realized that by including the m factor everywhere I invalidated these equations for anything massless. So just assume we're dealing with a timelike massive test particle and it all works

 

[michael879]

^bump, Nobodies got any insight on this?

 

[michael879]

Comon... 400 views and not one response? The Q=0 case is interesting, but I'm looking for the MOST general geodesic formula for a test particle near a black hole

 

[michael879]

Last shot at this and I'm giving up

 

[sardar]

out.Thank you for sharing your work on the Kerr-Newman charged geodesics. The equations you have derived and the potential you have included are indeed important for understanding the motion of charged particles in the Kerr-Newman geometry. It is not uncommon for different sources to have slightly different equations or approaches, and it is always valuable to try to derive them yourself.

Regarding your question about the separability of the Hamilton-Jacobi equation, it is important to note that it is only separable for certain cases. Specifically, the Hamilton-Jacobi equation is separable in a coordinate system where the metric is diagonal and the potential is independent of the coordinates. This is not the case for the Kerr-Newman metric, where the potential depends on both the time and azimuthal coordinates. Therefore, the Hamilton-Jacobi equation is not separable in this case.

However, as you have noted, the equations can still be solved for specific values of the charge (q=0) or for specific symmetries (such as the equatorial plane). In these cases, the equations can be separated and solved for the EOM of \dot{r} and \dot{\theta}.

Additionally, it is important to keep in mind that the Kerr-Newman metric and potential are valid for a black hole with parameters M, a, and Q. For other values of these parameters, the equations may not be separable or may require different approaches.

I hope this helps clarify your understanding of the Kerr-Newman charged geodesics. Keep up the good work in your research!