# Kerr–Newman metric

1. Oct 12, 2012

### Orion1

Kerr–Newman metric:
$$c^{2} d\tau^{2} = - \left(\frac{dr^2}{\Delta} + d\theta^2 \right) \rho^2 + (c \; dt - \alpha \sin^2 \theta \; d\phi)^2 \frac{\Delta}{\rho^2} - ((r^2 + \alpha^2) d\phi - \alpha c \; dt)^2 \frac{\sin^2 \theta}{\rho^2}$$
I used the Kerr–Newman metric equation form listed on Wikipedia for the purpose of isolating the Einstein tensor metric element functions for this particular metric. I expanded all the terms, combined all similar terms, then factored all the terms, and the result was this solution:
$$c^{2} d\tau^{2} = \frac{(\Delta - \alpha^2 \sin^2 \theta)}{\rho^2} \; c^2 \; dt^2 - \left(\frac{\rho^2}{\Delta} \right) dr^2 - \rho^2 d\theta^2 + (\alpha^2 \Delta \sin^2 \theta - r^4 - 2 r^2 \alpha^2 - \alpha^4) \frac{\sin^2 \theta \; d\phi^2}{\rho^2} - (\Delta - r^2 - \alpha^2) \frac{2 \alpha \sin^2 \theta \; c \; dt \; d\phi}{\rho^2}$$
Is there anyone here qualified to verify this solution?

Reference:
Kerr-Newman metric - Wikipedia

Last edited: Oct 12, 2012
2. Oct 12, 2012

### PAllen

Looks right to me.

3. Oct 12, 2012

### Mentz114

You've probably got this but it's well worth a mention,

The Kerr spacetime: A brief introduction
Matt Visser

http://arxiv.org/abs/0706.0622