The electric field of charge in a event horizon

Question

What if the charge is in a black hole? I mean, if charged mass collapse into a singularity(or ringularity), there will be charge inside a event horizon. At this case, is there a electric field outside a event horizon?

edit) I think my question can be described in other way. Does Gauss' law(or Green Theorem) still established when the black hole is in the gaussian closed surface?

Answer 1

Electric charge falling into a black hole creates a charged black hole. This object is described by the Reissner–Nordström metric if it's not rotating (https://en.wikipedia.org/wiki/Reissner%E2%80%93Nordstr%C3%B6m_metric) or the Kerr-Newman metric if it's rotating (https://en.wikipedia.org/wiki/Kerr%E2%80%93Newman_metric).

In the Reissner–Nordström case, the four-potential of a black hole with charge $Q$ is, in spherical coordinates, $A_\mu=(Q/r,0,0,0)$, similar to an ordinary point charge.

In the Kerr-Newman case, the four-potential of a black hole is given in Boyer-Lindquist coordinates by

$$A_\mu=\left(\frac{rr_Q}{\rho},0,0,-\frac{c^2arr_Q\sin^2\theta}{\rho^2GM}\right)$$

where:

• $\rho=\sqrt{r^2+a^2\cos^2\theta}$,
• $a=\frac{J}{Mc}$,
• $r_Q=\frac{Q}{c^2}\sqrt{\frac{G}{4\pi\epsilon_0}}$, and
• The black hole has mass $M$, angular momentum $J$, and charge $Q$.

You can see that the Kerr-Newman black hole has a "magnetic" component to its potential, unlike the Reissner–Nordström black hole.

Translating these four-potentials into electric and magnetic fields is a bit more complicated than it is in flat space; for a procedure on how to do this, consult any reference on Maxwell's equations in curved spacetime, such as https://en.wikipedia.org/wiki/Maxwell%27s_equations_in_curved_spacetime.

Answer 2

Yes, a black hole can have a non-zero charge - charge is one of the three characteristics of a black hole, the other two being mass and angular momentum.

And if a black hole has a non-zero charge then, yes, this will create be an electric field outside of the event horizon.