# The electric field of charge in a event horizon

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?

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.

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.

• Thanks for your answer! If black hole does create electric field, what is the strength of the field?(Let the charge of the singularity is Q) Oct 2, 2019 at 12:48
• Far away from the black hole the electric field will be the same as that of a point charge of magnitude $Q$. Near to the black hole the answer is much more complicated because you have to take into account general relativity, and the answer will also depend on whether or not the black hole is spinning. Oct 2, 2019 at 13:08