# Black holes and unusual matter

Something has been bugging the hell out of me about black holes for the last week or so.

The “no hair” theorem says that from the outside, a black hole can be described completely by its mass, charge, and angular momentum. Now the issue of charge is somewhat irrelevant in most cases because, as far as we know, all of the “natural” methods of creating black holes involve normal (anti)matter. That is, the matter that formed the black hole would consist either of chargeless particles like (anti)neturons in a (anti)neutron star, and/or a roughly equal mix of protons & electrons or antiprotons and antielectrons (positrons). So any black hole we encounter is probably going to have pretty darn close to a net neutral charge.

But let’s say that we find a normal (neutrally charged) blackhole of, I don’t know…6 solar masses. Suppose that we started firing a stream of electrons into this black hole. A lot of electrons. Like, 20 solar masses of them. I don’t know that 20 is the magic number, but at some point – since the charge of an electron contributes much more significantly to the strength of its electromagnetic field than its mass does to its gravitational field – the charge of the black hole would result in an electromagnetic field whose magnitude at the event horizon would be exactly equal to the strength of its gravitational field. If we increase the electromagnetic field any more…what happens?

If we cranked up the electromagnetic field any more, then there will be a region inside of the event horizon where any electron will experience a greater electromagnetic force trying to push it out of the black hole than the gravitational force trying to suck it farther in. In theory, wouldn’t it be ejected from the black hole? For that matter, so would any electrons already inside the black hole that hadn’t yet reached the singularity (if there is one). At the singularity, the two field strengths would both be infinite, but in opposite directions. This is a mathematical pissing contest between two different infinities, so I won’t even pretend to have any idea what this would imply for charges that occupied the singularity. Maybe they would get ejected, and maybe not. But at the very least, we could now fire an electron towards the black hole and it would be slowed down as it approached. With just the right velocity, it should come to a stop somewhere inside of the event horizon, but before reaching the singularity. It seem inevitable that it would then be accelerated away from the singularity and be able to return from beyond the event horizon.

But then thinking on it a little harder, doesn’t this raise some questions about the no hair theorem itself? How can a black hole’s charge effect anything in the first place? The photon is the carrier particle for the electromagnetic force, right? That is, two electrons repulse one another in effect by shooting virtual photons at each other. But if a black hole is defined as a region of space whose gravitational field is strong enough that not even light can escape, then how can charge inside of a black hole affect anything on the outside? I’m vaguely aware that there’s a difference between virtual photons and real photons, but aren’t they bound by the same rules? I don’t see how a charge inside of a black hole can ever interact with one outside of the black hole to begin with. A charge sitting exactly on the event horizon…maybe. But not one inside of it, and certainly not one within a singularity.

But you could generalize that even further. When a photon climbs out of a gravity well, it loses kinetic energy and downshifts its frequency. That is, it becomes red-shifted. Or you can prefer to think of it as a consequence of time passing slower in the gravity field. Either way is fine. Similarly, light moving into the gravity field gains kinetic energy and becomes blue-shifted. So if you have a charge on earth and a charge somewhere in orbit around the earth, wouldn’t the electromagnetic force between them behave asymetrically? Virtual particles emitted towards the orbiting charge would have less energy on collision than those emitted towards the earth charge from orbit.

This is making my head hurt.

Oh, and just to make things even more interesting:

Position two observers on opposite sides of a black hole. Give one an anti-matter gun and the other a matter gun. Have them aim at the center of the black hole (i.e. at each other) and commence fire. Does the black hole gain mass?

I say yes to this one. Annihilating particles and anti-particles won’t destroy their mass, even if it converts them into massless particles. The system still has mass, even if the individual particles are massless photons. The mass of the system is just the energy content in the “center of momentum” frame (the frame of reference where the system has zero momentum).

Recall from Einstein:
E[sup]2[/sup] = p[sup]2[/sup]c[sup]2[/sup] + m[sup]2[/sup]c[sup]4[/sup]

If p = 0, then E = mc[sup]2[/sup]

This works even for photons (but you need at least two photons to have a frame where the momentum is zero but the energy is non-zero).

To understand how two photons can have mass, even though a single photon can’t, consider this:

The frequency of light is red-shifted if the source of the light is moving away from us. We can shift our frame of reference so the source of the light moves away from us faster and faster. As the speed increases, the red-shift increases, and the frequency goes to zero. The energy of light is proportionate to its frequency, so the energy also goes to zero. No energy means no mass.

That works for one photon. But imagine two photons traveling in opposite directions. If we make the source of one move towards us, we make the source of the other move away from us. So it’s impossible to simultaneously red-shift the frequency of both photons to zero. This means even in the frame that minimizes the energy, the energy is still non-zero. This minimum energy is the mass of the system (times c[sup]2[/sup]).

Regarding the question of what happens when the Coulomb force is on par with the gravitational force, I’ll have to think about it a bit. But I bet Stranger, or Chronos, or Pasta or someone will be along to answer shortly. Lots of smart physics people on this board.

The electrical field has energy, and thus mass. The reason black holes form in the first place is that beyond a certain point, any possible repulsive force would negate it’s repulsion by the gravitational attraction of it’s own energy.

Wait, then what do people mean when they talk about “extremal black holes”?

From the always reliable wikipedia:

The point seems to be that there’s a limit on how much charge or angular momentum you can add to a black hole (assuming naked singularities can’t occur.) So I guess after a certain point the electrical repulsion would overwhelm the gravitational attraction and prevent any more charge from being added.

But general relativity is definitely not my strong suit when it comes to physics knowledge, so maybe I’m confused.

Hawking radiation and/or Schwinger pair production will not allow the charge to mass ratio of a BH to exceed a maximum that otherwise would cause the hole to become a naked singularity.

No. Virtual photons are not constrained by the speed of light and therefore can escape the hole.