Here +Fraser Cain
gives an excellent explanation of what would happen if you collided an ordinary black hole with a black hole made out of antimatter. The answer is that they wouldn't annihilate each other: you would just end up with a single, bigger black hole.
There's a lot of interesting science behind it, but the two most important things to know for this are a pair of acronyms: CPT symmetry and ADM equations. The first tells us what antimatter is; the second, what black holes look like. And importantly, we have to understand the idea that "black holes have no hair."
CPT stands for "Charge, Parity, Time Reversal." These are three separate switches you could flip on any piece of matter you encountered. If you flipped the charge, you would replace + charges with - charges, and also you would flip some of the more exotic (non-electrical) charges that other kinds of matter had. If you flipped the parity, you would exchange your left hand for your right hand; importantly, for some particles (like neutrinos) their direction of rotation about their axis is related to their direction of motion by a "right-hand rule" (point your right thumb in the direction of motion; your fingers show the direction of rotation) or a "left-hand rule." That would get flipped. And if you flipped the direction of time, you would literally play back the laws of physics backwards, having everything go the opposite way.
These seem like three rather random changes you could make to the universe, but what's important about them is that if you made all three
changes – flipped all the charges in the universe, exchanged right and left, and played everything backwards – it turns out that the laws of physics would be exactly the same. That's called "CPT Symmetry," and it's very important in quantum mechanics.
But what would that look like? Imagine you started with an electron moving along its merry way, say moving from your left to your right, and you CPT'ed it. You would end up with something that had the same mass as the electron, but a positive charge (C). P doesn't do anything to electrons (or rather, what it does is subtle and complicated to explain and doesn't really add anything here), but T would mean that the positively-charged electron would be "moving backwards in time," which would look like it was moving from your right to your left. The CPT rule tells you that our same laws of physics should describe this reversed situation. What that means from the perspective of quantum field theory, whose job it is to explain all of the various particles we see in the universe, is that if there's a particle that looks like an electron – having a charge of -1, a mass of 511MeV, and so on – then the same laws of physics should also describe this "flipped electron," with a charge of +1, a mass of 511MeV, and so on.
These "flipped particles" are what we call antimatter. Basically, a good way to think about it is to ignore time reversal; a particle moving from right to left is now moving from left to right, but for our purposes it's easiest to just note that yes, it's a particle, and it moves around. (It turns out that this actually works out well at the level of the math as well) For every particle, if you flip all its charges and its handedness, there should be a second particle in the universe that has those properties.
For an electron, this "flipped particle" is called a positron; for other matter particles, their name comes from giving them the prefix "anti." So just as three quarks (two ups and a down) make up a proton, three antiquarks (two anti-ups and an anti-down) make up an antiproton, and the antiproton has a charge of -1 and the same mass as a proton. Photons are their own antiparticles: they have no charge and come in all handednesses. So there's no "antiphoton," nor is there an "antigraviton."
So that's CPT, and it's all about very small things. ADM (named after its inventors, Arnowitt, Deser, and Misner) is a set of equations for describing behaviors in general relativity, our best theory of gravity.
The ADM equations are actually a very general tool, but what's important about them for our purpose is that they tell us that whenever you have an object that's bounded in space (i.e., that doesn't either stretch off to infinity in some direction, or distort spacetime so badly that even infinitely far away they have an effect), you can meaningfully talk about the "total energy" or "total mass" in that object. Basically, if there's any meaningful sense in which you can "get far away" from an object – so that you're basically in empty, flat space with some object far away – then you can talk about measuring the properties of that object in some way.
This is true of black holes, just like it is of basically every other configuration of matter, and so we can talk about a black hole's "ADM mass," its total mass measured using this method. Through a very similar argument, we can see that it's not just mass you can measure this way: you can also measure the total momentum of the black hole (if it's moving), its angular momentum (if it's spinning), its electrical charge (if it has any), and in fact the exact same set of charges that we talked about when we talked about CPT.
(That's actually not a coincidence: these charges aren't picked at random. Probably the most important theorem in mathematical physics is what's called Noether's Theorem, which says that there is a conserved quantity associated with a system if and only if there's a symmetry of the system, and those are related in a particular way. For example, if the system would be the same if you shifted its position by any amount, the associated conserved quantity is momentum. Shifting things forward and backwards in time leads to energy conservation; rotating them, angular momentum. The symmetries associated with electric charge and the like are more complicated, but they're fundamental to how physics works. The ADM "total energy" calculation basically works because you can zoom out until you're far enough away from the system that it's basically empty space with something in the middle, then use that symmetry and pull out the value of the conserved quantity using Noether's technique. Applying that to each symmetry in turn, you can measure total energy, momentum, and so on.)
And now another interesting fact kicks in: "Black holes have no hair." This is a fact about general relativity which takes some serious proving, but what it means is that the only
properties which black holes have that are measurable from the outside are exactly these global conserved charges. In particular, black holes have a shape completely determined by these charges (a perfect sphere unless they have angular momentum, in which case they distort in a particular way from rotation); they don't have filaments, curves, hairballs, or anything else like that.
So now, let's combine these ideas. Say you had two black holes: one matter and one antimatter. The matter one has some mass, some charge, some angular momentum, and so on, and so does the antimatter one. But as we know from understanding what antimatter is, the anti-black hole still has positive mass; it just got built up out of antiparticles, so maybe it has an opposite charge.
When the two black holes merge, the only thing that matters (by the no-hair theorem) is those total amounts, and they simply add up. So you end up with one bigger black hole.
In fact, what that means is that there's no such thing as an "anti-black hole" at all: no matter what went in to forming the black hole, at the end all that matters is the total mass, charge, and so on. You could have gotten a charge from anything – positrons or ordinary protons – and it would come out the same.
Here's another way to think about it. Imagine that we weren't just colliding black holes, we were also manufacturing
them. One of them we're going to build by crushing a big lump of matter; the other, by crushing a big lump of antimatter. Now, we know from the above that if we first build them, and then collide them, we end up with a single black hole. But what would happen if we first collided them and then built them? Would it matter (so to speak) if the black hole formed just before or just after the collision?
The answer is no. When matter and antimatter react, they get converted into energy, but all of those charges are conserved. (Remember that matter is energy, so the mass of the particles gets turned into energy as well) For example, an electron and a positron can react to turn into two photons, but the total charge stays the same (-1 from the electron, +1 from the positron, 0 for the two photons), and so does the energy: 511MeV worth for each particle, plus whatever kinetic energy they had, turns into the energy of the emitted photons. (You may have noticed, if you know some physics notation, that I actually gave the mass of the electron in units of energy; that’s very common in physics, since you go back and forth between the two all the time anyway)
So if the matter bundle and the antimatter bundle collided before forming into a black hole, they’d react dramatically, explode and so on, and produce a total energy which is exactly equal to the energy (and mass!) that went into them, with a total charge which is exactly equal to the charge that went into them, and so on. And if that were to form a black hole, it would form the exact same black hole that would have formed if they had made two black holes and then merged!
This is a fundamental idea that shows up over and over in science: conserved quantities are really conserved, no matter what, but everything else tends to get mushed up fairly easily. As we saw above, black holes “have no hair:” the only
properties they have are conserved charges. Both matter and antimatter obey this rule, and the two differ in properties like their handedness and their charges, so that when they react, while they may turn into energy (rather explosively), that energy (and thus mass) is exactly the same amount that went in.
And that’s why there’s no such thing as an “anti-black hole.”