Do Black Holes Really Exist?

Introduction

The purpose of this article is to discuss the title question from several different viewpoints, to show that it isn’t as simple as it looks. We will look at some common misconceptions that lead people to think the answer must be “no”, and we will look at some of the issues involved that prevent the answer from being a simple “yes”.

Defining a black hole

The first order of business is to define what we mean by the term “black hole”. The common pop-science definition, “a region from which nothing, not even light, can escape”, is a pretty good starting point (which is not always the case with pop science definitions). The key thing we need to tighten it up into something rigorous is to define exactly what “escape” means. In the usual idealized model of a black hole, it is viewed as being surrounded by empty space, and the hole’s gravity gets weaker and weaker as we move farther and farther away from it in this empty space, so the geometry of spacetime gets closer and closer to being flat. The technical term for a spacetime like this is “asymptotically flat”.

We might think that this is enough: “escape” just means that spacetime is asymptotically flat, and whatever it is that is escaping can get arbitrarily far away from the hole, into the region where the geometry of spacetime is arbitrarily close to being flat. The usual shorthand expression for this is “escape to infinity”. Only something outside the hole’s horizon can do this.

But it turns out that there is an additional wrinkle here: there is more than one possible “infinity” in an asymptotically flat spacetime. There are five; they are called future and past timelike infinity, future and past null infinity, and spacelike infinity. Why is this? Because spacetime includes time as well as space, so there are three kinds of curves, timelike, null, and spacelike, and the first two kinds have two different directions, future, and past. (Technically speaking, the light cone at every event in spacetime has two interior regions–future and past–which are disconnected, but only one exterior region–the spacelike region.) Each of these, if extended indefinitely, ends up at a different “infinity”.

So the question now is, which of the five infinities do we pick to define “escape”? The answer turns out to be future null infinity. If you think about it, this makes sense: “escape” should be to the future, not the past, and light moves faster than anything else, so if light rays can’t reach future null infinity from some region of spacetime, it seems evident that timelike objects certainly won’t be able to reach future timelike infinity either. That turns out to be the case when we do the math, so we arrive at our rigorous definition of a black hole: it is a region of spacetime that cannot send light signals to future null infinity. Or, in somewhat more technical language, a black hole is a region of spacetime that is not in the causal past of future null infinity.

Misconceptions About Black Hole Formation

Now we can look at the first common misconception about black holes, which is that they can’t be formed at all because it takes an infinite amount of time for an object to fall to the event horizon. The usual counterargument to this is to point out that “infinite time” as it is used here is coordinate-dependent: the “time” in question is not an invariant, and so it doesn’t in itself have any physical meaning. We can compute invariants, such as the proper time needed to reach the horizon by the infalling object’s clock, and show that they are finite.

But armed with the above definition, we have a much simpler response: is there a region of spacetime that is not in the causal past of future null infinity? If there is, a black hole is present, regardless of whether there is some coordinate chart in which it takes an infinite amount of time for something to fall into it. And can such a region form in a spacetime that starts not containing one? Yes, it can; this has been demonstrated by explicitly constructing models that have this property (the first was the classic Oppenheimer-Snyder model published in their 1939 paper). So the first common misconception is indeed a misconception: there are self-consistent solutions of the Einstein Field Equations that contain black hole regions, and objects that can form such regions by collapse, even though coordinate charts exist for these solutions in which coordinate time on the horizon becomes infinite (where “becomes infinite” is a sloppy way of saying “is not well-defined”).

So far we have been discussing “classical” black holes, without considering any quantum effects. A well-known theorem by Hawking says that such a black hole can never decrease in size (where “size” here means the area of its horizon); and a well-known argument by Bekenstein says that this is just an application of the second law of thermodynamics to black holes, with the horizon area being the hole’s entropy. But Hawking also discovered something else: if we include quantum effects, the area theorem can be violated, and a black hole can lose mass, in a process known as Hawking radiation. (Note that the second law still holds in this case; the hole’s entropy decreases, but the entropy of the radiation must be included as well, and when it is, the total entropy still increases.)

This brings us to the second common misconception about black holes, which is that, when Hawking radiation is included in the picture, a black hole can’t form because it would evaporate before anything had a chance to fall into it. Again, counterarguments are saying that, for an evaporating black hole, it no longer takes an infinite time, even by the coordinate charts referred to above, for something to fall in, and when the coordinates are adapted properly, objects fall in before the hole evaporates away. But again, armed with our definition above, there is a much simpler response: are there self-consistent models that include a region that is not in the causal past of future null infinity, even if such a region later disappears due to evaporation? Yes, there are; the simplest one is due to Hawking himself. So again, this common misconception is a misconception. There are self-consistent solutions that show the whole process, from the formation of a black hole by gravitational collapse where none existed before, all the way to the hole finally evaporating away due to Hawking radiation.

You might have noticed, though, that so far I have been careful to use the word “model” when describing what our physical theories say about black holes. Models are not reality. The real question is, do we have evidence that the mathematical models I described above are realized somewhere in our universe? That is what it would mean for black holes to “really exist”, and the fact that common misconceptions about the models are wrong does not in itself prove that the models describe nature.

The honest answer to this last question above is that we don’t know for sure. The main reason is that we don’t (yet) have a good theory of quantum gravity, so we don’t know whether Hawking radiation is the only significant change to the classical black hole model that we have to deal with. There may be other quantum effects that, when properly taken into account, will prevent a true black hole from ever forming–i.e., will prevent any region of spacetime from ever truly leaving the causal past of future null infinity. Currently, there are two schools of thought about this:

(1) The general heuristic that many physicists use to determine when quantum gravity effects should become important is that the spacetime curvature has to be very large–large enough to be equivalent (via the Einstein Field Equation) to a density approaching the Planck density–one Planck mass per Planck volume, or about $10^{94}$ times the density of water. But for any black hole we would expect to detect by astronomy (which would be roughly the mass of the Sun or larger), the spacetime curvature at and well inside the horizon is much, much smaller than this. So by this heuristic, we would expect classical GR to be a good approximation at and well inside the horizon, meaning that we would not expect quantum corrections to prevent true black holes–regions not in the causal past of future null infinity–from forming, even if quantum corrections did change what happened deep inside those regions.

(2) However, there is another rule which, at least in the view of the quantum physicists who make the argument, is much more than a general heuristic–it’s a law of nature, part of the bedrock of quantum mechanics. This law is called “unitarity”, and it means that quantum information can’t be created or destroyed. But at least in the simple model of a black hole, even an evaporating one, any quantum information that falls inside the horizon does get destroyed, when it hits the singularity. So on this view, at the very least, quantum effects must prevent a singularity from ever forming. But when you look at the structure of the evaporating black hole models, you see that it’s very hard, if not impossible, to remove the singularity without also removing the horizon–in other words, without changing the spacetime structure to something that does not have a region which is not in the causal past of future null infinity. So on this view, quantum effects must end up preventing true black holes from ever forming, even if we don’t understand quite how they would do this.

It’s important to note that position (2) above does not necessarily imply that there can’t be horizons at all, only that there can’t be true event horizons. But there is another kind of horizon called an “apparent horizon”, which is a surface at which, heuristically speaking, radially outgoing light does not move outward but stays in the same place. (The technical definition is that the expansion of the congruence of radially outgoing null geodesics is zero.) This does not necessarily make the apparent horizon a true event horizon, because “stays in the same place” is only local–radially outgoing light that is staying in the same place at one event might, at some future event, start moving outward, so it would end up ultimately escaping to future null infinity. (Note that, for this to happen, the area theorem must be violated; in pure classical GR, where black holes can never decrease in mass, any apparent horizon will always have an event horizon at or outside it. The latter will be the case if the matter is falling into the hole: the event horizon increases in area smoothly, while the apparent horizon “jumps” suddenly outward.)

The reason this is important is that all of the methods we have for actually testing, observationally, for the presence of horizons can’t tell us whether the horizon we think we have detected is a true horizon or only an apparent horizon. The only way to know for sure would be to know the entire future of the universe, which, of course, we can’t know. So the fact that we have observed several compact regions in which there appear to be horizons (basically because things fall into them and don’t come out and they’re too compact to be anything else) does not, in itself, allow us to test positions (1) vs. (2). We have to find other, indirect ways of exploring the issue, and the field is simply too young for there to have been much time to do so.

(I should also note that there are proposed mechanisms, referred to by terms like “firewalls”, by which quantum effects would destroy infalling objects before they ever reach a horizon, preventing any violation of unitarity; and there are also proposed mechanisms by which quantum effects would prevent even apparent horizons from forming, by basically pumping enough energy into collapsing matter via quantum effects to reverse the collapse and make it explode before it had a chance to become compact enough to form an apparent horizon. These proposals do not appear to be holding up very well under scrutiny, so I won’t say more about them here, but it’s important to be aware that they exist.)

Conclusion

So to summarize: the answer to the title question may end up being “no”, but if so, it won’t be for any of the simplistic reasons associated with common misconceptions about black holes–i.e., it won’t be because they take an infinite time to form, or because they would evaporate away before anything had a chance to fall in, or anything like that. We don’t know for sure whether the answer is “yes” or “no” at this point, but we expect to learn a lot more about the subject as we continue research into quantum gravity.

(Addendum: In the PF comments on the original version of this post, it was brought up that our universe as a whole is not asymptotically flat, so the concept of “black hole” that I use in the article, strictly speaking, does not apply to our universe. The usual way of addressing that is to say that the asymptotically flat region doesn’t have to be at infinity, it just has to be at a very large distance from the hole, relative to the hole’s horizon radius. But given that our universe, as best we can tell, is what we might call “asymptotically de Sitter” since it has a positive cosmological constant and therefore a cosmological horizon, we have a better answer: we can simply use Schwarzschild-de Sitter spacetime as our model of a black hole, instead of ordinary Schwarzschild spacetime. In this model, the cosmological horizon plays the role of future null infinity, so a black hole is a region of spacetime that cannot send light signals to the cosmological horizon. This doesn’t change anything substantive in the article.)