Why Black Stars Aren’t A Thing.

Not a black star,
but about equally real.

I came to physics by accident. I had studied mathematics, but the math department was broke. When I asked the mathematicians for a job, they sent me to the other side of the building. “Ask the physicists,” they said.

The physicists didn’t only give me a job. They also gave me a desk, a computer, and before I knew I had a topic for a diploma thesis. I was supposed to show that black holes don’t exist.

I didn’t know at that time, but it was my supervisor’s shtik, the black-holes-don’t-exist-speech. Prof Dr Dr hc mult Walter Greiner, who passed away two years ago, was the department head when I arrived. His argument against black holes was that “God wouldn’t separate himself from part of the universe.” Yo. He mostly worked on heavy ion physics.

I had made pretty clear to him that I wasn’t interested in heavy ion physics. Really I wasn’t sure I wanted to graduate in physics at all; it wasn’t even my major. But I was the math person, so certainly I could prove that black hole’s weren’t, no?

It was either that or computer simulations of big nuclei or back to the broke mathematicians. I picked the black holes.

That was 1997. Back then, measurements of the motion of stars around Sag A* were running, but they would not be published until 1998. And even after Sag A* proved to be dark, small, and heavy enough so that it should be a black hole, it took ten more years to demonstrate that indeed it doesn’t have hard surface, thus providing evidence for a black hole horizon.

We now know that Sag A* is a supermassive black hole, and that such black holes are commonly found in galactic centers. But when I was a student the case was not settled.

Greiner had explained to me why he thought black holes cannot form in stellar collapse. Because we know that black holes emit radiation, the famous “Hawking radiation.” So, when a star collapses it begins emitting all this radiation and it loses mass and the horizon never forms. That was his great idea. Ingenious! Why had no one thought of this before?

After some months digging in the literature, it became clear to me that it had been tried before. Not once, but several times, and equally many times it had been shown not to work. This was laid out in various publications, notably in Birrell and Davies’ textbook, but Greiner’s interest in the topic didn’t go far enough to look at this. Indeed, I soon found out that I wasn’t the first he had put on the topic, I was the third. The first delivered a wrong proof (and passed), the second left. Neither option seemed charming.

Black hole with accretion disk
and jet. Artist's impression.
[Image Source]

The argument for why Greiner’s idea doesn’t work is a shitload of math, but it comes down to a very physical reason: You can’t use Hawking radiation to prevent black holes from forming because that’s in conflict with the equivalence principle.

The equivalence principle is the main tenet of general relativity. It says that a freely falling observer should not be able to tell the presence of a gravitational field using only data from their vicinity, or “locally” as the terminology has it.

Hawking radiation obeys the equivalence principle – as it should. This means most importantly that an observer falling through the black hole horizon does not notice any radiation (or anything else that would indicate the presence of the horizon). The radiation is there, but its wavelengths are so long – of the size of the horizon itself – that the observer cannot measure the radiation locally.

If you want to know how Hawking-radiation affects the black hole you must calculate the total energy and pressure which the quantum effects creates. These are collected in what’s called the (renormalized) stress-energy-tensor. Turns out it’s tiny at the black hole horizon, and the larger the black hole, the smaller it is.

All of this is perfectly compatible with the equivalence principle. And that’s really all you need to know to conclude you can’t prevent the formation of black holes by Hawking-radiation: The contribution to the energy-density from the quantum effects is far too small, and it must be small because else an infalling observer would notice it, screwing over the equivalence principle.

What normally goes wrong when people argue that Hawking-radiation can prevent the formation of black hole horizons is that they use the result for the Hawking radiation which a distant observer would measure. Then they trace back this radiation’s energy to the black hole horizon. The result is infinitely large. That’s because if you want to emit anything at the horizon that can escape at all, you must give it an infinite amount of energy to start with. This is nonsense because Hawking radiation is not created at the black hole horizon. But it’s this infinity that has led many people to conclude that a collapsing star may be able to shed all of its energy in Hawking radiation.

But whenever you do physics and the math gives you an infinity, you should look for a mistake. Nothing physically real can be infinite. And indeed, the infinity which you get here cannot be observed. It is is cancelled by another contribution to the stress-energy which comes from the vacuum polarization. Collect all terms and you conclude, again, that the effects at the horizon are tiny. Done correctly, they do, of course, obey the equivalence principle.

In summary: Yes, black holes evaporate. But no, the energy-loss cannot prevent the formation of black hole horizons.

That was the status already in the late 1970s. Walter Greiner wasn’t the first but also not the last to try using quantum effects to get rid of the black hole horizon. I come across one or the other variation of it several times a year. Most recently it was via a piece on Science Daily, which also appeared PhysOrg, Science Alert, Gizmodo, BigThink, and eventually also Scientific American, where we read:

Black Hole Pretenders Could Really Be Bizarre Quantum Stars

New research reveals a possible mechanism allowing “black stars” and “gravastars” to exist

These articles go back to a press release from SISSA about a paper by Raúl Carballo-Rubio which was recently published in PRL (arXiv version here).

Carballo-Rubio doesn’t actually claim that black holes don’t form; instead he claims – more modestly – that “there exist new stellar configurations, and that these can be described in a surprisingly simple manner.”

These new stellar configurations, so his idea, are stabilized by strong quantum effects in a regime where general relativity alone predicts there should be nothing to prevent the collapse of matter. These “black stars” do not actually have a horizon, so the quantum effects never actually become infinitely large. But since the pressure from the quantum effects would get infinitely large if the mass were compressed into the horizon, the radius at which it stabilizes must be outside the horizon.

In other words, what stabilizes these black stars is the same effect that Greiner thought prevents black holes from forming. You can tell immediately it’s in conflict with the equivalence principle for there is nothing locally there, at the horizon or close by it, from which the matter would know when to stop collapsing. At horizon formation, the density of matter can be arbitrarily low, and the matter doesn’t know – cannot know! – anything about redshift from there to infinity. The only way this matter can know that something is supposed to happen is by using global information, ie by violating the equivalence principle.

Indeed that’s what Carballo-Rubio does, though the paper doesn’t really spell out where this assumption comes in, so let me tell you: Carballo-Rubio assumes from the onset that the system is static. This means the “quantum star” has no time-dependence whatsoever.

This absence of time-dependence is an absolutely crucial point that you are likely to miss if you don’t know what to look for, so let me emphasize: No stellar object can be truly static because this means it must have existed forever and will continue to exist for all eternity. A realistic stellar object must have formed somewhen. Static solutions do not exist other than as math.

The assumption that the system be static is hence a global assumption. It is not something that you can reach approximately, say, at the end of a collapse. Concretely the way this enters the calculation is by choice of the vacuum state.

Yes, that’s right. There isn’t only one vacuum state. There are infinitely many. And you can pick one. So which one do you pick?

Before we get there, allow me a digression. I promise it will make sense in a minute. Do you recall when Walter Wagner sued CERN because turning on the LHC might create tiny black holes that eat up earth?



It is rare for black hole physics to become a matter of lawsuits. Scientists whose research rarely attracts any attention were suddenly in the position of having to explain why these black holes, once created, would be harmless.

On the face of it, it’s not a difficult argument. These things would have interaction-probabilities far smaller than even neutrinos. They would readily pass through matter, leaving no trace. And being created in highly-energetic collisions, they’d be speedy, fly off to outer space and be gone.

But then, these tiny black holes would have a small but nonzero probability to become trapped in Earth’s gravitational field. They would then keep oscillating around the center of the planet. And if they stuck around for sufficiently long, and there were sufficiently many of them, they could grow and eventually eat up Earth inside-out. Not good.

That, however, the scientists argued, could not happen because these tiny black holes evaporate in a fraction of a second. If you believe they evaporate. And suddenly theoretical physicists had to very publicly explain why they are so sure black holes evaporate because otherwise the LHC might not be turned on and their experimentalist friends would never forgive them.

Rather unsurprisingly, there had been one-two people who had written papers about why black holes don’t evaporate. Luckily, these claims were easy to debunk. The court dismissed the lawsuit. The LHC turned on, no black holes were created, and everyone lived happily ever after.

For me the most remarkable part of this story isn’t that someone would go try to sue CERN over maybe destroying the world. Actually I have some understanding for that. Much more remarkable is that I am pretty sure everyone in the field knows it’s easy enough to find a theoretical reason for why black holes wouldn’t evaporate. All you have to do is postulate they don’t. This postulate is physical nonsense, as I will explain in a moment, so it would merely have complicated the case without altering the conclusion. Still I think it’s interesting no one even brought it up. Humm-humm.

So what’s that nonsense postulate that can keep black holes from evaporating? You choose a vacuum state in which they don’t. Yes, you can do that. Perfectly possible. It’s called the “Boulware state.” The price you pay for this, however, is that the energy created by quantum effects at the black hole horizon goes to infinity. So it’s an unphysical choice and no one ever makes it.

Ah! I hear you say. But not very loudly, so let me summarize this in plain terms.

You can assume a black hole doesn’t evaporate on the expense of getting an infinite amount of stress-energy in the horizon region. That’s an unphysical assumption. And it’s the same assumption as postulating the system does not change in time: Nothing in, nothing out.

And that – to tie together the loose ends – is exactly what Carballo-Rubio did. He doesn’t actually have a horizon, but he uses the same unphysical vacuum-state, the Boulware state. That’s the reason he gets such a large quantum pressure, hence violating the equivalence principle. It comes from the assumption that the system is static, has always been static, and will always remain static.

Let me be clear that Carballo-Rubio’s paper is (for all I can tell) mathematically sound. And the press-release is very carefully phrased and accurate. But I think he should have been clearer in pointing out that the assumption about time-independence is global and therefore he is describing a physically impossible situation that is not even approximately realistic.

If you followed my above elaborations, it should be clear that the details don’t matter all that much. The only way you can prevent a horizon from forming is to violate the equivalence principle. And worse, this violation must be possible when space-time curvature is arbitrarily small, as small or even smaller than what we have here on Earth.

Of course you can postulate whatever you want and calculate something. But please let us be clear that all these black stars and gravastars and quantum stars  and what have you require throwing out general relativity in regions where there is no local measure whatsoever that would call for such a breakdown. Doesn’t matter how much math you pour over it, it’s still in conflict with what we know about gravity.

The realistic situation is one in which matter collapses under its gravitational pull. In this case you have a different vacuum state (the Unruh state), which allows for evaporation. And that brings you full circle to the above argument for why the stress-energy is too small to prevent horizon formation. There’s no way to avoid the formation of a black hole. Nope, there isn’t. Black holes really exist.

As to my diploma. I simply wrote my thesis about something else but didn’t mention that until after the fact. I think Greiner never forgave me. A few years later he fired me, alas, unsuccessfully. But that’s a different story and shall be told another time.

That was a long post, I know. But I hope it explains why I think black stars and gravastars and qantum stars and so on are nonsense. And why I happen to know more about the topic than I ever wanted to know.