Are Black Holes Necessary?

Roger Penrose, Reinhard Genzel, and Andrea Ghez have won the 2020 Nobel Prize in Physics. The prize is divided half to Penrose for theoretical work and half to Genzel and Ghez for finding a convincing and appreciably large practical example.

Today we congratulate the winners and give further musings on the nature of knowledge and the role of theory.

The physics Nobel has always had the rule that it cannot be for a theory alone, no matter how beautiful and how many mathematical discoveries follow from its development. Stephen Hawking’s theory of black-hole radiation is almost universally accepted, despite its association with paradox, yet it was said that only an empirical confirmation such as mini-black holes being discovered to explode in an accelerator core would have brought it a Nobel. The official citation to Sir Roger says that his prize is:

“for the discovery that black hole formation is a robust prediction of the general theory of relativity.”

What is a “robust” prediction? The word strikes us as having overtones of necessity. Necessary knowledge is the kind we deal with in mathematics. The citation to Genzel and Ghez stays on empirical grounds:

“for the discovery of a supermassive compact object at the centre of our galaxy.”

The “object” must be a black hole—given relativity and its observed gravitational effects, it cannot be otherwise. Among many possible witnesses for the reality of black holes—one being the evident origin of the gravitational waves whose detection brought the 2017 Nobel—the centers of galaxies are hefty examples. The combination of these citations opens several threads we’d like to discuss.

The Proof Horizon of a Black Hole

Dick and I are old enough to remember when black holes had the status of conjecture. One of my childhood astronomy books stated that the Cygnus X-1 X-ray source was the best known candidate for a black hole. In 1974, Hawking bet Kip Thorne that it was not a black hole. The bet lasted until 1990, when Hawking conceded. He wrote the following in his famous book, A Brief History of Time:

This was a form of insurance policy for me. I have done a lot of work on black holes, and it would all be wasted if it turned out that black holes do not exist. But in that case, I would have the consolation of winning my bet. … When we made the bet in 1975, we were 80% certain that Cygnus X-1 was a black hole. By now [1988], I would say that we are about 95% certain, but the bet has yet to be settled.

In the 1980s, I was a student and then postdoc in Penrose’s department, so I was imbued with the ambience of black holes and never had a thought about doubting their existence. I even once spent an hour with John Wheeler, who coined the term “black hole,” when Penrose delegated me to accompany Wheeler to Oxford’s train station for his return to London. But it seems from the record that the progression to regarding black holes as proven entities was as gradual as many argue the act of crossing a large black hole’s event horizon to be. Although the existence of a central black hole from data emanating from Sagittarius had been proposed at least as far back as 1971, the work by Ghez and then Genzel cited for their prize began in 1995. The official announcement for Riccardo Giacconi’s share of the 2002 physics Nobel stated:

“He also detected sources of X-rays that most astronomers now consider to contain black holes.”

This speaks lingering doubt at least about where black holes might be judged to exist, if not their existence at all.

However their time of confirmation might be pinpointed, it is the past five years that have given by far the greatest flood of evidence, including the first visual image of a black hole last year. The fact of their presence in our universe is undeniable. But necessity is a separate matter, and with Penrose this goes back to 1964.

Relativity and Necessity

We have mentioned Kurt Gödel’s solution to the equations of general relativity (GR) in which time travel is possible. This does not mean that time travel must be possible, or that it is possible in our universe. A “solution” to GR is more like a model in logic: it may satisfy a theory’s axioms but have other properties that are contingent (unless the theory is categorical, meaning that all of its models are isomorphic). Gödel’s model has a negative value for Einstein’s cosmological constant; the 2011 physics Nobel went to the discovery that in our universe the constant has a tiny positive value. GR also allows solutions in which some particles (called tachyons) travel faster than light.

That GR has solutions allowing black holes had been known from its infancy in work by Karl Schwarzschild and Johannes Droste. There are also solutions without black holes; a universe with no mass is legal in GR in many ways besides the case of special relativity. Penrose took the opposite tack, of giving minimal conditions under which black holes are necessary. Following this article, we list them informally as follows:

1. Sufficiently large concentrations of mass exerting gravity exist.
2. Gravity always attracts, never repels.
3. No physical effect can travel faster than light.
4. Gravity determines how light bends and moves.
5. The space-time manifold is metrically complete.

Penrose showed that any system obeying these properties and evolving in accordance with GR must develop black holes. He showed this without any symmetry assumptions on the system. Thus he derived black holes as a prediction with the force of a theorem derived from minimal axioms.

His 1965 paper actually used a proof by contradiction. He derived five properties needed in order for the system to avoid forming a singularity. Then he showed they are mutually inconsistent—a proof by contradiction. Here is the crux of his paper:

[ Snip from paper ]

In the diagram, time flows up. The point in a nutshell—a very tight nutshell—is that once a surface flows inside the cylinder at the Schwarzschild radius then light and any other motion from it can go only inward toward a singularity. The analysis is possible without the kind of symmetry assumption that had been used to tame the algebraic complexity of the equations of GR. The metric completeness mandates a singularity apart from any symmetries; a periodic equilibrium is ruled out by analysis of Cauchy surfaces.

Necessary For Us?

Like Richard Feynman’s famous diagrams for quantum field theory, Penrose developed his diagrams as tools for shortcutting the vicissitudes of GR. We could devote entire other posts to his famous tiles and triangle and other combinatorial inventions. His tools enable quantifying black-hole formation from observations in our universe.

The question of necessity, however, pertains to other possible universes. Let us take for granted that GR and quantum theory are facets of a physical theory that governs the entire cosmos—the long-sought “theory of everything”—and let us also admit the contention of inflationary theorists that multiple universes are a necessary consequence of any inflation theory. The question remains, are black holes necessary in those universes?

It is possible that those universes might not satisfy axiom 1 above, or might have enough complexity for existence of black holes but not large-scale formation of them. The question then becomes whether black holes must exist in any universe rich enough for sentient life forms such as ourselves to develop. This is a branch of the anthropic principle.

Lee Smolin proposed a mechanism via which black holes engender new universes and so propagate the complexity needed for their large-scale formation. Since complexity also attends the development of sentient life forms, this would place our human existence in the wake of consequence, as opposed to the direction of logic when reasoning by the anthropic principle.

A Little More About Science

The 2020 Nobel Prize in Chemistry was awarded this week to Jennifer Doudna and Emmanuelle Charpentier for their lead roles in developing the CRISPR gene-editing technology, specifically around the protein Cas9.

We argue that two more different types of results cannot be found:

Penrose shows that black holes and general relativity are connected, which is a math result. We still cannot create black holes in a lab to experiment with—or maybe we could but should be very afraid of going anywhere near doing so. It was not clear that there could ever be a real application of this result.

Charpentier and Doudna discover that an existing genetic mechanism could be used to edit genetic material. Clearly this can and was experimented on in labs. Also clear that there are applications of this result. Actually it is now a standard tool used in countless labs. There even are patent battles over the method.

We like the fact that Nobels are given for such diverse type of research. It is not just that one is for astrophysics and one for chemistry. It is that Nobels can be given for very different types of research. We think this is important.

But wait. These results do have something in common, something that sets them apart from any research we can do in complexity theory. Both operate like this:

Observe something important from nature. Something that is there independent of us. Then in Penrose’s case explain why it is true. Then in Charpentier and Doudna’s case, use it to solve some important problems.

We wonder if anything like this could be done in our research world—say in complexity theory?

Open Problems

Besides our congratulations to all those mentioned in this post, Ken expresses special thanks to Sir Roger among other Oxford Mathematical Institute fellows for the kindness recorded here.

[changed note about massless universe]