Singularities in Space-Time Prove Hard to Kill

Two blind spots torture physicists: the birth of the universe and the center of a black hole. The former may feel like a moment in time and the latter a point in space, but in both cases the normally interwoven threads of space and time seem to stop short. These mysterious points are known as singularities.

Singularities are predictions of Albert Einstein’s general theory of relativity. According to this theory, clumps of matter or energy curve the space-time fabric toward themselves, and this curvature induces the force of gravity. Pack enough stuff into a small enough spot, and Einstein’s equations seem to predict that space-time will curve infinitely steeply there, such that gravity grows infinitely strong.

Most physicists don’t believe, however, that Einstein’s theory says much about what really happens at these points. Rather, singularities are widely seen as “mathematical artifacts,” as Hong Liu, a physicist at the Massachusetts Institute of Technology, put it, not objects that “occur in any physical universe.” They are where general relativity malfunctions. The singularities are expected to vanish in a more fundamental theory of gravity that Einstein’s space-time picture merely approximates — a theory of quantum gravity.

But as physicists take steps toward that truer and more complete theory by merging general relativity and quantum physics, singularities are proving hard to erase. The British mathematical physicist Roger Penrose won the Nobel Prize in Physics for proving in the 1960s that singularities would inevitably occur in an empty universe made up entirely of space-time. More recent research has extended this insight into more realistic circumstances. One paper established that a universe with quantum particles would also feature singularities, although it only considered the case where the particles don’t bend the space-time fabric at all. Then, earlier this year, a physicist proved that these blemishes exist even in theoretical universes where quantum particles do slightly nudge space-time itself — that is, universes quite a bit like our own.

This trilogy of proofs challenges physicists to confront the possibility that singularities may be more than mere mathematical mirages. They hint that our universe may in fact contain points where space-time frays so much that it becomes unrecognizable. No object can pass, and clocks tick to a halt. The singularity theorems invite researchers to grapple with the nature of these points and pursue a more fundamental theory that can clarify what might continue if time truly stops.

Space-Time’s Fatal Flaws

Karl Schwarzschild first discovered an arrangement of space-time with a singularity in 1916, just months after Einstein published general relativity. The bizarre features of the “Schwarzschild solution” took years for physicists to understand. Space-time assumes a shape analogous to a whirlpool with walls that swirl more and more steeply as you go farther in; at the bottom, the curvature of space-time is infinite. The vortex is inescapable; it has a spherical boundary that traps anything falling inside, even light rays.

It took decades for physicists to accept that these inconceivable objects, eventually dubbed black holes, might actually exist.

J. Robert Oppenheimer and Hartland Snyder calculated in 1939 that if a perfectly spherical star gravitationally collapses to a point, its matter will become so dense that it will stretch space-time into a singularity. But real stars bubble and churn, especially while imploding, so physicists wondered whether their nonspherical shapes would stop them from forming singularities.

Penrose eliminated the need for geometric perfection in 1965. His landmark proof relied on two assumptions. First, you need a “trapped surface” inside of which light can never escape. If you cover this surface in light bulbs and switch them on, their light rays will fall inward faster than they can travel outward. Crucially, this shell of light will shrink regardless of whether it started out as a perfect sphere, a dimpled golf ball, or something more misshapen.

Second, space-time should always curve in such a way that light rays bend toward each other but never diverge. In short, gravity should be attractive, which is the case so long as energy is never negative.

With these two stipulations, Penrose proved the mortality of at least one of the trapped light rays. Its otherwise eternal journey through space and time must terminate in a singularity, a point where the space-time fabric ceases to exist, where there is no future for the light ray to travel into. This was a new definition of a singularity, distinct from the infinite curvature of the Schwarzschild solution. Its generality enabled Penrose to prove in three scant pages of math that, under his two assumptions, singularities will inevitably form.

“Penrose’s paper was probably the most important paper in general relativity ever written, other than Einstein’s original paper,” said Geoff Penington, a physicist at the University of California, Berkeley.

Stephen Hawking soon extended Penrose’s argument to the early universe, proving that a cosmos described by general relativity must have sprung from a singular point during the Big Bang. This cosmological singularity resembles a black hole in that, if you imagine rewinding the history of the universe, light rays will run into a wall at the beginning of time.

Over the years, physicists have accumulated heaps of evidence that black holes exist, and that the universe began with an event that looks very much like a Big Bang. But do these phenomena truly represent space-time singularities?

Many physicists find the actual existence of such points unthinkable. When you try to calculate the fate of a particle approaching the singularity, general relativity glitches and gives impossible, infinite answers. “The singularity means a lack of predictability,” Liu said. “Your theory just breaks down.”

But the particle in the real world must have a fate of some sort. So a more universal theory that can predict that fate — very likely a quantum theory — must take over.

General relativity is a classical theory, meaning that space-time takes on one, and only one, shape at every moment. In contrast, matter is quantum mechanical, meaning it can have multiple possible states at once — a feature known as superposition. Since space-time reacts to the matter in it, theorists expect that any matter particles in a superposition of occupying two different locations should force space-time into a superposition of two distortions. That is, space-time and gravity should also follow quantum rules. But physicists haven’t yet worked out what those rules are.

Into the Onion

Theorists approach their quest for a quantum theory of gravity the way they might peel an onion: layer by layer. Each layer represents a theory of a universe that imperfectly approximates the real one. The deeper you go, the more of the interplay between quantum matter and space-time you can capture.

Penrose worked in the outermost layer of the onion. He used the general theory of relativity and ignored quantumness entirely. In effect, he proved that the space-time fabric has singularities when it is completely devoid of any quantum matter.

Physicists aspire to someday reach the onion’s core. In it, they’ll find a theory describing both space-time and matter in all their quantum glory. This theory would have no blind spots — all calculations should yield meaningful results.

But what about the middle layers? Could physicists resolve Penrose’s singularities by moving to something a little more quantum, and therefore a little more realistic?

“It was the obvious speculation, that somehow quantum effects should fix the singularity,” Penington said.

They first tried to do so in the late 2000s. The assumption that had confined Penrose’s proof to the outermost layer was that energy is never negative. That’s true in everyday, classical situations, but not in quantum mechanics. Energy goes negative, at least momentarily, in quantum phenomena such as the Casimir effect, where (experiments show) two metal plates attract each other in a vacuum. And negative energies play a role in the way black holes are thought to radiate particles, eventually “evaporating” entirely. All the deeper, quantum layers of the onion would feature this exotic energetic behavior.

The physicist who peeled the top layer was Aron Wall, then based at the University of Maryland and now at the University of Cambridge. To cut into the quantum realm and abandon Penrose’s energy assumption, Wall latched on to a theoretical discovery made in the 1970s by Jacob Bekenstein.

Bekenstein knew that for any given region of space, the contents of the region grow more mixed up as time goes on. In other words, entropy, a measure of this mixing, tends to increase, a rule known as the second law of thermodynamics. While considering a region that contains a black hole, the physicist realized that the entropy comes from two sources. There’s the standard source — the number of ways that quantum particles in the space around the black hole could be arranged. But the black hole has entropy too, and the amount depends on the black hole’s surface area. So the total entropy of the region is a sum: the surface area of the black hole plus the entropy of nearby quantum stuff. This observation became known as the “generalized” second law.

Wall “made it his mission to understand the generalized second law,” said Raphael Bousso, a physicist at Berkeley. “He was thinking about it in much clearer and much better ways than everybody else on the planet.”

Reaching the quantum layers of the onion would mean accommodating negative energy and the presence of quantum particles. To do so, Wall reasoned that he could take any surface area in general relativity and add to it the entropy of those particles, as the generalized second law suggested. Penrose’s proof of his singularity theorem had involved the trapped surface. So Wall upgraded it to a “quantum trapped surface.” And when he reworked Penrose’s singularity theorem in this way, it held. Singularities form even in the presence of quantum particles. Wall published his findings in 2010.

“Aron’s paper was a seminal breakthrough in combining quantum mechanics and gravity in a more precise way,” Penington said.

Having peeled back the classical outer layer of the onion, where energy is always positive, Wall reached a lightly quantum layer — a context physicists call semiclassical. In a semiclassical world, space-time guides the journeys of quantum particles, but it cannot react to their presence. A semiclassical black hole will radiate particles, for instance, since that’s a consequence of how particles experience a space-time warped into a black hole shape. But the space-time — the black hole itself — will never actually shrink in size even as the radiation leaks energy into the void for all eternity.

That’s almost, but not exactly, what happens in the real universe. You could watch a black hole radiate particles for a century without seeing it shrink a single nanometer. But if you could watch for longer — many trillions upon trillions of years — you would see the black hole waste away to nothing.

The next onion layer beckoned.

Dialing Up the Quantumness

Bousso recently revisited Wall’s proof and found that he could cut a little deeper. What about the world where black holes shrink as they radiate? In this scenario, the space-time fabric can react to quantum particles.

Using more refined mathematical machinery developed by Wall and others since 2010, Bousso found that, despite the intensified quantumness of his scenario, singularities continue to exist. He posted his paper, which has not yet been peer-reviewed, in January.

The world of Bousso’s new theorem still departs from our universe in notable ways. For mathematical convenience, he assumed that there’s an unlimited variety of particles — an unrealistic assumption that makes some physicists wonder whether this third layer matches reality (with its 17 or so known particles) any better than the second layer does. “We don’t have an infinite number of quantum fields,” said Edgar Shaghoulian, a physicist at the University of California, Santa Cruz.

Still, for some experts, Bousso’s work delivers a satisfying denouement to the Penrose and Wall singularity story, despite its unrealistic abundance of particles. It establishes that singularities can’t be avoided, even in space-times with mild reactions to quantum matter. “Just by adding small quantum corrections, you can’t prevent the singularity,” Penington said. Wall and Bousso’s work “answers that pretty definitively.”

The Real Singularity

But Bousso’s theorem still doesn’t tell us exactly what form singularities will take at the core of the onion — that is, in the ultimate quantum theory of gravity.

There is still a possibility that the dead ends do somehow go away. What seems like a singularity could actually connect to somewhere else. In the case of a black hole, perhaps those light rays end up in another universe. “A baby universe could form inside a black hole,” Bousso said. “You fall into a black hole, it looks like everything’s collapsing, but whoosh! Everything starts expanding again, and there’s a new world.”

And a lack of a Big Bang singularity might imply that our universe began with a “Big Bounce.” The idea is that a previous universe, as it collapsed under the pull of gravity, somehow dodged the formation of a singularity and instead bounced into a period of expansion. Physicists who are developing bounce theories often work in the second layer of the onion, using semiclassical physics that exploits negative-energy quantum effects to get around the singularity required by the Penrose and Hawking theorems. In light of the newer theorems, they will now need to burrow deeper into the onion and argue that the truer theories they find there can violate the requirements of the generalized second law.

One physicist pursuing bounces, Surjeet Rajendran of Johns Hopkins University, says he is undaunted. He points out that not even the generalized second law is gospel truth. Any loopholes would make singularities avoidable and continuations of space-time possible.

Bousso and like-minded physicists, however, suspect that the generalized second law holds throughout the onion, and therefore that the dead ends should persist in the core theory and in our universe. The beginning of the cosmos and the hearts of black holes would truly mark edges of the map where clocks can’t tick and space stops.

“Inside of black holes, I am positive there is some notion of singularity,” said Netta Engelhardt, a physicist at MIT who has worked with Wall.

In that case, the still-unknown fundamental theory of quantum gravity would not kill singularities but demystify them. This truer theory would allow physicists to ask questions and calculate meaningful answers, but the language of those questions and answers would change dramatically. Space-time quantities like position, curvature and duration might be useless for describing a singularity. There, where time ends, other quantities or concepts might have to take their place. “If you had to make me guess,” Penington said, “whatever quantum state describes the singularity itself does not have a notion of time.”