What If Singularities DO NOT Exist? (Script) (Patreon)

It's not too often that a giant of physics threatens to overturn an idea held to be self-evident by generations of physicists. Well, that may be the fate of the famous Penrose Singularity Theorem if we're to believe a recent paper by Roy Kerr. Long story short, the terrible singularity at the heart of the black hole may be no more.

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Around 400 years ago a guy named Isaac Newton figured out how gravity works. Suddenly a lot of mysterious things made a lot more sense–from the reason apples fell from trees to the motion of the planets and the stars. But the discovery also birthed a new and stranger mystery–it hinted at the possibility that the gravity of a sufficiently dense object could produce an event horizon–a surface of no escape, able to hold prisoner light itself. It raised the specter of black holes–whose paradoxical nature plagues physics to this day. 

Einstein’s update to Newtonian gravity seemed to confirm the theoretical prospect of the black hole, and it also revealed something even more challenging for physics. The first solution to the equations of general relativity–the Schwarzschild solution–hinted that at the very center of the black hole there is a singularity. At this hypothetical point of infinite density and infinite gravity, GR comes into terrible conflict with quantum mechanics. At the size scale of the singularity our two supreme theories of the physical world prove themselves mutually incompatible.

Many, perhaps most physicists were seriously uncomfortable with the idea of black hole singularities. And they got even more uncomfortable when, in 1965, British physicist Sir Roger Penrose showed that singularities really are unavoidable in general relativity. His black hole singularity theorem–for which he won the 2020 Nobel prize–claims that as long as an event horizon exists, so must a singularity.

Perhaps the most important implication of the singularity theorem is to show that general relativity really is in fundamental conflict with quantum mechanics. If that’s the case, then the only salvation from the paradox of the singularity is some greater theory combining quantum mechanics and general relativity, in which the singularity will evaporate away–as if it was just the bad dream of ignorant 20th century physics. But just recently we’ve had a ray of hope from a completely unexpected direction. In a paper released in December, Roy Kerr–one of the greatest black hole theorists of all time–may have shown that we can avoid the black hole singularity without quantum mechanics after all. 

In order to get to this radical new result, we need to build up some understanding, so let’s refresh our knowledge of black holes and review the Penrose singularity theorem. Then maybe we can decide if Roy Kerr really did destroy the singularity. So, to start with Roger Penrose didn’t exactly prove the existence of singularities–not explicitly anyway. He demonstrated that spacetime paths must terminate inside a black hole. 

Anything moving in spacetime under only gravity follows something called a geodesic. This is a path through spacetime that minimizes the combined spatial and temporal distance traveled. Before the Penrose Singularity Theorem, it was generally held that geodesics had no end. An object might travel along a segment of a geodesic–for example,  a ball being thrown on a parabola–but the geodesic itself can be traced both backwards and forwards beyond the ball’s trajectory. Forward forever into the expanding universe, or backwards to the beginning of the universe. 

Roger Penrose showed that inside a black hole, geodesics have to converge at the center and end there. When a spacetime admits geodesics which don’t go on forever we say the spacetime is “geodesically incomplete”. You can imagine geodesics as the gridlines of spacetime, forming a smooth, if sometimes quite warped fabric on which the laws of physics work nicely. Geodesic incompleteness means there are pinched-off regions where infinities appear and the laws of physics break down.

Kerr’s objection to this argument depends on the subtle interpretation of geodesic incompleteness, so let’s dig a little deeper. When Penrose said a geodesic captured by a black hole “ends” at its center, he meant something very specific, mathematically. He meant that the “geodesic parameter” is bounded–so the mathematical variable we use to describe the evolution of something along a geodesic terminates. Similar to how your latitude terminates if you travel to the s outh pole–you can’t go further south once your southness is maxed out.

For the geodesics describing the paths of matter, the geodesic parameter can be, and usually is, taken to be the “proper time” –that’s the time measured by someone moving along that trajectory. So if the parameter for matter geodesics is bounded then it would imply a singularity because there’s no way to trace a flow of time through it. There’s no meaningful way to define “after the singularity”, in either space or time. These are dead-ends in spacetime.

Now Penrose constructed his argument using the paths of light, not of matter, and it turns out the difference is crucial, as we’ll see. However the general argument that geodesic incompleteness equals a singularity was a convincing enough argument that for nearly 60 years almost all of us agreed that pure general relativity demands singularities. Stephen Hawking even used Penrose’s arguments to show that in pure GR the Big Bang was also a singularity–all geodesics traced backwards in time had to converge and end at one point.

But Roy Kerr had his doubts, to put it mildly. Kerr is a New Zealand physicist who, in 1963, came up with the Kerr metric–the mathematical descriptio n of a rotating black hole. This was the second black hole solution to the Einstein equations to be discovered–47 years after Karl Schwarzschild solution, which describes the much simpler case of a non-rotating black hole. Now we have good reason to believe that essentially all real black holes have some rotation, so the Kerr solution is kind of a big deal. As is Roy Kerr. And Roy Kerr vehemently disagrees that singularities exist, nor even that the Penrose Singularity Theorem has anything to say about their existence.

Let’s get straight to the heart of his objection. So I told you that geodesic incompleteness has been taken to mean that spacetime paths terminate, which in turn has been taken to mean that singularities are real. But there’s a catch to this argument. Penrose constructed his argument using a particular type of geodesic–the null geodesic. These are the spacetime paths traveled by massless, lightspeed objects. A null geodesic represents the shortest path between two points in curved space. OK, so what does it mean for a null geodesic to terminate? It means its geodesic parameter has to be bounded and not increase forever. 

In the case of massive particles we used proper time to trace these regular geodesics. But things traveling at the speed of light don’t experience time. Their clocks remain frozen and so proper time doesn’t increase along a null geodesic. To describe the geodesic motion of light, we need a different measure. We use something called an “affine parameter” which is a slightly complicated thing, but the main thing is that it increases in a nice clean way to track progress along a null geodesic. 

Penrose’s theorem shows very convincingly that affine parameters are bounded inside black holes, and so null geodesics end. He then took this to infer the inevitability of singularities as dead-ends in the grid of spacetime. But Kerr points out that these affine parameters DON’T track time in a meaningful way, and so don’t imply that the grid of spacetime falls apart at the termination of a null geodesic. To illustrate this crudely: the affine parameter could be an exponential of coordinate time. This function is bounded from below even though time can go from  minus to plus infinity. So, that limit of the affine parameter doesn’t mean that time itself comes to a halt, argues Roy Kerr. He also argues that this invalidates any arguments about the inevitability of the singularity due to coordinate system dead ends.

Kerr’s paper is quite fun to read. He is snarky to put it mildly, excoriating the physics community over and over for blindly following a conclusion that he states is “built on a foundation of sand”. I linked the paper in the description for your amusement.

Another important part of Kerr’s argument is about the difference between real black holes and the idealized black holes analyzed in the Penrose paper. Essentially all–and perhaps literally all–real black holes must have some rotation. Real astrophysical black holes will obey the Kerr metric, not the Schwarzschild, and the same argument can be extended for charged black holes. Kerr black holes do not have a point-like singularity in their center. In the Kerr metric, the point singularity is stretched out into a ring singularity–a looped strand of infinite curvature. But Kerr insists that even this isn’t a real singularity. 

The other cool thing about the Kerr metric is that collapse towards the singularity is not inevitable as it is in the Schwarzschild metric. There’s a region just below the Kerr event horizon where collapse really is unavoidable. Across the event horizon all paths lead down, just like in a Schwarzschild black hole. But not to the center. In a rotating black hole, the centrifugal effect of the spinning spacetime counteracts gravity, resulting in this inner region of almost normal spacetime. In the Kerr black hole there’s an inner horizon, and once you cross it you’re free to move in any direction, even back up. 

So what’s this ring singularity? Kerr implies that it’s a mathematical fiction. It’s just a convenient way to represent the gravitational field generated by a rotating object. And he suggests that a true collapsed star would exist in an extended, physical form inside the inner horizon.

Kerr nails down his argument by demonstrating that, contrary to the conclusion of the Penrose Singularity Theorem, NOT all null geodesics terminate at a singularity in the Kerr black hole, even if their affine parameter is finite. He reveals these families of geodesics that pass the inner event horizon of a Kerr black hole and continue to exist forever and trace out really any path inside the black hole without having to hit the supposed singularity and “stop existing”. This is contrary to the previous belief that light crossing any black hole's event horizon has to end up at the supposed singularity at the center. That’s if the ring singularity in the Kerr black hole is even a meaningful entity rather than a mathematical convenience, as Roy Kerr believes.

So what does this all really mean for the existence of singularities? Well, it is important to understand that Kerr’s argument isn’t necessarily saying that singularities don’t exist, it is saying that the conclusions of the singularity theorem proof may be incorrect. That bounded affine parameters for null geodesics don’t imply a singularity, contrary to the common interpretation of the Penrose Singularity Theorem.

Not too many physicists really believe that black hole singularities exist, but most thought that we’d have to bring quantum mechanics into the picture to figure out why. That’s why Kerr’s paper is a surprise–it may give us a path to defeating the singularity without having to wait for the elusive theory of quantum gravity. 

There’s still work to be done to see if Kerr’s ideas hold up to scrutiny–and we have no doubt that there’ll be some excited arguments from both sides. But in the meantime we now have reason to be less scared of the interiors of black holes—from a theoretical standpoint. Without singularities, perhaps we can start formulating sensible physics about what happens in their interiors. Perhaps, with Roy Kerr’s new ideas physicists can travel a bit safer in theoretical space thanks to a singularity-free spacetime.