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. A few hundred years ago 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 Penrose 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. So the argument of Penrose was that
geodesic incompleteness means singularity. But Kerr has an objection to
the argument and it depends on a subtle interpretation
of geodesic incompleteness. So let’s dig a little deeper. When Penrose said a geodesic captured by a black hole “ends” at the
black hole 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 just the time measured by someone moving along that trajectory. So if the parameter for a matter geodesics is
bounded that 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, and that one just 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 finally get 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 singularities 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 even exists 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's saying that the conclusions
of the singularity theorem proof may be incorrect. It's saying 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. Hey Everyone! We wanted to say thank you
to all the community who support this show, whether it’s buying merch from the merch store
or simply liking, commenting and subscribing. But in particular we wanted to say thank
you to our long time Patreon supporters. Patreon supporters are a huge reason why we’ve
been able to make the show as it is. So as a special thank you we’re adding in a permanent
10% merch store discount to Patreon members of all levels and we’ll be opening our next
Patreon Livestream AMA to all paid patreon levels. And of course if you join us on Patreon,
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