I think Roy Kerr’s recent paper is important. It’s called Do black holes have singularities? I think it’s important because it challenges an orthodoxy that’s been taken for granted, and because Kerr has the authority to get some attention. That’s because he was in on the Golden Age of General Relativity, and because the EHT collaboration described both M87* and Sagittarius A* as matching the Kerr metric. To set the scene, black hole physics has been around for a long time, but a major development was Karl Schwarzschild’s 1915 Schwarzschild metric. This is said to describe “the geometry of spacetime around an uncharged, spherically symmetric, and non-rotating body”.
A typical Schwarzschild black hole depiction, this one is from Universe review
The general idea of the Schwarzschild metric is that when you concentrate a lot of matter together, it all falls inwards. If you have enough matter, the result is a black hole with an event horizon at the Schwarzschild radius rs = 2GM / c². The G here is of course the gravitational constant, whilst M is the mass of the matter, and c is the speed of light.
It’s saying a black hole has a point singularity at its centre
There are said to be two singularities in the Schwarzschild metric, both of which are associated with a division by zero. I will play a straight bat for now, so here’s a quote from Wikipedia: “The solution contained terms of the form 1 − rs / r and 1 / (1 – rs / r), which becomes singular at r = 0 and r = rs respectively. The rs has come to be known as the Schwarzschild radius. The physical significance of these singularities was debated for decades. It was found that the one at r = rs is a coordinate singularity, meaning that it is an artifact of the particular system of coordinates that was used; while the one at r = 0 is a spacetime singularity and cannot be removed”. What it’s saying is the singularity at the event horizon isn’t a real singularity, but the point singularity at the centre is.
Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics
See the Wikipedia Kerr metric article where you can read that “the exact solution for an uncharged, rotating black hole, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr”. The Kerr metric is said to describe “the geometry of empty spacetime around a rotating uncharged axially symmetric black hole”. Check out Kerr’s 1963 paper in Physical Review Letters. The title is Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. It’s only a page and a half long, it doesn’t mention “black hole” or “singularity”, and it doesn’t describe the Kerr black hole as it’s currently described. There’s no mention of the outer ergosphere, the outer horizon, the inner ergosphere, the inner horizon, or the ring singularity:
CC by SA 4.0 Kerr Surfaces image by Yukterez (Simon Tyran, Vienna), see Wikimedia commons
Instead it refers to “the third-order Einstein-Infeld-Hoffmann approximation for a spinning particle”. This relates to the Einstein–Infeld–Hoffmann equations which describe “the approximate dynamics of a system of point-like masses due to their mutual gravitational interactions”. The equations are “valid in the limit where the velocities of the bodies are small compared to the speed of light and where the gravitational fields affecting them are correspondingly weak”. Interesting stuff. Follow the trail to the 1937 paper on The Gravitational Equations and the Problem of Motion by Einstein, Leopold Infeld, and Banesh Hoffman. This refers to “the ordinary Maxwell equations for empty space, in which electrical particles are regarded as point singularities of the field”. It also referred to “the well-known theory of Helmholtz on the motion of vortices”. I am reminded of the problem of infinities that plagued quantum electrodynamics because it employed Yakov Frenkel’s point-particle electron. What a pity Infeld didn’t lean on Einstein to use the Born-Infeld model, which talked about the electron’s inner angular momentum influenced by an external field. What a pity Einstein wasn’t working with Erwin Schrödinger who spoke of a charged particle as wave in a closed path.
Discovering the Kerr and Kerr-Schild metrics
But we are where we are. Kerr’s recent paper obviously refers to his 1963 paper. It also refers to two papers he wrote with Alfred Schild in 1965. One was A new class of vacuum solutions of the Einstein field equations, the other was Some algebraically degenerate solutions of Einstein’s gravitational field equations. I’m afraid I can’t find an online copy of the latter, it’s in an AMS symposia collection and it’s paywalled. But for some good background reading take a look at Kerr’s 2007 historical preprint Discovering the Kerr and Kerr-Schild metrics. See the footnote on page 2 and have a laugh at my expense, because Kerr said this: “There is a claim spread on internet that we were employed to develop an antigravity engine to power spaceships. This is rubbish!” Also see the footnote on page 8: “I spent many years trying to write up this research but, unfortunately, I could never decide whether to use spinors or a complex tetrad, and thus it did not get written up until Kerr and Debney (1969). George Debney also collaborated with Alfred Schild and myself on the Kerr-Schild metrics in Debney et al. (1970)”. You can find the titles of the papers in the bibliography. One was the 1969 JMP paper Solutions of the Einstein and Einstein-Maxwell equations by Kerr, Schild, and Debney. Another was the 1970 JMP paper Einstein Spaces with Symmetry Groups by Kerr and Debney. Yet another was the 1979 General Relativity and Gravitation paper Singularities in the Kerr-Schild metrics by Kerr and W B Wilson. I’m not sure what to make of these papers, because they’re all mathematics and no physics. And I’m not sure what a spinor has to do with a rotating mass, but nevermind that for now:
Spinor gif by John F Lindner from Spinors | Wooster Physicists.
What’s more important is that there are 21 instances of “singular” in the 2007 historical article, and 15 instances of “ring”. So you’d think Kerr is a total advocate of the ring singularity that’s supposed to be inside a rotating black hole. But see this on page 24: “any collapsing star can only form a black hole if the angular momentum is small enough: a < m. This seems to be saying that the body cannot rotate faster than light, if the final picture is that the mass is located on the ring radius a. However, it should be remembered that this radius is purely a coordinate radius, and that there is no way that the final stage of such a collapse is that all the mass is located at the singularity. The reason for the last statement is that if the mass were to end on the ring then there would be no way to avoid the second asymptotically flat sheet where the mass appears negative”. In 2007 Kerr was saying there’s no place where a mass appears negative, so there’s no ring singularity in a Kerr black hole.
Hence Kerr’s recent paper Do black holes have singularities? isn’t something new. He’s obviously been thinking about this for quite some time. Now at tender age of 89, he has decided to speak out. His abstract is fighting talk: “There is no proof that black holes contain singularities when they are generated by real physical bodies. Roger Penrose claimed sixty years ago that trapped surfaces inevitably lead to light rays of finite affine length (FALL’s). Penrose and Stephen Hawking then asserted that these must end in actual singularities. When they could not prove this they decreed it to be self evident. It is shown that there are counterexamples through every point in the Kerr metric”. What Kerr is saying is that light trapped inside a Kerr black hole does not necessarily end up in a singularity, and that this proves that Penrose–Hawking singularity theorems are wrong.
There is no direct proof that black holes contain singularities
This is heavy stuff, because Roger Penrose was awarded a half share in the 2020 Nobel prize in physics. His award was for “the discovery that black hole formation is a robust prediction of the general theory of relativity”. The blurb says “Einstein did not himself believe that black holes really exist”, but in 1965 Penrose “proved that black holes really can form and described them in detail; at their heart, black holes hide a singularity”. Check out the Nobel web page on The Singularity Theorem, Penrose’s paper, and Wikipedia for more. It says “light rays are always focused together by gravity, never drawn apart, and this holds whenever the energy of matter is non-negative”. Hence all the inward light rays allegedly end up at one point, the singularity.
NASA ESO black hole image from First Image of a Black Hole – NASA Science
Kerr says this just isn’t proven. See page 2 of his paper where he says there is no direct proof that black holes contain singularities. Just “papers by Penrose outlining a proof that all Einstein spaces containing a ‘trapped surface’ automatically contain FALLs”. Kerr then said “it was then decreed, without proof” that these finite affine length light-rays must end in actual points where the metric is singular. He went on to effectively say the Kerr black hole featured radial light rays which come down the axis through the North pole and die asymptotically on the inner horizon on the opposite side. So here we have light rays of finite affine length, and a scenario which refutes the claim that all black holes must contain a singularity.
He said he was going to get the Kool-Aid treatment
Kerr was similarly scathing on page 5 of his paper when he talked about a rapidly-spinning neutron star accreting mass and contracting. He described how you could initially launch a rocket from the surface, but as the mass increases and the radius decreases, an event shell and therefore a black hole forms. Kerr then said “Why do so many believe that the star inside must become singular at this moment? Faith, not science! Sixty years without a proof, but they believe!” These are strong words and Kerr knows it. That’s why he said he was going to get the Kool-Aid treatment at the end of this video. He even used the word “cult”.
Screenshot from the Elephant in the Room Zoom meeting featuring Roy Kerr.
Hence it’s generated a reaction in the physics community. See the video by Sabine Hossenfelder Backreaction: Black Hole Singularities “Faith, not science!” Prominent Physicist Claims. She was generally supportive of Kerr’s argument. So was Ethan Siegel, see Singularities don’t exist, claims black hole pioneer Roy Kerr. I didn’t like the way he said “space itself is flowing: like a waterfall”, or the way he used the Penrose diagram showing the parallel antiverse, but IMHO he gave Kerr a fair appraisal. Another broadly supportive article was Unitary Flow: Roy Kerr vs. the singularities by Klaas Landsman. OK I didn’t like the way he spoke of the Hawking and Ellis “bible”, but that’s one for another day.
A Humpty Dumpty statement
Meanwhile when I search the internet I can see a question on Quora and a discussion on Cosmoquest that were both supportive of what Kerr was saying. Ditto for a brief article Then24 written by Peggy McColl. However a discussion on Physicsforums was not supportive. Mentor Peter Donis gave a Humpty Dumpty statement saying “the term ‘singularity’ is defined in the GR literature as ‘that FALL thing’”. He also said “the claim Kerr claims to be rebutting, as far as I can tell, is a claim that nobody in the GR community actually makes”. Answers to a question on Physics Stack Exchange were similarly negative, and thought-police moderator ACuriousMind deleted Kerr’s reply:
Screenshot from physics stack exchange
Peter Woit hasn’t said anything about it, nor has Sean Carroll, and nor has Matt Strassler. In addition I haven’t seen any news articles about it, or any statements from “experts in the field”. I will be interested to see how this one develops. Will Penrose pipe up? We will see, but don’t hold your breath.
Neither ray crosses the event horizon in the original Schwarzschild coordinates
What do I think of it? I feel very sympathetic towards Roy Kerr, and I wish him every success. However I say that with some sadness. Because on page 6 of his paper Kerr talked about the Schwarzschild black hole, saying it appeared to have two singularities, one at the centre where the curvature tensor was infinite, the other at the event horizon. He then said “for several years it was thought that the latter was real”, but “Eddington and Finklestein showed that this was false by writing the metric in different coordinate systems where the only singularity was at the centre. They also showed that any object that crossed the horizon would quickly fall to this point”. Then on page 7 he talked about inward and outward light rays, and said “neither ray crosses the event horizon in the original Schwarzschild coordinates”. Then on page 9 he said “many have said to the author “What about the Kruskal-Szekeres extension?” as if this makes a difference to any singularities”. Roy oh Roy. Kruskal-Szekeres coordinates don’t make any different to any singularities, and nor do Eddington-Finkelstein coordinates. Take a look at page 828 of MTW where Box 31.2 says Eddington and Finkelstein used free-falling photons as the foundation of their coordinate system:
Screenshot from Gravitation by Misner Thorne and Wheeler
Now take a look at the Wikipedia article on Eddington–Finkelstein coordinates. It says this: “they are named for Arthur Stanley Eddington and David Finkelstein, even though neither ever wrote down these coordinates or the metric in these coordinates. Roger Penrose seems to have been the first to write down the null form but credits it (wrongly) to the above paper by Finkelstein, and, in his Adams Prize essay later that year, to Eddington and Finkelstein”. Penrose again. All this is despite the fact that by 1907 Einstein was saying light curves in a gravitational field because the speed of light varies. He said it again in 1911, 1912, 1913, 1914, 1915, and 1916. To ram it home he said it again in 1920: “the law of the constancy of the speed of light no longer holds, according to the general theory of relativity, in spaces that have gravitational fields. As a simple geometric consideration shows, the curvature of light rays occurs only in spaces where the speed of light is spatially variable”. He reiterated the point in 1939 in his paper on a stationary system with spherical symmetry consisting of many gravitating masses. That’s where Einstein said light rays “take an infinitely long time (measured in “coordinate time”) in order to reach the point r = μ/2 when originating from a point r > μ/2. In this sense the sphere r = μ /2 constitutes a place where the field is singular”. Irwin Shapiro reiterated it again in 1964: “according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path”. However people who ought to know better claim it doesn’t. They flatly contradict Einstein by making the speed of light constant by definition:
Screenshot from Comments on “Note on varying speed of light theories” by Joao Magueijo,and John W. Moffat
Sadly Kerr was taught the incorrect Bergmann version, hence his 1969 paper said “units are chosen such that the speed of light c and the Newtonian gravitational constant G are equal to unity: c=G=1”. I say sadly because the speed of light varies vertically in a gravitational field, as demonstrated by NIST optical clocks. The more it varies, the stronger the gravitational field. The stronger the gravitational field, the more the ascending light beam speeds up, and the descending light beam slows down. This is the exact opposite of what Penrose’s partner-in-crime Hawking said in his 1966 paper singularities and the geometry of spacetime: “such a strong gravitational field that even the ‘outgoing’ light rays from it are dragged back”. Penrose and Hawking didn’t have a clue about how gravity works. They flatly contradicted Einstein, who spoke of “the refraction of light rays” in a gravitational field. Light curves like any other waves curve when the properties of the medium vary, causing the wave speed to vary. It varies because of a concentration of energy in the guise of a massive star conditions the surrounding space, making it neither homogeneous nor isotropic in a non-linear fashion, which we label as spacetime curvature. Then matter falls down because it’s a wave in a closed path, like Schrödinger said. The horizontal component is refracted downwards resulting in a downward displacement.
At the event horizon, there is no gravitational field
Only guess what? At the event horizon, the speed of light is zero, and it can’t go lower than that. So that’s the end of your spacetime curvature. So at the event horizon, there is no gravitational field. So there is no collapse to a point singularity at the centre. So the black hole is a frozen star, like Wheeler said, and that’s the end of that. When light is stopped you can’t make it move via a coordinate transformation using time intervals of infinite duration that take you into some never-never land beyond the end of time. It’s a schoolboy error to think you can, and another schoolboy error to believe the black hole charlatans who tell you that “to an outside observer the contraction to r = 2m appears to take an infinite time”, and then tell you that “the body contracts and continues to contract”. It takes an infinite time for all observers. Or would do if the infalling body didn’t come to grief long before the event horizon. Because a falling body doesn’t slow down, and because matter can’t go faster than the light from which it’s made. That’s why we have gamma ray bursts. Poor old Roy. On page 15 of his paper he talked of rotation at near the speed of light. But that counts for nothing when the speed of light is zero. On page 14 of his paper he said physics is needed, not just mathematics. How right he was. We need evidence, not mathematical “discoveries”. He’s also right that there are no point singularities, but for the wrong reason. Unfortunately the right reason means there’s no Kerr black holes either. That won’t please people like Kip Thorne, who’s been peddling wormhole time-travel woo for decades. But it will please those of us who know that gravitational physics has been lost in maths for sixty years, and is a tottering tower of fantasy. Those of us who know that we are living in a dark age.