Do black holes actually have a singularity in them? Good question. Don’t be too sure you know the answer.

Albert Einstein [1879-1955] introduced his spectacularly successful, but also spectacularly complicated, theory of general relativity, in 1915. But this first publication was not much more than an edited set of lecture notes, essentially incomprehensible to anyone except Einstein. His real publication a real science paper, came along in 1916, and this one is the official first paper introducing general relativity to the world, in its final & corrected form.

But hot on the heels of Einstein’s 1915 paper, German astronomer & mathematician Karl Schwarzschild [1873-1916] derived the first known solutions to Einstein’s equations, also in 1915 (but first published in 1916). The Schwarzschild solution (or Schwarzschild metric) describes spacetime outside a spherical mass, assuming that both electric charge and angular momentum are zero. It’s the simplest form of solution.

Unfortunately, Schwarzschild died in 1916, from an autoimmune disease he caught while on the Russian front, for the German army, during World War I. Nevertheless, this simplest solution to Einstein’s equations introduced the concept of a black hole, due to a singularity in the equation. But of course, zero angular momentum is not a physically realistic solution.

Fast forward 48 years later, to 1963: New Zealand mathematician Roy Kerr [b. 1934] finally solved the complicated problems posed by non-zero angular momentum. The Kerr metric ignores electric charge, as did Schwarzschild, but not angular momentum. So, it describes the spacetime around an axially symmetry (but non-spheroidal) mass. Unlike Schwarzschild’s solution, the Kerr metric is physically realistic. And it has finally been verified by observations. The Event Horizon Telescope obtained millimeter wave images for the supermassive black holes in both the Milky Way, and the giant elliptical galaxy M87. And in both cases, the observations matched predictions created using the Kerr metric.

Fast forward again to 2020: Roger Penrose [b. 1931] received half of the 2020 Nobel Prize in Physics, “for the discovery that black hole formation is a robust prediction of the general theory of relativity”. The citation is a reference to the Penrose-Hawking singularity theorems, wherein Penrose (1965) proves the inevitability of singularities in the formation of black hole, and Stephen Hawking [1942-2018] (1972) proves the inevitability of a singularity at the spacetime point of origin for the universe. Or so they claim.

And that brings us to December 2023, and the whole point of this epistle on general relativity. Enter the very same Roy Kerr we saw in 1963, still in fighting trim. In a new paper, Kerr claims that neither Penrose nor Hawking, ever actually proved anything. Simply put, he claims that the geodetic paths that Penrose & Hawking bring to a singularity, are actually asymptotic to an event horizon.

It’s better perhaps in Kerr’s own words, from the abstract to his paper:

“There is no proof that black holes contain singularities when they are generated by real physical bodies. Roger Penrose [1] claimed sixty years ago that trapped surfaces inevitably lead to light rays of finite affine length (FALL’s). Penrose and Stephen Hawking [2] 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. These are asymptotic to at least one event horizon and do not end in singularities.”

And from his conclusion:

“In conclusion, I have tried to show that whatever the Penrose and Hawking theorems prove has nothing to do with Physics breaking down and singularities appearing. Of course, it is impossible to prove that these cannot exist, but it is extremely unlikely and goes against known physics.”

In between the abstract and conclusion are a 17 pages of material which will be nigh onto impossible to follow, unless you are a mathematician or physicist, with experience in this peculiar branch of theoretical or mathematical physics. I have yet to see a response from Penrose. But I would expect one at some point. Penrose is not particularly shy about such things.

But all of this does make a point about the history of physics. Isaac Newton [1643-1727] & Albert Einstein [1879-1955] introduced classical physics, and relativistic physics, respectively. Both of them were thinking well beyond any but a few of their contemporaries. Yet neither of them came close to understanding how incredibly deep & intellectually rich their own discoveries really are. In this case, 69 years after Einstein is gone, some of the most advanced mathematicians & physicists in the world, are still trying to grasp the details of a theory he invented 109 years ago. In this way, physics marches on.

The images posted here are slides from a talk retired Tim Thompson recently gave on black holes. The first image shows that the Event Horizon Telescope image of the M87 black hole matches the theoretical simulation of he photon ring, based on the Kerr metric. The ring of light has nothing to do with an accretion disk around the black hole. Rather, it is made from photons escaping from the photon sphere, a shell of photons in orbit around the black hole. Some photons eventually escape, and others fall into the black hole. The escaping photons always create a photon ring effect, no matter from what direction you see the black hole. The inner edge of the visible photon ring is about 150 AU from the center of the image, roughly corresponding to the current distance to the Voyager spacecraft, in our own solar system.

The other is an image of the black hole as computed for the movie Interstellar (2014), by the Caltech team led by Kip Thorne [b. 1940], who shared the 2017 Nobel Prize in Physics, for his role in the detection of gravitational waves. Note that the left side of the image is relatively brighter than the right side. That’s because the left side is rotating towards you. A relativistic Doppler effect brightens the side coming towards you, and dims the side going away. In this case, we are seeing the accretion disk, nearly edge on. So, we can see the disk in front of the black hole, while the part behind the black hole would normally be hidden from sight behind it. However, light from the top surface of the “hidden” accretion disk is gravitationally lensed over the top of the black hole, forming the upper vertical loop. Meanwhile, light from the bottom surface of the accretion disk is gravitationally lensed below the black hole, creating the lower vertical loop.

Various sources & original papers are linked here. Kerr's 2023 paper, on the arXiv server is open access. Most of the others are not, and some are in German as well.

Kerr, 2023:

“Do Black Holes have Singularities?”

R.P. Kerr; arXiv, November 2023

Hawking, 1972:

“Black holes in general relativity”

Stephen Hawking; Communications in Mathematical Physics 25(2): 152-166, June 1972

Penrose, 1965:

“Gravitational Collapse and Space-Time Singularities”

Roger Penrose; Physical Review Letters 14(3): 57-59, January 1965

Kerr, 1963:

“Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics”

Roy P. Kerr; Physical Review Letters, vol. 11, Issue 5, pp. 237-238, September 1963

Schwarzschild, 1916a:

“Über das Gravitationsfeld eines Massenpunktes nach der Einsteinschen Theorie”

Karl Schwarzschild; Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 189-196, 1916

Schwarzschild, 1916b:

“Über das Gravitationsfeld einer Kugel aus inkompressibler Flüssigkeit nach der Einsteinschen Theorie”

Karl Schwarzschild; Sitzungsberichte der Königlich Preussischen Akademie der Wissenschaften zu Berlin, Phys.-Math. Klasse, 424-434, 1916

Einstein, 1916:

“Die Grundlage der allgemeinen Relativitätstheorie”

Albert Einstein; Annalen der Physik, vol. 354, Issue 7, pp.769-822, 1916

Einstein, 1915:

“Die Feldgleichungen der Gravitation”

Albert Einstein; Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 844-847, 1915

https://en.wikipedia.org/wiki/Spacetime (Spacetime - Wikipedia)

https://en.wikipedia.org/wiki/Black_hole (Black hole - Wikipedia)

https://en.wikipedia.org/wiki/General_relativity (General relativity - Wikipedia)

https://en.wikipedia.org/wiki/Schwarzschild_metric (Schwarzschild metric - Wikipedia)

https://en.wikipedia.org/wiki/Kerr_metric (Kerr metric - Wikipedia)

Originally posted by Tim Thompson, NASA JPL Senior Astronomer (Retired)

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