I read Sabine Hossenfelder’s latest blog post yesterday. The title was A brief history of black holes. I left a couple of comments. One was a reply to Louis Marmet, and referred to Oppenheimer’s 1939 frozen star black hole. I said I think the black hole grows like a hailstone, from the inside out. The other was addressed to Hossenfelder, and referred to Einstein talking about the variable speed of light. I said that IMHO this had to mean Penrose/Hawking singularity theorems were wrong. I found that neither comment appeared. Hence I thought I’d spend a little time going through the blog post giving my thoughts. I’m black, with blue hyperlinks. Hossenfelder is green:
The possibility that gravity can become so strong that it traps light appears already in Newtonian gravity, but black holes were not really discussed by scientists until it turned out that they are a consequence of Einstein’s theory of general relativity.
Like physicsFAQ editor Don Koks says, “light speeds up as it ascends from floor to ceiling”. In a strong gravitational field, it speeds up all the more, so strong gravity doesn’t trap light. That’s not to say black holes don’t exist. But they don’t exist because gravity is strong. They exist because they’re a place where the “coordinate” speed of light is zero. Note though that Einstein talked of the speed of light, not the coordinate speed of light. Light doesn’t behave like a ball that slows on the way up and goes faster on the way down. By the by, the local speed of light is not always c. A stopped observer doesn’t see light moving at c. He sees nothing. Ever. Zero divided by zero is not one.
General Relativity is a set of equations for the curvature of space and time, called Einstein’s field equations. And black holes are one of the possible solutions to Einstein’s equations. This was first realized by Karl Schwarzschild in 1916. For this reason, black holes are also sometimes called the “Schwarzschild solution”.
This is also wrong. General Relativity is a theory of gravity, not a set of equations for the curvature of spacetime. Curved spacetime is a curvature of the “metric”. Gravity is real, but the metric is an abstract thing, associated with measurement. Think of it as a curvature in your plot of your measurements of space and time. Imagine you could place a 15 x 15 array of optical clocks throughout a horizontal slice of space around the Earth. Then you plot all the clock rates, such that the lower slower clock rates generate data points lower down in a 3D image, and the higher faster clock rates generate data points higher up in the 3D image. When you join the dots, your plot looks like this:
CCASA image by Johnstone, see Wikipedia
That’s your typical Riemann curvature tensor image, the “rubber-sheet” depiction of curved spacetime. But it’s derived from optical clock rates, so what it’s really plotting is the variable speed of light. It isn’t plotting curved space and curved time. That’s why Einstein said a gravitational field is a place where space is “neither homogeneous nor isotropic”. Not a place where space is curved.
Schwarzschild of course was not actually looking for black holes. He was just trying to understand what Einstein’s theory would say about the curvature of space-time outside an object that is to good precision spherically symmetric, like, say, our sun or planet earth. Now, outside these objects, there is approximately no matter, which is good, because in this case the equations become particularly simple and Schwarzschild was able to solve them.
Hooray. The curvature of spacetime. Not the curvature of space and the curvature of time. A gravitational field is a place where space is “neither homogeneous nor isotropic”. Not a place where space is curved.
What happens in Schwarzschild’s solution is the following. As I said, this solution only describes the outside of some distribution of matter. But you can ask then, what happens on the surface of that distribution of matter if you compress the matter more and more, that is, you keep the mass fixed but shrink the radius. Well, it turns out that there is a certain radius, at which light can no longer escape from the surface of the object, and also not from any location inside this surface. This dividing surface is what we now call the black hole horizon. It’s a sphere whose radius is now called the Schwarzschild radius.
There’s nothing wrong with that. Apart from the explanation for why the light can’t escape. It’s because the speed of light at the event horizon is zero. So the vertical light beam doesn’t ascend from the event horizon. And remember this: the speed of light can’t go lower than zero. That’s important. Light can’t go slower than stopped.
Where the black hole horizon is, depends on the mass of the object, so every mass has its own Schwarzschild radius, and if you could compress the mass to below that radius, it would keep collapsing to a point and you’d make a black hole.
Who says it would keep collapsing to a point? It wasn’t Einstein. It wasn’t Oppenheimer either. It was Penrose and Hawking. Two guys who were doing their own thing, and had clearly never read what Einstein said.
But for most stellar objects, their actual radius is much larger than the Schwarzschild radius, so they do not have a horizon, because inside of the matter one has to use a different solution to Einstein’s equations. The Schwarzschild radius of the sun, for example, is a few miles*, whereas the actual radius of the sun is some hundred-thousand miles. The Schwarzschild radius of planet Earth is merely a few millimeters.
No problem. The Schwarzschild radius of the Sun is circa 1.9 miles, and the Schwarzschild radius of the Earth is circa 9 millimetres.
Now, it turns out that in Schwarzschild’s original solution, there is a quantity that goes to infinity as you approach the horizon. For this reason, physicists originally thought that the Schwarzschild solution makes no physical sense. However, it turns out that there is nothing physically wrong with that. If you look at any quantity that you can actually measure as you approach a black hole, none of them becomes infinitely large. In particular, the curvature just goes with the inverse of the square of the mass. I explained this in an earlier video. And so, physicists concluded, this infinity at the black hole horizon is a mathematical artifact and, indeed, it can be easily removed.
It can only be “easily removed” by using tortoise coordinates. That’s seconds of infinite length, which is total crap. You cannot get rid of infinite gravitational time dilation by introducing smoke-and-mirror seconds of infinite length. Not unless you’re a mathematical quack peddling total garbage.
With that clarified, physicists accepted that there is nothing mathematically wrong with black holes, but then they argued that black holes would not occur in nature because there is no way to make them. The idea was that, since the Schwarzschild solution is perfectly spherically symmetric, the conditions that are necessary to make a black hole would just never happen.
That’s what Einstein said in his 1939 paper on a stationary system with spherical symmetry consisting of many gravitating masses. For some reason he missed the significance of Oppenheimer and Snyder’s 1939 frozen star paper on continued gravitational contraction. Maybe it was because war was looming, and he had other things on his mind. I don’t know. But I do know he should have predicted gamma ray bursters. Because falling bodies don’t slow down. And because those falling bodies are falling because the speed of light is reducing. He missed the trick. Maybe that was his greatest blunder. The irony is that the detection of gamma ray bursters reawakened interest in general relativity in the 1960s. In fact, that’s why Hawking proposed his Hawking radiation. See his 1974 Nature paper on black hole explosions?
But this too turned out to be wrong. Indeed, it was proved by Stephen Hawking and Roger Penrose in the 1960s that the very opposite is the case. Black holes are what you generally get in Einstein’s theory if you have a sufficient amount of matter that just collapses because it cannot build up sufficient pressure. And so, if a star runs out of nuclear fuel and has no new way to create pressure, a black hole will be the outcome. In contrast to what physicists thought previously, black holes are hard to avoid, not hard to make.
This is incorrect in that Penrose and Hawking proved nothing, because they did their own thing and ignored Einstein. But it is correct in that black holes are what you get if a big star runs out of fuel. The issue of course is this: is there a point singularity in the middle of a black hole? The answer has to be no, because light can’t go slower than stopped. At the event horizon there is no more gradient in the speed of light, so no gravitational field. So now you know why all that stuff about an elephant in two places at once is absolute nonsense:
Image from the New Scientist article the elephant and the event horizon
The bottom line is this: if there’s a paradox, there’s something wrong somewhere. Probably something seriously wrong, right at the beginning.
So this was the situation in the 1970s. Black holes had turned from mathematically wrong, to mathematically correct* but non-physical, to a real possibility. But there was at the time no way to actually observe a black hole. That’s because back then the main mode of astrophysical observation was using light. And black holes are defined by the very property that they do not emit light.
There’s some rewriting of history here. See the 1971 Physics Today article introducing the black hole by Remo Ruffini and John Wheeler who said “in this sense the system is a frozen star”. This sort of thing is par for the course in contemporary physics. When you’ve read the old papers, you realise how bad it is.
However, there are other ways of observing black holes. Most importantly, black holes influence the motion of stars in their vicinity, and the other stars are observable. From this one can infer the mass of the object that the stars orbit around and one can put a limit on the radius.
Animation by Andrea Ghez and research team at UCLA
There’s something there with a mass that’s circa 4.28 million times the mass of the Sun. But it’s at most thirty times bigger than the Sun in terms of spatial extent. There’s only one thing it can be, and that’s a black hole. Hence we’re confident that black holes exist. As to their exact nature, that’s another story.
Black holes also swallow material in their vicinity, and from the way that they swallow it, one can tell that the object has no hard surface.
You can’t tell what’s inside the black hole from observations of infalling bodies. However you can surely say something about gamma ray bursts. See the 2013 AMPS paper an apologia for firewalls. Tucked away in the conclusion is footnote 31, containing a reference 87 to Friedwardt Winterberg’s 2001 paper gamma ray bursters and Lorentzian relativity. Winterberg talked about the direct conversion of an entire stellar rest mass into gamma ray energy: “if the balance of forces holding together elementary particles is destroyed near the event horizon, all matter would be converted into zero rest mass particles which could explain the large energy release of gamma ray bursters”. I’m sure Winterberg is essentially correct. Unfortunately he gets no publicity. He gets “studiously ignored” instead whilst people like Hossenfelder pretend to be the expert and suck up to Penrose.
The first convincing observations that our own galaxy contains a black hole came in the late 1990s. About ten years later, there were so many observations that could only be explained by the existence of black holes that today basically no one who understands the science doubts black holes exist.
Again, no problem with that, but see Peter Erwin’s comment about Cygnus X-1. The issue is the nature of black holes. I say they’re frozen stars, others say they’re point singularities. I say point singularities contradict Einstein’s general relativity.
What makes this story interesting to me is how essential it was that Penrose and Hawking understood the mathematics of Einstein’s theory and could formally prove that black holes should exist. It was only because of this that black holes were taken seriously at all. Without that, maybe we’d never have looked for them to begin with. A friend of mine thinks that Penrose deserves a Nobel Prize for his contribution to the discovery of black holes. And I think that’s right.
I don’t. I think Penrose did his own thing and appealed to Einstein’s authority whilst flatly contradicting the guy. That was his MO. See MTW, which refers to Eddington-Finkelstein coordinates. Box 31.2 on page 828 says Eddington and Finkelstein used free-falling photons as the foundation of their coordinate system. Only those photons aren’t falling faster and faster, they descend slower and slower. Also see the Wikipedia article, which 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 was making it up as he went along. Yea verily. If that isn’t enough, have you ever sat down and taken a long hard look at Penrose diagrams which plot the route to the parallel antiverse?
Penrose diagram from the CERN courier article Physics in the multiverse, image credit Andrew Hamilton
FFS. The parallel antiverse. The word that springs to is charlatan. As for Hawking, see his singularities and the geometry of spacetime dating from 1966. On page 76 he talked of “such a strong gravitational field that even the ‘outgoing’ light rays from it are dragged back”. It’s clear Hawking didn’t understand the first thing about gravity, and had never read what Einstein said. He was winging it, and getting an easy ride on account of the wheelchair.
* Unfortunately, a mistake in the spoken text.
No problem. But what is a problem, is so-called physicists like Hossenfelder peddling horseshit on the internet and censoring those who refer to bona-fide papers and point out the issues. This sort of thing is endemic. There isn’t just something rotten in the state of QED. There’s something rotten in the state of physics. Safe spaces and no-platforming is how these guys have been operating for fifty years. It truly is the trouble with physics.
PS: If all this is new to you, you might want to take a look at this: