We can trace black holes all the way back to John Michell in 1783. He’s the man who devised the torsion balance used by Henry Cavendish to determine the mass of the Earth. Michell was something of an expert on gravity. He talked about “dark stars”, and said this: “if there should really exist in nature any bodies, whose density is not less than that of the sun, and whose diameters are more than 500 times the diameter of the sun, since their light could not arrive at us… we could have no information from sight”. I think that was pretty good for 1783. As was this: “if any other luminous bodies should happen to revolve about them we might still perhaps from the motions of these revolving bodies infer the existence of the central ones with some degree of probability”. That’s exactly what we’ve done to establish the existence of a supermassive black hole in the centre of our galaxy.
Check out Sagittarius A*. It’s part of Sagittarius A, which is a radio source in the middle of the Milky Way. Sagittarius A* is arguably the site of an accretion disk or a relativistic jet rather than the central black hole itself, but either way there’s something very small and very massive at the heart of our galaxy. We’re confident of this because of many years of work by many good men and women. Different people have joined and left the various groups over the years, but in 2002 Rainer Schödel, Thomas Ott, Reinhard Genzel, and twenty other authors reported on the orbital motion of star S2 over a ten year period. In 2008 Stefan Gillessan, Frank Eisenhauer, Sascha Trippe, Tal Alexander, Reinhard Genzel, Fabrice Martins, and Thomas Ott published a paper on the orbits of nearby stars over a sixteen year period. Another noteworthy paper is an update on monitoring stellar orbits in the galactic center. It’s dated November 2016, and is by twelve authors mainly from the Max Planck Institute but also from the Racah Institute and Berkeley. It uses a 25-year dataset derived from VLT and Keck observations. There’s more people involved, too many to mention. But also see the animations produced by Andrea Ghez and team at the UCLA Galactic Center Group using Keck datasets:
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.
The speed of light varies with gravitational potential
The story starts with Einstein saying the speed of light is not constant, and instead varies with gravitational potential. That’s the speed of light in vacuo, which I will shorten to the speed of light. It varies with altitude. Some people will tell you that Einstein said this in 1907 or 1911, then stopped saying it, but he didn’t. He was still saying it in 1913, in 1914, in 1915, in 1916, and in 1920. He never ever stopped saying it. He made it crystal clear that “a curvature of rays of light can only occur in a place where the speed of light is spatially variable”. You can find Irwin Shapiro saying much the same thing in 1964: “according to the general theory, the speed of a light wave depends on the strength of the gravitational potential along its path”. Also see what Don Koks the PhysicsFAQ editor said in 2014: “light travels faster near the ceiling than near the floor”. Hence light curves as per Huygen’s principle, like sonar waves curve downwards when the speed of sound decreases with depth. Matter is similarly affected because of the wave nature of matter. That’s how gravity works. Light doesn’t curve because spacetime is curved. That thing called “curved spacetime” is an abstract thing. It isn’t curved space, it’s a curved “metric”, associated with measurement. It’s effectively a curved plot. A curved plot of measurements of the speed of light made using optical clocks:
Image from Ethan Siegel’s blog starts with a bang
The tidal force at some location relates to the curvature of the plot at that location. This curvature is the second derivative of potential, and is effectively the spacetime curvature or Riemann curvature. The force of gravity at some location relates to the gradient of the plot at that location, the first derivative of potential. The force of gravity is greatest where the gradient is greatest, not where the spacetime curvature is greatest. If there is no gradient, light doesn’t curve and there is no gravity. The tilted light-cones in the Stanford singularities and black holes article are another way of depicting this. Alternatively you can emulate this gradient or tilt with a piece of stiff board. Lift one side up, and roll a marble across it. The path of the marble curves because the board is tilted, not because the board is curved. It’s similar for the room you’re in. The force of gravity is 9.8 m/s² at the floor and at the ceiling, so there’s no detectable tidal force, and so no detectable spacetime curvature. But your pencil still falls down. That’s detectable, as is a difference in NIST optical clock rates. The lower clock goes slower because light goes slower when it’s lower. The bottom line is that the speed of light varies in the room you’re in. If it didn’t, your pencil wouldn’t fall down.
But many physicists say it’s constant
See what David Wineland of NIST says: “if one clock in one lab is 30cm higher than the clock in the other lab, we can see the difference in the rates they run at”. An optical clock goes slower when it’s lower. This is hard scientific evidence for gravitational time dilation. It’s also hard scientific evidence that light goes slower when it’s lower. However despite what Einstein and others said, and despite the hard scientific evidence of optical clocks, many physicists think the speed of light is constant. As to why, I’m not sure. But the Wikipedia variable speed of light article talks about Peter Bergmann who was Einstein’s research assistant in Princeton between 1936 and 1941. He wrote the first textbook on general relativity in 1942. After Einstein died ”Bergmann wrote a new book in 1968 claiming that vector light velocity could change direction but not speed. This has become a prevailing opinion in science”. However that prevailing opinion is wrong, and I’m afraid to say that as a result of that, the nature of black holes is generally misunderstood.
Einstein didn’t believe black holes could form
Einstein wrote a paper on black holes in 1939. It was on a stationary system with spherical symmetry consisting of many gravitating masses. He said “g44 = (1 – μ/2r / 1 + μ/2r)² vanishes for r = μ/2. This means that a clock kept at this place would go at the rate zero. Further it is easy to show that both light rays and material particles 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”. That fits with the speed of light being spatially variable. Einstein also said this: “In this sense the sphere r = μ/2 constitutes a place where the field is singular”. He thought of the thing we now call the event horizon as the black hole singularity. That fits with the speed of light reducing to zero. So far so good. But sadly Einstein concluded that “the ‘Schwarzschild singularity’ does not appear for the reason that matter cannot be concentrated arbitrarily”. He said that this was “due to the fact that otherwise the constituting particles would reach the velocity of light”. He said this even though he knew that the material particle falls faster and faster because the speed of light gets slower and slower. If he had ridden his material particle like he rode the light beam he would surely have known that something had to give and predicted gamma ray bursters. Or if he’d recalled his own words about the energy of the gravitational field acting gravitatively he might have sidelined the material particles and focused on light and energy. Then he would surely have been talking frozen-star black holes. Perhaps with Robert Oppenheimer and Hartland Snyder who’d just written their paper on continued gravitational contraction. But this was 1939, a time of Nazis and war, a time when Einstein had somehow lost his confident intuition that had served him so well. It was not to be.
The baby and the bathwater
Twenty years later the “golden age” physicists threw out the baby with the bathwater. See the Wikipedia black hole article where you can read that David Finkelstein identified the Schwarzschild surface as an event horizon, “a perfect unidirectional membrane: causal influences can cross it in only one direction”. The Wikipedia article says “this did not strictly contradict Oppenheimer’s results”, but it did. It utterly contradicted it, and Einstein too. Finkelstein plus Martin Kruskal didn’t say Einstein missed the trick wherein the black hole formed from the inside out. They discarded the Schwarzschild singularity and the frozen star, and the very essence of Einstein’s general relativity. They discarded the speed of light is spatially variable. In doing this they threw out the very reason light curves and why matter falls down. They discarded the very reason the black hole is black. What they discarded, is why doesn’t the light get out?
Why doesn’t the light get out?
It’s one of the simplest questions in cosmology, and one of the most important. But the answer is usually wrong. As for the right answer, imagine you’re standing on a gedanken planet shining a laser beam straight up into space. The light goes straight up. It doesn’t curve, and it doesn’t fall back down. Now imagine it’s a denser more massive planet. The light still goes straight up. It still doesn’t curve, and it still doesn’t fall back down. Let’s make it a really massive planet. That light still goes straight up. It still doesn’t curve, and it still doesn’t fall back down:
Public domain image courtesy of NASA, with light beam added by me
But when we make our gedanken planet so massive that it’s a black hole, all of a sudden light can’t escape. Why? Why doesn’t the light get out? Some will tell you that the light curves back to the event horizon. When you challenge that by saying the light didn’t start curving on the ever-more massive planet, they’ll change tack and say it’s because spacetime is curved. Then when you challenge that by saying light curves because of the gradient rather than the spacetime curvature they’ll change tack again and tell you about the waterfall analogy.
The waterfall analogy
The waterfall analogy is where space is falling inward so the light beam doesn’t make any upward progress. It’s derived from Gullstrand-Painlevé coordinates, which Einstein rejected for good reason. The waterfall analogy may have been publicised on Horizon by Max Tegmark, but it is not in accord with the general theory of relativity. In no sense is space falling inwards in a gravitational field. We do not live in some Chicken-Little world where the sky is falling in. A gravitational field alters the motion of light through space, it doesn’t suck space down into some cosmic plughole. Because as Einstein said in his 1920 Leyden Address a gravitational field is a place where space is “neither homogeneous nor isotropic”. You can find modern authors saying more or less the same thing. See inhomogeneous vacuum: an alternative interpretation of curved spacetime dating from 2008. That’s where Xing-Hao Ye and Qiang Lin talk about the propagation of light in a medium with a graded refractive index. They are essentially correct. We don’t call it gravitational lensing for nothing. Einstein referred to refraction, as did Newton, see Opticks query 20. The contest between Einstein plus Newton versus Chicken Little is no contest at all.
Why the light doesn’t get out
So why doesn’t the light get out? PhysicsFAQ editor Don Koks tells it like it is: “light speeds up as it ascends from floor to ceiling, and it slows down as it descends from ceiling to floor; it’s not like a ball that slows on the way up and goes faster on the way down”. Somewhat counter intuitively, the ascending light beam speeds up, and the descending light beam slows down. Because light goes slower when it’s lower. That’s why optical clocks go slower when they’re lower. So if light goes slower when it’s lower, how much slower can it go? Have a google on infinite gravitational time dilation. What comes up time and time again? Black holes. Gravitational time dilation goes infinite at the black hole event horizon. An optical clock at the event horizon doesn’t tick at all, like Einstein said. And when you understand the nature of time, you know why. It isn’t because some abstract thing called time stops. It’s because light stops. Because the speed of light at the event horizon is zero. That’s why the light doesn’t get out of the black hole. Not because of some mystic curvature that makes vertical light beams bend back round. Not because the sky is falling in. But because at that location, the speed of light is zero. The light isn’t moving, so it doesn’t go up. It doesn’t get out because it is effectively “frozen”. That’s why a black hole is black. Because it’s a frozen star.
If you google on frozen star and Robert Oppenheimer you can find ample references to the frozen-star black hole. Such as 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”. However if you google on frozen star alone, what tends to come up is articles about some new hypothetical stars, or a TV program. Or you get redirected to black holes which feature a central point singularity. It’s like the original “frozen star” has been airbrushed away and replaced with something else. Something that contradicts Einstein but doesn’t say so. At least Andrew Hamilton says so on his JILA website. He says Einstein misunderstood how black holes work and thought the Schwarzschild geometry had a singularity at the event horizon. Hamilton isn’t alone in thinking that. On the Wikipedia Schwarzschild metric article you can read that “the singularity at r = rs is an illusion”, that it’s “an instance of what is called a coordinate singularity”, and that it “arises from a bad choice of coordinates or coordinate conditions”.
Flamm Paraboloid (exterior Schwarzschild solution) CCASA image by AllenMcC, see Wikipedia
I think most physicists would concur with that, and with the Wikipedia article on the propagation of light in non-inertial reference frames. This says “at the event horizon of a black hole the coordinate speed of light is zero”. There’s nothing wrong with that. However the article then says the proper speed is c, and “the local instantaneous proper speed of light is always c”. There’s a problem there of biblical proportions. You can see where it goes in Kevin Brown’s mathpages article the formation and growth of black holes. The article refers to the frozen star interpretation, saying this gives “a serviceable account of phenomena outside the event horizon”. It also says a clock runs increasingly slowly as it approaches the event horizon, and “the natural limit of this process is that the clock asymptotically approaches full stop (i.e., running at a rate of zero). It continues to exist for the rest of time, but it’s frozen”. That fits with Einstein’s thinking, and Oppenheimer’s. However the article favours a “geometrical interpretation” which it incorrectly attributes to Einstein, saying that’s where “all clocks run at the same rate, measuring out real distances along worldlines in curved spacetime”. That’s a surprise given what Einstein said about zero-rate clocks. It’s also a surprise given the hard scientific evidence: NIST can demonstrate two clocks running at different rates when one is a mere 2cm above the other. Hence it contradicts “Einstein and the evidence”. Even more surprising is this: “rather than slowing down as it approaches the event horizon, the clock is following a shorter and shorter path to the future time coordinates. In fact, the path gets shorter at such a rate that it actually reaches the future infinity of Schwarzschild coordinate time in finite proper time”. That’s saying the clock reaches the end of time. And get this: “the object goes infinitely far into the “future” (of coordinate time), and then infinitely far back to the “present”. The clock doesn’t just go to the end of time, it goes to the end of time and back again.
To the end of time and back again
You might be tempted to dismiss Kevin Brown’s mathspages article as some kind of crackpot outlier. Don’t. Because it’s in line with Gravitation by Charles Misner, Kip Thorne, and John Wheeler. Dating from 1973, Gravitation is known as MTW and is a mighty 1,279 pages long. It’s considered to be the bible of general relativity. You can perhaps find a pdf online. On page 848 you can see figure 32.1, which shows free-fall Schwarzschild coordinates on the left:
Image copyright W H Freeman and company, publishers of Gravitation
The horizontal axis denotes distance, and the vertical axis denotes time. The vertical dashed line is at r = 2M and denotes the event horizon. The curve on the right denotes the path of an infalling body outside the event horizon. It gets closer and closer to the event horizon as the time t increases. Note though that the time axis is truncated, obscuring the way the infalling body somehow manages to cross the event horizon at time t = infinity. Then it comes back down the chart, tracing out the curve to the left of the vertical dashed line. It ends up in the central point singularity at r = 0 at proper time tau τ = 35.1 M. Yes, according to MTW an infalling body goes to the end of time and back. But that’s not all. If you look horizontally across the Schwarzschild chart at time t = 45, you will notice that the infalling body is at two locations at the same time t. It’s outside the event horizon with a proper time τ = 33.3 M, and at the same time it’s inside the event horizon with a proper time of circa τ = 34.3 M. That’s why you read about the elephant and the event horizon, where the elephant is in two places at once.
Eddington-Finkelstein and Kruskal-Szekeres coordinates
The issues go on. MTW also refers to Eddington-Finkelstein coordinates. See box 31.2 on page 828, which says Eddington and Finkelstein used free-falling photons as the foundation of their coordinate system. However the Wikipedia article 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”. The article also says “one advantage of this coordinate system is that it shows that the apparent singularity at the Schwarzschild radius is only a coordinate singularity and is not a true physical singularity”. Eddington-Finkelstein coordinates employ a “tortoise coordinate” which is like Zeno’s paradox in reverse. The time unit used in the coordinate system gets bigger and bigger as you approach the black hole. Hence it allegedly cancels out the gravitational time dilation, increasing to infinity at the event horizon. Kruskal-Szekeres coordinates are similar. Hence the chart on the right of MTW page 848 has done away with the troublesome trip to the end of time and back. It has done a hop skip and a jump over the end of time and swept the problem under the carpet. The conclusion is then that a star collapses to a singular point of infinite density in finite time:
Fair use excerpt from Misner Thorne Wheeler’s Gravitation
There’s just one little problem with that: light curves and an infalling body falls down because the speed of light is spatially variable. Your pencil falls down because the speed of light near the floor is lower than the speed of light near the ceiling, and there’s a gradient in between. But at the event horizon the speed of light is zero and it can’t go lower than that. So there is no more gravity. If we could somehow place our gedanken observer at the event horizon, he wouldn’t fall down. Nor would anything else. There is no gradient in gravitational potential, so there’s no further collapse, and no tidal forces. But the speed of light is zero so there’s no observing either. This is a crucial point.
Hawking did not understand gravity
Unfortunately many physicists are unaware of it. For example, one of Stephen Hawking’s “seminal” papers was singularities and the geometry of spacetime dating from 1966. On page 26 he referred to the Schwarzschild metric and the “apparent” singularity at r=2m. He said it was “simply due to a bad choice of coordinates”. On page 76 Hawking talked of “such a strong gravitational field that even the ‘outgoing’ light rays from it are dragged back”. It’s clear Hawking had never read Einstein’s fundamental ideas and methods of the theory of relativity. That’s where Einstein explained why light curves. It’s clear that Hawking did not understand that a gravitational field is a place where “the speed of light is spatially variable”. He didn’t know that in a gravitational field, “light speeds up as it ascends from floor to ceiling”. He didn’t know that in a strong gravitational field, outgoing light rays aren’t dragged back. They speed up even more. Hawking didn’t understand the first thing about gravity, so he didn’t understand the first thing about black holes either. If you think that’s bad, it gets worse. A lot worse.
He sees nothing unusual
If we could lower a gedanken observer into a black hole such that we could watch him approach the event horizon through some gedanken camera, we’d see his optical clock going slower and slower until it stopped. We’d see him stop too. Kruskal-Szekeres coordinates try to cancel out the stopped clock with the stopped observer, who is somehow supposed to see the clock ticking normally “in his frame”. It is said that he sees nothing unusual. But Kruskal-Szekeres coordinates contain a schoolboy error, because at the event horizon the speed of light is zero. So the gedanken observer can’t see. His light is stopped, and because of the wave nature of matter, so is he. Electrochemical signals in his nerves and brain do not move. So instead of seeing nothing unusual, he sees nothing. Ever. Which means Kruskal-Szekeres coordinates are like some dead parrot sketch, where the shopkeeper swears that a dead customer sees the dead parrot squawking normally. Which means the singularity at r = rs is not just a coordinate singularity. You can’t get past it by inventing a fantasy coordinate system where seconds last forever. Gravitational time dilation goes infinite, so there are no more events, so there isn’t any time, so proper time isn’t proper at all. Yes, in general relativity we talk of coordinate independence and say all coordinate systems are equal. But when light has stopped we can’t measure seconds and metres, so that’s where all coordinate systems end. They are all equal, but there is no never-never land beyond the end of time.
What a black hole is
We now start to get a clearer picture of what a black hole is. The central point singularity has gone, and in its place we have a place where you can’t go. As such it’s akin to the gravastar, featuring a “gravitational vacuum”, a void in the fabric of space and time. This fabric is like some gin-clear ghostly elastic, which is why the stress-energy momentum tensor features a shear stress term. The black hole is a hole in this fabric, so the frozen-star black hole is more of a hole than the point-singularity black hole. In simulated images it even looks like a hole. Think of a blue-grey party balloon, somewhat translucent, with a starscape painted on it. Now add a bullethole whilst keeping the balloon intact. What you have is Alain Riazuelo’s black hole depiction:
CCASA simulated image of a stellar black hole by Alain Riazuelo see Wikipedia
But whilst it looks like a hole in space, the frozen-star black hole is like solid space too. Rock solid, because it’s a place where c is zero, so there can be no motion. And if there can be no motion there can be no angular momentum. There can be no spin. We can find articles that say a black hole spins at nearly the speed of light. But the speed of light at the event horizon is zero, so that creates a problem for the Kerr black hole. Since however Kerr black holes are associated with negative space and wormholes and other universes and time travel, I don’t think that’s a problem myself. In similar vein a charged particle has a Poynting-vector energy flow going around and around at the speed of light. But at the event horizon this speed is zero, so that creates a problem for the Reissner-Nordstrom black hole. Since however Reissner-Nordstrom black holes are associated with naked singularities and one-way wormholes that connect to white holes in another space and time, I don’t think that’s a problem myself. Particularly because of the mass inflation and the infinite blueshift. Conservation of energy rules this out. When you drop a 511kev photon into a black hole, you would say the black hole mass increases by 511keV/c². So the photon didn’t gain any E=hf energy or increase in frequency as it descended. You and your clock go slower when you’re lower, so you measure the selfsame frequency to be higher, that’s all. It’s similar for gravitational redshift. Einstein said this in 1917: “an atom absorbs or emits light at a frequency which is dependent on the potential of the gravitational field in which it is situated”. The frequency doesn’t reduce as the photon ascends, it was already lower when the photon was emitted. The ascending photon does not reduce in frequency. It does not lose energy. There is no magical mechanism by which a photon in space loses energy. There is no magic, there is no time travel, and despite what the Penrose diagrams say, there is no wormhole, or parallel universe, or parallel antiverse:
Penrose diagram by Andrew Hamilton, cropped by me
Some would say that the frozen-star black hole cannot be correct because nothing passes through the event horizon, and therefore black holes cannot grow. After all, that’s more or less what Einstein said. But think in terms of a hailstone. A hailstone doesn’t grow because water molecules pass through its surface.
Black holes grow like hailstones
Imagine you’re a water molecule. You alight upon the surface of the hailstone. You can’t pass through this surface. But you are presently surrounded by other water molecules, and eventually buried by them. So whilst you can’t pass through the surface, the surface can pass through you. So the frozen-star black hole grows like a hailstone. The event horizon expands outwards from the centre of a collapsing star. Which consists of matter, falling down because of the wave nature of matter and because light curves where the speed of light varies. You can think of an electron as light going around and around, then you can simplify it to light going round a square path, then you can imagine it’s in a gravitational field. The vertical parts of the path stay vertical, but the horizontal parts bend down a little, so the electron falls down. The reducing speed of light is transformed into the downward motion of the electron. It’s all rather simple and straightforward when you know how gravity works. There are no messenger particle. There is no magical mysterious action at a distance. There is no mystery to it. The mystery is how a black hole falls down, and how LIGO could have detected a black hole merger. There’s another mystery too. Something that grew out of gamma ray bursts, which we’ll come back to another day. Something that used to be called the Hawking effect. It’s nowadays called Hawking radiation.