# Black Holes Revisited

To mark the anniversary of the death of Karl Schwarzschild, on 11May 1916

Adapted from my book Companion to the Cosmos

A concentration of matter which has a gravitational field strong enough to curve spacetime completely round upon itself so that nothing can escape, not even light, is said to be a black hole.  This can happen either if a relatively modest amount of matter is squeezed to very high densities (for example, if the Earth were to be squeezed down to about the size of a pea), or if there is a very large concentration of relatively low mass material (for example, a few million times the mass of our Sun in a sphere as big across as our Solar System, equivalent to about the same density as water).
The first person to suggest that there might exist “dark stars” whose gravitation was so strong that light could not escape from them was John Michell, a Fellow of the Royal Society whose ideas were presented to the Society in 1783.  Michell based his calculations on Isaac Newton’s theory of gravity, the best available at the time, and on the corpuscular theory of light, which envisaged light as a stream of tiny particles, like miniature cannon balls (now called photons).  Michell assumed that these particles of light would be affected by gravity in the same way as any other objects.  Ole Ro/mer had accurately measured the speed of light a hundred years earlier, and Michell was able to calculate how large an object with the density of the Sun would have to be in order to have an escape velocity greater than the speed of light.
If such objects existed, light could not escape from them, and they would be dark.  The escape velocity from the surface of the Sun is only 0.2 per cent of the speed of light, but if you imagine successive larger objects with the same density as the Sun the escape velocity increases rapidly.  Michell pointed out that such an object with a diameter 500 times the diameter of the Sun (roughly as big across as the Solar System) would have an escape velocity greater than the speed of light.
The same conclusion was reached independently by Pierre Laplace, and published by him in 1796.  In a particularly prescient remark, Michell pointed out that although such objects would be invisible, “if any other luminiferous 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”.  In other words, he suggested that black holes would most easily be found if they occurred in binary systems.  But the notion of dark stars was forgotten in the 19th century and only revived in the context of Albert Einstein’s general theory of relativity, when astronomers realised that there was another way to make black holes.
One of the first people to analyse the implications of Einstein’s theory was Karl Schwarzschild, an astronomer serving on the eastern front in World War I.  The general theory of relativity explains the force of gravity as a result of the way spacetime is curved in the vicinity of matter.  Schwarzschild calculated the exact mathematical description of the geometry of spacetime around a spherical mass, and sent his calculations to Einstein, who presented them to the Prussian Academy of Sciences early in 1916.  The calculations showed that for any mass there is a critical radius, now called the Schwarzschild radius, which corresponds to such an extreme distortion of spacetime that if the mass were to be squeezed inside the critical radius space would close around the object and pinch it off from the rest of the Universe.  It would, in effect, become a selfcontained universe in its own right, from which nothing (not even light) could escape.
For the Sun, the Schwarzschild radius is 2.9 km; for the Earth, it is 0.88 cm.  This does not mean that there is what we now call a black hole (the term was first used in this sense only in 1967, by John Wheeler) of the appropriate size at the centre of the Sun or of the Earth.  There is nothing unusual about spacetime at this distance from the centre of the object.  What Schwarzschild’s calculations showed was that if the Sun could be squeezed into a ball less than 2.9 km across, or if the Earth could be squeezed into a ball only 0.88 cm across, they would be permanently and cut off from the outside Universe in a black hole.  Matter can still fall in to such a black hole, but nothing can escape.
For several decades this was seen simply as a mathematical curiosity, because nobody thought that it would be possible for real, physical objects to collapse to the states of extreme density that would be required to make black holes.  Even white dwarf stars, which began to be understood in the 1920s, contain about the same mass as our Sun in a sphere about as big as the Earth, much more than 3 km across.  And for a time nobody realised that you can also make a black hole, essentially the same as the kind of dark star envisaged by Michell and Laplace, if you have a very large amount of matter at quite ordinary densities.  The Schwarzschild radius corresponding to any mass M is given by the formula 2GM/c^2, where G is the constant of gravity and c is the speed of light.
In the 1930s, Subrahmanyan Chandrasekhar showed that even a white dwarf could be stable only if it had a mass less than 1.4 times the mass of the Sun, and that any heavier dead star would collapse further.  A few researchers considered the possibility that this could lead to th formation of neutron stars, typically with a radius only one sevenhundredth of that of a white dwarf, just a few kilometers across.  But the idea was not widely accepted until the discovery of pulsars in the mid1960s showed that neutron stars really did exist.
This led to a revival of interest in the theory of black holes, because neutron stars sit on the edge of becoming black holes.  Although it is hard to imagine squeezing the Sun down to a radius of 2.9 km, neutron stars with about the same mass as the Sun and radii less than about 10 km were now known to exist, and it would be a relatively small step from there to a black hole.
Theoretical studies show that a black hole has just three properties that define it  its mass, its electric charge, and its rotation (angular momentum).  An uncharged, nonrotating black hole is described by the Schwarzschild solution to Einstein’s equations, a charged, nonrotating black hole is described by the ReissnerNordstrom solution, an uncharged but rotating black hole is described by the Kerr solution, and a rotating, charged black hole is described by the KerrNewman solution.  A black hole has no other properties, summed up by the phrase “a black hole has no hair”.  Real black holes are likely to be rotating and uncharged, so that the Kerr solution is the one of most interest.
Both black holes and neutron stars are now thought to be produced in the death throes of massive stars that explode as supernovas.  The calculations showed that any compact supernova remnant with a mass less than about three times the mass of the Sun (the OppenheimerVolkoff limit) could form a stable neutron star, but any compact remnant with more than this mass would collapse into a black hole, crushing its contents into a singularity at the centre of the hole, a mirror image of the Big Bang singularity in which the Universe was born.  If such an object happened to be in orbit around an ordinary star, it would strip matter from its companion to form an accretion disk of hot material funneling in to the black hole. The temperature in the accretion disk might rise so high that it would radiate Xrays, making the black hole detectable.
In the early 1970s, echoing Michell’s prediction, just such an object was found in a binary system.  An Xray source known as Cygnus X1 was identified with a star known as HDE 226868.  The orbital dynamics of the system showed that the source of the Xrays, coming from an object smaller than the Earth in orbit around the visible star, had a mass greater than the OppenheimerVolkoff limit.  It could only be a black hole.  Since then, a handful of other black holes have been identified in the same way, and in 1994 a system known as V404 Cygni became the best black hole “candidate” to date when it was shown to be made up of a star with about 70 per cent as much mass as our Sun in orbit around an Xray source with about 12 times the Sun’s mass.  But such confirmed identifications may be much less than the tip of the proverbial iceberg.
Such “stellar mass” black holes can only be detected if they are in binary systems, as Michell realised.  An isolated black hole lives up to its name  it is black, and undetectable (but see gravitational lens).  But very many stars should, according to astrophysical theory, end their lives as neutron stars or black holes.  Observers actually detect about the same number of good black hole candidates in binary systems as they do binary pulsars, and this suggests that the number of isolated stellar mass black holes must be the same as the number of isolated pulsars.  This supposition is backed up by theoretical calculations.     There are about five hundred active pulsars known in our Galaxy today.  But theory tells us that a pulsar is only active as a radio source for a short time, before it fades into undetectable silence.  So there should be correspondingly more “dead” pulsars (quiet neutron stars) around.  Our Galaxy contains a hundred billion bright stars, and has been around for thousands of million of years.  The best estimate is that there are around four hundred million dead pulsars in our Galaxy today, and even a conservative estimate would place the number of stellar mass black holes at a quarter of that figure  one hundred million.  If so, and the black holes are scattered at random across the Galaxy, the nearest one is probably just 15 light years away.  And since there is nothing unusual about our Galaxy, every other galaxy in the Universe must contain a similar profusion of black holes.
They may also contain something much more like the kind of “dark star” originally envisaged by Michell and Laplace.  These are now known as “supermassive black holes”, and are thought to lie at the hearts of active galaxies and quasars, providing the gravitational powerhouses which explain the source of energy in these objects.  A black hole as big across as our Solar System, containing a few million solar masses of material, could swallow matter from its surroundings at a rate of one or two stars a year.  In the process, a large fraction of the star’s mass would be converted into energy, in line with Einstein’s equation E = mc^2.  Quiescent supermassive black holes may lie at the centres of all galaxies, including our own.     In 1994, observers using the Hubble Space Telescope discovered a disc of hot material, about 150 thousand parsecs across, orbiting at speeds of about two million kilometers per hour (about 3 x 10^7 cm/sec, almost 1 per cent of the speed of light) around the central region of the galaxy M87, at a distance of about 15 million parsecs from our Galaxy.  A jet of hot gas, more than a kiloparsec long, is being shot out from the central “engine” in M87.  The orbital speeds in the accretion disk at the heart of M87 is conclusive proof that it is held in the gravitational grip of a supermassive black hole, with a mass that may be as great as three billion times the mass of our Sun, and the jet is explained as an outpouring of energy from one of the polar regions of the accretion system.
Also in 1994, astronomers from the University of Oxford and from Keele University identified a stellar mass black hole in a binary system known as V404 Cygni.  The orbital parameters of the system enabled them to “weigh” the black hole accurately, showing that it has about 12 times as much mass as our Sun and is orbited by an ordinary star with about 70 per cent of the Sun’s mass.  This is the most precise measurement so far of the mass of a “dark star”, and is therefore the best individual proof that black holes exist.
A more speculative suggestion is that tiny black holes, known as mini black holes or primordial black holes, may have been produced in profusion in the Big Bang and could provide a significant fraction of the mass of the Universe.  Such mini black holes would typically be about the size of an atom and each have a mass of perhaps a hundred million tonnes (10^11 kilograms).  There is no evidence that such objects really exist, but it would be very hard to prove that they do not exist.

.