Attempting to avoid the absurd
Black holes were first proposed in 1783 by the English rector and scientist John Michell and independently in 1798 by the French mathematician Pierre-Simon Laplace. Michell and Laplace reasoned that, just as a ball thrown upwards is slowed down and pulled back to the ground by the Earth’s gravity, it is conceivable that there exist objects that exert such a strong gravitational pull they could trap light.
An object flung upwards from the surface of the Earth must have a speed in excess of 11 kilometres per second to escape into deep space. This is known as Earth’s escape velocity. The gravitational pull at the Sun’s surface is much stronger, and the escape velocity is correspondingly higher at 620 kilometres per second. At the surface of a neutron star, the escape velocity can approach an appreciable fraction of the speed of light.‡ Laplace calculated that a body with a density comparable to the Earth but with a diameter 250 times larger than the Sun would have a gravitational pull so great that the escape velocity would exceed the speed of light, and therefore ‘the largest bodies in the Universe may thus be invisible by reason of their magnitude’.3 This was a fascinating idea and ahead of its time. Imagine a spherical shell in space touching the surface of one of Laplace’s giant dark stars. The escape velocity from the shell would be the speed of light. Now make the star a little denser. The stellar surface would shrink inwards, but the imaginary shell would remain in place, marking out a boundary in space. If you hovered on the shell, now above the surface of the star, and shone a torch outwards, the light would go nowhere. It would remain forever frozen, unable to escape. This boundary is the event horizon. Inside the shell, the torchlight would be turned around and pulled back onto the star. Only outside of the shell could light escape.
Michell and Laplace imagined these dark stars as huge objects, perhaps because they could not conceive of the alternative. But an object doesn’t have to be big to have a strong gravitational pull at its surface. It can also be very small and very dense; a neutron star, for example. For an object of any mass, one can use Isaac Newton’s laws to calculate the radius of the region of no escape that would form around it if it were compressed sufficiently:
where G is Newton’s gravitational constant, which encodes the strength of gravity, and c is the speed of light. If we crush anything with mass M into a ball smaller than this radius, we will have created a dark star. Putting the mass of the Sun into this equation, we find that the radius is approximately 3 kilometres. For the Earth, it’s just under 1 centimetre. It is difficult to imagine the Earth being crushed to the size of a pebble, which is probably why Michell and Laplace didn’t consider the possibility. Fantastical as they are, however, there would seem to be nothing particularly troublesome or absurd about dark stars, should they exist. They would trap light but, as Laplace pointed out, that would just mean that we wouldn’t be able to see them.
This simple Newtonian argument gives us a feel for the idea of a black hole – gravity can get so strong that light cannot escape – but Newton’s law of gravitation is not applicable when gravity is strong and Einstein’s theory must be used. General relativity also allows for objects whose gravitational pull is so great that light cannot escape, but the consequences are very different and most definitely troublesome and absurd. As in the Newtonian case, if any object is compressed below a certain critical radius, it will trap light. In general relativity this radius is known as the Schwarzschild radius, because it was first calculated in 1915, very shortly after general relativity was published, by the German physicist Karl Schwarzschild. Coincidentally, the expression for the Schwarzschild radius in general relativity is precisely the same as the Newtonian result above. The Schwarzschild radius is the radius of the event horizon of a black hole.
We will learn more about the Schwarzschild radius in Chapter 4, when we have the machinery of general relativity at our disposal, but we can catch a glimpse of some of the absurdities to come. We will learn that black holes affect the flow of time in their vicinity. As an astronaut falls towards a black hole, their time will tick more slowly as measured on clocks far away in space. That’s interesting, but not absurd. The absurd-sounding result is this: according to the far away clocks, time grinds to a halt on the event horizon. As viewed from the outside, nothing is ever seen to fall into a black hole, which means an astronaut falling towards a black hole will remain frozen on the horizon for all eternity. This also applies to the surface of a star collapsing inwards through the horizon to form the black hole. At first sight, it seems the theory of general relativity predicts a nonsense. How can a star collapse through the event horizon to form a black hole if its surface is never seen to cross the horizon? Observations like this troubled Einstein and the early pioneers, and this is only one in a blizzard of apparent paradoxes.
For Einstein, and the majority of physicists until the 1960s, such worries led to the conclusion that Nature would find a way out, and research into black holes was primarily concerned with demonstrating that they could not exist. Perhaps it is not possible to compress a star without limit and thereby generate an event horizon. This doesn’t seem unreasonable, given that a sugar-cube-sized lump of neutron star material would weigh at least 100 million tonnes. Perhaps we don’t fully understand how matter behaves at such extreme densities and pressures.
Stars are large clumps of matter fighting gravitational collapse, and when they run out of nuclear fuel their fate depends on their mass. In 1926, Eddington’s Cambridge colleague R. H. Fowler published an article ‘On Dense Matter’ in which he showed that the newly-discovered quantum theory provided a way for an old collapsing star to avoid forming an event horizon due to an effect known as ‘electron degeneracy pressure’.4 This was the first glimpse of the ‘quantum jiggling’ we referred to earlier in the context of neutron stars. His conclusion appeared to be an unavoidable consequence of two of the cornerstones of quantum theory: Wolfgang Pauli’s Exclusion Principle and Werner Heisenberg’s Uncertainty Principle.
The Exclusion Principle states that particles like electrons cannot occupy the same region of space. If lots of electrons are squashed together by gravitational collapse, they will separate themselves into their own individual tiny volumes inside the star in order to stay away from each other. Heisenberg’s Uncertainty Principle now comes into play. It states that as a particle is confined to a smaller volume, its momentum becomes larger. In other words, if you confine an electron it will jiggle around, and the more you try to confine it, the more it will jiggle. This creates a pressure in much the same way that the heat from nuclear fusion reactions earlier in the star’s life causes its atoms to jiggle and halt the collapse. Unlike the pressure from fusion reactions, however, electron degeneracy pressure requires no energy release to power it. It seemed a star could resist the inward pull of gravity indefinitely.
Astronomers knew of such a star, known as a white dwarf. Sirius B is a faint companion of Sirius, the brightest star in the heavens. Sirius B was known to have a mass close to that of our Sun, but a radius comparable to the Earth. Its density, using the measurements of the time, was estimated to be around 100 kg/cm3, which, as Fowler notes ‘has already given rise to most interesting theoretical considerations’. In his book The Internal Constitution of Stars, Eddington wrote, ‘I think it is generally considered proper to add the conclusion “which is absurd”.’ Modern measurements put the density over ten times higher. Absurd as this exotic planet-sized star appeared, however, Fowler had discovered a mechanism that explained how it could resist gravity. This seems to have offered a great deal of relief to the physicists of the day because it stopped the unthinkable happening. Thanks to Fowler, it appeared that stars end their lives as white dwarfs. Supported by the quantum jiggling of electrons, they will not collapse inside the Schwarzschild radius and an event horizon will not form.