6
Some of the best days of my life have been spent travelling the world. I vividly remember a trip to the Grand Canyon when I was a teenager; the colours, the heat, the sheer scale of the thing took my breath away. I crept ever closer to the cliff edge, just to catch a better glimpse of what it had to offer. The closer I got to the edge, the more I could see of the canyon itself; the strange rock formations and the water meandering through its base. Of course, being a teenager at the time, I could not be trusted; my parents reminded me every five minutes not to get too close to the edge, and being a teenager I terrorised them and did it anyway. But, as parents often are, they were right to warn me (not just because I am possibly the clumsiest person they’ve ever known), because if I’d taken just one step too close to that edge, I would have fallen a long way down.
Let’s presume I’d have survived such a fall to the bottom of the Grand Canyon – I would’ve then been stranded at the bottom of the valley without enough energy to claw my way back up the cliff face. Now, I realise I’ve spent the last few chapters convincing you that black holes aren’t holes but mountains instead, but you can picture what’s known as the ‘event horizon’ around a black hole as the edge of the Grand Canyon – it’s the point at which you’ve gone too far and neither you nor anything else in the entire Universe has enough energy to claw its way back out.
As you get closer to a black hole, the escape velocity needed increases until it reaches the speed of light. This point is what we call the event horizon, and it only exists because of that ultimate speed limit in the Universe – the speed of light. The event horizon is often described as the ‘point of no return’ – but it’s not a point at all. The event horizon is a three-dimensional sphere around whatever lies inside, and it’s what we describe as the ‘size’ of a black hole, known as the Schwarzschild radius.
Karl Schwarzschild was a German physicist and astronomer, who at the outbreak of the First World War was director of the Astrophysical Observatory at Potsdam.51 Despite being exempt from mandatory service in the German army due to his age (he was pushing forty-one), he volunteered and served on both fronts. For Schwarzschild, though, war did not put a stop to his science, because in the middle of the First World War, in 1915, Einstein announced his theory of general relativity to the world, including the equations that described how space and time were affected when matter was present. These equations are notoriously tricky beasts to solve,52 and even Einstein himself didn’t think they had exact solutions, having made many approximations himself to get answers (for example to explain the orbit of Mercury). But that didn’t put off German army artillery lieutenant Karl Schwarzschild, who is a figure from history who could definitely be described as ‘non-stop’.53
During his time serving on the Eastern Front (and despite suffering from a rare, painful autoimmune disease), Schwarzschild wrote three scientific papers in his ‘downtime’, two of which were on general relativity.54 He worked out an exact solution to Einstein’s field equations for the strength of gravity around a spherical, non-rotating object using the simple trick of employing a different coordinate system (instead of normal x, y, z coordinates, he used polar coordinates of radius and angle, like you would use to describe a position on the Earth in terms of latitude and longitude). After he figured out this solution to the equations, he wrote Einstein a letter on 22 December 1915, while he was still on the Eastern Front. There’s a fantastic line in it, which in its original German reads: ‘Wie Sie sehen, meint es der Krieg freundlich mit mir, indem er mir trotz heftigen Geschützfeuers in der durchaus terrestrischer Entfernung diesen Spaziergang in dem von Ihrem Ideenlande erlaubte.’ He is thanking Einstein here, saying that the war has treated him kindly enough despite all the heavy gunfire, as it has given him the opportunity to take a walk through Einstein’s ideas about gravity published in his theory of general relativity. This line is especially poignant with the knowledge that Schwarzschild died just five months later in May 1916, at the age of just forty-two.
Schwarzschild wasn’t trying to solve these equations for a black hole; instead his solutions describe any sphere of mass, whether a star or a diffuse nebula of gas scattered across a huge region of space. But there was something about his solution that had people worried for decades afterwards, because in this solution there were two points at which the strength of gravity became infinite. Because Schwarzschild used polar coordinates, the equation he got for the strength of gravity depended on the distance away from a certain central point. So, the solution is the same for all points at this distance, i.e. a sphere defined by a certain radius. When that radius was equal to zero, the strength of gravity became infinite. But it also occurred at a larger distance too; a distance that depended on the mass.
The places where this happened were known as ‘singularities’. This is a fancy mathematics word that means ‘we can’t tell you what happens here’. It’s a point that is undefinable, that’s usually reviled by mathematicians because to work out the strength of gravity at a radius of zero, you have to *takes deep breath* divide by zero (I think I just involuntarily shuddered). Dividing by zero is mathematically impossible, but us physicists don’t dwell on it for too long. If you take ever smaller and smaller numbers and divide by them, the answer you get grows and grows (like with an object’s momentum as it travels closer to the speed of light). So us physicists are very happy to divide by zero and say that we get infinity, something mathematicians will debate the philosophy of endlessly. Now, these singularities weren’t an issue for most objects, like stars, as the larger radius, where the other singularity appeared, is very small and normal stars are very large. This radius is now known as the Schwarzschild radius, but it wasn’t until the 1960s that it would be recognised for what it truly was: an event horizon.
It was Austrian-born physicist Wolfgang Rindler who we have to thank for the term ‘event horizon’. At the age of just fourteen, Rindler was evacuated from Austria to England in the Kindertransport rescue of Jewish children before the outbreak of the Second World War. He finished school and went to university in the UK, before being offered a job at Cornell University in New York state in 1956. Once at Cornell, Rindler managed to publish the results from his PhD research at the University of London, and the world was introduced to the idea of an event horizon. He defined a ‘horizon’ as ‘a frontier between things observable and things unobservable’, in the same way you can’t see anything beyond the Earth’s horizon when looking into the distance. An event horizon therefore divides events into those that can be seen, and those that can’t. Or, to put it in Rindler’s much more poetic words: ‘those [events] that are forever outside [our] possible powers of observation’.
The Schwarzschild radius is the event horizon of a black hole; it marks the region where we no longer get any information out of the black hole because light can no longer reach us. It is the region where the escape velocity becomes greater than the speed of light. It’s not a true singularity, because if you use a different coordinate system you can still define the strength of gravity at that point, even if you can’t get any real information from beyond it. But the Schwarzschild radius still represents something physical about black holes. Schwarzschild’s solution to Einstein’s general relativity equations essentially tells us the size of the event horizon, or the size of the black hole itself. The size is only dependent on the mass of the black hole (and the speed of light and the overall strength of gravity, but those are constant values that don’t change, as far as we know). Essentially, the bigger the black hole, the bigger the event horizon.
I remember first learning the derivation of Schwarzschild’s solution while I was studying for my undergraduate degree in physics at Durham University. Obviously, once armed with the equation to calculate the size of a black hole, the first thing I did was work out how large of a black hole I personally would be. Having consumed a fair bit more cheese since my university days I’ve had to re-do the calculation for this book, but in case you were wondering, if we had the ability to squish an average human at around 62 kilograms down into a black hole, they would have an event horizon with a radius of about 0.09 yoctometres (0.00000000000000000000000009 m; that’s twenty-five zeros after the decimal place!). That’s smaller than an atom. Smaller than a proton that makes up the nuclei of atoms. Smaller even than the quarks that make up protons.
Admittedly, it’s a number that is a little bit small for our brains to comprehend, so let’s try something bigger: the entire Earth, for example. If you could take the Earth and turn it into a black hole it would have a radius of just 0.9 cm, smaller than your fingernail. Whereas if you could turn the Sun into a black hole it would have a radius of 2.9 km. The actual radius of the Sun is 696,342 km, much larger than its Schwarzschild radius. But no matter the size, whether 0.09 yoctometres, 0.9 cm or 2.9 km, the black holes we could make out of the Earth and Sun would behave in exactly the same way, with escape velocities higher than that finite speed limit of the Universe: the speed of light.
But what about the other singularity in Schwarzschild’s solution? The one that appears at r=0. The Schwarzschild radius isn’t a real singularity, it’s what’s known as a ‘coordinate singularity’ (it only exists because of whatever system of coordinates you’ve solved your problem in), but the one at r=0 is a genuine physical singularity known as a ‘gravitational singularity’. It is completely undefinable and unknowable. The curvature of space at that point, and therefore the strength of gravity, cannot be defined. In fact, the point itself is not even considered to be a part of normal ‘spacetime’ anymore; you can’t define where (or even when!) the point is.
Again, this isn’t a big deal for objects that are much larger than the Schwarzschild radius, like for a star whose mass is nice and evenly distributed. We don’t need to know the value at r=0 and we can say that the strength of gravity is nicely described by Schwarzschild’s solution to Einstein’s equation, as long as r is greater than 0. It is a big deal, though, when we think about the end of a star’s life, when there’s so much mass in the core that nothing is able to resist the crush of gravity. Not electron degeneracy pressure, nor neutron degeneracy pressure. The star keeps collapsing, getting ever smaller, until it becomes smaller than the Schwarzschild radius. What happens to it then? We don’t know, because the star’s collapse is now an event that is occurring beyond the event horizon: forever outside our powers of observation.
There is no process or form of matter that we know of in all of physics that can resist gravity to stop the collapse of the star. As far as we know, it keeps collapsing down to an ever smaller size until all the mass is contained in an infinitely dense, infinitesimally small undefinable point at r=0: the singularity. At least this is the mathematical description. The event horizon shrouds the true nature of what’s ‘inside’ the black hole from our view, due to the nature of light itself: what do these dark stars truly look like?
Light is how we observe the Universe around us; recording the brightness of stars or the positions of planets reflecting light from the Sun. We send information encoded on radio waves of light through the air which get decoded into sound at the other end. We do the same with infrared light through fibre-optic cables so we can access the internet. We communicate and receive information with light. This means black holes are not only prisons for light, but prisons for information and data. Under the laws of physics, as we understand them right now, we might be able to run the maths beyond the event horizon as much as we like, but we can never test those predictions because we can never receive any information from beyond the event horizon of a black hole.
No data equals very sad scientists. Imagine the feeling if you got closer to the cliff edge of the Grand Canyon and yet you still couldn’t see into the spectacular canyon itself. It’s utterly infuriating. But it’s something us astronomers have had to resign ourselves to. However, unlike the Grand Canyon, which has a very obvious and clear cliff edge that has likely been making parents nervous for thousands of years, the event horizon is not obvious at all. There’s no cliff edge around a black hole. No line drawn in the sand. No Schwarzschild dressed as a referee with a spray can drawing a line on the pitch. An event horizon is completely and utterly invisible, you wouldn’t even know it was there unless you were paying close attention . . . adventurous space travellers beware!