as a mountain.
Imagine, for a moment, this exotic star system. The white dwarf and neutron star are very close together; they orbit around each other at a distance of 830,000 kilometres – that’s around twice the distance to the Moon – once every 2 hours and 27 minutes. That’s an orbital velocity of around 2 million kilometres per hour. The neutron star is twice the mass of our Sun, around 10 kilometres in diameter, and spins on its axis 25 times a second. This is a star system of unbelievable violence. Einstein’s Theory of General Relativity predicts that the two stars should spiral in towards each other because they lose energy by disturbing spacetime itself, emitting what are known as gravitational waves. The loss of energy is minuscule, resulting in a change in orbital period of eight millionths of a second per year. In a triumph of observational astronomy, using the giant Arecibo radio telescope in Puerto Rico, the Effelsberg telescope in Germany and the European Southern Observatory’s VLT in Chile, astronomers measured the rate of orbital decay of PSR J0348+0432 in 2013 and found it to be precisely as Einstein predicted. This is quite remarkable. Einstein could never have dreamt of the existence of white dwarfs and neutron stars when he had his happiest thought in 1907, and yet by thinking carefully about falling off a roof he was able to construct a theory of gravity that describes, with absolute precision, the behaviour of the most exotic star system accessible to twenty-first-century telescopes. And that, if I really need to say it, is why I love physics.
Einstein’s Theory of General Relativity has, at the time of writing, passed every precision test that scientists have been able to carry out in the century since it was first published. From the motion of feathers and bowling balls in the Earth’s gravitational field to the extreme astrophysical violence of PSR J0348+0432, the theory comes through with flying colours.
There is rather more to Einstein’s magisterial theory than the mere description of orbits, however. General Relativity is fundamentally different to Newton’s theory because it doesn’t simply provide a model for the action of gravity. Rather, it provides an explanation for the existence of the gravitational force itself in terms of the curvature of spacetime. It’s worth writing down Einstein’s field equations, because they are (to be honest) deceptively simple.
Here, the right-hand side describes the distribution of matter and energy in some region of spacetime, and the left-hand side describes the shape of spacetime as a result of the matter and energy distribution. To calculate the orbit of the Earth around the Sun one would put a spherical distribution of mass with the radius of the Sun into the right-hand side of the equation, and (roughly speaking) out would pop the shape of spacetime around the Sun. Given the shape of spacetime, the orbit of the Earth can be calculated. It’s not completely trivial to do this by any means, and the notation above hides great complexity. But the point is simply that, given some distribution of matter and energy, Einstein’s equations let you calculate what spacetime looks like. But here is the remarkable point that draws us towards the end of our story. Einstein’s equations deal with the shape of spacetime – the fabric of the universe. The first thing to note is that we are dealing with spacetime, not just space. Space is not a fixed arena within which things happen with a big universal clock marking some sort of cosmic time upon which everyone agrees. The fabric of the universe in Einstein’s theory is a dynamical thing. Very importantly, therefore, Einstein’s equations don’t necessarily describe something that is static and unchanging. The second thing to note is that nowhere have we restricted the domain of Einstein’s theory to the region of spacetime around a single star, or even a double star system such as PSR J0348+0432. Indeed, there is no suggestion in Einstein’s theory that such a restriction is necessary. Einstein’s equations can be applied to an unlimited region of spacetime. This implies that they can, at least in principle, be used to describe the shape and evolution of the entire universe.
Storytelling is an ancient and deeply embedded human impulse; we learn, we communicate, we connect across generations through stories. We use them to explore the minutiae of human life, taking delight in the smallest things. And we tell grander tales of origins and endings. History is littered with stories about the creation of the universe; they seem as old as humanity itself. Multifarious gods, cosmic eggs, worlds emerging from chaos or order, from the waters or the sky or nothing at all – there exist as many creation myths as there are cultures. The impulse to understand the origin of the universe is clearly a powerful unifying idea, although the very existence of many different mythologies continues to be a source of division. It is an unfortunate testament to the emotional power of creation narratives that so much energy is spent arguing about old ones rather than using the increasingly detailed observational evidence available to twenty-first-century citizens to construct new ones. We live in a very privileged and exciting time in this sense, because observational evidence for creation stories was scant even a single lifetime ago. When my grandparents were born in Oldham at the turn of the twentieth century, there was no scientific creation story. Astronomers were not even aware of a universe beyond the Milky Way, which makes it all the more remarkable that the modern scientific approach to the description of the universe emerged almost fully formed from Einstein’s Theory of General Relativity before Edwin Hubble published the discovery of his Cepheid variable star in Andromeda and settled Shapley and Curtis’s Great Debate.
One of the beautiful things about mathematical physics is that equations contain stories. If you think of equations in terms of the nasty little things you used to solve at school on a damp autumn afternoon, then that may sound like a strange and abstract idea. But equations like Einstein’s field equations are much more complex animals. Recall that Einstein’s equations will tell you the shape of spacetime, given some distribution of matter and energy. That shape is known as a solution of the equations, and it is these solutions that contain the stories. The first exact solution to Einstein’s field equations was discovered in 1915 by the German physicist Karl Schwarzschild. Schwarzschild used the equations to calculate the shape of spacetime around a perfectly spherical, non-rotating mass. Schwarzschild’s solution can be used to describe planetary orbits around a star, but it also contains some of the most exotic ideas in modern physics; it describes what we now know as the event horizon of a black hole. The well-known tales of astronauts being spaghettified as they fall towards oblivion inside a supermassive collapsed star are to be found in Schwarzschild’s solution. The calculation was a remarkable achievement, not least because Schwarzschild completed it whilst serving in the German Army at the Russian Front. Shortly afterwards, the 42-year-old physicist died of a disease contracted in the trenches.
There were two ways of arriving at the truth; I decided to follow them both.
Georges Lemaître
The most remarkable stories waiting to be found inside Einstein’s equations reveal themselves when we take an audacious and seemingly reckless leap. Instead of confining ourselves to describing the spacetime around spherical blobs of matter, why not think a little bigger? Why not try to use Einstein’s equations to tell us about all of spacetime? Why can’t we apply General Relativity to the entire universe? Einstein noticed this as a possibility very early in the development of his theory, and in 1917 he published a paper entitled ‘Cosmological Considerations of the General Theory of Relativity’. It’s a big step, of course, from thinking about someone falling off a roof to telling the story of the universe, and Einstein appears to have been uncharacteristically wobbly. In a letter to his friend Paul Ehrenfest a few days before he presented his paper to the Prussian Academy, he wrote ‘I have … again perpetrated something about gravitation theory which somewhat exposes me to the danger of being confined in a madhouse.’
The universe modelled in Einstein’s 1917 paper is not the one we inhabit, but the paper is of interest for the introduction of what Einstein later came to view as a mistake. Einstein tried
