Most of the time, books are better at conveying complex ideas than television. There are many reasons for this, some of which I’ll discuss in a future autobiography when my time on TV is long over. But when done well, television pictures can convey ideas with an elegance and economy unavailable in print. Human Universe contains, I hope, some of these moments, but there is one sequence in particular that I think fits into this category.
NASA’s Plum Brook Station in Ohio is home to the world’s largest vacuum chamber. It is 30 metres in diameter and 37 metres high, and was designed in the 1960s to test nuclear rockets in simulated space-like conditions. No nuclear rocket has ever been fired inside – the programme was cancelled before the facility was completed – but many spacecraft, from the Skylab nosecone to the airbags on Mars landers, have been tested inside this cathedral of aluminium. To my absolute delight, NASA agreed to conduct an experiment using their vacuum chamber to demonstrate precisely what motivated Einstein to his remarkable conclusion. The experiment involves pumping all the air out of the chamber and dropping a bunch of feathers and a bowling ball from a crane. Both Galileo and Newton knew the result, which is not in question. The feathers and the bowling ball both hit the ground at the same time. Newton’s explanation for this striking result is as follows. The gravitational force acting on a feather is proportional to its mass. We’ve already seen this written down in Newton’s Law of Gravitation. That gravitational force causes the feather to accelerate, according to Newton’s other equation, F=ma. This equation says that the more massive something is, the more force has to be applied to make it accelerate. Magically, the mass that appears in F=ma is precisely the same as the mass that appears in the Law of Gravitation, and so they precisely cancel each other out. In other words, the more massive something is, the stronger the gravitational force between it and the Earth, but the more massive it is, the larger this force has to be to get it moving. Everything cancels out, and so everything ends up falling at the same rate. The problem with this explanation is that nobody has ever thought of a good reason why these two masses should be the same. In physics, this is known as the equivalence principle, because ‘gravitational mass’ and ‘inertial mass’ are precisely equivalent to each other.
Einstein’s explanation for the fact that both the feathers and the bowling ball fall at the same rate in the Plum Brook vacuum chamber is radically different. Recall Einstein’s happiest thought. ‘Because for an observer falling freely from the roof of a house there exists … no gravitational field’. There is no force acting on the feathers or the ball in freefall, and therefore they don’t accelerate! They stay precisely where they are: at rest, relative to each other. Or, if you prefer, they stand still because we are always able to define ourselves as being at rest if there are no forces acting on us. But, you are surely asking, how come they eventually hit the ground if they are not moving because no forces are acting on them? The answer, according to Einstein, is that the ground is accelerating up to meet them, and hits them like a cricket bat! But, but, but, you must be thinking, I’m sitting on the ground now and I’m not accelerating. Oh yes you are, and you know it because you can feel a force acting on you. It’s the force exerted by the chair on which you may be sitting, or the ground on which you are standing. This is obvious – if you stand up long enough then your feet will hurt because there is a force acting on them. And if there is a force acting on them, then they are accelerating. There is no sleight of hand here. The very beautiful thing about Einstein’s happiest thought is that, once you know it, it’s utterly obvious. Standing on the ground is hard work because it exerts a force on you. The effect is precisely the same as sitting in an accelerating car and being pushed back into your seat. You can feel the acceleration viscerally, and if you switch off your common sense for a moment, then you can feel the acceleration now. The only way you can get rid of the acceleration, momentarily, is to jump off a roof.
This is wonderful reasoning, but of course it does raise the thorny question of why, if there is no such thing as gravity, the Earth orbits the Sun. Maybe Aristotle was right after all. The answer is not easy, and it took Einstein almost a decade to work out the details. The result, published in 1916, is the General Theory of Relativity, which is often cited as the most beautiful scientific theory of them all. General Relativity is notoriously mathematically and conceptually difficult when you get into the details of making predictions that can be compared with observations. Indeed, most physics students in the UK will not meet General Relativity until their final year, or until they become postgraduates. But having said that, the basic idea is very simple. Einstein replaced the force of gravity with geometry – in particular, the curvature of space and time.
Imagine that you are standing on the surface of the Earth at the equator with a friend. You both start walking due north, parallel to each other. As you get closer to the North Pole, you will find that you move closer together, and if you carry on all the way to the Pole you will bump into each other. If you don’t know any better, then you may conclude that there is some kind of force pulling you both together. But in reality there is no such force. Instead, the surface of the Earth is curved into a sphere, and on a sphere, lines that are parallel at the equator meet at the Pole – they are called lines of longitude. This is how geometry can lead to the appearance of a force.
Einstein’s theory of gravity contains equations that allow us to calculate how space and time are curved by the presence of matter and energy and how objects move across the curved spacetime – just like you and your friend moving across the surface of the Earth. Spacetime is often described as the fabric of the universe, which isn’t a bad term. Massive objects such as stars and planets tell the fabric how to curve, and the fabric tells objects how to move. In particular, all objects follow ‘straight line’ paths across the curved spacetime that are known in the jargon as geodesics. This is the General Relativistic equivalent of Newton’s first law of motion – every body continues in a state of rest or uniform motion in a straight line unless acted upon by a force. Einstein’s description of the Earth’s orbit around the Sun is therefore quite simple. The orbit is a straight line in spacetime curved by the presence of the Sun, and the Earth follows this straight line because there are no forces acting on it to make it do otherwise. This is the opposite of the Newtonian description, which says that the Earth would fly through space in what we would intuitively call a ‘straight line’ if it were not for the force of gravity acting between it and the Sun. Straight lines in curved spacetime look curved to us for precisely the same reason that lines of longitude on the surface of the Earth look curved to us; the space upon which the straight lines are defined is curved.
This is all well and good, but there may be a question that has been nagging away in your mind since I told you that the ground accelerated up and hit the feathers and the bowling ball at Plum Brook like a cricket bat. How could it possibly be that every piece of the Earth’s surface is accelerating away from its centre, and yet the Earth stays intact as a sphere with a fixed radius? The answer is that if a little piece of the Earth’s surface at Plum Brook were left to its own devices, it would do precisely the same thing as the feather and the bowling ball; it would follow a straight line through spacetime. These straight lines point radially inwards towards the centre of the Earth. This is the ‘state of rest’, if you like – the natural trajectory that would be followed by anything. The geodesics point radially inwards because of the way that the mass of the Earth curves spacetime. So a collapsing Earth would be the natural state of things without any forces acting – one in which, ultimately, all the matter would collapse into a little black hole. The thing that prevents this from happening is the rigidity of the matter that makes up the Earth, which ultimately has its origin in the force of electromagnetism and a quantum mechanical effect called the Pauli Exclusion Principle. In order to stay as a big, spherical, Earth-sized ball, a force must act on each little piece of ground and this must cause each piece of gr
ound to accelerate. Every piece of big spherical things like planets must continually accelerate radially outwards to stay as they are, according to General Relativity.
From what I’ve said so far, it might seem that General Relativity is simply a pleasing way of explaining why the Earth orbits the Sun and why objects all fall at the same rate in a gravitational field. General Relativity is far more than that, however. Very importantly, it makes precise predictions about the behaviour of certain astronomical objects that are radically different from Newton’s. One of the most spectacular examples is a binary star system known rather less than poetically as PSR J0348+0432. The two stars in this system are exotic astrophysical objects. One is a white dwarf, the core of a dead star held up against the force of gravity by a sea of electrons. Electrons behave according to the Pauli Exclusion Principle, which, roughly speaking, states that electrons resist being squashed together. This purely quantum mechanical effect can halt the collapse of a star at the end of its life, leaving a super-dense blob of matter. White dwarfs are typically between 0.6 and 1.4 times the mass of our Sun, but with a volume comparable to that of the Earth. The upper limit of the mass of a white dwarf is known as the Chandrasekhar limit, and was first calculated by the Indian astrophysicist Subrahmanyan Chandrasekhar in 1930. The calculation is a tour de force of modern physics, and relates the maximum mass of these exotic objects to four fundamental constants of nature – Newton’s gravitational constant, Planck’s constant, the speed of light and the mass of the proton. After almost a century of astronomical observations, no white dwarf has ever been discovered that exceeds the Chandrasekhar limit. Almost all the stars in the Milky Way, including our Sun, will end their lives as white dwarfs. Only the most massive stars will produce a remnant that exceeds the Chandrasekhar limit, and the vast majority of these will produce an even more exotic object known as a neutron star. In the PSR J0348+0432 system, quite wonderfully, the white dwarf has a neutron star companion, and this is what makes the system so special.
If the remains of a star exceed the Chandrasekhar limit, the electrons are squashed so tightly onto the protons in the star that they can react together via the weak nuclear force to produce neutrons (with the emission of a particle called a neutrino). Through this mechanism, the whole star is converted into a giant atomic nucleus. Neutrons, just like electrons, obey the Pauli Exclusion Principle and resist being squashed together, leading to a stable dead star. Neutron stars can have masses several times that of our Sun, but quite astonishingly are only around 10 kilometres in diameter. They are the densest stars known; a teaspoonful of neutron star matter weighs as much 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.
A DAY WITHOUT YESTERDAY
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 Shap
ley 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
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