THE LARGE NUMBER N: GRAVITY IN THE COSMOS
Who could believe an ant in theory?
A giraffe in blueprint?
Ten thousand doctors of what’s possible
Could reason half the jungle out of being.
John Ciardi
NEWTON’S ‘CLOCKWORK’
If we were establishing a discourse with intelligent beings on another planet, it would be natural to start with gravity. This force grips planets in their orbits and holds the stars together. On a still larger scale, entire galaxies – swarms of billions of stars – are governed by gravity. No substance, no kind of particle, not even light itself escapes its grasp. It controls the expansion of the entire universe, and perhaps its eventual fate.
Gravity still presents deep mysteries. It is more perplexing than any of the other basic forces of nature. But it was the first force to be described in a mathematical fashion. Sir Isaac Newton told us in the seventeenth century that the attraction between any two objects obeys an ‘inverse square law’. The force weakens in proportion to the square of the distance between the two masses: take them twice as far away and the attraction between them is four times weaker. Newton realized that the force that makes apples fall and governs a cannon-ball’s trajectory is the same that locks the Moon in its orbit around the Earth. He proved that his law accounted for the elliptical orbits of the planets – a compelling demonstration of the power of mathematics to predict the ‘clockwork’ of the natural world.
Newton’s great work, the Principia, published in 1687, is a three-volume Latin text, laced with elaborate theorems of a mainly geometric kind. It is a monument to the pre-eminent scientific intellect of the millennium. Despite the forbidding austerity of his writings (and his personality), Newton’s impact was immense, on philosophers and poets alike. And that influence percolated to a wider public as well: for instance, a book entitled Newtonianism for Ladies was published in 1737. The essence of his theory of gravity appeared in a more accessible book called The System of the World.
In this latter work, a key idea is neatly illustrated by a picture showing cannon-balls fired horizontally from a mountain-top. The faster they’re flung, the further they go before hitting the ground. If the speed is very high, the earth ‘falls away’ under the projectile’s trajectory, and it goes into orbit. The requisite speed (about eight kilometres per second) was of course far beyond the cannons of Newton’s time, but today we’re familiar with artificial satellites that stay in orbit for just this reason. Newton himself showed that the same force holds the planets in their elliptical orbits round the Sun. Gravity acts on a grander scale in clusters of stars; and in galaxies, where billions of stars are held in orbit around a central hub.
In the Sun and other stars like it, there is a balance between gravity, which pulls them together, and the pressure of their hot interior, which, if gravity didn’t act, would make them fly apart. In our own Earth’s atmosphere, the pressure at ground level, likewise, balances the weight of all the air above us.
GRAVITY ON BIG AND SMALL SCALES
Our Earth’s gravity has more drastic effects on big objects than small ones. When producers of ‘disaster movies’ use a model to depict (for example) a bridge or dam collapsing, they must make it not of real steel and concrete but of very flimsy material that bends or shatters when dropped from table-top height. And the film has to be shot fast and replayed in slow motion to look realistic. Even when this is carefully done, there may be other give-away clues that we are viewing a miniature version rather than the real thing – for instance, small wavelets in a water tank are smoothed by surface tension (the force that holds raindrops together), but this effect is negligible in a full-scale turbulent river or in ocean waves. Surface tension allows spiders to walk on water, but we can’t.
Being the right size is crucial in the biological world. Large animals are not just blown-up versions of small ones: they are differently proportioned, with, for instance, thicker legs in relation to their height. Imagine you doubled the dimensions of an animal, but kept its shape the same. Its volume and weight would become eight (23) times larger, not just twice as large; but the cross-section of its legs would only go up by a factor of four (22) and would be too weak to support it. It would need a redesign. The bigger they are, the harder they fall: ‘godzillas’ would need legs thicker than their bodies, and would not survive a fall; mice, on the other hand, can climb vertically, and are unharmed even when dropped from many times their own height.
Galileo (who died in the same year that Newton was born) was the first clearly to realize these constraints on size. He wrote:
Nor could Nature make trees of immeasurable size, because their branches would eventually fall of their own weight . . . When bodies are diminished, their strengths do not proportionally diminish; rather, in very small bodies the strength grows in greater ratio, and I believe that a little dog might carry on his back two or three dogs of the same size, whereas I doubt that a horse could carry even one horse his size.
Similar arguments limit the size of birds (the constraints are more stringent for humming birds that can hover than for albatrosses that glide); but the limits are more relaxed for floating creatures, allowing leviathans in the ocean. In contrast, being too small leads to problems of another kind: a large area of skin in proportion to weight, whereby heat is lost quickly; small mammals and birds must eat and metabolize fast in order to stay warm.
There would be analogous limits on other worlds. For example, the physicist Edwin Salpeter has speculated, along with Carl Sagan, on the ecology of hypothetical balloon-like creatures that could survive in the dense atmosphere of Jupiter. Each new generation would face a race against time: it would have to inflate large enough to achieve buoyancy before gravity pulled it to destruction in the dark high-pressure layers deeper down.
THE VALUE OF N AND WHY IT IS SO LARGE
Despite its importance for us, for our biosphere, and for the cosmos, gravity is actually amazingly feeble compared with the other forces that affect atoms. Electric charges of opposite ‘sign’ attract each other: a hydrogen atom consists of a positively charged proton, with a single (negative) electron trapped in orbit around it. Two protons would, according to Newton’s laws, attract each other gravitationally, as well as exerting an electrical force of repulsion on one another. Both these forces depend on distance in the same way (both follow an ‘inverse square’ law), and so their relative strength is measured by an important number, N, which is the same irrespective of how widely separated the protons are. When two hydrogen atoms are bound together in a molecule, the electric force between the protons is neutralized by the two electrons. The gravitational attraction between the protons is thirty-six powers of ten feebler than the electrical forces, and quite unmeasurable. Gravity can safely be ignored by chemists when they study how groups of atoms bond together to form molecules.
How, then, can gravity nonetheless be dominant, pinning us to the ground and holding the moon and planets in their courses? It’s because gravity is always an attraction: if you double a mass, then you double the gravitational pull it exerts. On the other hand, electric charges can repel each other as well as attract; they can be either positive or negative. Two charges only exert twice the force of one if they are of the same ‘sign’. But any everyday object is made up of huge numbers of atoms (each made up of a positively charged nucleus surrounded by negative electrons), and the positive and negative charges almost exactly cancel out. Even when we are ‘charged up’ so that our hair stands on end, the imbalance is less than one charge in a billion billion. But everything has the same sign of ‘gravitational charge’, and so gravity ‘gains’ relative to electrical forces in larger objects. The balance of electric forces is only slightly disturbed when a solid is compressed or stretched. An apple falls only when the combined gravity of all the atoms in the Earth can defeat the electrical stresses in the stalk holding it to the tree. Gravity is important to us because we live on the heavy Earth.
We can quantify this. In Chapter 1, we envisaged a set of pictures, each being viewed from ten times as far as the last. Imagine now a set of differently sized spheres, containing respectively 10, 100, 1000, . . . atoms, in other words each ten times heavier than the one before. The eighteenth would be as big as a grain of sand, the twenty-ninth the size of a human, and the fortieth that of a largish asteroid. For each thousandfold increase in mass, the volume also goes up a thousand times (if the spheres are equally dense) but the radius goes up only by ten times. The importance of the sphere’s own gravity, measured by how much energy it takes to remove an atom from its gravitational pull, depends on mass divided by radius,1 and so goes up a factor of a hundred. Gravity starts off, on the atomic scale, with a handicap of thirty-six powers of ten; but it gains two powers of ten (in other words 100) for every three powers (factors of 1000) in mass. So gravity will have caught up for the fifty-fourth object (54 = 36 × 3/2), which has about Jupiter’s mass. In any still heavier lump more massive than Jupiter, gravity is so strong that it overwhelms the forces that hold solids together.
Sand grains and sugar lumps are, like us, affected by the gravity of the massive Earth. But their self-gravity – the gravitational pull that their constituent atoms exert on each other, rather than on the entire Earth – is negligible. Self-gravity is not important in asteroids, nor in Mars’s two small potato-shaped moons, Phobos and Deimos. But bodies as large as planets (and even our own large Moon) are not rigid enough to maintain an irregular shape: gravity makes them nearly round. And masses above that of Jupiter get crushed by their own gravity to extraordinary densities (unless the centre gets hot enough to supply a balancing pressure, which is what happens in the Sun and other stars like it). It is because gravity is so weak that a typical star like the Sun is so massive. In any lesser aggregate, gravity could not compete with the pressure, nor squeeze the material hot and dense enough to make it shine.
The Sun contains about a thousand times more mass than Jupiter. If it were cold, gravity would squeeze it a million times denser than an ordinary solid: it would be a ‘white dwarf’ about the size of the Earth but 330,000 times more massive. But the Sun’s core actually has a temperature of fifteen million degrees – thousands of times hotter than its glowing surface, and the pressure of this immensely hot gas ‘puffs up’ the Sun and holds it in equilibrium.
The English astrophysicist Arthur Eddington was among the first to understand the physical nature of stars. He speculated about how much we could learn about them just by theorizing, if we lived on a perpetually cloud-bound planet. We couldn’t, of course, guess how many there are, but simple reasoning along the lines I’ve just outlined could tell us how big they would have to be, and it isn’t too difficult to extend the argument further, and work out how brightly such objects could shine. Eddington concluded that: ‘When we draw aside the veil of clouds beneath which our physicist is working and let him look up at the sky, there he will find a thousand million globes of gas, nearly all with [these] masses.’
Gravitation is feebler than the forces governing the microworld by the number N, about 1036. What would happen if it weren’t quite so weak? Imagine, for instance, a universe where gravity was ‘only’ 1030 rather than 1036 feebler than electric forces. Atoms and molecules would behave just as in our actual universe, but objects would not need to be so large before gravity became competitive with the other forces. The number of atoms needed to make a star (a gravitationally bound fusion reactor) would be a billion times less in this imagined universe. Planet masses would also be scaled down by a billion. Irrespective of whether these planets could retain steady orbits, the strength of gravity would stunt the evolutionary potential on them. In an imaginary strong-gravity world, even insects would need thick legs to support them, and no animals could get much larger. Gravity would crush anything as large as ourselves.
Galaxies would form much more quickly in such a universe, and would be miniaturized. Instead of the stars being widely dispersed, they would be so densely packed that close encounters would be frequent. This would in itself preclude stable planetary systems, because the orbits would be disturbed by passing stars – something that (fortunately for our Earth) is unlikely to happen in our own Solar System.
But what would preclude a complex ecosystem even more would be the limited time available for development. Heat would leak more quickly from these ‘mini-stars’: in this hypothetical strong-gravity world, stellar lifetimes would be a million times shorter. Instead of living for ten billion years, a typical star would live for about 10,000 years. A mini-Sun would burn faster, and would have exhausted its energy before even the first steps in organic evolution had got under way. Conditions for complex evolution would undoubtedly be less favourable if (leaving everything else unchanged) gravity were stronger. There wouldn’t be such a huge gulf as there is in our actual universe between the immense timespans of astronomical processes and the basic microphysical timescales for physical or chemical reactions. The converse, however, is that an even weaker gravity could allow even more elaborate and longer-lived structures to develop.
Gravity is the organizing force for the cosmos. We shall see in Chapter 7 how it is crucial in allowing structure to unfold from a Big Bang that was initially almost featureless. But it is only because it is weak compared with other forces that large and long-lived structures can exist. Paradoxically, the weaker gravity is (provided that it isn’t actually zero), the grander and more complex can be its consequences. We have no theory that tells us the value of N. All we know is that nothing as complex as humankind could have emerged if N were much less than 1,000,000,000,000,000,000,000,000,000,000,000,000.
FROM NEWTON TO EINSTEIN
More than two centuries after Newton, Einstein proposed his theory of gravity known as ‘general relativity’. According to this theory, planets actually follow the straightest path in a ‘space-time’ that is curved by the presence of the Sun. It is commonly claimed that Einstein ‘overthrew’ Newtonian physics, but this is misleading. Newton’s law still describes motions in the Solar System with good precision (the most famous discrepancy being a slight anomaly in Mercury’s orbit that was resolved by Einstein’s theory) and is adequate for programming the trajectories of space probes to the Moon and planets. Einstein’s theory, however, copes (unlike Newton’s) with objects whose speeds are close to that of light, with the ultra-strong gravity that could induce such enormous speeds, and with the effect of gravity on light itself. More importantly, Einstein deepened our understanding of gravity. To Newton, it was a mystery why all particles fell at the same rate and followed identical orbits – why the force of gravity and the inertia were in exactly the same ratio for all substances (in contrast to electric forces, where the ‘charge’ and ‘mass’ are not proportionate) – but Einstein showed that this was a natural consequence of all bodies taking the same ‘straightest’ path in a space-time curved by mass and energy. The theory of general relativity was thus a conceptual breakthrough – especially remarkable because it stemmed from Einstein’s deep insight rather than being stimulated by any specific experiment or observation.
Einstein didn’t ‘prove Newton wrong’; he transcended Newton’s theory by incorporating it into something more profound, and with wider applicability. It would actually have been better (and would have obviated widespread misunderstanding of its cultural implications) if his theory had been given a different name: not ‘the theory of relativity’ but ‘the theory of invariance’. Einstein’s achievement was to discover a set of equations that can be applied by any observer and incorporate the remarkable circumstance that the speed of light, measured in any ‘local’ experiment, is the same however the observer is moving.
The development of any science is marked by increasingly general theories, that subsume previously unrelated facts and extend the scope of those that precede them. The physicist and historian Julian Barbour offers a mountaineering metaphor,2 which I think rings true:
The higher we climb, the more comprehensive the view. Each new vantage point yields a better understanding of the interconnection of things. What is more, gradual accumulation of understanding is punctuated by sudden and startling enlargements of the horizon, as when we reach the brow of a hill and see things never conceived of in the ascent. Once we have found our bearings in the new landscape, our path to the most recently attained summit is laid bare and takes its honourable place in the new world.
Experience shapes our intuition and common sense: we assimilate the physical laws that directly affect us. Newton’s laws are in some sense ‘hardwired’ into monkeys that swing confidently from tree to tree. But far out in space lie environments differing hugely from our own. We should not be surprised that commonsense notions break down over vast cosmic distances, or at high speeds, or when gravity is strong.
An intelligence that could roam rapidly through the universe – constrained by the basic physical laws but not by current technology – would extend its intuitions about space and time to incorporate the distinctive and bizarre-seeming consequences of relativity. The speed of light turns out to have very special significance: it can be approached, but never exceeded. But this ‘cosmic speed limit’ imposes no bounds to how far you can travel in your lifetime, because clocks run slower (and on-board time is ‘dilated’) as a spaceship accelerates towards the speed of light. However, were you to travel to a star a hundred light-years away, and then return, more than two hundred years would have passed at home, however young you still felt. Your spacecraft cannot have made the journey faster than light (as measured by a stay-at-home observer), but the closer your speed approached that of light, the less you would have aged.
These effects are counterintuitive simply because our experience is limited to slow speeds. An airliner flies at only a millionth of the speed of light, not nearly fast enough to make the time dilation perceptible: even for the most inveterate air traveller it would be less than a millisecond over an entire lifetime. This tiny effect has, nevertheless, now been measured, and found to accord with Einstein’s predictions, by experiments using atomic clocks accurate to a billionth of a second.
A related ‘time dilation’ is caused by gravity: near a large mass, clocks tend to run slow. This too is almost imperceptible here on Earth because, just as we are only used to ‘slow’ motions, we experience only ‘weak’ gravity. This dilation must, however, be allowed for, along with the effects of orbital motion, in programming the amazingly accurate Global Positioning Satellite (GPS) system.
A measure of the strength of a body’s gravity is the speed with which a projectile must be fired to escape its grasp. It takes 11.2 kilometres per second to escape from the Earth. This speed is tiny compared with that of light, 300,000 kilometres per second, but it challenges rocket engineers constrained to use chemical fuel, which converts only a billionth of its so-called ‘rest-mass energy’ (Einstein’s mc2 – see Chapter 4) into effective power. The escape velocity from the Sun’s surface is 600 kilometres per second – still only one fifth of one per cent of the speed of light.
‘STRONG GRAVITY’ AND BLACK HOLES
Newtonian theory works, with only very small corrections, everywhere in our Solar System. But we should prepare for surprises when gravity is far stronger. And astronomers have discovered such places: neutron stars, for instance. Stars leave these ultra-dense remnants behind when they explode as supernovae (discussed further in the next chapter). Neutron stars are typically 1.4 times as massive as the Sun, but only about twenty kilometres across; on their surface, the gravitational force is a million million times fiercer than on Earth. More energy is needed to rise a millimetre above a neutron star’s surface than to break completely free of Earth’s gravity. A pen dropped from a height of one metre would impact with the energy of a ton of TNT (although the intense gravity on a neutron star’s surface would actually, of course, squash any such objects instantly). A projectile would need to attain half the speed of light to escape its gravity; conversely, anything that fell freely onto a neutron star from a great height would impact at more than half the speed of light.
Newton’s theory cannot cope when gravity is as powerful as it is around neutron stars; Einstein’s general relativity is needed. Clocks near the surface would run ten to twenty per cent slower compared with those far away. Light from the surface would be strongly curved, so that, viewing from afar, you would see not just one hemisphere but part of the backside of the neutron star as well.
A body that was a few times smaller, or a few times heavier, than a neutron star would trap all the light in its vicinity and become a black hole; the space around it would ‘close up’ on itself. If the Sun were squeezed down to a radius of three kilometres, it would become a black hole. Fortunately, Nature has done such experiments for us, because the cosmos is known to contain objects that have collapsed, ‘puncturing’ space and cutting themselves off from the external universe.
There are many millions of black holes in our galaxy, of about ten solar masses each, which are the terminal state of massive stars or perhaps the outcome of collisions between stars. When isolated in space, such objects are very inconspicuous: they can be detected only by the gravitational effect that they exert on other bodies or light rays that pass close to them. Easier to detect are those with an ordinary star orbiting around them to make a binary system. The technique is similar to that used to infer planets from the motion they induce in their parent star; but in this case the task is easier because the visible star is of lower mass than the dark object (instead of being a thousand or more times heavier), and so gyrates in a larger and faster orbit.
Astronomers are always specially interested in the most ‘extreme’ phenomena in the cosmos, because it is through studying these that we are most likely to learn something fundamentally new. Perhaps most remarkable of all are the amazingly intense flashes called ‘gamma-ray bursts’. These events, so powerful that for a few seconds they outshine a million entire galaxies of stars, are probably black holes caught in the act of formation.
Much larger black holes lurk in the centres of galaxies. We infer their presence by observing intense radiation from gas swirling around them at close to the speed of light, or by detecting the ultra-rapid motions of stars passing close to them. The stars very close to the centre of our own galaxy are orbiting very fast, as though feeling the gravity of a dark mass: a black hole with a mass of 2.5 million Suns. The size of a black hole is proportionate to its mass, and the hole at the Galactic Centre has a radius of six million kilometres. Some of the real monsters in the centres of other galaxies, weighing as much as several billion suns, are as big as our entire Solar System – although they are nonetheless still very small compared with the galaxies in whose cores they lurk.
Peculiar and counterintuitive though they are, black holes are actually simpler to describe than any other celestial object. The Earth’s structure depends on its history, and on what it’s made of; similarly sized planets orbiting other stars would assuredly be very different. And the Sun, basically a huge globe of gas exhibiting continuous turbulence and flaring on its surface, would look different if it contained a different ‘mix’ of atoms. But a black hole loses all ‘memory’ of how it was formed and quickly settles down to a standard smooth state described just by two quantities: how much mass went into it, and how fast it is spinning. In 1963, long before there was any evidence that black holes existed – before, indeed, the American physicist John Archibald Wheeler introduced the name ‘black hole’ – a theorist from New Zealand, Roy Kerr, discovered a solution of Einstein’s equations that represented a spinning object. Later work by others led to the remarkable result that anything that collapses would settle down into a black hole that was exactly described by Kerr’s formula. Black holes are as standardized as elementary particles. Einstein’s theory tells us exactly how they distort space and time, and what shape their ‘surface’ is.
Around black holes, our intuitions about space and time go badly awry. Light travels along the ‘straightest’ path, but in strongly warped space this can be a complicated curve. And near them, time runs very slowly (even more slowly than near a neutron star). Conversely, if you could hover, or orbit, very close to a black hole, you would see the external universe speeded up. There is a well-defined ‘surface’ around a black hole, where, to an observer at a safe distance, clocks (or an in-falling experimenter) would seem to ‘freeze’ because the time dilation becomes almost infinite.
Not even light can escape from inside this surface: the distortions of space and time are even worse. It is as though space itself is being sucked in so fast that even an outwardly directed light ray is dragged inwards. In a black hole, you can no more move ‘outwards’ in space than you can move backwards in time.
A spinning black hole distorts space and time in a more complicated way. To envisage it, imagine a whirlpool in which water spirals towards a central vortex. Far away from the vortex, you can navigate as you wish, either going with the flow or making headway against it. Closer in, the water swirls faster than your boat’s speed: you are constrained to go round with the flow, although you can still move outwards (on an outward spiral) as well as in. But, closer still, even the inward flow becomes faster than your boat. If you venture within some ‘critical radius’ you have no choice about your fate, and are sucked in towards destruction.
A black hole is shrouded by a surface that acts like a one-way membrane. No signals from inside can be transmitted to colleagues watching from a safe distance. Anyone who passes inside the ‘surface’ is trapped, and fated to be sucked inward towards a region where, according to Einstein’s equations, gravity ‘goes infinite’ within a finite time, as measured by their own clock. This ‘singularity’ actually signifies that conditions transcend the physics that we know about, just as we believe they did at the very beginning of our universe. Anyone falling into a black hole thus encounters ‘the end of time’. Is this a foretaste of the Big Crunch that could be the ultimate fate of our universe? Or does our universe have a perpetual future? Or could some still-unknown physics protect us from this fate?
Einstein’s theory was, famously, triggered by his ‘happy thought’ that gravity was indistinguishable from accelerated motion and would be undetectable in a freely falling lift. Non-uniformities in gravity cannot, however, be eliminated. If a phalanx of kamikaze astronauts were in free fall towards the Earth in regular formation, then the horizontal spacings between them would shrink but the vertical spacings would increase. This is because their trajectories all converge towards the centre of the Earth, and the gravitational force pulls more strongly on those lower down in the formation and hence nearer to the Earth. And there would be a similar effect between the different parts of each astronaut’s body: falling feet-first, the astronaut would feel a vertical stretching and a sideways compression. This ‘tidal’ force, imperceptible for astronauts in the Earth’s gravity, becomes catastrophically large in a black hole, leading to shredding and ‘spaghettification’ before the central ‘singularity’ is reached. An astronaut falling towards a stellar-mass black hole would feel severe tidal effects even before reaching the hole’s surface; thereafter, only a few milliseconds would remain (as measured by the astronaut’s clock) before encountering the singularity. But tidal effects are more gentle around the supermassive black holes in the centres of galaxies: even after passing inside the surface of one of these, several hours would remain for leisured exploration before getting close enough to the central singularity to be severely discomforted.3
ATOMIC-SCALE BLACK HOLES
Black holes are a remarkable theoretical construct, but they are more than that. Evidence that they actually exist is now compelling. They are implicated in some of the most spectacular phenomena we observe in the cosmos – quasars and explosive outbursts. There are still lively debates about exactly how they formed, but there is no mystery about how gravity could have overwhelmed all other forces in a dead star, or within a cloud of gas in the centre of a galaxy. These formation processes require them to be at least as massive as a star, because we’ve seen that, for asteroids and planets, gravity can’t compete with other forces. Indeed, a physicist on a cloud-bound planet could have predicted that if stars existed, then so probably did stellar-mass black holes.
The scale of stars, which determines the mass of black holes that can actually form today, stems, as we’ve seen, from a balance between gravitational and atomic forces. But nothing in Einstein’s theory picks out any special mass. Black holes are made from the fabric of space itself. Insofar as space is a smooth continuum, nothing apart from a simple scaling distinguishes whether a hole (once it has formed) is as big as an atom, or as big as a star, or as big as our observable universe.
Even a hole that was only the size of an atom would have the mass of a mountain. Black holes are, by definition, objects where gravity has overwhelmed all other forces. For an atom-sized black hole to form, 1036 atoms must be squeezed into the dimensions of one. This forbidding requirement is another consequence of the hugeness of our cosmic number N, which measures the weakness of gravity on the atomic scale. What about black holes even smaller than an atom? Here there is an eventual limit (which will reappear in Chapter 10) due to an inherent graininess of space on the tiniest scale.
Atomic-scale black holes could have formed, if at all, only in the immense pressures that prevailed in the earliest instants of the universe. If they actually existed, such mini-holes would be extraordinary ‘missing links’ between the cosmos and the microworld.