For example, if our moon were alone in space, it would not sit still but rather move in a straight line without changing its speed. (Of course, if it were truly all alone, there would be no way to tell it was doing this, nothing to which we could relate its motion.) But the moon isn’t all alone. A force known as gravity acts on the moon to change its speed and direction. Where does that force come from? It comes from a nearby voting bloc of particles (a massive object) known as the Earth. The moon resists the change. It tries to keep moving in a straight line. How well it’s able to resist depends on how many votes are in it, how massive it is. Meanwhile, the moon’s gravity also affects the Earth. The most obvious result is the ocean tides.
Newton’s theory tells us that the amount of mass a body has affects how strong the pull of gravity is between it and another body. Other factors remaining equal, the greater the mass, the greater the attraction. If the Earth were double its present mass, the attraction of gravity between the Earth and the moon would be double what it currently is. Any change in the mass of either the Earth or the moon would change the strength of the gravitational pull between them. Newton also discovered that the farther apart bodies are, the weaker the pull between them is. If the moon were twice its present distance from the Earth, the pull of gravity between the Earth and the moon would be only one-fourth as strong. Newton’s theory is usually stated: Bodies attract each other with a force that is proportional to their mass and inversely proportional to the square of the distance between them.
Newton’s theory of gravity is an extremely successful theory. It wasn’t improved on for over two hundred years. We still use it, though we now know that it fails in some circumstances, such as when gravitational forces become enormously strong (near a black hole, for instance) or when bodies are moving at near light speed.
Albert Einstein, early in the twentieth century, saw a problem with Newton’s theory. If the strength of gravity between two objects is related to the distance between them, then if someone takes the sun and moves it farther from the Earth, the force of gravity between the Earth and the sun should change instantly. Is this possible?
Einstein’s theory of special relativity recognized that the speed of light measures the same no matter where you are in the universe or how you’re moving and that nothing can move faster than the speed of light. Light from the sun takes about eight minutes to reach Earth. We always see the sun as it was eight minutes ago. So, move the sun farther from the Earth; the Earth won’t find out this has happened and feel any effect of the change for eight minutes. For eight minutes we’ll continue to orbit just as though the sun hadn’t moved. In other words, the effect of the gravity of one body on the other cannot change instantaneously because gravity can’t move faster than the speed of light. Information about how far away the sun is cannot move instantaneously across space. It can move no faster than about 186,000 miles (300,000 kilometres) per second.
It’s obvious then that when we talk about things moving in the universe, it’s not realistic to talk in terms of only the three dimensions of space. If no information can travel faster than the speed of light, things out there at astronomical distances simply don’t exist for us or for each other without a time factor. Describing the universe in three dimensions is as inadequate as describing a cube in two. Instead we must recognize the time dimension, admit there are really four dimensions, and talk of spacetime.
Einstein spent several years developing a theory of gravity that would work with what he’d discovered about light and motion at near light speed. In 1915 he introduced his theory of general relativity, requiring us to think of gravity not as a force acting between bodies but in terms of the shape, the curvature, of four-dimensional spacetime itself. In general relativity gravity is the geometry of the universe.
Bryce DeWitt, at the University of Texas, suggested we begin thinking about this curvature by imagining someone who believes the Earth is flat trying to draw a grid on the Earth:
The result can be seen from an airplane on any clear day over the cultivated regions of the Great Plains. The land is subdivided by east–west and north–south roads into square-mile sections. The east–west roads often extend in unbroken lines for many miles, but not the north–south roads. Following a road northward, there are abrupt jogs to the east or west every few miles. The jogs are forced by the curvature of the Earth. If the jogs were eliminated, the roads would crowd together, creating sections of less than a square mile. In the three-dimensional case one can imagine building a giant scaffold in space out of straight rods of equal length joined at angles of precisely 90 degrees and 180 degrees. If space is flat, the construction of the scaffold would proceed without difficulty. If space is curved, one would eventually have to begin shortening the rods or stretching them to make them fit.13
According to Einstein the curvature is caused by the presence of mass or energy. Every massive body contributes to the curvature of spacetime. Things going ‘straight ahead’ in the universe are forced to follow curved paths. Imagine a trampoline (Figure 5.1). In its centre lies a bowling ball, which causes a depression in the rubber sheet. Try to roll a golf ball in a straight line past the bowling ball. The golf ball will certainly change direction slightly when it meets the depression caused by the bowling ball. It will probably do more than that: it may even describe an ellipse and roll back in your direction. Something like that happens as the moon tries to continue in a straight line past the Earth. The Earth warps spacetime as the bowling ball warps the rubber sheet. The moon’s orbit is the nearest thing to a straight line in warped spacetime.
Figure 5.1. A bowling ball bends a rubber sheet where it lies. If you try to roll a smaller ball past the bowling ball, the path of the smaller ball will be bent when it encounters the depression caused by the bowling ball. In a similar manner, mass bends spacetime. Paths of objects in spacetime are bent when they encounter the curvature caused by a more massive object.
Einstein was describing the same phenomenon that Newton described. To Einstein a massive object warps spacetime. To Newton a massive object sends out a force. The result, in each case, is a change in the direction of a second object. According to the theory of general relativity, ‘gravitational field’ and ‘curvature’ are the same thing.
If you calculate planetary orbits in our solar system using Newton’s theories and then calculate them again using Einstein’s, you get almost precisely the same orbits, except in the case of Mercury. Because Mercury is the nearest planet to the sun, it’s affected more than the others by the sun’s gravity. Einstein’s theory predicts a result of this nearness which is slightly different from the result predicted by Newton’s theory. Observation shows that Mercury’s orbit fits Einstein’s prediction better than Newton’s.
Einstein’s theory predicts that other things besides moons and planets are affected by the warp of spacetime. Photons (particles of light) have to travel a warped path. If a beam of light is travelling from a distant star and its path takes it close to our sun, the warping of spacetime near the sun causes the path to bend inwards towards the sun a bit, just as the path of the golf ball bends inwards towards the bowling ball in our model. Perhaps the path of light bends in such a way that the light finally hits the Earth. Our sun is too bright for us to see such starlight except during an eclipse of the sun. If we see it then and don’t realize the sun is bending the path of the star’s light, we’re going to get the wrong idea about which direction the beam of light is coming from and where that star actually is in the sky (Figure 5.2). Astronomers make use of this effect. They measure the mass of objects in space by measuring how much they bend the paths of light from distant stars. The greater the mass of the ‘bender’, the greater the bending.
Figure 5.2. Because mass causes curvature of spacetime, the path of light travelling from a distant star bends as it passes a massive body like the sun. Notice the difference between the position of the star as we see it from the Earth and its true position.
We’ve been
talking about gravity in terms of what we observe on the large scale. That, of course, is the scale on which gravity becomes conspicuous – in stars, galaxies, even the entire universe – and that is the scale Hawking was dealing with in the late sixties. However, recall from Chapter 2 that gravity can also be looked at in terms of the very small, the quantum level. In fact, unless we can study it there, we will never get it unified with the other three forces, two of which work exclusively on that level. The quantum-mechanical way of looking at the gravitational attraction between the Earth and the moon is to picture it as an exchange of gravitons (the bosons, or messenger particles, of the gravitational force) between the particles that make up those two bodies.
With that background, we’ll treat ourselves to a little science fiction.
A Disastrous Day for Earth
Remind yourself what the force of gravity feels like on Earth (Figure 5.3a), then pretend you go off on a vacation in space. While you’re away something drastic happens to the Earth: it gets squeezed to only half its original size. It still has the same mass, but that mass is pressed together much more tightly. Returning from your vacation, your spacecraft hovers for a while at the place in space where the Earth’s surface used to be before the squeezing. You feel as heavy there as you did before you went away. The pull of the Earth’s gravity there hasn’t changed, because neither your mass nor that of the Earth has changed, and you are still the same distance as before from the Earth’s centre of gravity. (Remember Newton!) The moon, out beyond you, still orbits as before. However, when you land on the new surface (a much smaller radius, quite a bit nearer the Earth’s centre of gravity), the gravity on that new surface is four times what you remember on the Earth’s surface before the squeezing. You feel much heavier (5.3b).
What if something far more dramatic happened? What if the Earth were squeezed to the size of a pea – all the mass of the Earth, billions of tons, squeezed into that tiny space? Gravity on its surface would be so strong that escape velocity would be greater than the speed of light. Even light couldn’t escape. The Earth would be a black hole. However, at the radius out in space where the surface of the Earth was before any squeezing, the pull of the Earth’s gravity would still feel just the same as it does to us today (Figure 5.3c). The moon would still be orbiting just as before.
As far as we know, that story can’t happen. Planets don’t become black holes. However, there’s a good chance some stars do. Let’s retell the story, this time with a star.
Begin with a star that has a mass about ten times that of the sun. The star’s radius is about 3 million kilometres, about five times that of the sun. Escape velocity is about 1,000 kilometres per second. Such a star has a life span of about a hundred million years, during which a life-and-death struggle goes on within it.
On one side of the contest is gravity: the attraction of every particle in the star for every other. It was gravity that pulled particles in a gas together to form the star in the first place. The pull is even more powerful now that the particles are closer together. Gravity tries to make the star collapse.
The pressure of the gas in the star opposes gravity. This pressure comes from heat released when hydrogen nuclei in the star collide and merge to form helium nuclei. The heat makes the star shine and creates enough pressure to resist gravity and prevent the star from collapsing.
Figure 5.3. The day the Earth gets squeezed.
Figure 5.4. A star collapses and becomes a black hole.
For a hundred million years the contest continues. Then the star runs out of fuel: no more hydrogen to convert into helium. Some stars then convert helium into heavier elements, but that gives them only a short reprieve. When there’s no more pressure to counteract gravity, the star shrinks. As it does, the gravity on its surface becomes stronger and stronger, in the same way that gravity on the Earth’s surface did in the shrinking Earth story. It won’t have to shrink to the size of a pea to become a black hole. When the 10-solar-mass star’s radius is about 20 miles (30 kilometres), escape velocity on its surface will have increased to 186,000 miles (300,000 kilometres) per second, the speed of light. When light can no longer escape, the star becomes a black hole (Figure 5.4).fn1
Figure 5.5. In (a), particles from space move towards a star. The paths of particles 1, 2 and 3 are bent as they pass the star. The closer to the star, the greater the bending. Particles 4 and 5 hit the surface of the star. In (b), we see the same particles moving towards the star after it has become a black hole. The paths of particles 1, 2 and 3 are bent exactly as before, because the spacetime outside a star is the same as the spacetime outside a black hole of the same mass. (Recall the shrinking Earth.) Particle 4 circles the black hole and then escapes. It might circle it many times. Particle 5 is captured by the black hole.
After the escape velocity on its surface is greater than the speed of light, we don’t have to ask whether the star goes on shrinking. Even if it doesn’t, we still have a black hole. Remember how gravity at the original radius never changed in the Earth-shrinking story. Whether our star goes on shrinking to a point of infinite density or stops shrinking just within the radius where escape velocity reaches the speed of light, gravity at that radius is going to feel the same, as long as the star’s mass doesn’t change. Escape velocity at that radius is the speed of light and will stay the speed of light. Light coming from the star will find escape impossible. Nearby beams of light from distant stars won’t only be bent; they may curl around the black hole several times before escaping or falling in (Figure 5.5). If the light enters the black hole, it cannot escape. Nothing can achieve a greater velocity than the speed of light. What a profound ‘blackout’ we have! No light, no reflection, no radiation of any kind (radio, microwave, X-ray and so on), no sound, no sight, no space probe, absolutely no information can escape. A black hole indeed!
The radius where escape velocity is the speed of light becomes the border of the black hole, the radius-of-no-return: the ‘event horizon’. Hawking and Penrose, in the late 1960s, suggested defining a black hole as an area of the universe, or a ‘set of events’, from which it’s impossible for anything to escape to a distance. That has become the accepted definition. A black hole, with its event horizon for an outer boundary, is shaped like a sphere, or, if it’s rotating, a bulged-out sphere that looks elliptical when seen from the side (or would, if you could see it). The event horizon is marked by the paths in spacetime of rays of light that hover just on the edge of that spherical area, not being pulled in but unable to escape. Gravity at that radius is strong enough to stop their escape but not strong enough to pull them back. Will you see them as a great orb shimmering in space? No. If the photons can’t escape from that radius, they can’t reach your eyes. In order for you to see something, photons from it have to reach your eyes.
Classical black hole theory tells us that there are only three secrets a black hole divulges: its mass, its electric charge (if it has any), and its angular momentum or speed of rotation (if it is rotating). John Wheeler, who liked to draw helpful pictures on the chalk-board for his students, drew a television set, a flower, a chair, ‘known particles’, gravitational and electromagnetic waves, angular momentum, mass and even ‘particles as yet undetected’ falling into a black hole, shown as a funnel, and nothing coming out at the bottom of the funnel except mass, electric charge and angular momentum. Part of Hawking’s work in the early 1970s14 would help to show that, as Wheeler summed it up: ‘Black holes have no hair.’
It’s the mass of the black hole that determines its size. If you want to calculate the radius of a black hole (the radius at which the event horizon forms), take the solar mass of the black hole (the same as for the star that collapsed to form it unless that star lost mass earlier in the collapse) and multiply by 2 for miles or 3 for kilometres. You’ll find that a 10-solar-mass black hole (that is, a black hole whose mass is ten times the mass of our sun) has its event horizon at a radius of 20 miles (30 kilometres). It’s clear that if the mass changes, t
he radius where the event horizon is also changes. The black hole changes in size. We’ll talk more about this possibility later.
Having drawn the curtain at the event horizon, the star has complete privacy, while any light it emits (any picture of itself that otherwise would be viewed from elsewhere in the universe) is pulled back in. Penrose had wanted to know whether the star would go on collapsing – or just what would happen to it. He had discovered that a star collapsing as we’ve described has all its matter trapped inside its own surface by the force of its own gravity. Even if the collapse isn’t perfectly spherical and smooth, the star does go on collapsing. The surface eventually shrinks to zero size, with all the matter still trapped inside. Our enormous 10-solar-mass star is then confined not just in a region with a 20-mile (30-kilometre) radius (where its event horizon is), but rather in a region of zero radius – zero volume. Mathematicians and physicists call that a singularity. At such a singularity the density of matter is infinite. Spacetime curvature is infinite, and beams of light aren’t just curled around: they’re wound up infinitely tightly.
General relativity predicts the existence of singularities, but in the early 1960s few took this prediction seriously. Physicists thought that a star of great enough mass undergoing gravitational collapse might form a singularity. Penrose had shown that if the universe obeys general relativity, it must.
fn1 Stars less massive than about 8 solar masses probably don’t shrink all the way to become black holes. Only more massive stars become black holes.
6
‘There is a singularity in our past’
PENROSE’S DISCOVERY THAT a star of great enough mass undergoing gravitational collapse must form a singularity set fire to Hawking. With Robert Geroch and Penrose, he began to extend ideas about singularities to other physical and mathematical cases.1 He was certain the discovery had significant implications for the beginning of the universe. This was exhilarating work, with the ‘glorious feeling of having a whole field virtually to ourselves’.2 Hawking realized that if he reversed the direction of time so that the collapse became an expansion, everything in Penrose’s theory would still hold. If general relativity tells us that any star that collapses beyond a certain point must end in a singularity, then it also tells us that any expanding universe must have begun as a singularity. For this to be true the universe must be like what scientists call a Friedmann model. What is a Friedmann model of the universe?
Stephen Hawking, His Life and Work Page 8
