The Book of Nothing Page 17

  “Pure mathematics consists entirely of such asseverations as that, if such and such a proposition is true of anything, then such and such another proposition is true of that thing. It is essential not to discuss whether the first proposition is really true, and not to mention what the anything is of which it is supposed to be true … If our hypothesis is about anything and not about some one or more particular things, then our deductions constitute mathematics. Thus mathematics may be defined as the subject in which we never know what we are talking about, nor whether what we are saying is true.”38

  Pure mathematics became the first of the ancient subjects to free itself of metaphysical shackles. Pure mathematics became free mathematics. It could invent ideas without recourse to correspondence with anything in the worlds of science, philosophy or theology. Ironically, this renaissance emerged most forcefully not with the plurality of zeros that it spawned, but with the plethora of infinities that Georg Cantor unleashed upon the unsuspecting community of mathematicians. The ancient prejudice that there could be potential infinities, but never actual infinities, was ignored. Cantor introduced infinities without end in the face of howls of protest by conservative elements in the world of mathematics. Cantor was eventually driven into the deep depression that overshadowed the end of his life, yet he vigorously maintained the freedom of mathematicians to invent what they will:

  “Because of this extraordinary position which distinguishes mathematics from all other sciences, and which produces an explanation for the relatively free and easy way of pursuing it, it especially deserves the name of free mathematics, a designation which I, if I had the choice, would prefer to the customary ‘pure’ mathematics.”39

  These free-spirited developments in mathematics marked the beginning of the end for metaphysical influences on the direction of the mathematical imagination. Nothingness was unshackled from zero, leaving the vagueness of the void and the vacuum behind. But there were more surprises to come. The exotic mathematical structures emerging from the world of pure mathematics may have been conceived free from application to Nature, but something wonderful and mysterious was about to happen. Some of those same flights of mathematical fancy, picked out for their symmetry, their neatness, or merely to satisfy some rationalist urge to generalise, were about to make an unscheduled appearance on the stage of science. The vacuum was about to discover what the application of the new mathematics had in store for time and space and all that’s gone before.

  “You cannot have first space and then things to put into it, any more than you can have first a grin and then a Cheshire cat to fit on to it.”

  Alfred North Whitehead

  DEALING WITH ENTIRE UNIVERSES ON PAPER

  “I always think love is a little like cosmology. There’s a Big Bang, a lot of heat, followed by a gradual drifting apart, and a cooling off which means that a lover is pretty much the same as any cosmologist.”

  Philip Kerr1

  The most spectacular intellectual achievement of the twentieth century is Einstein’s theory of gravity. It is known as the ‘general theory of relativity’ and supersedes Newton’s three-hundred-year-old theory. It is a generalisation of Newton’s theory because it can be used to describe systems in which objects move at a speed approaching that of light and in gravitational fields which are extremely strong.2 Yet, when applied to environments where speeds are low and gravity is very weak, it looks like Newton’s theory. In our solar system the distinctive differences between Newton and Einstein are equal to just one part in one hundred thousand, but these are easily detectable by astronomical instruments. Far from Earth, in highdensity astronomical environments, the differences between the predictions of Newton and Einstein are vastly larger, and so far our observations have confirmed Einstein’s predictions to an accuracy that exceeds the confirmation of any other scientific theory. Remarkably, the picture that Einstein has given us of the way in which gravity behaves, locally and cosmically, is the surest guide we have to the structure of the Universe and the events that occur within it.

  From this short prologue one could be forgiven for thinking that Einstein’s gravity theory is just a small extension of Newton’s, a little tweaking of his claim that the force between two masses falls off in proportion to the square of the distance separating their centres. Nothing could be further from the truth. Although in some situations the differences between the predictions of Einstein and Newton are very small, Einstein’s conceptions of space and time are radically different. For Newton, space and time were absolutely fixed quantities, unaffected by the presence of the bodies contained within them. Space and time provided the arena in which motion took place; Newton’s laws gave the marching orders.

  When gravity attracted different masses, it was supposed to act instantaneously through the space between them regardless of their separation. No mechanism was proposed by which this notorious ‘action at a distance’ could occur. Newton was as aware of this lacuna as anybody, but pushed ahead regardless with his simple and successful law of gravity because it worked so well, giving accurate predictions of the tides, the shape of the Earth, together with an explanation of many observed lunar, astronomical and terrestrial motions. Indeed, one could go along sweeping this problem under the rug, secure in the knowledge that it wasn’t creating any crises for human thought elsewhere, right up until the discovery of the special theory of relativity. Relativity predicted that it should be impossible to send information faster than the speed of light in a vacuum.3

  In 1915, Einstein solved the problem of how gravity acts in a novel way. He proposed that the structure of space and time is not fixed and unchanging like a flat table top; rather, it is shaped and distorted by the presence of mass and energy4 distributed within it. It behaves like a rubber sheet that forms undulations when objects are placed upon it. When mass and energy are absent, the space is flat. As masses are added, the space curves. If the masses are large, the distortion of the flat surface of space is large near to the mass but decreases as one moves far away. This simple analogy is quite suggestive. It implies that if we were to wiggle a mass up and down at a point on the rubber sheet so as to produce ripples, as on the surface of a pool of water, then the ripples would travel outwards like waves of gravity. Also, if one were to rotate a mass at one point on the rubber sheet, it would twist the sheet slightly, further away, dragging other masses around in the same direction. Both these effects occur in Einstein’s theory and have been observed.5 Einstein discovered two important sets of mathematical equations. The first, called the ‘field equations’, enable you to calculate what the geometry of space and time6 is for any particular distribution of matter and energy within it. The other, called the ‘equations of motion’, tell us how objects and light rays move on the curved space. And what they tell is beautifully simple. Things move so that they take the quickest route over the undulating surface prescribed by the field equations. It is like following the path taken by a stream that meanders down from the mountain top to the river plain below.

  This picture of matter curving space and curvaceous space dictating how matter and light will move has several striking features. It brings the non-Euclidean geometries that we talked about in the last chapter out from the library of pure mathematics into the arena of science. The vast collection of geometries describing spaces that are not simply the flat space of Euclid are the ones that Einstein used to capture the possible structures of space distorted by the presence of mass and energy. Einstein also did away with the idea of a gravitational force (although it is so ingrained in our intuitions that astronomers still use it as a handy way of describing the appearances of things), and with it the problematic notion of its instantaneous action at a distance. You see, in Einstein’s vision, the motions of bodies on the curved space are dictated by the local topography that they encounter. They simply take the quickest path that they can. When an asteroid passes near the Sun it experiences a region where the curvature of space is significantly distorted by the Sun’s presenc
e and will move towards the Sun in order to stay on a track that will minimise its transit time (see Figure 6.1). To an observer just comparing their relative positions it looks as if the planet is attracted to the Sun by a force. But Einstein makes no mention of any forces: everything moves as if acted upon by no forces and so moves along a path that is the analogue of a straight line in flat Euclidean space. Moving objects take their marching orders from the local curvature of space, not from any mysterious long-range force of gravity acting instantaneously without a mechanism.

  Figure 6.1 Bodies that move take the quickest route between two points on a curved surface.

  Einstein’s theory had a number of spectacular successes soon after it was first proposed. It explained the discrepancy between the observed motion of the planet Mercury and that predicted by Newton’s theory, and successfully predicted the amount by which distant starlight would be deviated by the Sun’s gravity en route to our telescopes. Yet its most dramatic contribution to our understanding of the world was the ability it gave us to discuss the structure and evolution of entire universes, even our own.

  Every solution of Einstein’s field equations describes an entire universe – what astronomers sometimes call a ‘spacetime’. At each moment of time a solution tells us what the shape of space looks like. If we stack up those curved slices then they produce an unfolding picture of how the shape of space evolves in response to the motion and interaction of the mass and energy it contains. This stack is the spacetime.7 The field equations tell us the particular map of space and the pattern of time change created by a given distribution of mass and energy. Thus a ‘solution’ of the equations gives us a matching pair: the geometry that is created by a particular distribution of mass and energy, or conversely, the curved geometry needed to accommodate a specified pattern of mass and energy. Needless to say, Einstein’s field equations are extremely difficult to solve and the solutions that we know always describe a distribution of matter and a geometry that has certain special and simplifying properties. For example, the density of matter might be the same everywhere (we say it is homogeneous in space), or the same in every direction (we say it is isotropic), or assumed to be unchanging in time (static). If we don’t make one of these special assumptions we have to be content with approximate solutions to the equations which are valid when the distribution is ‘almost’ homogeneous, ‘almost’ isotropic, almost ‘static’ or changes in a very simple way (rotating at a steady speed, for example). Even these simpler situations are mathematically very complicated and make Einstein’s theory extremely difficult to use in all the ways one would like. Often, supercomputing capability is required to carry out studies of how very realistic configurations, like pairs of stars, will behave. This complexity is, however, not a defect of Einstein’s theory in any sense. It is a reflection of the complexity of gravitation. Gravity acts on all forms of mass and energy, but energy comes in a host of very different forms that behave in peculiar ways that were not known in Newton’s day. Worst of all, gravity gravitates. Those waves of gravity that spread out, rippling the curvature of space, carry energy too and that energy acts as a source for its own gravity field. Gravity interacts with itself in a way that light does not.8

  VACUUM UNIVERSES

  “… and he shall stretch out upon it the line of confusion, and the stones of emptiness. They shall call the nobles thereof to the kingdom, but none shall be there.”

  Isaiah9

  The fact that the solutions of Einstein’s theory describe whole universes is striking. Some of the first solutions that were found to his field equations provided excellent descriptions of the astronomical universe around us that telescopes would soon confirm. They also highlighted a new concept of the vacuum.

  We have seen that Einstein’s equations provide the recipe for calculating the curved geometry of space that is created by a given distribution of mass and energy in the Universe. From this description one might have expected that if there were no matter or energy present – that is, if space was a perfectly empty vacuum in the traditional sense – then space would be flat and undistorted. Unfortunately, things are not so simple. A geometry that is completely flat and undistorted is indeed a solution to the equations when there is no mass and energy present, as one would expect. But there are many other solutions that describe universes containing neither mass nor energy but which have curved spatial geometries.

  These solutions of Einstein’s equations describe what are called ‘vacuum’ or ‘empty’ universes. They describe universes with three dimensions of space and one of time, but they can be imagined more easily if we forget about one of the dimensions of space and think of worlds with just two dimensions of space at any moment of time, like a table top, but not necessarily flat, so rather like a trampoline. As time flows the topography of the surface of space can change, becoming flatter or more curved and contorted in some places. At each moment of time we have a different ‘slice’ of curved space.10 If we stack them all up in a pile then we create the whole spacetime, like making a lump of cheese out of many thin slices (see Figure 6.2). If one picked any old collection of slices and stacked them up, they would not fit together in a smooth and natural way that would correspond to a smooth flow of events linked by a chain of causes and effects. That’s where Einstein’s equations come in. They guarantee that this stacking will make sense if the ingredients solve the equations.11

  This is all very well, but having got a picture of how Einstein’s theory works by imagining that the presence of mass and energy creates curvatures in the geometry of space and changes in the rate of flow of time, shouldn’t empty universes all be flat? If they contain no stars, planets and atoms of matter, how can space be curved? What is there to do the curving?

  Figure 6.2 Spacetime is composed of a stack of slices of space, each one labelled by a moment of time. Only two of the dimensions of space are shown.

  Einstein’s theory of gravity is much larger than Newton’s. It does away with the idea that the effects of gravity are instantaneously communicated from one side of the Universe to the other and incorporates the restriction that information cannot be sent at speeds faster than that of light. This allows gravity to spread its influence by means of waves travelling at the speed of light. These gravitational waves were predicted to exist by Einstein and there is little doubt that they do exist. Although they are too weak to detect directly on Earth today, their indirect effects have been observed in a binary star system containing a pulsar. The pulsar is like a lighthouse beam spinning at high speed. Every time it comes around to face us we see a flash. Its rotation can be very accurately monitored by timing observations of its periodic pulses. Twenty years of observations have shown that the pulsing of the binary pulsar is slowing at exactly the rate predicted if the system is losing energy by radiating gravitational waves at the rate predicted by Einstein’s theory (see Figure 6.3).

  In the next few years, ambitious new experiments will attempt to detect these waves directly. They are like tidal forces in their effects. When a gravitational wave passes through the page that you are reading it will slightly stretch the book sideways and squeeze it longways without changing its volume. The effect is tiny but with elaborate apparatus, similar to the interferometer used by Michelson to test the existence of the ether, we may be able to detect gravitational waves from violent events far away in our own galaxy and beyond. The prime candidates for detection are waves from very dense stars or black holes that are in the final throes of circling each other in orbits that are getting closer and closer to each other. In the end they will spiral together and merge in a cataclysmic event that produces huge amounts of outgoing light and gravitational waves. In the far future the binary pulsar will collapse into this state and provide a spectacular explosion of gravitational waves.

  Figure 6.3 The Binary Pulsar PSR1913+16, one of about 50 known systems of this type. It contains two neutron stars orbiting around one another. One of the neutron stars is a pulsar and emits pulses of radio
waves which can be measured to high precision. These observations show that the orbital period of the pulsar is changing by 2.7 parts in a billion per year. This is the change predicted by the general theory of relativity due to the loss of energy by the radiation of gravitational waves from the neutron stars.12

  If we imagine a space that has had its geometry distorted by the presence of a large mass then we can see how gravitational waves can alter the picture. Suppose that the mass starts changing shape in a way that makes it non-spherical. The changes create ripples in the geometry which spread out through the geometry, moving away from the mass. The further away one is from the source of this disturbance the weaker will be the effect of the ripples when they reach you. Although we talk about these waves as if they are a form of energy, like sound waves, that have been introduced into the Universe, they are really rather different in character. They are an aspect of the geometry of space and time. If we take away the changing mass that is generating the ripples in the geometry of space we can still have such ripples present. The whole Universe can be expanding in a non-spherical way, slightly faster in one direction than in another, and very long-wavelength gravitational waves will be present to support the overall tension in the expanding ‘rubber sheet’ that is the universe of space at any moment.