The Book of Nothing Read online

  The fact that Einstein’s theory of gravity allows one to find precise descriptions of universes which expand, like our own, but which contain no matter does not mean that such universes are realistic. Einstein’s theory is remarkable in that it describes an infinite collection of possible universes of all shapes and sizes according to the distribution and nature of the matter you care to put into them. One of the simplest solutions of Einstein’s equations, which does contain matter, and expands at the same rate in every direction and at every place, gives an extremely accurate description of the behaviour of our observed Universe. The biggest problem facing cosmologists is to explain why this solution is selected to pass from being a mere theoretical possibility to real physical existence. Why this simple universe and not some other solution of Einstein’s equations?

  We expect that there is more to the Universe than we can learn from Einstein’s equations alone. Linking Einstein’s theory up to our understanding of the most elementary particles of matter may place severe restrictions on which curved spaces are physically possible. Or it may be that strange forms of matter existed in the early stages of the Universe’s history which ensure that all, or almost all, of the complicated universes which Einstein’s equations permit end up looking more and more like the simple isotropically expanding state that we see today if you wait for billions of years.

  ERNST MACH – A MAN OF PRINCIPLE

  “It is easy to be certain. One has only to be sufficiently vague.”

  Charles Sanders Peirce13

  Einstein’s own views about empty universes played an important role in his conception and creation of the general theory of relativity. Over a long period of time, his thinking about the defects in Newton’s theory and how to repair them had been much influenced by the physicist and philosopher of science, Ernst Mach (1838–1916). Mach had wide interests and made important contributions to the study of sound. Aerodynamicists invariably label high velocities by their ‘Mach number’, that is the value of the speed in units of the speed of sound (about 750 miles per hour). Yet in some respects Mach was something of a Luddite and opposed the concept of atoms and molecules as basic components of matter on philosophical grounds even after there was direct experimental evidence for their existence. Nonetheless, Einstein had been greatly impressed by Mach’s famous text on mechanics14 and it played an important role in guiding him to formulate both the special and general theories of relativity in the ways that he did. As a consequence, Einstein was much influenced by another of Mach’s convictions about the origin of the inertia of local objects, a view that has since become known as ‘Mach’s Principle’. Mach believed that the inertia and mass of the objects we see around us should be a consequence of the collective effect of the gravitational field of all the mass in the Universe. When Einstein conceived of his general theory of relativity, with the presence of mass and energy creating curvature, he hoped and believed that Mach’s idea was automatically built into it. Alas, it was not. Mach’s Principle boiled down to requiring that there were no vacuum solutions of Einstein’s theory: no universes where the geometry of space and time was curved by gravitational waves alone, rather than by the presence of mass and energy. Gravitational waves were allowed to exist, but they had to arise from the movement of irregular distributions of matter. There could not exist wavelike ripples in the geometry of space that were built into the Universe when it came into being or which were associated solely with disparities in its expansion rate from one direction to another.

  The most dramatic type of motion of the Universe, not associated with matter, that Mach and Einstein needed to veto was an overall cosmic rotation. For a long time Einstein believed that his theory ensured this, but he got a surprise. In 1952, the logician Kurt Gödel, his colleague at the Institute for Advanced Study in Princeton, discovered a completely unexpected solution of Einstein’s equations that described a rotating universe. More dramatic still, this possible universe permitted time travel to take place! Subsequent investigation showed that this solution of Einstein’s equations was peculiar and could not describe our Universe. However, the genie was out of the bottle. Maybe there were other solutions which possessed the same properties but which were much more realistic? Or maybe Mach was right and we just haven’t found the right way to formulate his ‘Principle’ when we look for solutions of Einstein’s equations. For no overall rotation of the Universe has ever been detected. Some years ago, some of us15 used astronomical observations of the isotropy of the intensity of radiation in the Universe to show that if it is rotating then it must be rotating at a rate that is between one million and ten million million times slower than the rate at which the Universe is expanding.16

  Mach’s Principle reflects older ideas about the undesirability of a vacuum. It is largely ignored in modern cosmology, not least because it is rather difficult to get everyone to agree on a precise statement of the Principle. Many scientists have tried to modernise it to see if it can be used as a way of selecting out some of the solutions of Einstein’s equations as physically realistic but none of the proposals has caught on. Even if they did, it is not clear what Mach’s Principle would tell us that we could not learn in other ways. It is all very well to say that gravitational fields must all arise from sources of matter, with no free gravitational waves left over from the Big Bang, but why should such a state of affairs exist? If our Universe was dominated by the presence of very strong sourceless gravitational waves then its expansion would behave very differently. It would expand at quite different rates in different directions and it might rotate as fast as it expands. Our observations show us that neither of these scenarios exists in the Universe today. The expansion of the Universe is the same in every direction to an accuracy of one part in one hundred thousand.

  Mach’s Principle faded from the stage because it could not supply an answer to the question ‘Why is the Universe like it is today?’ Later, we shall see that other ideas have been able to come up with more compelling reasons for the lack of measurable effects by sourceless gravitational waves in the universes today. They do not stipulate that those waves cannot exist, as Mach would have decreed, but show that they are inevitably very weak when the universe is old and have negligible effects upon the overall expansion of the Universe.

  LAMBDA – A NEW COSMIC FORCE

  “If an elderly but distinguished scientist says that something is possible he is almost certainly right, but if he says that it is impossible he is very probably wrong.”

  Arthur C. Clarke

  When Albert Einstein first began to explore the cosmological consequences of his new theory of gravity, in 1915, our knowledge of the scale and diversity of the astronomical universe was vastly smaller than it is today. There was no reason to believe that there existed galaxies other than our own Milky Way. Astronomers were interested in stars, planets, comets and asteroids. Einstein wanted to use his equations to describe our whole Universe but they were too complicated for him to solve without some simplifying assumptions. Here he was very fortunate. He assumed something about the Universe that certainly makes life easy for the mathematician but which might well not have been an appropriate assumption to make about the real Universe. The observational evidence simply did not exist. Einstein’s simplifying assumption was that the Universe is the same in every place and in every direction at any moment of time. We say that it is homogeneous and isotropic. Of course, it is not exactly so. But the assumption is that it is so close to being so that the deviations from perfect uniformity are too small to make any significant difference to the mathematical description of the whole Universe.17

  As Einstein continued he found that his equations were telling him something very peculiar and unexpected: the Universe had to be constantly changing. It was impossible to find a solution for a universe which contained a uniform distribution of matter, representing the distant stars, which remained on average the same for long periods of time. The stars would attract one another by the force of their gravity. In orde
r to avoid a contraction and pile-up of matter in a cosmic implosion, there would need to be an outward motion of expansion to overcome it – an ‘expanding’ universe.

  Einstein didn’t like either of these alternatives. They were both contrary to the contemporary conception of the Universe as a vast unchanging stage on which the motions of the celestial bodies were played out. Stars and planets may come and go, but the Universe should go on for ever. Faced with this dilemma of a contracting or an expanding universe, he returned to his equations and searched for an escape clause. Remarkably, he found one.

  To see how this happened we must first see something of what led Einstein to his original equations. His equations relating the geometry of curved space to the material content of space have a particular form:

  {geometry} = {distribution of mass and energy}.

  All sorts of formulae describing the shapes of surfaces are possible in principle on the left-hand side of this equation. But if they are going to be equated to realistic distributions of matter and radiation, with properties like density, velocity and pressure, then they must reflect the fact that quantities like energy and momentum have to be conserved in Nature. They can be reshuffled and redistributed in all sorts of ways when interactions occur between different objects, but when all the changes are complete and all the energies and momenta are finally added up they must give the same sums that they did at the start. This requirement, that energy and momentum be conserved in Nature, was enough to guide Einstein to the simplest geometrical ingredients on the left-hand side of his equations.

  Everything seemed to fit together beautifully. If he looked at the situation where gravity was very weak and speeds were far less than that of light, so that the deviations in the geometry of space from perfect Euclidean flatness were tiny, then these complex equations miraculously turned into the self-same law of gravity that Newton had discovered more than 230 years earlier. This law was called the ‘inverse-square law’ because it dictated that the gravitational force between two masses falls inversely as the square of the distance between their centres.

  Unfortunately, it was this elegant picture that stubbornly refused to allow the Universe to be unchanging. Faced with an expanding universe, Einstein saw a way out. His desire to make his theory turn into Newton’s when gravity became very weak, and space was nearly flat, had led him to ignore a strange possibility. The parts of his equations storing the in-formation about the geometry allowed another simple piece to be added to them without altering the requirement that they allow energy and momentum to be conserved in Nature. When one looked at what this new addition would do to Newton’s description of weak gravitational fields, the result looked very odd. It said that Newton’s inverse-square law was only half the story; there was really another piece to be added to it: a force between all masses that increased in proportion to the distance of their separation. As one looked out to astronomical distances this extra force of gravity should overwhelm the effects of Newton’s decreasing inverse-square law.

  Einstein introduced the Greek symbol lambda, Λ, to denote the strength of this force in his equations, so that schematically they became:

  {geometry} + {Λ force} = {distribution of mass and energy}.

  Nothing in his theory could tell him how large a number lambda was, or even whether lambda was positive or negative. Indeed, an important reason to keep it in his equations was that, equally, there was no reason why its value should be zero either. Lambda was a new constant of Nature, like Newton’s gravitation constant, G, which determined the strength of the attractive, inverse-square part of the gravitational force. Einstein called lambda the ‘cosmological constant’.

  Einstein saw that if lambda was positive then its repulsive contribution to the overall force of gravity would be opposite to the attractive character of Newton’s force. It would cause distant masses to repel one another. He realised that if its value was chosen appropriately it could exactly counterbalance the gravitational attraction of the inverse-square law and so allow a universe of stars to be static, neither expanding nor contracting. The fact that we did not see any evidence on Earth for this lambda force was easily explained. The value of lambda required to keep the Universe static was very small, so small that its consequences on Earth would be far too small to have any perceptible effect on our measurements of gravity. This situation arose because the force increased with distance. It could be large over astronomical dimensions where it controlled the overall stability of the Universe, yet be very small over the small distances encountered on the surface of the Earth or in the solar system.

  What happened next was something of an embarrassment for Einstein. He believed that his static universe was the only type of solution that his new equations permitted for the Universe. However, he was not the only person studying his equations.

  Alexander Friedmann was a young meteorologist and applied mathematician working in St Petersburg. He followed new developments in mathematical physics closely and was one of the very first scientists to understand the mathematics behind Einstein’s new theory of gravity. This was a remarkable achievement. Einstein’s theory used parts of mathematics that were highly abstract and which had never been used in physics before. Astronomers were, for the most part, practically inclined physicists rather than mathematical specialists, and ill equipped to understand Einstein’s theory at a level that enabled them to check his calculations and go on and do new ones. Friedmann was different. He assimilated the mathematics required very quickly and was soon finding new solutions of Einstein’s equations which Einstein himself had missed.18 He found the expanding and contracting solutions that Einstein had tried to suppress by introducing the lambda term. The three varieties of expanding universe are shown in Figure 6.4. But he also found something more interesting. Even with Einstein’s lambda force added to the equations, the Universe would not remain static. The solution that Einstein found in which the attractive force of gravity exactly balanced the new repulsive lambda force did exist. But it would not persist. It was unstable. Like a needle balanced on its point, if nudged in any direction, it would fall. If Einstein’s static universe possessed the slightest irregularity in its density, no matter how small, it would begin to expand or contract. Friedmann confirmed this by showing that even when the lambda force was present there were solutions to Einstein’s equations which described expanding universes. Following these calculations to their logical conclusion, Friedmann made the greatest scientific prediction of the twentieth century: that the whole Universe should be expanding.19

  Friedmann wrote to Einstein to tell him that there were other solutions to his equations but Einstein didn’t pay close attention, believing Friedmann’s calculations to be mistaken. Soon afterwards one of Friedmann’s more senior colleagues went to Berlin on a lecture tour with the added purpose of discussing Friedmann’s calculations with Einstein. Einstein was rapidly persuaded that it was he, rather than Friedmann, who was mistaken; he had completely overlooked the new solutions to his equations. Einstein wrote to announce that Friedmann was correct and the static universe was dead. Years later, Einstein would describe his invention of the cosmological constant to sustain his belief in a static universe as ‘the biggest blunder of my life’.

  Figure 6.4 The three universes discovered by Friedmann. The open and critical cases increase in spatial extent for ever; the closed case eventually collapses back to a state of maximum compression. The critical trajectory is the dividing line between infinite and finite future histories.

  In 1929, astronomers finally established that the Universe is indeed expanding, just as Friedmann had predicted, and Friedmann’s solutions of Einstein’s equations, both with and without the lambda force, still provide the best working descriptions of that expansion today. Friedmann never lived to see how far-reaching his ideas had been. Tragically, he died when only thirty-five years old after failing to recover from the effects of a highaltitude balloon flight to gather meteorological data.20

  Despit
e the debacle of the static universe, the lambda force lived on. Einstein’s logic that led to its inclusion in his equations was inescapable, even if the desideratum of a static universe was not. Lambda might be so small in value that its effects are negligible even over astronomical distances and its presence ignored for all practical purposes, but there was no reason just to leave it out of the theory. Observations soon showed that if it existed it must be very small. But why should it be so small? Einstein’s theory told astronomers nothing about its magnitude or its real physical origin. What could it be? These were important questions because their answers would surely tell us something about the nature of the vacuum. For even if we expunged all the matter in the Universe the lambda force could still exist, causing the Universe to expand or contract. It was always there, acting on everything but unaffected by anything. It began to look like an omnipresent form of energy that remained when everything that could be removed from a universe had been removed, and that sounds very much like somebody’s definition of a vacuum.