The Book of Nothing: Vacuums, Voids, and the Latest Ideas about the Origins of the Universe - John D. Barrow (2002)

Chapter 6. Empty Universes

“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 presence 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.

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 order 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

Despite 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.

DEEP CONNECTIONS

“I love cosmology: there’s something uplifting about viewing the entire universe as a single object with a certain shape. What entity, short of God, could be nobler or worthier of man’s attention than the cosmos itself? Forget about interest rates, forget about war and murder, let’s talk about space.”

Rudy Rucker21

The first person to suggest that the cosmological constant might be linked to the rest of physics was the Belgian astronomer and Catholic priest, Georges Lemaître. Lemaître was one of the first scientists to take the idea of the expanding Universe seriously as a problem of physics. If the Universe was expanding, then he realised that it must have been hotter and denser in the past: matter would be transformed into heat radiation if cosmic events were traced far enough into the past.

Lemaître rather liked Einstein’s lambda force and found several new solutions of Einstein’s equations in which it featured. He was persuaded that it needed to be present in Einstein’s theory but, unlike Einstein, who tried to forget about it, and some other astronomers, who assumed that even if it existed it was negligible, he wanted to reinterpret it. Lemaître realised22 that although Einstein had added the lambda force to the geometrical side of his equations, it was possible to shift it across to the matter and energy side of the equation

{geometry} = {distribution of mass and energy} − {Λ energy},

and reinterpret it as a contribution to the material content of the Universe,

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

If you do this then you have to accept that the Universe always contained a strange fluid whose pressure is equal to minus its energy density. A negative pressure is just a tension which is not unusual, but the lambda tension is as negative as it could possibly be and this means that it exerts a gravitational effect that is repulsive.23

Lemaître’s insight was very important because he saw that by interpreting the cosmological constant in this way it might be possible to understand how it originated by studying the behaviour of matter at very high energies. If those investigations could identify a form of matter which existed with this unusual relation between its pressure and its energy density then it would be possible to link our understanding of gravity and the geometry of the Universe to other areas of physics. It was also important for the astronomical concept of the vacuum. If we ignored the possibility of Einstein’s cosmological constant then it appeared that there could exist vacuum universes devoid of any ordinary matter. But if the cosmological constant is really a form of matter that is always present then there really are no true vacuum universes. The ethereal lambda energy is always there, acting on everything but remaining unaffected by the motion and presence of other matter.

Unfortunately, no one seems to have taken any notice of Lemaître’s remark even though it was published in the foremost American science journal of the day. The early nuclear and elementary-particle physicists never found anything in their theories of matter that looked compellingly like the lambda stress. Its image amongst cosmologists ebbed and flowed. The Second World War intervened and changed the direction of physics towards nuclear processes and radio waves. Soon after it ended the interest of cosmologists was captured by the novel steady-state theory of the Universe first proposed by Fred Hoyle, Hermann Bondi and Thomas Gold. Like Friedmann’s universes the steady-state universe expanded, but its density did not diminish with time. In fact, none of its gross properties changed with time. This steadiness was achieved by means of a hypothetical ‘creation’ process that produced new matter everywhere at a rate that exactly counterbalanced the dilution due to expansion. The rate required is imperceptibly small, just a few atoms appearing in each cubic metre every ten billion years. In contrast to the Big Bang24models, the steady-state theory had no apparent beginning when everything came into being at once. Its creation was continual.

At first, it looked as if this cosmological theory required a new theory of gravity to supersede Einstein’s. It needed to include a new ‘creation field’ that could generate the steady trickle of new atoms and radiation needed to maintain the constant density of the Universe. In 1951, William McCrea,25 a British astrophysicist, showed that nothing so radical was required. The creation field could be added into Einstein’s equations as an extra source of energy and mass. And when it was, it looked just like the lambda term. No continual creation was needed.

Sadly for its enthusiastic inventors, the steady-state universe was soon consigned to the history books. It was a good scientific theory because it made very definite predictions: the Universe should look, on average, the same at all epochs. This made it extremely vulnerable to observational test. In the late 1950s, astronomers started to amass evidence that the Universe was not in a steady state. The population of galaxies of different sorts changed significantly over time. Quasars were discovered to populate the Universe more densely in the past than today. Finally, in 1965, the remnant heat radiation from a hot past Big Bang state was detected by radio astronomers and modern cosmology was born.

During the mid-1960s, when the first quasars were discovered with redshifts clustered around a single value, it was proposed that a large enough lambda stress might have been able to slow the expansion of the Universe temporarily in the past when it was about a third of its present extent. This could have led to a build-up of quasar formation close to this epoch. However, this idea faded away as more and more quasars were found with larger redshifts and it began to be appreciated how the apparent confinement of their redshifts to lie below a particular value was an artefact of the methods used to search for them.

Since that time observational astronomers have been searching for definitive evidence to determine whether the Universe is expanding fast enough to continue expanding for ever or whether it will one day reverse into contraction and head for a big crunch. If a lambda force exists that is large enough to dominate the attractive force of gravity over very large extragalactic distances, then it should affect the expansion of the Universe in the way shown in Figure 6.5. The most distant clusters of galaxies should be accelerating away from one another rather than continually decelerating as they expand.

Figure 6.5 The effect of lambda on the expansion of the Universe. When it becomes larger than the inverse square force of gravity it causes the expansion of the Universe to switch from deceleration to acceleration.

The search for this tell-tale cosmic acceleration needs ways to measure the distances to faraway stars and galaxies. By looking at the change in the pattern of light colours coming from these objects we can easily determine how fast they are expanding away from us. This can now be done to an accuracy of a few parts in a million. But it is not so easy to figure out how far away they are. The basic method is to exploit the fact that the apparent brightness of a light source falls off as the inverse square of its distance away from you, just like the effect of gravity. So if you had a collection of identical 100-watt light bulbs located at different distances from you in the dark, then their apparentbrightnesses would allow you to determine their distances from you, assuming there is no intervening obscuration. If you didn’t know the intrinsic brightness of the bulbs, but knew that they were all the same, then by comparing their apparent brightnesses you could deduce their relative distances: nine times fainter means three times further away.

This is just what astronomers would like to be able to do. The trouble is Nature does not sprinkle the Universe with well-labelled identical light bulbs. How can we be sure that we are looking at a population of light sources that have the same intrinsic brightnesses so that we can use their apparent brightnesses to tell us their relative distances?

Astronomers try to locate populations of objects which are easily identifiable and which have very well-defined intrinsic properties. The archetypal example was that of variable stars which possessed a pattern of change in their brightness that was known theoretically to be linked to their intrinsic brightness in a simple way. Measure the varying light cycle, deduce the intrinsic brightness, measure the apparent brightness, deduce the distance away, measure the spectral light shift, deduce the speed of recession and voilà, you can trace the increase of speed with distance and see the expansion of the Universe, as Edwin Hubble first did in 1929 to confirm Friedmann’s prediction from Einstein’s theory that the Universe is expanding, as shown in Figure 6.6.

Unfortunately, these variable stars cannot be seen at great distances and ever since Hubble’s work, the biggest problem of observational astronomy has been determining distances accurately. It is the twentieth-century analogue of John Harrison’s27 eighteenth-century quest to measure time accurately so that longitude could be determined precisely at sea. Until quite recently, it has made any attempt to map out the expansion of the Universe over the largest extragalactic dimensions too inaccurate to use as evidence for or against the existence of Einstein’s lambda force. We could not say for sure whether or not the present-day expansion of the Universe is accelerating. Absence of evidence was taken as evidence of absence – and in any case it seemed to require a huge coincidence if the lambda force just started to accelerate the Universe at an epoch when human astronomers were appearing on the cosmic scene. Moreover, this would require lambda to have a fantastically small value. Better, they argued, to assume that it is really zero and keep on looking for a good reason why.

Figure 6.6 Hubble’s Law:26 the increase of the speed of recession of distant sources of light versus their distance from us.

In the last year things have changed dramatically. The Hubble Space Telescope (HST) has revolutionised observational astronomy and, from its vantage point above the twinkling distortions of the Earth’s atmosphere, it is now possible to see further than ever before. Telescopes on the ground have also advanced to achieve sensitivities undreamt of in Hubble’s day. New electronic technologies have replaced the old photographic film with light recorders that are fifty times more sensitive at catching light than film. By combining the capability of ground-based telescopes to survey large parts of the sky and the HST’s ability to see well-targeted, small, faint sources of light with exquisite clarity, a new measure of distance has been found.

Observers use powerful ground-based telescopes to monitor nearly a hundred pieces of the night sky, each containing about a thousand galaxies, at the time of the New Moon, when the sky is particularly dark. They return three weeks later and image the same fields of galaxies, looking for stars that have brightened dramatically in the meantime. They are looking for faraway supernovae: exploding stars at the ends of their life cycles. With this level of sky coverage, they will typically catch about twenty-five supernovae as they are brightening. Having found them, they follow up their search with detailed observations of the subsequent variation of the supernova light, watching the increase in the brightness to maximum and the ensuing fall-off back down to the level prior to explosion, as shown in Figure 6.7. Here, the wide sky coverage of ground-based telescopes can be augmented by the HST’s ability to see faint light and colours.

The detailed mapping of the light variation of the supernovae enables the astronomers to check that these distant supernovae have the same light signature as ones nearby that are well understood. This family resemblance enables the observers to determine the relative distances of the distant supernovae with respect to the nearby ones from their apparent peak brightnesses, because their intrinsic brightnesses are roughly the same. Thus a powerful new method of determining the distances to the supernovae is added to the usual Doppler shift measurements of their spectra from which their speeds of recession are found. This gives a new and improved version of Hubble’s law of expansion out to very great distances.

Figure 6.7 A supernova light-curve. The variation in the observed brightness of a supernova, showing the characteristics to a maximum and gradual fall back to the level prior to the explosion.

The result of these observations of forty distant supernovae by a combination of observations from the Earth and by the Hubble Space Telescope by two separate international teams of astronomers is to provide strong evidence that the expansion of the Universe is accelerating. The striking feature of the observations is that they require the existence of the cosmological constant, or lambda force. The probability that these observations could be accounted for by an expanding universe that is not accelerating is less than one in a hundred. The contribution of the vacuum energy to the expansion of the Universe is most likely28 to be fifty per cent more than that of all the ordinary matter in the Universe.

The variation of the redshifting of light with distance for the sources of light cannot be made to agree with the pattern predicted if lambda does not exist. The only way of escape from lambda is to appeal to a mistake in the observations or the presence of an undetected astronomical process creating a bias in the observations, changing the apparent brightnesses of the supernovae so that they are not true indicators of distance, as assumed. These last two possibilities are still very real ones and the observers are probing every avenue to check where possible errors might have crept in. One worry is that the assumption that the supernovae are intrinsically the same as those we observe nearby is wrong.29 Perhaps, when the light began its journey to our telescopes from these distant supernovae, there were other varieties of exploding star which are no longer in evidence. After all, when we look at very distant objects in the Universe we are seeing them as they were billions of years ago when the light first left them en route to our telescopes. In that distant past the Universe was a rather denser place, filled with embryonic galaxies and perhaps rather different than it appears today. So far, none of these possibilities has withstood detailed cross-checking.

If these possible sources of error can be excluded and the existing observations continue to be confirmed in detail by different teams of astronomers using different ways of analysing different data, as is so far the case, then they are telling us something very dramatic and unexpected: the expansion of the Universe is currently controlled by the lambda stress and it is accelerating. The implications of such a state of affairs for our understanding of the vacuum and its possible role in mediating deep connections between the nature of gravity and the other forces of Nature are very great.

So far we have seen what the astronomers thought about lambda and its possible role as the ubiquitous vacuum energy that Lemaître suggested. During the last seventy years, the study of the subatomic world has gathered pace and focus. It was also in search of the vacuum and its simplest possible contents. The discovery of a cosmic vacuum energy by astronomical telescopes turns out to have profound implications for that search too, and it is to this thread of the story that our attention now turns. It will start in the inner space of elementary particles and bring us, unexpectedly, full turn back to the outer space of stars and galaxies in our quest to understand the vacuum and its properties.