Cosmology_A Very Short Introduction

by Peter Coles

  Following this lead, we can ask what kind of path light rays follow according to the general theory of relativity. In Euclidean geometry, light travels on straight lines. We can take the straightness of light paths to mean essentially the same thing as the flatness of space. In special relativity, light also travels on straight lines, so space is flat in this view of the world too. But remember that the general theory applies to accelerated motion, or motion in the presence of gravitational effects. What happens to light in this case?

  Let us go back to the thought experiment involving the lift. Instead of a spring with a weight on the end, the lift is now equipped with a laser beam that shines from side to side. The lift is in deep space, far from any sources of gravity. If the lift is stationary, or moving with constant velocity, then the light beam hits the side of the lift exactly opposite to the laser device that produces it. This is the prediction of the special theory of relativity. But now imagine the lift has a rocket which switches on and accelerates it upwards. An observer outside the lift who is at rest

  3. The bending of light. In (a), our lift is accelerating upwards, as in Fig. 2(c). Viewed from outside, a laser beam follows a straight line. In (b), viewed inside the lift, the light beam appears to curve downwards. The effect in a stationary lift situated in a gravitational field is the same, as we see in (c).

  sees the lift accelerate away, but if he could see the laser beam from outside it would still be straight. On the other hand, a physicist inside the lift notices something strange. In the short time it takes light to travel from one side of the lift to the other, the lift’s state of motion has changed. It has accelerated so it is moving faster when the light ends its journey than it was when the light started out. This means that the point at which the laser beam hits the other wall is slightly below the starting point on the other side. Seen by an observer inside, acceleration has ‘bent’ the light ray downwards.

  Now remember the case of the spring and the equivalence principle. What happens when there is no acceleration but there is a gravitational field is very similar to the accelerated lift. Consider now a lift standing on the Earth’s surface. The light ray must do much the same thing as in the accelerating lift: it bends downward. The conclusion we are led to is that gravity bends light. And if light paths are not straight but bent, then space is not flat but curved.

  One of the reasons we find curved space hard to understand is that we don’t observe it in everyday life. This is because gravity is so weak in commonplace circumstances. Even on the scale of our Solar System, gravity is so weak that the curvature it causes is negligible and light travels in lines that are so nearly straight that we can’t tell the difference. In these situations, Newton’s laws of motion are very good approximations to what happens. There are cases, however, where we must be prepared to deal with strong gravity and all that implies.

  Black holes and the Universe

  One example where Newton’s gravity breaks down is when a very large amount of matter is concentrated in a very small region of space. When this happens the action of gravity is so strong, and space so warped, that light is not merely bent but is trapped. Such an object is a black hole.

  4. The curvature of space. In the absence of a source of gravitation, light travels in a straight line. If a massive object is placed near the light path, the distortion of space produces a bent light ray.

  The idea that black holes might exist in nature dates back to John Michell, an English clergyman, in 1783, and was also discussed by Laplace. Such objects are, however, most commonly associated with Einstein’s theory of general relativity. Indeed, one of the first mathematical solutions of Einstein’s equations obtained, describes such an object. The famous ‘Schwarzschild’ solution was obtained only a year after the publication of Einstein’s theory in 1916 by Karl Schwarzschild, who died soon after on the eastern front in the First World War. The solution corresponds to a spherically symmetric distribution of matter, and it was originally intended that this could form the basis of a mathematical model for a star. It was soon realized, however, that for an object of any mass the Schwarzschild solution implied the existence of a critical radius (now called the Schwarzschild radius). If a massive object lies entirely within its Schwarzschild radius then no light can escape from the surface of the object. For the mass of the Earth the critical radius is only 1 cm, whereas for the Sun it is about 3 km. Making black holes involves compressing material to a phenomenal density.

  Since the pioneering work of Schwarzschild, research on black holes has been intense. Although there is as yet no truly watertight direct evidence for the existence of black holes in nature, there is a mountain of circumstantial evidence suggesting they might be lurking in many kinds of astronomical object. The intense gravitational field surrounding a black hole of about 100 million times the mass of the Sun is thought to be the engine that drives the enormous luminosity of certain types of galaxies. More recent observational studies of the dynamics of stars near the centre of galaxies indicate very strong mass concentrations that are usually identified with black holes with masses similar to this figure. It is now thought to be a serious possibility that nearly all galaxies have a black hole in their core. Black holes of much smaller mass may be formed at the end of the life of a star, when its energy source fails and it collapses in on itself.

  There is a great deal of interest nowadays in black holes, but they are not central to the development of cosmology so I shall not discuss them further in this book. Instead, in the next chapter I’ll discuss the role Einstein’s theory has played in understanding the behaviour of the Universe as a whole.

  Chapter 3

  First principles

  Einstein published his general theory of relativity in 1915. Almost immediately he sought to exploit this new theoretical framework to explain the large-scale behaviour of the entire cosmos. He was handicapped in his pursuit of this goal by the lack of information available to him about what it was that he was attempting to explain. What was the Universe really like? Einstein’s knowledge of astronomy was scanty, but he needed to know the answers to some fundamental questions before he could proceed. He knew that pure thought alone could not tell him what the Universe should look like and how it should behave. Observations and guesswork would have to guide him.

  Simplicity and symmetry

  There is no doubt that the general theory furnishes an elegant conceptual framework, as I have tried to explain using thought experiments and pictures. The harsh truth, however, is that it involves some of the most difficult mathematics ever applied to a description of nature. To give some idea of the complexity involved, it is useful to compare Einstein’s theory with the older Newtonian approach.

  In Newton’s theory of motion there is basically one mathematical equation to solve. This equation is ‘F = ma’, and it relates the force F on an object to the acceleration a of that object. It sounds simple enough, but in practice it can be overwhelmingly complicated to describe gravity using this approach. The reason is that every piece of matter in the Universe exerts a gravitational force on every other. It’s relatively easy to apply this idea to the motion of two interacting bodies, such as the Earth and the Sun, but if you start to add more bodies then things get very sticky. Indeed, while there is an exact mathematical solution to Newton’s theory for two orbiting bodies, there is no known general solution for any situation more complicated than this. Not even three bodies. Applying Newton’s theory to systems comprising large collections of gravitating objects is very difficult and usually requires the use of powerful computers to understand what is happening. The only exception is when the system involves some simplifying symmetry, such as a sphere, or has components that are distributed uniformly through space.

  Newton’s gravity is hard enough to apply in realistic situations, but Einstein’s theory is an absolute nightmare. For one thing, instead of Newton’s one equation, Einstein has no less than ten, which must all be solved simultaneously. And each separate equation is
much more complicated than Newton’s simple inverse-square law. Because of the equivalence between mass and energy given by E = mc2, all forms of energy gravitate. The gravitational field produced by a body is itself a form of energy, and it also therefore gravitates. This kind of chicken-and-egg problem is called ‘non-linearity’ by physicists, and it often leads to unmanageable mathematical complexity when it comes to solving the equations. This is the case for general relativity. Exact mathematical solutions of Einstein’s equations are very few and far between. Even with special symmetry the theory poses grand challenges for mathematicians and computers alike.

  Einstein knew that his equations were hard to solve, and that he would not be able to make much progress unless he assumed the Universe had some simplifying symmetry or uniformity. In 1915 relatively little was known for sure about the way in which the contents of the Universe were distributed. Many astronomers felt that the Milky Way was an ‘Island Universe’; others believed that it was just one of many such objects scattered more or less uniformly throughout space. The latter possibility appealed most to Einstein. The Milky Way is an ugly slab of gas, dust, and stars that would be very difficult to describe properly if it were the whole Universe. The second option was better in that it allowed a rough-and-ready description in which the Milky Way and other galaxies were the fine details in a largely smooth distribution of material. Einstein also had philosophical reasons for preferring large-scale smoothness, stemming from an idea called Mach’s principle. If the Universe were the same everywhere he could set his cosmological theory on a solid footing by allowing the distribution of matter to define a special reference frame that would help him deal with the effects of gravity.

  So, with precious little observational evidence to go on, Einstein decided that he would simplify the Universe he described by making it homogeneous (i.e. the same in every place); at least on scales much larger than the observed lumpy bits (i.e. the galaxies). He also assumed the Universe to be isotropic (i.e. looking the same in every direction). These twin assumptions together form the Cosmological Principle.

  The Cosmological Principle

  The twin assumptions of homogeneity and isotropy are related but not equivalent. Isotropy does not necessarily imply homogeneity without the additional assumption that the observer is not in a special place. One would observe isotropy in any spherically symmetric distribution of matter, but only if one were in the middle. A circular carpet with a pattern consisting of a series of concentric rings would look isotropic only to an observer standing in the centre of the pattern. The principle that we do not live in a special place in the Universe is called the Copernican Principle, an indication of the debt that modern cosmology owes to history. Observed isotropy, together with the Copernican Principle implies the Cosmological Principle. The Milky Way is clearly not isotropic, as anyone will know who has looked at the night sky. It occupies a distinct band across the heavens. A Universe consisting only of the Milky Way could therefore not be consistent with the Cosmological Principle.

  Although the name ‘Cosmological Principle’ sounds grand, one should have no illusions about its origin. More often than not, Principles are introduced in order to allow some progress to be made when one has no data to go on. Cosmology was no exception to this rule. It is now known that this guess was basically correct. In the 1920s it was established that the nebulae were definitely outside the Milky Way, and more recent observational studies of the large-scale distribution of galaxies and the cosmic microwave background (discussed in Chapter 7) seem to indicate the Universe is smooth on large scales as required by this idea. It is only more recently that astrophysicists have come up with a reasonably convincing argument as to why the Universe has this special symmetry. The mysterious origin of large-scale smoothness has been called the horizon problem and it is one of the issues addressed by the idea of cosmic inflation discussed in Chapter 8.

  Einstein’s biggest blunder

  Armed with the Cosmological Principle, Einstein was able to construct self-consistent mathematical models of the Universe. Immediately, however, he ran into a problem. It was an unavoidable consequence of his theory that, in any solution of his equations in which the Cosmological Principle applies, space-time must be dynamic. This meant that it was impossible for him to construct a model of a cosmos that is static and unchanging with time. His theory required the Universe to be either expanding or contracting, although it didn’t say which of these two possibilities would be the case. Einstein didn’t have a great knowledge of astronomy, but he had asked experts about the motions of distant stars. Perhaps because he asked the wrong question, he got the answer that on average the stars were neither approaching nor receding from the Sun. This is actually true inside our galaxy, but we now know it is not the case for other galaxies.

  Einstein was so convinced that the Universe should be static that he went back to his original equations. He realized that he could retain their essential character but introduce a slight modification that would counteract the tendency for his cosmological models to expand or contract with time. The modification he introduced was called the ‘cosmological constant’. This new term in the theory represents an alteration of the behaviour of gravity on the very largest scales. The cosmological constant allows space itself to possess a tendency to expand or contract, and it can be adjusted in the theory so that it exactly balances the expansion or contraction the Universe would otherwise be forced to possess.

  Satisfied with this fix for the time being, Einstein went on to construct a static cosmological model, which he published in 1917. Some years later, in 1929, Hubble published the results that led to the acceptance of the idea that the Universe was not static after all, but expanding. Einstein’s original model is now only of historical interest. Without the need to prevent global expansion, there was no need for him to have introduced the cosmological constant. In his later years, Einstein referred to this episode as the greatest blunder he had made in science. This comment is usually taken to refer to the cosmological constant itself, but the true blunder was to have failed to predict the expansion of the Universe.

  Although until recently most cosmologists were happy not to include it in their models, the cosmological constant never really went away. It lurked in the background like a mad relative living in the attic. Now, as we shall see in later chapters, it has broken free from obscurity and again plays a leading role. For the remainder of this chapter, however, I shall put it to one side.

  The Friedmann models

  Einstein was not the only scientist to turn to cosmology in the years immediately following the publication of the general theory of relativity in 1915. One of the others was an obscure Russian physicist by the name of Alexander Friedmann. It was not Einstein but Friedmann who developed the mathematical models of an expanding Universe, which form the basis of the modern Big Bang cosmology. His achievements in this respect are all the more remarkable because he performed his calculations in conditions of extreme hardship during the siege of Petrograd. Friedmann died in 1925, before his work (published in 1922) had achieved any international recognition. Stalin later liquidated the institute he had worked in. Somewhat later a Belgian priest, Georges Lemaître, independently obtained the same results, and it is through Lemaître that these ideas were explored and amplified in Western Europe.

  The simplest Friedmann models are the special family of solutions to Einstein’s equations obtained by requiring that the Cosmological Principle holds, and assuming that there is no cosmological constant. The Cosmological Principle plays a big part in these models. In relativity theory, time and space are not absolutes. The mathematical description of these two aspects of events (the ‘when’ and the ‘where’) involves a complicated four-dimensional ‘space-time’, which is hard to conceptualize. In general, Einstein’s theory does not give an unambiguous way of separating space and time. Different observers can disagree about the time elapsed between events, depending on their motion and on the gravitational fields they ha
ve experienced. If the Cosmological Principle applies then there is a special way to think about time that makes all this much simpler. If the Universe has the same density everywhere (which it must if it is homogeneous) then the density of matter itself defines a kind of clock. If the Universe expands, then the space between particles increases and the density of matter consequently goes down. The later the time, the lower the density of matter. Likewise, a higher density implies an earlier time. Observers anywhere in the Universe can set their clocks according to the local density of matter so that all these clocks will be perfectly synchronized. and a perfect synchronization can therefore be achieved. The measure of time that results is usually called ‘cosmological proper time’.

  Because the density is the same in every place, and it is the density of matter and/or energy that determines the curvature of space through the Einstein equations, the Cosmological Principle also simplifies the way space can curve in response to gravity. Space can be warped, but it must be warped in the same way at every point. There are in fact only three ways in which this can happen.

  The obvious way of having the same curvature at every point is to have no curvature at every point. This is usually called the flat universe. In a flat universe, light travels in straight lines and all the laws of Euclidean geometry apply just as they did in the ‘normal’ world. But if space isn’t curved, what has happened to gravity? There is matter in a flat universe so why does it not warp space? The answer is that the mass of the universe does warp space, but this is exactly counterbalanced by energy contained in the expansion of the Universe; matter and energy conspire to negate each other’s gravitational effects. And in any case, even though space may be flat, space–time is still curved.