by Peter Coles
The flat universe is clearly special because it requires an exact balance between the expansion and the gravitational pull of the matter. When these are not in balance, there are two other alternatives. If the universe has a high matter density then the gravitational effect of the mass within it wins, and it pulls space in on itself like a three-dimensional version of the surface of a sphere. Mathematically, the curvature of space is positive in such a situation. In this case, the closed universe, light rays actually converge on each other. While a flat universe can extend indefinitely in all directions, the closed universe is finite. Go off in one direction and you will come back to where you started from. The other alternative is the open universe. This too is infinite, but is harder to visualize than the closed version because the space curvature is negative. Light rays diverge in this example, as illustrated in the two-dimensional example shown in the Figure 5.
The behaviour of space in these models mirrors the way they evolve in time. A closed universe is a finite space, but it also has a finite duration. If the universe is expanding at any time and is closed, the expansion will slow down in the future. Eventually the universe will stop expanding and recollapse. The open and flat models will expand for ever. Gravity always fights the expansion of the universe in the Friedmann models, but only in the closed model does it actually win.
The Friedmann models underpin much of the modern Big Bang theory, but they also contain the key to its greatest weakness. If we use these calculations to reverse the expansion of the Universe and turn the clock back on the current state of the cosmos, we find the universe gets ever denser the earlier we go. If we try to go back too far, the mathematics fall apart at a singularity.
The singular nature of gravity
In mathematics, a singularity is a pathological property wherein the numerical value of a particular quantity becomes infinite during the course of a calculation. To give a very simplified example, consider the calculation of the Newtonian force due to gravity exerted by a massive body on another particle. This force is inversely proportional to the square of the distance between the two bodies, so that if one tried to calculate the force for objects at zero separation, the result would be infinite. Singularities are not always signs of serious mathematical problems. Sometimes they are simply caused by an inappropriate choice of coordinates. For example, something strange and akin to a singularity happens in the standard maps to be found in an atlas. These maps look quite sensible until you look very near the poles. In a
5. Open, flat, and closed spaces in two dimensions. In a flat two-dimensional space (middle) the laws of Euclidean geometry hold true. In this case the sum of the angles of a triangle is 180 degrees. In a closed space such as a sphere (bottom), the angles of a triangle come to somewhat more than 180 degrees, whereas for an open space (such as the saddle shape shown) it is less than 180 degrees.
standard equatorial projection, the North Pole does not appear as a point as it should, but is spread out from a point to a straight line along the top of the map. But if you were to travel to the North Pole you would not see anything catastrophic there. The singularity that causes this point to appear is an example of a coordinate singularity, and it can be transformed away by using a different kind of projection. Nothing particularly odd will happen to you if you attempt to cross this kind of singularity.
Singularities occur with depressing frequency in solutions of the equations of general relativity. Some of these are coordinate singularities like the one I just discussed. These are not particularly serious. However, Einstein’s theory is special in that it predicts the existence of real singularities where real physical quantities that should know better, such as the density of matter or the temperature, become infinite. The curvature of space-time can also become infinite in certain situations. The existence of these singularities suggests to many that some fundamental physics describing the gravitational effect of matter at extreme density is absent from our understanding. It is possible that a theory of quantum gravity might enable physicists to calculate what happens deep inside a black hole without having all mathematical quantities becoming infinite. Indeed, Einstein himself wrote in 1950:
The theory is based on a separation of the concepts of the gravitational field and matter. While this may be a valid approximation for weak fields, it may presumably be quite inadequate for very high densities of matter. One may not therefore assume the validity of the equations for very high densities and it is just possible that in a unified theory there would be no such singularity.
Probably the most famous example of a singularity lies at the centre of a black hole. This appears in the original Schwarzschild solution corresponding to a hole with perfect spherical symmetry. For many years, physicists thought that the existence of a singularity of this kind was merely due to the rather artificial special nature of this spherical solution. However, in a series of mathematical investigations, Roger Penrose and others have shown that no special symmetry is required and that singularities arise whenever any objects collapse under their own gravity.
As if to apologize for predicting these singularities in the first place, general relativity does its best to hide them from us. A Schwarzschild black hole is surrounded by an event horizon that effectively protects outside observers from the singularity itself. It seems likely that all singularities in general relativity are protected in this way, and so-called naked singularities are not thought to be physically realistic.
In the 1960s, however, Roger Penrose’s work on mathematical properties of the black hole singularity came to the attention of Stephen Hawking, who had the idea of trying to apply them elsewhere. Penrose had considered what would happen in the future when an object collapses under its own gravity. Hawking was interested to know whether these ideas could be applied instead to the problem of understanding what had happened in the past to a system now known to be expanding, i.e. the Universe! Hawking contacted Roger Penrose about this, and they worked together on the problem of the cosmological singularity, as it is now known. Together they showed that expanding-universe models predict the existence of a singularity at the very beginning, where the temperature and density become infinite. No matter whether the universe is open, closed, or flat, there is a fundamental barrier to our understanding. In the beginning, there was infinity.
Most cosmologists interpret the Big Bang singularity in much the same way as the black hole singularity discussed above, i.e. as meaning that Einstein’s equations break down at some point in the early Universe due to the extreme physical conditions present there. If this is the case, then the only hope for understanding the early stages of the expansion of the Universe is through a better theory. Since we don’t have such a theory, the Big Bang is incomplete. In particular, since we need to know the total energy budget of the Universe to know whether it is open or closed, we cannot determine by theory alone which of these alternatives is the ‘correct’ description. This shortcoming is the reason why the word ‘model’ is probably more appropriate than ‘theory’ for the Big Bang. The problem of not knowing about the initial conditions of the Universe is the reason why cosmologists still cannot answer some basic questions, such as whether the Universe will expand forever.
Chapter 4
The expanding Universe
So far I have concentrated on the way in which developments in theoretical physics, particularly the general theory of relativity, led to major developments in cosmological theory in the 1920s. But these new ideas only gained acceptance when improved observational facilities allowed astronomers to begin making reliable estimates of the distances to and motions of galaxies. In this chapter I will discuss these observations, and how they fit into the theoretical framework.
Hubble’s Law
The nature of the expansion of the Universe is encapsulated in one simple equation, now known as the Hubble Law. This states that the apparent velocity v of a galaxy away from the observer is proportional to its distance d. Nowadays the constant of proportionality is
known as the Hubble constant and is given the symbol H or Ho. The Hubble Law is thus written ‘v = Hod’. The relationship between v and d is called a linear relationship because if you plot a graph (like Hubble did) of the measured velocities and distances of a sample of galaxies, you find they lie on a straight line. The slope of this line is Ho. The Hubble Law basically means that galaxies twice as far away from the observer are moving away twice as quickly. Those three times away move three times as fast, and so on.
6. Hubble’s Law. As observed from the central point, Hubble’s Law states that the apparent recession velocity of distant galaxies is proportional to their distance, so the further away they are the quicker they recede. The expansion does not have a centre: any point can be treated as the origin.
Hubble published the discovery of his famous law in 1929, which resulted from a study of the spectra of a sample of galaxies. The American astronomer Vesto Slipher also deserves a large part of the credit for the discovery. As early as 1914 Slipher had obtained spectra of a group of nebulae (as galaxies were then called) that also displayed this relationship, although his distance estimates were very rough. Unfortunately, Slipher’s early results, presented at the 17th Meeting of the American Astronomical Association in 1914, were never published; history has never adequately acknowledged the contribution Slipher made.
So how did Hubble obtain his law? The technique he used is called spectroscopy. Light from a galaxy contains a mixture of colours, produced by all the stars within it. A spectroscope splits light up into its component hues so that its precise mixture of colours can be analysed separately. A prism is a simple way of achieving the same end. With a prism ordinary white light can be split into a spectrum that resembles a rainbow. But as well as having different colours, astronomical spectra also contain sharp features called emission lines. These lines are produced in the gas contained in an object by electrons shifting between different energy levels. These transitions occur at definite wavelengths depending on the chemistry of the source; these wavelengths can be measured accurately in laboratory experiments. Hubble was able to identify emission lines in many of his galaxies. But comparing their position in the measured spectrum to where the lines should be, he found they were usually in the wrong place. In fact, the lines were almost always shifted to the red end of the spectrum, towards longer wavelengths. Hubble interpreted this as a Doppler shift.
7. The Hubble diagram. Hubble’s original velocity-distance plot published in 1929. Notice that some nearby galaxies are actually approaching the galaxy, and there is considerable scatter in his plot.
Doppler shift
The Doppler effect was originally introduced to physics with a fanfare in the 1840s. In fact this is literally true, because the first experimental demonstration of this effect involved several trumpeters moving on a steam train. The application in that instance was to the properties of sound waves when there is relative motion between the source of the sound and the receiver. We are all familiar with the effect from everyday experience: an approaching police siren has a higher pitch than a receding one. The easiest way to understand the effect is to remember that the pitch of sound depends on the wavelength of the waves from which it is made. High pitch means short wavelength. If a source is travelling near the speed of sound, it tends to catch up the waves it emits in front, thus reducing their apparent wavelength. Likewise, it tends to rush ahead of the waves it emits behind, increasing the gap between the waves and thus lowering their apparent pitch.
In the astronomical setting, the Doppler effect applies to light. Usually the effect is very small, but it becomes appreciable if the velocity of a source is a significant fraction of the velocity of light. (The Doppler effect for sound is small unless the speed of the car is reasonably large, the relevant scale being set by the speed of sound.) A moving source of emission tends to produce light of shorter wavelength if it is approaching the observer and longer wavelength if it is receding. In these cases the light is shifted towards the blue and red parts of the spectrum, respectively. In other words, there is a blueshift (approaching source) or a redshift (receding source).
If the source is emitting white light, however, one would not be able to see any kind of shift. Suppose each line were redshifted by an amount x in wavelength. Then light emitted at a wavelength y would be observed at wavelength y + x. But the same amount of light would still be observed at the original wavelength y, because light originally emitted at wavelength y – x would be shifted there to fill the gap. White light therefore still looks white, regardless of the Doppler shift. To see an effect, one has to look at emission lines, which occur at discrete frequencies so that no such compensation can occur. A whole set of lines will be shifted one way or the other in the spectrum, but the lines will keep their relative spacing and it is therefore quite easy to identify how far they have shifted relative to a source which is at rest in a laboratory.
Hubble measured a larger redshift for the more distant galaxies in his sample than for the nearby ones. He assumed that what he was seeing was a Doppler shift, so he converted the shifting of the spectral lines into a measure of velocity. When he plotted this ‘apparent recession velocity’ against the distance, he got his famous linear relationship. Although Hubble’s Law is now taken to represent the expansion of the Universe, Hubble himself never made this interpretation of his results. Lemaître was probably the first theorist to explain Hubble’s Law in terms of the expansion of the entire Universe. Lemaître’s paper, published in 1927, prefiguring Hubble’s classic paper of 1929, had made little impression at the time because it was written in French and published in an obscure Belgian journal. It was not until 1931 that the British astronomer Arthur Stanley Eddington had Lemaître’s paper published (in English) in the more influential Monthly Notices of the Royal Astronomical Society. The identification of the Hubble Law with the cosmic expansion is one of the main supporting pillars of the Big Bang theory, so Lemaître too deserves great credit for making this important step.
Interpreting the Hubble Law
The fact that galaxies are observed to be moving away from us suggests that we must be at the centre of the expansion. Doesn’t this violate the Copernican Principle and put us in a special place? The answer is ‘no’. Any other observer would also see everything moving away. In fact, every point in the Universe is equivalent as far as the expansion is concerned. Moreover it can be proved mathematically that Hubble’s Law must apply in a homogeneous and isotropic expanding universe, i.e. one in which the Cosmological Principle holds. It is the only way such a universe can expand.
It may help to visualize the situation by reducing the three dimensions of space to the two-dimensional surface of a balloon (this would be a closed universe, but the geometry does not particularly matter for this illustration). If one paints dots onto the surface of the balloon and then blows it up, each dot sees all the other dots moving away as if it were the centre of expansion. This analogy has a problem, however, in that one tends to be aware that the two-dimensional surface is embedded in the three dimensions of our ordinary space. One therefore sees the centre of the space inside the balloon as the real centre of expansion. This is inaccurate. One must think of the balloon as being the entire Universe. It is not embedded in another space and there is no such global centre. Every point in the balloon is the centre. This difficulty is often also confused in one’s mind with the question of where the Big Bang actually happened: are we not moving away from the site of the original explosion? Where was this explosion situated? The answer to this is the explosion happened everywhere and everything is moving away from it. But in the beginning, at the Big Bang singularity, everywhere and everything was in the same place.
More than seventy years after Lemaître, the Hubble Law still poses some difficulties of interpretation. Hubble had not measured velocities but redshifts. The redshift, usually given the symbol z in cosmology, measures the fractional change in wavelength of an observed line relative to its expected position. Hubble’s
Law is sometimes stated as a linear relationship between redshift z and distance d, rather than between recession velocity v and d. If the velocities concerned are much smaller than the speed of light c then there is no problem because in this case the redshift is roughly the velocity of the sources expressed as a fraction of the speed of light. So if z and d are proportional and so are z and v, then v and d are also. But when the redshifts are large this relationship breaks down. What, then, is the correct form to use? In the Friedmann models, the interpretation of Hubble’s Law is amazingly simple. The linear relationship between recession velocity v and distance d, is exact even when the velocity is arbitrarily large. This may worry some of you, because you will have heard that it is not possible for objects to move faster than light. In a Friedmann universe the more distant is the object, the greater its velocity away from the observer. The velocity of the object can exceed the speed of light by any amount you please. It does not violate any principle of relativity, however, because the observer cannot see it; it is infinitely redshifted.
There is also a potential problem in what is meant by d and how to measure it. Astronomers cannot usually measure the distance of an object directly. They cannot extend a ruler to a distant galaxy and cannot usually use triangulation like surveyors do because the distances involved are too large. They have instead to make measurements using light emitted by the object. Since light travels with a finite speed and, as we know thanks to Hubble, the Universe is expanding, things are not at the same position now as they were when light set out from them. Astronomers are therefore forced to use indirect distance measurements, and to attempt to correct for the expansion of the Universe to locate where the object actually is.
