The Geometry of the Universe:
A Point-Form Summary.
This section of the course notes, and the associated PowerPoint presentation, makes the following critical points:
the universe has no center, but developing an understanding of this (especially in the case of a finite universe) requires some careful visualisation and an appreciation of the importance and effects of spatial curvature. The finite universe requires you to visualise a universe which 'folds back on itself' so that there is no edge or outside
a reasonable analogy is provided by a curved object like the surface of a globe, except that you have to imagine yourself confined to the surface with absolutely no way of even imagining the directions "up" and "down", perpendicular to the surface (as we can see with our three-dimensional insights). In particular, you have to imagine light itself being restricted to travel along the curved surface
there are ways of measuring the curvature, some of which are fairly local. (You don't have to walk all the way around the globe.) Likewise, there are, in principle at least, ways of determining the curvature of the universe by observing the distribution of galaxies around us, or comparing the motions of distant galaxies to those nearby
the problem is made more difficult by the fact that we are looking back in time as we look out into space, and the galaxies we see were almost certainly different in the past (when they were younger). Unless we know these 'evolutionary corrections', we can't figure out the geometry unambiguously by studying the galaxies
for these reasons, decades of endeavour left unanswered the question of the eventual fate of the universe. (The sense of the curvature, positive or negative, is intimately related to the fate: recollapse or infinite expansion.) Recent developments (to be described in the next section) may now have resolved the issue
despite the uncertainties over the future, astronomers have been confident for some decades over the essential correctness of the view that the universe began in the 'hot big bang.' There are several successes, including the good agreement between the Hubble time and the ages of old stars; the discovery of the CMBR; and the agreement of the abundances of helium and several light elements as predicted by nuclear reactions during the early moments
Associated Readings from the Text.Please look at: Chapter 20, pages 641-642. Chapter 23, pages 707-712.No Center.In last day's lecture, I said that we are not at the center of the universe, although everything appears to be rushing away from us. I hope I persuaded you that a uniform expansion will cause all observers to see things rushing away from them, no matter where they are located, but you may still think that there has to be a center somewhere. Let us address this in stages. First of all, look at the figure on page 21 of your text, and imagine a huge loaf of raisin bread floating in the middle of a hot oven. (I don't want it to sit in a metal pan on a shelf because that will affect the way it can expand. Try to visualise it as free-floating in the air.) Now as the loaf expands uniformly during the cooking, the expanding dough carries all the raisins with it. From the point of view of any particular raisin, all other raisins are being carried away. Moreover, a `Hubble law' will apply: raisins twice as far away seem to be `moving' twice as fast as raisins nearby. You will rightly object that there is a center to this activity. There is, after all, a center to the loaf, and outer edges. If you were an intelligent creature on one of the raisins, you would even be able to figure uout if you were at the center by counting the distribution of raisins around you. If you see equal numbers of raisins on all sides, you are at the center of the loaf; if there are many raisins below you, but only a few above you, you are near the top edge of the loaf; and so on. In the universe, if the Cosmological Principle - our fundamental working hypothesis - is correct there is no center. If there were, the observer at the very center is in a privileged location. That observer would not necessarily be us, by the way: we would still see a `Hubble law' from a non-central location if the expansion is uniform. Let us explore this further by considering two obvious possibilities.An Infinite Universe.If the universe is infinite, there is clearly no center. This is fairly straightforward to visualise: the universe may stretch forever in all directions, with no end, and be approximately the same everywhere (the Cosmological Prinicple). This does not mean, by the way, that distant parts of the universe will look the same to us as the nearby parts do! Don't forget that we see the remote parts as they used to be, since it has taken a long time for the light to reach us. So if the universe changes in description as time passes, we will see different features at large distances than nearby. The same will be true, of course, for all observers. You may find it fairly straightforward to visualise an infinite homogeneous raisin loaf expanding as it cooks. If you do, you are ready for the next challenge.A Finite Universe.Imagine a raisin loaf which is finite in size in the sense that it contains some fixed, countable number of raisins and some finite amount of dough. Imagine yourself as an ant on a raisin, bravely setting off to visit other raisins in some arbitrary order, moving randomly from one to the next. You would be very surprised to be told that the ant might never reach a boundary, no point beyond which there are no more raisins and dough. In what kind of a loaf would an ant always see raisins equally distributed around itself in all directions, no matter what raisin it was on at a given moment? This is hard to imagine! Yet there is a very simple example of such a geometry right under your feet. Imagine living on a globe like the Earth which is completely smooth and uniform, but covered everywhere with trees - one every ten meters, say. You could walk all over and around it, coming to one tree after another, and never run into an edge beyond which you see no more trees: there is no fence or boundary. The globe has a finite, unbounded surface. No matter what tree you are standing beside, there will be trees stretching off in all directions around you; there is no `middle' tree. Indeed, you could mark each tree you come to, and eventually discover that you had visited every tree on the planet. But nothing prevents you travelling in any direction you like, as far as you like. You simply wind up coming back on yourself. The reason, of course, is that the globe has a two-dimensional surface (that is you can move left or right, forward or back) which, because of its curvature, folds back onto itself to form a surface of finite size and extent. We can see this curvature because we can sit outside the globe, in a real three-dimensional space which surrounds it. It would be obvious to an outside observer that the person walking from tree to tree is following a curved trajectory, but if the globe is big enough (as it surely is, in the case of the Earth) the ground locally seems flat. Still, there are ways of telling that we are on a curved surface. One way is to look out at the stars. Aristotle noted, for instance, that the constellations change in altitude as we move North or South. This tells us that the Earth is ball-shaped. But what would happen if no light reached us from above or below, and if we had absolutely no sensation or knowledge of `up and down.' If light were forced to travel along the surface of the Earth , we would have no visible cues that we were even living inside a truly three-dimensional universe!Curvature in the Universe.You will remember that Einstein said that the presence of matter curves space. In fact, if the density of matter in the universe is sufficiently high, space will curve `right back on itself' so that the universe is closed and finite but unbounded. There will be no region of the universe we cannot get to, and it does not stretch to infinity; yet there is no boundary, and no `outside' at all. The universe is all there is! If the density of matter is lower than that critical value, the universe may be infinite in extent - which you may find a more appealing proposition. But whether you find it comfortable or not, it is entirely possible that we may live in a finite unbounded universe within which spatial curvature `folds' it back onto itself. I admit that coming to terms with this concept is difficult. The only way I can suggest is to think about the two-dimensional globe in three-dimensional space, and ask yourself how that surface would look to a creature on the surface if light were forced to follow the curvature of the globe. (This is what happens in the real universe. Light is forced to follow the curvature imposed by the distribution of matter.) Then try to raise this visualisation to one higher dimension!Measuring the Curvature of a Globe.Let us go back to our analogy of a globe, There are ways of determining whether or not we live on a curved surface. One way is to set out in a straight line and just keep on walking without changing direction. If the world is a ball, you will eventually get back to where you started. Of course, if the world is very large - say, billions of miles around - this might prove impractical. Can we make some more local measurement? Another possibility is the following. Send two friends out in orthogonal directions: say, one to the East and one to the South. After they have each travelled 100 meters, they will be about 140 meters apart. But after they have each travelled about 10,000 km, they will only be about 10,000 km apart! You could also imagine trying to paint the surface of the Earth. Suppose you stretch out a rope with a length of 100 meters, and use it to define a circle of that radius. If that circle takes a certain amount of paint to cover it, a circle with a radius of 200 meters will require four times as much. But a circle with a radius of 10,000 km will take less paint than you think because of the curvature. You could also look at distinct objects which appear to be in the distance. (This one only works if light is forced to follow the surface of the Earth, which it doesn't really of course -- but in the curved universe, the light does follow the curvature!) Suppose, for instance, you are standing at the North Pole and you look left and right. If you see two penguins in the remote distance, you will assume that they are distinct, and very far apart. But you may really be seeing two different aspects of the same penguin, which is standing at the South Pole. If the penguin does something interesting like flap its wings, we will see both images do that and deduce the truth. Indeed, you can see that you might even hope to see yourself in the light of photons which have travelled all the way around the globe. In other words, there are various ways in which we could test whether we live on a curved surface, ways which even a clever `flat' creature could carry out even without an understanding of `up and down' and even if light is forced to travel along the curved surface of the globe.Measuring the Curvature of the Universe.Similar (but less-easily visualised) experiments can be carried out in the universe to determine if it is indeed curved. I will mention only one directly. Imagine looking out into space to some large distance: say one hundred million light years. Count all the galaxies in that immense volume. Now look out to twice the distance. You would expect to see eight times as many galaxies, if space is `Euclidean' (the larger volume is twice as wide, twice as high, twice as long). But if space is curved, you may get some different answer. Such experiments are not easy for a number of reasons. For one thing, galaxies are not distributed entirely uniformly in space: there is large-scale structure, like clustering and superclustering. You have to look out far enough to `average out' these effects. But the farther out you look, the farther back in time you are looking, to a time when galaxies were not the same as at present. (There may, for instance, have been more of them in the early days, before a lot of them merged.) You have to know how galaxies themselves evolved to interpret the results. So the experiments are difficult, and no persuasive answer has yet been obtained on the basis of geometrical arguments of this sort. There are, however, other ways.How Mass Determines the Geometry and the Fate of the Universe.The curvature of the universe is determined by the presence of the matter, and the amount of matter determines the rate at which the present expansion of the universe is slowing down under the influence of gravity. As Einstein showed, there is in fact an intimate relationship between these, as follows: If the density of matter in the universe is rather high, the universal expansion will come to a halt and it will eventually recollapse to a dense, hot state (sometimes called the Big Crunch). In that case, the universe will be of positive curvature (a sphere is a two-dimensional example) and of finite extent and total mass, although unbounded. If the density of matter in the universe is rather low, the universal expansion will continue forever, more or less unchecked. In that case, the universe will be of negative curvature (a saddle surface is a two-dimensional example) and of infinite extent. If the density of matter in the universe is at some critical value, the universal expansion will continue forever but ever more slowly, until it becomes effectively static in the remote future. In that case, the universe is flat (having neither negative nor positive curvature). These considerations suggest two more ways of determining the nature and eventual fate of the universe.Measure the Local Density.The first approach is to see how dense the universe is in the vicinity of the Milky Way - a density which is averaged over some very large volume, however. We can then ask if the density of matter is enough to cause the expansion eventually to come to a halt and recollapse. One problem with this technique is that we have to take into account all the dark matter in the universe, and we aren't yet sure just how much there is!Measure the Slowdown.The second approach is to see if the expansion is slowing down at a rate which implies that it will eventually come to a halt. We cannot do this by watching a single object, like a remote galaxy, because any change in its velocity will be immeasureably small over countless human lifetimes. Instead, we compare the motions of nearby and remote galaxies. Since we see the remote galaxies as they were in the ancient past, when the expansion rate was faster than it is now, they will appear to be moving a little faster than expected. Careful analysis may tell us what this implies for the future of the cosmos.The Likely Future.Until recently, these various efforts provided no absolutely firm answer, despite decades of endeavour. The evidence could not unambiguously rule out any one of these possibilities with certainty, although the balance seemed to be in favour of an open universe (or possibly a flat one) of infinite extent, rather than a closed universe that is doomed to recollapse in the future. As we will see in the next (and last) section of these notes, however, there have been recent developments which have had a profound effect on our understanding -- most notably, the discovery that the universal expansion seems to be accelerating rather than decelerating! In any event, there is an interesting philosophical question you might like to consider. I wonder which you might prefer: a universe in which everything coasts apart forever (perhaps accelerating as it does so) until all the galaxies are left essentially isolated in an infinitely long, ever-colder void, or ``...a universe which grinds to ashes 50 to 100 billion years of galactic, stellar, planetary, biological and cultural evolution..'' (to quote Carl Sagan). On the scale of human lifetimes, of course, it does not matter.Why We Believe in the Big Bang.The Big Bang has several profound successes: It explains why the oldest stars we see are about 13-14 billion years old: they formed at about the time the first galaxies formed in the cooling Big Bang. The good agreement between the Hubble age (the expansion age) and the stellar-evolutionary models give us some confidence that this picture is right. The abundance of Helium and a couple of the light elements is understood as an immediate consequence of primodial nucleosynthesis in the Big Bang (see an attached figure, the details of which are not important). This finding, in the spirit of Gamow, is a second strong reason. The ubiquity and uniformity of the Cosmic Microwave Background Radiation seems to point unequivocally back to a hot dense phase of the universe. (It is, in fact, very difficult to come up with any alternative explanation for what that radiation might be!) Previous chapter:Next chapter0: Physics 016: The Course Notes, spring 2005. 1: The Properties of the Sun: 2: What Is The Sun Doing? 3: An Introduction to Thermonuclear Fusion. 4: Probing the Deep Interior of the Sun. 5: The Sun in More Detail. 6: An Introduction to the Stars. 7: Stars and Their Distances: 8: The HR Diagram: 9: Questions Arising from the HR Diagram: 10: The Importance of Binary Stars: 11: Implications from Stellar Masses: 12: Late in the Life of the Sun: 13: The Importance of Star Clusters in Understanding Stellar Evolution: 14: The Chandrasekhar Limit: 15: Supernovae: The Deaths of Massive Stars, 16: Pulsars: 17: Novae: 18: An Introduction to Black Holes: 19: Gravity as Geometry: 20: Finishing Off Black Holes: 21: Star Formation: 22: Dust in the Interstellar Medium: 23: Gas in the ISM: 24: The Size and Shape of Our Galaxy: 25: The Discovery of External Galaxies: 26: Galaxies of All Kinds: 27: The Expanding Universe: 28: Quasars and Active Galaxies: 29: The Hot Big Bang: 30: The Geometry of the Universe: 31: Closing Thoughts: Part 1:Part 2:Part 3: |
(Wednesday, 22 April, 2026.)
