**Table of Contents**

*.......The Elegant Universe*

**THE ELEGANT UNIVERSE,****Brian Greene,**1999, 2003

```(annotated and with added

**bold highlights by Epsilon=One**)

**Chapter 10 - Quantum Geometry**

A Cosmological Playground

According to the big bang model of cosmology, the whole of the universe violently emerged from a singular cosmic explosion, some 15 or so billion years ago. Today, as originally discovered by Hubble, we can see that the "debris" from this explosion, in the form of many billions of galaxies, is still streaming outward. The universe is expanding. We do not know whether this cosmic growth will continue forever or if there will come a time when the expansion slows to a halt and then reverses itself, leading to a cosmic implosion. Astronomers and astrophysicists are trying to settle this question experimentally, since the answer turns on something that in principle can be measured: the average density of matter in the universe.

If the average matter density exceeds a so-called

By carefully studying the distribution of galaxies throughout space, astronomers can get a pretty good handle on the average amount of visible matter in the universe. This turns out to be significantly less than the critical value. But there is strong evidence, of both theoretical and experimental origin, that the universe is permeated with dark matter. This is matter that does not participate in the processes of nuclear fusion that powers stars and hence does not give off light; it is therefore invisible to the astronomer's telescope. No one has figured out the identity of the dark matter, let alone the precise amount that exists. The fate of our presently expanding universe, therefore, is as yet unclear.

Just for argument's sake, let's assume that the mass density does exceed the critical value and that someday in the distant future the expansion will stop and the universe will begin to collapse upon itself. All galaxies will start to approach one another slowly, and then as time goes by, their speed of approach will increase until they rush together at blinding speed. You need to picture the whole of the universe squeezing together into an ever shrinking cosmic mass. As in Chapter 3, from a maximum size of many billions of light-years, the universe will shrink to millions of light-years, every moment gaining speed as

But when the distance scales involved are around the Planck length or less, quantum mechanics invalidates the equations of general relativity, as we are now well aware. We must instead make use of string theory. And so, whereas Einstein's general relativity allows the geometrical form of the universe to get arbitrarily small—in exactly the same way that the mathematics of Riemannian geometry allows an abstract shape to take on as small a size as the intellect can imagine—we are led to ask how string theory modifies the picture. As we shall now see, there is evidence that string theory once again sets a lower limit to physically accessible distance scales and, in a remarkably novel way, proclaims that the universe cannot be squeezed to a size shorter than the Planck length in any of its spatial dimensions.

Based on the familiarity you now have with string theory, you might be tempted to hazard a guess as to how this comes about. After all, you might argue that no matter how many points you pile up on top of each other—point particles that is—their combined volume is still zero. By contrast, if these particles are really strings, collapsed together in completely random orientations, they will fill out a nonzero-sized blob, roughly like a Planck-sized ball of entangled rubber bands. If you made this argument, you would be on the right track, but you would be missing significant, subtle features that string theory elegantly employs to suggest a minimum size to the universe. These features serve to emphasize, in a concrete manner, the new stringy physics that comes into play and its resultant impact on the geometry of spacetime.

To explain these important aspects, let's first call upon an example that pares away extraneous details without sacrificing the new physics. Instead of considering all ten of the spacetime dimensions of string theory—or even the four extended spacetime dimensions we are familiar with—let's go back to the Garden-hose universe. We originally introduced this two-spatial-dimension universe in Chapter 8 in a prestring context to explain aspects of Kaluza's and Klein's insights in the 1920s. Let's now use it as a "cosmological playground" to explore the properties of string theory in a simple setting; we will shortly use the insights we gain to better understand all of the spatial dimensions string theory requires. Toward this end, we imagine that the circular dimension of the Garden-hose universe starts out nice and plump but then shrinks to shorter and shorter size, approaching the form of Lineland—a simplified, partial version of the big crunch.

The question we seek to answer is whether the geometrical and physical properties of this cosmic collapse have features that markedly differ between a universe based on strings and one based on point particles.

If the average matter density exceeds a so-called

*critical density*of about a hundredth of a billionth of a billionth of a billionth (10^-29) of a gram per cubic centimeter—about five hydrogen atoms for every cubic meter of the universe—then a large enough gravitational force will permeate the cosmos to halt and reverse the expansion. If the average matter density is less than the critical value, the gravitational attraction will be too weak to stop the expansion, which will continue forever. (Based upon your own observations of the world, you might think that the average mass density of the universe greatly exceeds the critical value. But bear in mind that matter—like money—tends to clump. Using the average mass density of the earth, or the solar system, or even the Milky Way galaxy as an indicator for that of the whole universe would be like using Bill Gates's net worth as an indicator of the average earthling's finances. Just as there are many people whose net worth pales in comparison to that of Bill Gates, thereby diminishing the average enormously, there is a lot of nearly empty space between the galaxies that drastically lowers the overall average matter density.)By carefully studying the distribution of galaxies throughout space, astronomers can get a pretty good handle on the average amount of visible matter in the universe. This turns out to be significantly less than the critical value. But there is strong evidence, of both theoretical and experimental origin, that the universe is permeated with dark matter. This is matter that does not participate in the processes of nuclear fusion that powers stars and hence does not give off light; it is therefore invisible to the astronomer's telescope. No one has figured out the identity of the dark matter, let alone the precise amount that exists. The fate of our presently expanding universe, therefore, is as yet unclear.

Just for argument's sake, let's assume that the mass density does exceed the critical value and that someday in the distant future the expansion will stop and the universe will begin to collapse upon itself. All galaxies will start to approach one another slowly, and then as time goes by, their speed of approach will increase until they rush together at blinding speed. You need to picture the whole of the universe squeezing together into an ever shrinking cosmic mass. As in Chapter 3, from a maximum size of many billions of light-years, the universe will shrink to millions of light-years, every moment gaining speed as

*everything*is crushed together to the size of a single galaxy, and then to the size of a single star, a planet, and down to the size of an orange, a pea, a grain of sand, and further, according to general relativity, to the size of a molecule, an atom, and in a final inexorable cosmic crunch to*no size at all.*According to conventional theory, the universe began with a bang from an initial state of zero size, and if it has enough mass, it will end with a crunch to a similar state of ultimate cosmic compression.But when the distance scales involved are around the Planck length or less, quantum mechanics invalidates the equations of general relativity, as we are now well aware. We must instead make use of string theory. And so, whereas Einstein's general relativity allows the geometrical form of the universe to get arbitrarily small—in exactly the same way that the mathematics of Riemannian geometry allows an abstract shape to take on as small a size as the intellect can imagine—we are led to ask how string theory modifies the picture. As we shall now see, there is evidence that string theory once again sets a lower limit to physically accessible distance scales and, in a remarkably novel way, proclaims that the universe cannot be squeezed to a size shorter than the Planck length in any of its spatial dimensions.

Based on the familiarity you now have with string theory, you might be tempted to hazard a guess as to how this comes about. After all, you might argue that no matter how many points you pile up on top of each other—point particles that is—their combined volume is still zero. By contrast, if these particles are really strings, collapsed together in completely random orientations, they will fill out a nonzero-sized blob, roughly like a Planck-sized ball of entangled rubber bands. If you made this argument, you would be on the right track, but you would be missing significant, subtle features that string theory elegantly employs to suggest a minimum size to the universe. These features serve to emphasize, in a concrete manner, the new stringy physics that comes into play and its resultant impact on the geometry of spacetime.

To explain these important aspects, let's first call upon an example that pares away extraneous details without sacrificing the new physics. Instead of considering all ten of the spacetime dimensions of string theory—or even the four extended spacetime dimensions we are familiar with—let's go back to the Garden-hose universe. We originally introduced this two-spatial-dimension universe in Chapter 8 in a prestring context to explain aspects of Kaluza's and Klein's insights in the 1920s. Let's now use it as a "cosmological playground" to explore the properties of string theory in a simple setting; we will shortly use the insights we gain to better understand all of the spatial dimensions string theory requires. Toward this end, we imagine that the circular dimension of the Garden-hose universe starts out nice and plump but then shrinks to shorter and shorter size, approaching the form of Lineland—a simplified, partial version of the big crunch.

The question we seek to answer is whether the geometrical and physical properties of this cosmic collapse have features that markedly differ between a universe based on strings and one based on point particles.

Table of Contents

*.......The Elegant Universe*