Three Roads to Quantum Gravity (27 page)

The main idea on each side is the same. Einstein’s theory of general relativity is to be thought of as a macroscopic description, obtained by averaging over the atomic structure of spacetime, in exactly the same way that thermodynamics is obtained by applying statistics to the motion of atoms. Just as a gas is described roughly in terms of continuous quantities such as density and temperature, with no mention of atoms,
in Einstein’s theory space and time are described as continuous, and no mention is made of the discrete, atomic structure that may exist on the Planck scale.

Given this general picture, it is natural to ask whether the black hole’s entropy is a measure of the missing information that could be obtained from an exact quantum description of the geometry of space and time around a black hole. The fact that the entropy of a black hole is proportional to the area of its horizon should be a huge clue to its meaning. String theory and loop quantum gravity have each found a way to use this clue to construct a description of a quantum black hole.

In string theory, good progress has been made by conjecturing that the missing information measured by the black hole’s entropy is a description of how the black hole was formed. A black hole is a very simple object. Once formed, it is featureless. From the outside one can measure only a few of its properties: its mass, electric charge and angular momentum. This means that a particular black hole might have been formed in many different ways: for example, from a collapsing star, or - in theory at least - by compressing, say, a pile of science-fiction magazines to an enormous density. Once the black hole has formed there is no way to look inside and see how it was formed. It emits radiation, but that radiation is completely random, and offers no clue to the black hole’s origin. The information about how the black hole formed is trapped inside it. So one may hypothesize that it is exactly this missing information that is measured by the black hole’s entropy.

Over the last few years string theorists have discovered that string theory is not just a theory of strings. They have found that the quantum gravity world must be full of new kinds of object that are like higher-dimensional versions of strings in that they extend in several dimensions. Whatever their dimension, these objects are called branes. This is shortened from ‘membranes’, the term used for objects with two spatial dimensions. The branes emerged when new ways to test the consistency of string theory were discovered, and it was found that the theory can be made mathematically consistent only by including a whole set of new objects of different dimensions.

String theorists have found that in certain very special cases black holes could be made by bringing together a collection of these branes. To do this they make use of a feature of string theory, which is that the gravitational force is adjustable. It is given by the value of a certain physical field. When this field is increased or decreased, the gravitational force becomes stronger or weaker. By adjusting the value of the field it is possible to turn the gravitational force on and off. To make a black hole they begin with the gravitational field turned off. Then they imagine assembling a set of branes which have the mass and charge of the black hole they want to make. The object is not yet a black hole, but they can turn it into one by turning up the strength of the gravitational force. When they do so a black hole must form.

String theorists have not yet been able to model in detail the process of the formation of the black hole. Nor can they study the quantum geometry of the resulting black hole. But they can do something very cute, which is to count the number of different ways that a black hole could be formed in this way. They then assume that the entropy of the resulting black hole is a measure of this number. When they do the counting, they get, right on the nose, the right answer for the entropy of the black hole.

So far only very special black holes can be studied by this method. These are black holes whose electric charges are equal to their mass. This is to say that the electrical repulsions of two of these black holes are exactly balanced by their gravitational attractions. As a result, one can put two of them next to each other and they will not move, for there is no net force between them. These black holes are very special because their properties are strongly constrained by the condition that their charge balances their mass. This makes it possible to get precise results, and, when this is possible the results are very impressive. On the other hand, it is not known how to extend the method to all black holes. Actually, string theorists can do a bit better than this, for the methods can be used to study black holes whose charges are close to their masses. These calculations also give very impressive results: in particular, they reproduce every last factor of 2 and
π in the formula for the radiation emitted by these black holes.

A second idea about a black hole’s entropy is that it is a count, not of the ways to make a black hole, but of the information present in an exact description of the horizon itself. This is suggested by the fact that the entropy is proportional to the area of the horizon. So the horizon is something like a memory chip, with one bit of information coded in every little pixel, each pixel taking up a region 2 Planck lengths on a side. This picture turns out to be confirmed by calculations in loop quantum gravity.

A detailed picture of the horizon of a black hole has been developed using the methods of loop quantum gravity. This work started in 1995 when, inspired by the ideas of Crane, ’t Hooft, and Susskind, I decided to try to test the holographic principle in loop quantum gravity. I developed a method for studying the quantum geometry of a boundary or a screen. As I mentioned earlier, the result was that the Bekenstein bound was always satisfied, so that the information coded into the geometry on the boundary was always less than a certain number times its total area.

Meanwhile, Carlo Rovelli was developing a rough picture of the geometry of a black hole horizon. A graduate student of ours, Kirill Krasnov, showed me how the method I had discovered could be used to make Carlo’s ideas more precise. I was quite surprised because I had thought that this would be impossible. I worried that the uncertainty principle would make it impossible to locate the horizon exactly in a quantum theory. Kirill ignored my worries and developed a beautiful description of the horizon of a black hole which explained both its entropy and its temperature. (Only much later did Jerzy Lewandowksi, a Polish physicist who has added much to our understanding of loop quantum gravity, work out how the uncertainty principle is circumvented in this case.)

Kirill’s work was brilliant, but a bit rough. He was subsequently joined by Abhay Ashtekar, John Baez, Alejandro Corichi and other more mathematically minded people who developed his insights into a very beautiful and powerful description of the quantum geometry of horizons. The results
can be applied very widely, and give a general and completely detailed description of what a horizon would look like were it to be probed on the Planck scale.

While this work applies to a much larger class of black holes than can be addressed by string theory, it does have one shortcoming compared with string theory: there is one constant that has to be adjusted to make the entropy and temperature come out right. This constant determines the value of Newton’s gravitational constant, as measured on large scales. It turns out that there is a small change in the value of the constant when one compares its value measured on the Planck scale with the value measured at large distances. This is not surprising. Shifts like this occur commonly in solid state physics, when one takes into account the effect of the atomic structure of matter. This shift is finite, and has to be made just once, for the whole theory. (It is actually equal to the √3/log 2.) Once done it brings the results for all different kinds of black holes in exact agreement with the predictions by Bekenstein and Hawking that we discussed in Chapters 6 to 8.

Thus, string theory and loop quantum gravity have each added something essential to our understanding of black holes. One may ask whether there is a conflict between the two results. So far none is known, but this is largely because, at the moment, the two methods apply to different kinds of black hole. To be sure, we need to find a way of extending one of the methods so that it covers the cases covered by the other method. When we can do this we will be able to make a clean test of whether the pictures of black holes given by loop quantum gravity and string theory are consistent with each other.

This is more or less what we have been able to understand so far about black holes from the microscopic point of view. A great deal has been understood, although it must also be said that some very important questions remain unanswered. The most important of these have to do with the interiors of black holes. Quantum gravity should have something to say about the singular region in the interior of a black hole, in which the density of matter and the strength of the gravitational field become infinite. There are speculations that quantum effects will remove the singularity, and that one consequence of this
may be the birth of a new universe inside the horizon. This idea has been studied using approximation techniques in which the matter forming the black hole is treated quantum theoretically, but the geometry of spacetime is treated as in the classical theory. The results do suggest that the singularities are eliminated, and one may hope that this will be confirmed by an exact treatment. But, at least so far, neither string theory nor loop quantum gravity, nor any other approach, has been strong enough to study this problem.

Until 1995 no approach to quantum gravity could describe black holes in any detail. None could explain the meaning of the entropy of a black hole or tell us anything about what black holes look like when probed on the Planck scale. Now we have two approaches that are able to do all these things, at least in some cases. Every time we are able to calculate something about a black hole, in either theory, it comes out right. There are many questions we still cannot answer, but it is difficult to avoid the impression that we are finally understanding something real about the nature of space and time.

Furthermore, the fact that both string theory and loop quantum gravity both succeed in giving the right answers about quantum black holes is strong evidence that the two approaches may be revealing different sides of a single theory. Like Galileo’s projectiles and Kepler’s planets, there is more and more evidence that we are glimpsing the same world through different windows. To find the relation of his work to Kepler’s, Galileo would only have had to imagine throwing a ball far enough and fast enough that it became a moon. Kepler, from his point of view, could have imagined what a planet orbiting very close to the Sun might have looked like to people living on the Sun. In the present case, we only have to ask whether a string can be woven from a network of loops, or whether, if we look closely enough at a string, we can see the discrete structures of the loops. I personally have little doubt that in the end loop quantum gravity and string theory will be seen as two parts of a single theory. Whether it will take a Newton to find that theory, or whether it is something we mortals can do, is something that only time will tell.

B
ack in the 1970s there was a simple dream about how physics would end. A unified theory would be found that incorporated quantum theory, general relativity, and the various particles and forces known to us. This would not only be a theory of everything, it would be unique. We would discover that there was only one mathematically consistent quantum theory that unified elementary particle physics with gravity. There could be only one right theory and we would have found it. Because it was unique, this theory would have no free parameters - there would be no adjustable masses or charges. If there was anything to adjust, the theory would then come in different versions, and it would not be unique. There would be only one scale, against which everything was measured, which was the Planck scale. The theory would allow us to calculate the results of any experiment to whatever accuracy we desired. We would calculate the masses of the electron, proton, neutron, neutrinos and all the other particles, and our results would all be in exact agreement with experiments.

These calculations would have to explain certain very strange features of the observed masses of the particles. For example, why are the masses of the proton and neutron so very small in Planck units? Their masses are of the order of 10
^-19
Planck masses. Where do such terribly small numbers come from? How could they come out of a theory with no free parameters? If the fundamental atoms of space have the
Planck dimensions, then we would expect the elementary particles to have similar dimensions. The fact that protons and neutrons are nearly 20 orders of magnitude lighter than the Planck mass seems very hard to understand. But since the theory would be unique it would have to get this right.

String theory was invented with the hope that it would be this final theory. It was its potential uniqueness that made it worth studying, even as it became clear that it was not soon going to lead to predictions about the masses of particles or anything else that could be tested experimentally. After all, if there is one unique theory it does not need experiments to verify it - all that is needed is to show that it is mathematically consistent. A unique theory must automatically be proved right by experiments, so it does not matter if a test of the theory is several centuries away. If we accept the assumption that there is one unique theory, then it will pay to concentrate on the problem of testing that theory for mathematical consistency rather than on developing experimental tests for it.

The problem is that string theory did not turn out to be unique. It was instead found to come in a very large number of versions, each equally consistent. From our present-day perspective, taking into account only the results on the table, it seems that the hope for a unique theory is a false hope. In the current language of string theory, there is no way to distinguish between any of a very large number of theories: they are all equally consistent. Moreover, many of them have adjustable parameters, which could be changed to obtain agreement with experiment.