That said, experimental data is not the only reason to be interested in a topic. The quantum mechanics of gravity contains many rich conceptual questions. What is a black hole made of, and how many different ways are there to make the same black hole? Does anything special happen as you approach a black hole? What happened to space and time at the beginning of the universe? Can space itself tear and then reform? Are there fundamental quantum limits on the amount of information space can contain? What happens when particles approach each other at distances less than the Planck length? These may all be theoretical questions, but they are well posed and should admit of answers.
Interest in these and related questions has been one of the major reasons to study string theory – and string theory has in turn responded through provocative answers that provoke more questions. This is the subject of this chapter, which focuses on three particular issues to do with quantum gravity.
11.1 COUNTING BLACK HOLES
We have already encountered black holes several times in this book. They are special objects that are relevant for both the most abstract parts of formal theory and the grubby world of gastrophysics. They occur when there is so much matter in so small a region that all the matter collapses in on itself. This collapse produces an object that is so dense that even light cannot escape from it: a black hole.
Black holes appeared in general relativity as a theoretical curiosity, when the German astronomer Karl Schwarzschild discovered the first example, the Schwarzschild solution, while serving in the German army during the first world war.1 He died a year later, though, and relatively little attention was paid to this work for several decades. As interest revived, it would take until the 1970s before black holes began to be accepted as parts of the real universe: objects that actually existed in our galaxy and were in fact rather numerous. One of the first good candidates was the X-ray source Cygnus X-1, first discovered in 1964, over which Stephen Hawking had a bet with his friend Kip Thorne from Caltech. Hawking bet that Cygnus X-1 was not a black hole; Thorne that it was. The bet was four years subscription to the British satirical magazine Private Eye versus one years subscription to Penthouse. Hawking conceded in 1990, paying up ‘to the outrage of Kip’s liberated wife’.
Black holes in the galaxy are characterised by how heavy they are and how fast they are spinning. The typical mass of a galactic black hole is similar to or slightly larger than our sun. This is not surprising once you learn that they are usually formed from the inward collapse of large massive stars as they approach the end of their lives. There are much larger black holes at the centres of galaxies: our own Milky Way galaxy hosts a black hole at its centre with a mass of around one million suns. Such obese ‘supermassive’ black holes start small and then grow through consuming progressively more and more matter during the course of billions of years. Over this period, they turn into the galactic version of Michelin Man. There is typically one supermassive black hole per galaxy, playing a crucial role in galactic dynamics, as it is the centre around which all else revolves.
Beyond their role in the digestive processes of the galaxy, black holes are of interest in themselves. As discussed in chapter 8, the theoretical study of black holes reveals that they have a temperature. They are not entirely black. Not only do they absorb light, but they also emit it. Like any oven, they radiate energy. As we saw in chapter 8, this ‘Hawking radiation’ is a consequence of quantum mechanics: in a classical world, a black hole is indeed entirely black.
The temperature of a black hole is inversely proportional to its mass. A small black hole is hot. A large black hole is cool. The black holes that are present in the galaxy, with masses comparable to the sun, are very cold – with temperatures under one billionth of a degree. However, if we could compress a cubic kilometre of rock into a black hole, the result would be pleasantly toasty, with a temperature of a few hundred billion degrees and a power output of almost a gigawatt. As the lifetime of such a black hole would exceed the age of the universe, a collection of them in orbit around the earth would provide a permanent and clean solution to the energy problem.
Black holes also have an intrinsic spin. The spin is a rough measure of the internal rotation – how fast the black hole is, in some sense, rotating about itself. This rotation causes observers outside the black hole to find themselves dragged around the black hole instead of just falling straight in, and the larger the intrinsic spin, the larger this effect is.
In principle, black holes can also have an electric charge. This is a charge in the conventional sense: if a positively charged proton or a negatively charged electron falls into a black hole, the charge of that particle is added to that of the black hole. However this is insignificant for black holes in the galaxy, as the mutual attraction of positive and negative charges causes any non-zero charge to be rapidly neutralised.
Mass, spin and charge: these are the three quantities that define a black hole. From the values of these three quantities, it is possible to work out the entropy of the black hole. ‘Entropy’ is in effect a name for the number of possible internal arrangements of the object, or even more precisely the logarithm of the number of internal arrangements. It implicitly tells you how many ways there are to reorder the innards while the externals remain the same. The entropy of the gas in this room tells you how many ways there are to rearrange the molecules while leaving the pressure and temperature unaltered.
While we may not know exactly what is inside a black hole, the entropy tells us the total number of options. This expression for the entropy is a known formula, and was determined by Stephen Hawking to be exactly one quarter of the area of the black hole’s event horizon.2 The event horizon is the surface in the space around the black hole that marks the point of no return. Once you pass it, you are pulled unavoidably into the singularity at the centre of the black hole.
This poses a clean challenge to those who like clean problems. What are these internal arrangements (sometimes called microstates)? In a good theory of quantum gravity, clearly it should be possible to list these arrangements, count them, and confirm that they come in precisely the multiplicity anticipated by Hawking’s formula. Equally clearly, this result does not tell us about any experimental observable: it would be career-shortening to try and look inside a black hole and see what it is made from.
However, it does offer a precise question with a known answer, and an ability to get the answer right is a necessary part of any candidate theory of quantum gravity. As well as a consistency check, a good understanding of the innards of a black hole would also offer considerable conceptual insight into quantum gravity.
The best place to ask these questions is not for the actual black holes that are formed within the galaxy. Even from a theoretical viewpoint, the black holes that actually exist within the galaxy are messy. As the problem of understanding black hole entropy is a theorist’s problem, so it is best analysed in a theorist’s playground: an idealised universe that is the mathematical version of a laboratory cleanroom. As occurs often in string theory, this idealised universe is one with large amounts of supersymmetry. Supersymmetry both simplifies the calculations and protects the results by ensuring that many otherwise dangerous effects cancel to zero.
While this question of understanding black hole entropy had existed in the background for a long time, for string theory the real breakthrough occurred in January 1996. In that month, Andrew Strominger of Santa Barbara and Cumrun Vafa of Harvard published a paper showing how they could reproduce Hawking’s formula for the black hole entropy directly from a string theory calculation. The black holes they considered were certainly not ‘realistic’. Not least, they existed in an imaginary world with four and not three spatial dimensions, where Strominger and Vafa had made only five of the ten dimensions of string theory very small. The black holes also belonged to a special class of black hole called extremal black holes. Extremal black holes are a type of black hole where the effects of its charge are, in a sense that can be made precise, exactly
as large as the gravitational effects of its mass. As mentioned before, this certainly does not hold for the effectively chargeless black holes in our galaxy.
The property of extremality leads to several simplifications within the calculations. One reason Strominger and Vafa had to work in a world with four spatial dimensions is that, in a world with three spatial dimensions, extremality causes the area of the black hole horizon to go to zero. It vanishes entirely, and in principle you could approach as close as you dared to the black hole singularity without necessarily falling in. As the entropy is one quarter of this area, the entropy for such black holes vanishes – at least in three spatial dimensions.
The advantage of working in the fictional world of four spatial dimensions is that this statement is no longer true. In such a world, it is possible to have black holes that are both extremal and have a finite entropy. The advantage of extremality means that calculations are easier to do and under greater control. The advantage of finite entropy is that there is a meaningful target to try and reproduce – being able to reproduce an answer of zero is not hard and generally shows very little.
In this fictitious universe, Strominger and Vafa did two things within their paper. The first was to describe the five-dimensional extremal black holes they were using, work out the area of their event horizon and determine how it depended on the properties of the black hole. This part of the calculation was old: the techniques used descended from Hawking’s original work. What this gave them was an expression for the entropy that they could then try to reproduce. The second part was the breakthrough, which was to explain microscopically from string theory where this entropy came from by counting configurations.
They did the counting using D-branes. We met D-branes in chapter 5: extended membrane-like objects of two, three or more spatial dimensions. Although always present in string theory, it had taken a long time for their import to be realised. It is also true that D-branes are heavy objects that are charged under analogues of the electric force. The large amounts of supersymmetry present in Strominger and Vafa’s configuration made the D-branes extremal objects: appropriately measured, their charge and their mass are one and the same. As you put D-branes together, you assemble an object with both mass and charge. If you bind enough D-branes together, this object turns into a black hole.
How many ways are there of binding D-branes together? In the ideal universe of Strominger and Vafa, five of the extra dimensions are curled up. The details of the counting lie in the details of the curled-up geometry, and this curled-up geometry has many complicated properties. Asking how many ways there are to arrange the branes is a bit like asking how many ways there are of looping an elastic band around the quills of a porcupine: it requires both knowing the precise geometry and then counting carefully.
Strominger and Vafa had chosen their geometry carefully, and they were able to do the counting. The answer they found was precisely that of Hawking’s formula – the number of configurations gave an entropy of exactly one quarter of the area of the black hole’s event horizon, as required.
It is true that the black holes studied by Strominger and Vafa were artificial. It is not just that the dimensions of space were wrong. The calculations were also performed in a limit in which the interaction strength of strings was precisely zero. The world in which the calculations were done looked nothing like the real world of observations.
Nevertheless, for the first time there existed a calculation which explained, in a microscopic fashion, precisely where Hawking’s entropy formula came from. It enumerated all possible innards of a black hole: and that enumeration added up exactly to what Hawking’s formula required.
In the quest to understand quantum gravity, this result belongs to the good side. While clearly a triumph, it also leaves many questions to be explored. For example, what happens to black holes that are not extremal? Can the result also be extended to these black holes – and what about black holes in different backgrounds, black holes that are rotating, black holes that exist in smaller numbers of dimensions? There are many ways to take this result and extend it, and in doing so obtain an even better understanding of the quantum mechanics of black holes.
Since 1996, there have been something like one thousand papers that have done precisely this, using string theory to understand better the quantum mechanics of black holes from many directions. I am going to pick out one of these directions to focus on, which concerns subleading corrections to Hawking’s formula for the entropy of a black hole.
One of the things that makes the entropy of black holes an attractive topic for a theorist is that it connects both the physics of the very big and the physics of the very small. Classical black holes are large objects – indeed, there is no real limit within general relativity on how large they can be. They are large enough that the space outside them is classical, with no large perturbations from quantum effects. One does not need to know quantum gravity to determine the entropy of a black hole. The classical, large-scale macroscopic physics is sufficient to compute the entropy, and this is precisely what Hawking did in 1974. However, we do need to know quantum gravity to write down the microscopic configurations that the entropy actually counts, and this is where Strominger and Vafa’s contribution lay.
In fact, Hawking’s computation of entropy can be improved slightly. In the great tradition of physics, Hawking computed not the full expression for the entropy of black holes but only the most important part. We recall that Hawking’s formula was that the entropy of a black hole equalled one quarter of the area of its event horizon. The more massive the black hole, the larger its area, and the more true this result is. However, Hawking’s result is not exact. There are additional contributions to the entropy. These grow not with the area but with the logarithm of the area. These are less important – for an event horizon area of one hundred, one thousand and one million in some arbitrary units, the logarithm would be respectively two, three and six – but they are still there, and they can still be calculated.
Furthermore, they can also be calculated using methods that do not rely on knowing the full details of quantum gravity. Just as for Hawking’s original result, the subleading logarithmic corrections to a black hole’s entropy can be determined knowing only the almost classical physics of big black holes, with no need to know quantum gravity. A better expression for the entropy of a black hole is then one quarter of the area of its event horizon – plus a small correction that depends on the logarithm of this area.
What about the microphysics? Strominger and Vafa had reproduced the first term in the entropy formula – entropy grows with one quarter of area. What about the subleading part that depends on the logarithm? Here the calculations are much harder, but for several cases these logarithmic terms have also been successfully matched. This gives an even more acute check of the fact that in string theory the microscopic counting of the number of states that can contribute to the black hole agrees exactly with the determination of the entropy through large-scale, semi-classical calculations.
This is further evidence that the formalism works and that string theory really can give a correct description of the innards of a black hole – even if so far only for black holes that exist in mathematical universes.
11.2 SINGULARITIES AND TOPOLOGY CHANGE
It is a truism that the arena for physics is spacetime. Events take place in space and are counted by time. Spacetime is the backdrop against which we record all that happens – but it is also a dynamical backdrop. In Einstein’s theory of general relativity, the geometry of space and time themselves change under the influence of matter and energy. The more matter present, the more the geometry is deformed.
How far can this be taken? As the density of matter and energy increases, the geometry becomes more and more deformed – and eventually it breaks. In the language of general relativity, the geometry develops singularities. What this means is that within the equations that define the geometry, infinities start appearing. Sometimes in physic
s, infinities appear in equations that are not real infinities; they are simply an artefact of writing an equation in the wrong form, or choosing the wrong coordinates, and they disappear when the equations are rewritten correctly. The geometric singularities are not of this form. These infinities are real; spacetime becomes, according to the equations of general relativity, infinitely curved.
In this limit the equations of general relativity break down, signalling that Einstein’s theory is no longer valid. This does not mean physics breaks down. Theories that fail outside their realm of validity are a common theme in physics. Newtonian mechanics is not a good description for objects that move at speeds close to that of light, and classical physics is not a good description of atoms. An important task for any quantum theory of gravity is to understand how it can resolve these singularities. In a correct theory, we expect to get finite, sensible answers in the region where classical general relativity gives nonsense. As geometry changes, we should be able to follow it to the places where general relativity fails – and then back out again.
There is a related question that one would hope quantum gravity would answer: can space tear and reform? In general relativity, geometry becomes dynamical – but this does not mean that anything goes. For example, it is impossible in general relativity to change the topology of space and add handles or holes to it. Topology remains unaltered under smooth changes – which are the only changes allowed by general relativity. The route to modifications of topology is barred by singularities, and it is only a better theory of gravity that can breach this barrier. These questions are not questions of observation or experiment; rather, they are classic questions of quantum gravity and good reasons to think about string theory.
Why String Theory? Page 28