String theory has had several clear and uncontrovertible successes in this area, replacing singularities present in general relativity with smooth behaviour. As the geometry becomes singular, nothing happens in string theory.
How and why is this so? The most prosaic answer is that string theory is governed by the equations of string theory and not by the equations of general relativity. If you examine these equations as geometry moves towards (certain) singularities, nothing happens. For certain classes of singularities, technically called orbifold singularities, these equations remain just as harmless at the singularity as they were for a smooth, featureless geometry.3 This was recognised in the middle of the 1980s as work on string theory exploded.
There are other classes of singularities, called conifold singularities, which throughout the 1980s also appeared inconsistent within string theory. For these, even string theory did not appear to make sense. It was realised in the middle of the 1990s that this was due to a failure to include the effects of D-branes, whose importance had only been recognised in 1995. With the inclusion of D-branes, the conifold singularity also made sense in string theory.
While I have stated in bald prose that these two singularities are resolved by string theory, I have not explained why string theory should be different from general relativity: what comes in to save the day?
What is special about string theory? What is special is that it involves strings, and strings are extended objects. Some of the problems of singularities are associated with infinities concentrated at a point, and having a probe that is smeared out helps dilute the effects of these infinities.
However, there is a more precise and technical way that string theory helps de-smear the singularities. This is true for both orbifold and conifold singularities – although I warn that the following discussion may become a little difficult. A geometry approaching a singularity can be regarded as similar to a needle being progressively deformed so that it becomes ever sharper. The closer you get to the singularity, the sharper the point, and at the singular limit the point is infinitely sharp. The transition from a blunt needle to an infinitely sharp one is like the transition from a smooth geometry to a singular one.
The next step is to imagine a circular cross-section through the needle approaching the point. You see a circle that tapers and tapers in size, and it eventually reaches zero size only if the point is infinitely sharp. Now consider a tiny elastic band wrapped around the needle. The tension of the band contracts it around the needle. Far along the needle, the band still has a finite size coming from the width of the needle. For the singular geometry, at the point of the infinitely sharp needle, the contracted string shrinks to zero size. The string has zero energy – and so, according to Einstein’s identification of mass and energy, zero mass.
The key point to draw from this analogy is that geometric singularities in string theory result in new massless particles as strings (or branes) are able to contract down to zero size. This occurs only because string theory is a theory of extended objects whose mass-energy is given by tension times length. The presence of these new massless particles in the singular geometry is what distinguishes string theory from general relativity. Although it is by no means obvious, it is precisely the existence of these new massless particles that turns out to cure the infinities that these singularities produce within classical general relativity.
String theory is then able to excise some of the geometric singularities from general relativity. While general relativity does not make sense on these spaces, string theory does. Where general relativity gave infinite answers, string theory produces finite numbers.
This is a good reason to work on string theory. It should however be said that string theory is not able to resolve every singularity. There are examples where general relativity gives singular behaviour – and it is unknown if, and how, string theory is able to smooth it out.
One example is the behaviour of singularities at the very beginning of the universe, so-called cosmological singularities. As we move the universe back in time, general relativity tells us that the universe becomes denser and denser and denser – and according to general relativity, at the very earliest time it was infinitely small and infinitely dense. This is a singularity that cries out for help from a theory of quantum gravity, and any such theory should certainly produce a sensible answer for this epoch. However plaintively it may cry out, string theory has so far not been able to help. Even in the simplest of toy models, there is so far no string theory account of cosmological singularities.
It is not that this is necessarily impossible, or beyond the ken of the theory – it is just that systems that evolve rapidly in time are harder to deal with than those that do not. No-one has ever been able to solve the equations or come up with an insightful shortcut.
However, the spatial singularities that can be resolved also lead to another beautiful result – the topology of space itself can change smoothly in string theory. As has already been mentioned, topology refers to the properties of spaces that remain the same under any smooth change. Topology – by definition – remains the same no matter how much a geometry is pulled, twisted or contorted.
Topology can be modified in two ways. One way is by tearing something apart and then reforming it, as with children using playdough who turn necklaces into figurines. This first way is not realised, as far as we know, in string theory.
The other way is by realising that singularities can sit at the border between topologically distinct spaces. Two different geometries, topologically distinct from each other, can have the same singular geometry sitting as a gate between them. Provided you can pass through this gate, the topology of space can be changed.
However, the singularities at the gate are precisely the ones string theory can deal with. Resolving these singularities allows the very topology of space to be changed: from a geometric perspective, it is possible to go into the singularity from one direction and with one topology, and come out again in another direction with another topology. This provides a controlled realisation of topology change, which is attractive as reasonable expectation would suggest that topology change should be possible within quantum gravity.
In summary then, string theory has had a lot of success in resolving singularities in space but less success in resolving singularities involving time. This provides inspiration from the past and open problems for the future.
11.3 THE ULTIMATE COLLIDER
Sitting outside Geneva, CERN’s Large Hadron Collider is the most complicated machine ever built by humankind. It is a ring of superconducting magnets twenty-seven kilometres long, cooled to a temperature that is colder than space itself. Circulating protons within this ring, it can accelerate them to energies of six and a half trillion electronvolts in both clockwise and anticlockwise directions, before smashing them head on. The overall energy in the collision is thirteen trillion electronvolts. The volume around the collision vertices are surrounded by enormous multi-storey detectors, packed with wiring and sensors that record the details and debris of each event.
In terms of colliders that we can build, this is currently the best of the best. The growth in collider energies has also been impressive. Despite the technological demands, the energies involved in collisions at the LHC are over a thousand times greater than those that could be attained fifty years ago. Despite timescales of decades to design, excavate, build and operate colliders, their energies have continued to follow a particle physics version of Moore’s Law, the computing rule of thumb that says that processing power doubles every eighteen months.
How long can this continue? The energy of current collisions at the LHC is smaller than the Planck energy by a factor of almost a million billion. Suppose, cometh the revolution, that the new world government decides to devote all resources to the construction of the PanTerran collider: an accelerating ring of magnets that stretches the entire way around the earth, looping from the North Pole to the South Pole and then back again. As ambiti
on should be tempered with caution, we assume that the magnets used in the PanTerran collider are no stronger than those used in the LHC, and that as in the LHC the PanTerran collider is used to collide protons with protons.
The energies attained in the collisions would then be around ten quadrillion electron volts. This is a factor of one thousand larger than that occurring at the LHC, but still almost a factor of one trillion smaller than the quantum gravity scale. Even the use of alien technology to increase the size of the collider to the circumference of the earth’s orbit around the sun would leave collision energies still a factor of a million smaller than the quantum gravity scale.
We see that collisions at the Planck scale are a long way off. This is not to say that it will never be possible to probe the quantum gravity scale directly. Only fools and fortune-tellers bet against what may be technologically possible centuries from now. However, it will not be in the school textbooks in the near future. What happens when two particles are collided with Planck scale energies is a definite fact of nature – but it is also a fact that is beyond observational ken anytime soon.
It still remains a good theoretical question. What does happen when elementary particles are collided with Planck scale energies? Even more specifically, what happens when gravitons are collided with Planck scale energies? Gravitons are the elementary quanta of the gravitational field, being to spacetime what photons are to the electromagnetic field. They are the basic, irreducible, elementary quantum excitations of that field.
As the minimal perturbation of gravity, gravitons are at the furthest remove in the subject of quantum gravity from ideas such as black holes or topology change. While black holes involve large objects that make large dents in spacetime, the graviton is the simplest possible excitation. Comparing a graviton to a black hole is like comparing an individual photon to the magnetic field produced in a tokamak. Both are configurations of the field, but one is large and classical while the other is small and quantum.
We have never measured a graviton – ever. Gravitons interact through the gravitational force, and so unlike photons the effects of individual gravitons are far too small to be detectable. This can be understood by thinking about how hard it already is to detect the gravitational pull even of a large bowling ball – and then extrapolate down to a single particle.
However, it is a perfectly valid theoretical question to ask what would happen if two gravitons approached each other with energies close to that of the Planck scale. How would they interact and what would the possible results be? This question is well defined. It involves the quanta of the gravitational force. It also involves something that – in principle, if not in practice – could occur in this world.
This question is also interesting because general relativity does not give a good answer to it. For collision energies much less than the Planck scale, general relativity does allow us to compute what will happen when two gravitons approach each other. However, as the energies increase general relativity ceases to be a predictive theory. This is because general relativity is, in the parlance of chapter 3, a non-renormalisable theory.
As described there, for the quantum field theories of the Standard Model, the techniques of renormalisation allow for a finite number of infinities to be isolated and eliminated. For such renormalisable theories, it is possible to trade the appearance of infinities in calculations for measurements; by measuring enough quantities we can remove the infinities. The quantum mechanics of general relativity leads to a proliferation of infinities – indeed, it leads to an infinity of infinities – and these cannot be eliminated via renormalisation.
For gravitons that approach each other with Planck scale energies, general relativity is not a predictive theory. It is impossible to use general relativity to work out what will happen – for what is clearly a well-defined physical process. A clear task for quantum gravity is to be able to give finite, predictive answers for this process. Even if the experiment can never actually be performed, a quantum theory of gravity is required to provide an answer.
Historically, this was one of the leading motivations for working on string theory. The fact that string theory does give finite answers to this question has actually been known since before string theory was recognised as a theory of strings. In this case, the answer was known even before the question was posed. As we saw in chapter 5, in string theory graviton particles exist as oscillatory modes of a closed string. The question of how one can compute the scattering of gravitons is the same as the question of how one can compute the scattering of strings, and formulae for the scattering of strings existed even before string theory’ was recognised to be a theory of strings.
It is true that in string theory, the scattering of gravitons with energies close to the Planck scale is finite. By itself, this is only so interesting. However in string theory there is another distinctive feature of the scattering of gravitons at these energies: this is the softness of the scattering. As described in chapter 5, the scattering of objects can be viewed as either hard or soft. While there is not a rigid definition, hard scattering is the type of scattering produced by ball bearings. Ball bearings colliding at high velocity can easily ricochet and fly out again at large angles to their original direction of approach. Soft scattering is that produced by blobs of jelly, which are incapable of ricocheting and either disintegrate or continue broadly in their initial direction of motion.
When strings are collided, the equations of string theory tell us that they are more like blobs of jelly than ball bearings. Two strings, heading for each other at sufficiently high energies, are extremely unlikely to come off at right angles, and the more energetic they are, the less likely this is to happen. Instead, they can only be deflected by small amounts. Their scattering is soft, and it can only divert the strings by minor perturbations.
This is a characteristic feature, and a characteristic prediction, of string theory: when our far-off descendants build the Ultimate Collider and use it to smash together particles with energies at the scale of quantum gravity, they will find that these particles interact as strings and are therefore unable to scatter by large amounts.
This is clearly a prediction – but a theoretical rather than a practical one. Those who work on the applications of string theory to quantum gravity do not generally do so because they are hunting for falsifiable predictions and seeking an imminent confrontation with experimental data. Instead, they care about quantum gravity and the types of questions that arise when one tries to combine quantum mechanics and gravity. These questions are well removed from direct empirical probes, but no less valid for that.
This chapter has discussed three aspects of quantum gravity, and over the last four chapters we have now seen several good motivations for working on string theory. There are many different styles of doing science successfully, and this variety of topics appeals to the different varieties of physicist. We now turn from these technical topics to a more light-hearted discussion of the different kinds of physicists and the many ways of doing physics.
1Given the condition of first world war trenches, this provides an unusually apt example of the normally trite aphorism that we are all in the gutter, but some of us are looking at the stars.
2As a point of technical pedantry, the dimensions of this statement are correct only in ‘fundamental units’ where Planck’s constant ħ, the speed of light c and the Planck mass MP are all set to be equal to one.
3The simplest examples of orbifold singularities arise from taking a plane and identifying all points related by certain rotations – for example, identifying all points that are related by a one-hundred-and-twenty degree rotation. This leaves one fixed point, namely the coordinate origin, and it turns out that this fixed point of the symmetry has a singularity at it.
IV
Who?
CHAPTER 12
A Thousand Flowers Blooming: Styles of Science
Over the four previous chapters, I have described some of the many different reasons
for caring about string theory – what do all those people who work on ‘string theory’ actually do? The diversity of topics reflects a diversity of interests. The diversity of interests reflects the diversity of individuals. It also reflects very different styles of doing science.
These different styles are the topic of this chapter. The previous chapters have been heavy, and this is intended to be lighter. It will provide idealised caricatures or portraits of common examples of the genus theoretical physicist. None of these caricatures represent actual individuals. Everyone is unique, but this uniqueness co-exists with certain styles of approach to the subject that are common to many.
There is also a more serious purpose to these descriptions. Today the route to professional science starts out with the PhD degree. Eighty years ago, it may have been possible to have a scientific career without a doctorate, but those times are gone. However, the PhD is about more than just solving problems. For the aspirant student, an important part of this process is to discover what type of scientist they are and what makes them happy. Different styles are suited to different problems, and the subject advances most rapidly when square pegs fit into square holes. Like an author finding their mature voice, for young scientists independence comes from finding the niche and style that suits them.
The descriptions I offer below deliberately contain elements of caricature and exaggeration. They are also not always maximally respectful. There are several good reasons for this. The first is that as a rule satire is more truthful than hagiography. The prints of Hogarth are more instructive about 18th century London than the marmoreal memorials lining the walls of Georgian churches.
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