In the previous chapter we initiated the study of the 11- and ten-dimensional supergravity theories which arise in the low-energy limits of the various string theories and M theory. Our goal was to study the dualities that relate the various string theories and M theory using effective-field-theory actions as we did in Section 15.2 with T duality in the effective action of the common string sector. In the coming chapters we will study these dualities from the point of view of their effect on classical solutions of the effective actions that represent the classical long-range fields of perturbative and non-perturbative states of these theories, as we did in Section 15.3 with the solutions associated with string and winding modes.

In this chapter we are going to study the N = 2B, d = 10 (chiral) supergravity theory, the effective field theory of the type-IIB superstring, and how it is related to the N = 2A (nonchiral) theory after compactification on a circle (type-II T duality). Furthermore, we are also going to study the truncations to N = 1 theories, which are the effective-field theories of the type-I and heterotic superstrings, finding the field-theory version of the type-I/heteroticstring duality.

First, in Section 17.1, we will study the bosonic sector of the theory, giving a non-selfdual action from which one can derive equations of motion that have to be supplemented by the self-duality constraint of the 5-form field strength [111]. We will also give the supersymmetry transformation rules to lowest order in fermions. Then, in Section 17.2 we will study the S-duality symmetry of this theory, which becomes manifest only after several redefinitions.

Hands-on exercise: manifolds and coordinate patches

To complete this exercise you will need the following supplies:

• Two rulers or other rigid length-measuring sticks.

• Large flat surface like the top of a table or desk, similar to that used in the Chapter 1 hands-on exercise.

• Large spherical object (LSO), similar to that used in the Chapter 1 hands-on exercise.

• A device for measuring time.

Build a model of E2 (two-dimensional Euclidean space) with two rulers taped together at right angles to make rectangular coordinate axes. Place the coordinate axes on the surface of the table or desk that you used in the Chapter 1 hands-on exercise so that the corners are touching the surface. Move them around to see what it means for the surface to be both locally and globally like E2. Could you accurately measure the distance between any two points on this surface using only these straight axes, if they were long enough and couldn't bend?

Place the axes on the surface of the LSO that you used in the Chapter 1 handson exercise so that they are tangent to the surface. Move them around from point to point and ponder what it means for the surface to be locally but not globally like E2. Could you accurately measure the distance between any two points on this surface using only these straight axes, if they were long enough and couldn't bend?

Hands-on exercise: review of Galilean relativity

To complete this exercise you will need the following:

• A large flat level smooth surface, like the top of a table or desk, and some smooth round glass marbles or a pool table and some billiard balls.

The purpose of this exercise is to compare observations in the two coordinate frames most relevant to particle physics – the rest frame of a particle in a collision, and the center of momentum frame of two colliding particles. In the first case a moving marble or billiard ball moving with speed v strikes another one that is initially at rest. Repeat this a few times. Devise a method to determine the final speeds and directions of the two marbles or balls as accurately as you can from the point of view of an observer at rest. For the second case, the center of momentum (CM) frame, there are two possible approaches. You can try to put the two balls or marbles into simultaneous motion on a collision course with equal speeds. The alternative is only to have one ball in motion, as in the first part of the exercise, but to observe the collision in a frame that is moving with speed v/2 so that the two marbles or balls appear to have equal and opposite velocities. Try to make repeated observations for each of these two alternatives.

After the previous chapter's brief introduction to perturbative string theory and T duality we are going to discuss how one arrives at the low-energy string effective (field theory) actions, what their meaning is, and what their limits of validity are. We are going to start exploiting them to study T duality and find Buscher's T-duality rules, that relate different curved backgrounds that are equivalent from the string-theory point of view. These rules are some of the most powerful tools of string theory.

Effective actions and background fields

The low-energy string effective action describes the low-energy dynamics of a given string theory. Here low energy means energies lower than the relevant energy scale: the string mass ms. Thus, the low-energy limit is the ά→ 0 limit, heuristically, the limit in which the string length can be ignored and a theory of particles (a field theory) is recovered.

On the other hand, at low energies only the massless modes are relevant and their dynamics is described by a theory of the corresponding massless fields. The obvious way to find this field theory is to compute string amplitudes for the massless modes, take the ά→ 0 limit and then construct a field theory that reproduces these amplitudes. In principle, the effective field theory has an expansion in powers of ά, although usually only the lowestorder terms are considered. The terms of higher order in ά are also of higher order in derivatives. Also, string amplitudes are calculated order by order in string perturbation theory and the effective action can also be expanded in powers of the string coupling constant, which here is the exponential of the dilaton field.

In previous chapters we have encountered frequently the concept of symmetry of physical laws and symmetry of physical systems under mathematical or physical transformations. The primary examples of symmetry transformations that we have encountered were spatial rotations and Lorentz transformations. The concept of symmetry is very important in physics, not only in the context of transformations of space and time, but also in many others. As it happens, there is a beautiful branch of mathematics, called group theory, which is ideally suited to the description of symmetry. The application of group theory to physics, especially in the context of quantum mechanics, was worked out by Wigner, Weyl, and others in the 1920s and 1930s. By now it is part of the standard toolkit of an educated physicist. This chapter will introduce basic group theory with an emphasis on the application to the descriptions of rotational symmetry and Lorentz symmetry, which are the examples of greatest relevance here.

What is a group?

Rotations and Lorentz transformations are examples of transformations that one can imagine applying to physical objects, collections of objects, systems of equations (the Maxwell equations, for example), or even the entire Universe. When they are symmetries of the system in question, as in the case of the Maxwell equations, the transformed system is identical to the original one. However, it is also possible to transform a system that does not possess the symmetry, in which case the transformed system looks different from the original one.

Towards the end of the nineteenth century many physicists believed that all the fundamental laws that describe the physical Universe were known, and that all that remained to complete the understanding was an elaboration of details. The mind-boggling error of this viewpoint was laid bare within a few short years. Max Planck introduced the quantum in 1899 and Albert Einstein's breakthrough work on special relativity appeared in 1905. The ensuing relativity and quantum revolutions each led to surprising and unexpected concepts and phenomena that have profoundly altered our view of physical reality. The science and the history associated with each of these revolutions has been told many times before. But they are worth coming back to again and again with the added benefit of historical perspective. After all, they have changed the world scientifically, technologically, and philosophically. Perhaps due to the lesson from a century ago, very few people today are so foolish as to speak of an “end of science”. In fact, revolutionary advances in theoretical physics are currently in progress, and we seem to be a long way from achieving a settled and final picture of physical reality.

As the title indicates, this book is about the special theory of relativity. This theory overthrew the classical view of space and time as distinct and absolute entities that provide the backdrop on which physical reality is superimposed. In special relativity space and time must be viewed together (as spacetime) to make sense of the constancy of the speed of light and the structure of Maxwell's electromagnetic theory.

After the general introduction to extended objects of the preceding chapter, in this one we are going to study specifically the extended objects that appear in string theory. The existence of these objects is implied by our previous knowledge of existing objects (strings and Dp-branes) combined with duality. This path will be followed in Section 19.1, in which we will arrive at the diagrams in Figures 19.4.1 and 19.4.1 that represent, respectively, more- and less-conventional extended string/M-theory objects and their duality relations. The duality relations can be used to find the masses of all these objects compactified on tori (Tables 19.1–19.3) using as input the mass of a string wound once on a circle (i.e. the mass of a winding mode). To obtain consistent results (in particular for electric–magnetic dual branes to coexist satisfying the Dirac quantization condition), the ten-dimensional Newton constant has to have a specific value in terms of the string coupling constant and the string length that we will determine.

The next step (Section 19.2) will be to identify which are, among the general solutions of the p-brane a-model, those that represent the long-range fields of the basic extended objects of string and M theory that we found before. We will first identify families of solutions and then we will study one by one the most important solutions. In Section 19.3 we will check the values of the integration constants of those solutions against the masses and charges of the extended objects that we determined using duality arguments. Then, the duality relations between the solutions will be checked in Section 19.4.

As we saw in Part I, the weak-field limit of GR is just a relativistic field theory of a massless spin-2 particle propagating in Minkowski spacetime. In the absence of sources, by choosing the De Donder gauge Eq. (3.100), it can be shown that the gravitational field hμν satisfies the wave equation (3.101) and, correspondingly, there are wave-like solutions of the weak-field equations like the one we found in Section 3.2.3 associated with a massless pointparticle moving at the speed of light.

GR is, however, a highly non-linear theory and it is natural to wonder whether there are exact wave-like solutions of the full Einstein equations. The answer is definitely yes and in this chapter we are going to study some of them, the so-called pp-waves, which are especially interesting for us. In particular we are going to see that the linear solution we found in Section 3.2.3 is an exact solution of the full Einstein equations that has the same interpretation. We will use this solution many times in what follows to describe the gravitational field of Kaluza–Klein momentum modes, for instance.

pp-Waves

pp-waves (shorthand for plane-fronted waves with parallel rays) are metrics that, by definition, admit a covariantly constant null Killing vector field ℓμ:

The first spacetimes with this property were discovered by Brinkmann in [193]. To describe pp-wave metrics, we define light-cone coordinates u and ν in terms of the usual Cartesian coordinates

which are related to the null Killing vector by

i.e. ν is the coordinate we can make the metric independent of, the only non-vanishing components of ℓ are ℓu = ℓν = 1, and the metric describes a gravitational wave propagating in the positive…

Hands-on exercise

Make a two-dimensional coordinate grid on a piece of heavyweight paper that is stiff enough to bend into a circle without folding over. Make a time axis labeled τ and a space axis labeled x, with the origin of the coordinate system in the center of the sheet. Pick a direction on the τ axis to be the direction of increasing τ and label it.

Roll the paper into a cylinder with the x axis running lengthwise and lightly tape it shut. The circular axis should be the τ axis. Using a marking pen, draw two null lines starting the origin, towards the direction of increasing τ, which should be 45° lines in the (τ, x) grid. Draw a timelike line in one direction, starting at the origin. Keep drawing the lines around the cylinder until the end of the paper in both directions. Untape the paper and unroll it. Is there any area on this diagram that cannot be reached from the origin by a future-directed timelike or null world line? Is there any difference between the past and the future light cone of an event in such a spacetime? Are there any two events in this spacetime that cannot be connected by a null or timelike path?

What is time?

Flat spacetime appears on its face to be very simple. We have space in the form of D coordinates (x1, …, xD) and time in the form of the τ coordinate, and we have Lorentz transformations between the two.

The twentieth century gave birth to three major conceptual revolutions in human understanding of the fundamental physical principles that characterize how the Universe works. They are relativity (both special and general), quantum mechanics, and string theory. The first two arose in the early part of the century and are supported by overwhelming experimental and observational evidence. The latter (string theory) arose much later – around 1970 – and is much more speculative than relativity or quantum mechanics, though most practitioners are completely convinced that it is also an essential ingredient of the story. String theory is certainly every bit as mind-bending in its implications as relativity and quantum mechanics.

The primary purpose of this book, as its title indicates, is to explain and explore the special theory of relativity. However, this also gives us license to explore its interplay with other profound ideas. By confronting quantum theory with relativity physicists were led to uncover remarkable concepts and insights. A thorough treatment of relativistic quantum mechanics is the subject of quantum field theory and is beyond the scope of this book. This chapter will present an overview of the basics. (Why should all the fun be reserved for the specialists?) Our brief treatment will serve as a general introduction for students who have not studied these matters before. Those who have may find a few surprises.

The definition of conserved charges in GR (and, in general, in non-Abelian gauge theories) is a very important and rather subtle subject, which is related to the definition of the energy–momentum tensor of the gravitational field. As we saw in the construction of the SRFT of gravity, perturbatively (that is, for asymptotically flat, well-behaved gravitational fields), GR gives a unique energy–momentum (Poincaré) tensor. It is natural to ask whether there is a fully general-covariant energy–momentum tensor for the gravitational field that would reduce to this in the weak-field limit. Many people (starting from Einstein himself) have unsuccessfully tried to find such a tensor, the current point of view being that it does not exist and that we have to content ourselves with energy–momentum pseudotensors for the gravitational field, which are covariant only under a restricted group of coordinate transformations (in most cases, Poincaré's). This, in fact, would be one of the characteristics of the gravitational field tied to the PEGI (see e.g. the discussion in Section 2.7 of [242]) that says that all the physical effects of the gravitational field (and one should include amongst them its energy density) can be locally eliminated by choosing a locally inertial coordinate system.

The most important consequence of the absence of a fully general-covariant energy–momentum tensor for the gravitational field is the non-localizability of the gravitational energy: only the total energy of a spacetime is well defined (and conserved) because the integral of the energy–momentum pseudotensor over a finite volume would be dependent on the choice of coordinates. Some people find this unacceptable and, thus, the search for the general-covariant tensor goes on.

Introduction

In the previous chapters we have studied the upper-left- and upper-right-hand boxes of Figure 14.1 that concern the standard perturbative formulation of string theory and the effective actions of the ten-dimensional string theories (and M theory). We have also learned a bit about the existence of some non-perturbative states in the string spectrum, in particular D-branes and KK and winding modes in compactified theories (the lower-left-hand box of Figure 14.1). We have studied in the three cases the existence of dualities that related various theories and how these dualities are realized in the worldsheet action (when this is possible, i.e. for T duality) and in the effective actions. We have also mentioned that S dualities and T dualities imply the existence of new solitonic states in the string spectrum.

In this chapter and the next we are going to study systematically the lower-right-hand and central boxes of Figure 14.1, that is, the solitonic solutions of the string effective-field theories and their worldvolume actions. We will study the implications that the various dualities have for them (which are evidently related to the effects of dualities on the effective actions) and for the non-perturbative string spectrum. This chapter will be devoted to a general introduction to extended objects and in the next chapter we will deal specifically with those that occur in string/M theory.

These are subjects with many facets that are related in many ways to each other and to the subjects of the previous chapters. Therefore, it is hopeless to try to give a complete, or even half-complete, account of them in the space that we have at our disposal.

In the previous chapters we have occasionally included remarks about advanced topics. For example, we mentioned that the relativistic description of gravity requires an extension of special relativity, called general relativity. We also referred on occasion to the standard model of elementary particles and some of its properties. We described basic facts about supersymmetry, but we did not pursue how it might fit into a realistic theory. References were also made to string theory, which is the leading approach for constructing a fully unified quantum theory of particles and forces. In this chapter we will say a little more about each of these topics. The purpose is to convey a general impression and to whet the reader's appetite for exploring them further. The chapter concludes by discussing some important unsolved problems.

Relativity and gravity

When Einstein formulated special relativity he understood that a relativistic generalization of Newton's theory of gravity was not a simple matter. Rather, it requires new concepts and a new mathematical framework. Einstein dedicated himself to this project during the subsequent decade, which culminated in 1916 with a new theory of gravity, which he called general relativity. As will be discussed below, general relativity makes testable predictions for new phenomena that are not accounted for in Newton's theory. Some were verified right away and others in more recent times. At present, general relativity is tested to good precision (better than 1 percent in some cases) and no discrepancies have been observed.

In this chapter we start the study of the extended objects that appear in the non-perturbative spectrum of string theory, the subject of the third part of this book. In this part we will make use of all the techniques we have developed in the first and second parts, whose main goal was to serve as a preparation for the third.

In a certain sense, this third part also presents the synthesis and (it is hoped) culmination of the ideas presented in the previous two in the framework of string theory: on the one hand, string theory includes a presumably consistent theory of quantum gravity that contains the gravitons described at lowest order by the Fierz–Pauli theory we studied in Chapter 3 [833, 834]. There are two main differences from the non-renormalizable theory of GR: the presence of a dimensionless coupling constant different from the Planck length and the presence of terms of higher order in derivatives. Furthermore, consistent string theories have spacetime supersymmetry and, therefore, supergravity, which we studied in Chapters 5 and 13. On the other hand, string theory incorporates naturally extra dimensions that have to be compactified. Thus, the ideas of Kaluza and Klein studied in Chapter 11 are also integrated into the picture.

Finally, the Schwarzschild, Reissner–Nordström, pp-wave, etc. solutions studied in other chapters are also solutions of string theory and it is natural to try to use them to solve the puzzles that arise when one tries to do quantum mechanics in those backgrounds: the information and entropy problems.