ARECIBO RADAR OBSERVATORY

Hidden deep in the northwestern rain forests of Puerto Rico, near the coastal city of Arecibo, is one of the technological wonders of man – the Arecibo Observatory, part of the National Astronomy and Ionosphere Center, or NAIC for short. The observatory is roughly an hour and a half drive west of San Juan at the terminus of highway PR-625. The last few miles of the trip wind precariously through small neighborhoods on narrow roads. The roads follow the hill and hollow karst terrain as a roller coaster might – up and down, left and right – and the effect on those inclined to motion sickness is just as wretched.

Because the topography is so rugged, you don't see any part of the enormous antenna until you pull up to the gate. Even then, the full effect isn't felt until you walk the hill to the Visitor Center, or are invited to visit the control room on the edge of the dish. Once there, though, the sight of it is powerful. Like a cybernetic prosthesis, the antenna dish appears to fit organically within the terrain.

The dish itself is not a solid object. Instead, hundreds of steel cables, parallel but separated from each other by a meter or so, are strung across a natural hollow and allowed to hang, creating a concave web. Nearly 40,000 lightweight aluminum screens, each about one by two meters, are set on top of the cables to create a giant, floating mesh. Each screen is precisely aligned by hand so that the entire structure deviates from a true spherical shape by less than a few millimeters anywhere.

The gauze-like metallic mesh has a couple of advantages over solid antennas. The telescope is designed to detect microwaves with wavelengths of tens of centimeters. Waves that large cannot pass through a fine screen mesh – it might as well be solid.

Gravitational potential

Within the Earth itself, the gravitational potential at some point P due to the Sun (or any of the other celestial bodies) varies with the location of P, inversely as its distance from the body. The gravitational attraction on the mass elements of the Earth varies as the inverse square of the same distance. (We are using the word “attraction” here with the specific meaning of “the force exerted per unit mass.”) Suppose that the potential, as a function of the location of P, is separated into two parts: one that depends linearly on the position vector from the Earth's center of mass to P (including, in general, a term that is independent of the position) and the remainder of the potential function that is non-linear in the vector. The first part produces a uniform attraction on mass elements throughout the Earth, and is responsible for the relative orbital motion of the celestial body and the Earth. It is the second part that is responsible for the variation of the gravitational attraction over the volume of the Earth. This part of the potential is designated as the tide generating potential (TGP) or, for brevity, as the tidal potential. The reason for this nomenclature is simple: it generates the ocean tides as well as deformations of the solid Earth, referred to as solid Earth tides. (The term “solid Earth” is generally applied to the whole of the Earth excepting the fluid layers at the surface, namely, the oceans and the atmosphere.) It is the same tidal potential that generates Earth rotation variations too. Thus, the contents of this book relate to the effects of the TGP.

Axes associated with Earth rotation

Variations in the Earth's rotation are most naturally conceived of as variations in the direction or the magnitude (or both) of the Earth's angular velocity vector_, also called the rotation vector. (Throughout this book we use the bold notation for denoting vectors.)

THE AWAKENING

For two centuries, asteroids were discovered randomly by astronomers attempting to do other “real” work in astronomy. With the exception of the first decades after the discovery of Ceres, there does not appear to have been any systematic effort to discover more. Eros, the 433rd numbered asteroid, was not discovered until nearly a century after Ceres. And by the 1980s, well into the modern astronomical era, only a couple of thousand asteroids had been discovered; of those, only a handful were in reservoirs other than the main belt. Since then, though, discoveries have accelerated; at present there are more than 400,000 numbered asteroids. Why the sudden quickening? It is because we are searching scared, using dedicated telescopes, high-tech instruments, and increasingly sophisticated software to extract these tiny needles from the cosmic haystack.

Culturally, there has been a shift. Until recently, there was no urgency for this search task. But several events have frightened us and made the effort worthwhile to those who control purse strings. Perhaps the watershed event was the discovery and subsequent impact of Comet Shoemaker–Levy 9 into Jupiter in 1994. A few scientists, including Gene Shoemaker (1928–1997), had been warning us of the threat of an impact for years. Shoemaker's role in proving that Meteor Crater, Arizona, was from a relatively recent impact had convinced him and others. But still, events of 50,000 years ago – the estimated age of that impact – don't register with most of us. Our temporal horizons are much closer.

Shoemaker, his wife Carolyn, and friend David Levy had worked together for years surveying the sky for asteroids that could approach and potentially hit the Earth. In March 1993, the trio discovered their ninth periodic comet together, and following the naming convention, it was designated Shoemaker–Levy 9 (SL9).

For several years (about 20) both authors have been working in the domain of Earth rotation, and in particular on nutation. The Working Group on nutation, established by the International Astronomical Union and the International Union of Geodesy and Geophysics in 1994, was a starting point for bringing together scientists thinking about what was missing in the nutation series adopted by the International Astronomical Union. Collaboration between the scientists of the WG was very successful and, in particular, the authors' collaboration began at that time. Recently, it has appeared to us that there were no existing books yet dedicated to the subject, and the scientific community is looking for a suitable publication. The literature contains a lot of relevant articles, but many of them rely on previous work and do not give full details. This book aims at bringing everything together for the first time. The book is addressed to students or scientists who want to understand nutations. The aim of this book is to give a reasonably comprehensive introduction to the fundamental concepts, mathematical formalism, and methodology of the Earth's nutation. It is only assumed that the student or reader is familiar with the elementary principles of calculus, although we might have used in some parts short-cuts for reasons of simplicity, and with the underlying physical principles in the foreground. Another important aim of this book is to make a comprehensive list of the geophysical and astronomical processes involved in nutation, in order to be able to investigate the “next decimal place.”

The authors wish to take this opportunity to acknowledge all those who have aided in the preparation of this book.

Rotation and global shape of the Earth

At a very elementary level, the Earth is considered to be an axially symmetric ellipsoid, rotating with uniform angular speed about its symmetry axis, which is the polar axis passing through the Earth's center and its north and south poles; under the steady rotation, the direction of this axis maintains a fixed direction in space, i.e., relative to the directions of the “fixed stars.” (The celestial objects that come closest to the ideal of remaining “fixed in space” are the quasars, the most distant extragalactic celestial objects.) The direction of the symmetry axis is at an inclination of about 23.5° to the direction of the normal to the plane of the Earth's orbit around the Sun (or more precisely, about the solar system barycenter, i.e. the center of mass of the solar system).

An axially symmetric ellipsoidal shape, bulging at the equator and flattened at the poles, and an internal structure with the same symmetry, would result from the centrifugal force associated with uniform Earth rotation around the polar axis, counterbalanced by gravity. The ellipsoidal structure computed on the basis of this balance of forces, assuming that the material of the rotating body behaves like a fluid under the incessant action of forces acting over very long timescales (i.e., that the resistance of even solid regions to shear deformation is overcome under such conditions), is called the “hydrostatic equilibrium ellipsoid.” The Earth's figure (shape) does conform quite closely, though not perfectly, to that of such an ellipsoid. The equatorial radius of the Earth exceeds the polar radius by about 21 km; this is often described as the equatorial bulge. This bulge is to be viewed in relation to the mean radius of about 6371 km.

Orbit of the Earth

The force of gravitational attraction of the Sun on the total mass of the Earth holds the Earth in orbital motion around the Sun; similarly, the Earth’s gravitational force maintains the orbital motion of the Moon around the Earth.

In the previous chapter, we introduced increasingly complex theories of gravity, starting from GR, to accommodate fermions, and we saw that the generalizations of GR that we had to use could be thought of as gauge theories of the symmetries of flat spacetime.

A very important development of the past few decades has been the discovery of supersymmetry and its application to the theory of fundamental particles and interactions. This symmetry relating bosons and fermions can be understood as the generalization of the Poincaré or AdS groups, which are the symmetries of our background spacetime to the super-Poincaré or super-AdS (super-)groups, which are the symmetries of our background superspacetime, a generalization of standard spacetime that has fermionic coordinates.

It is natural to construct generalizations of the standard gravity theories that can be understood as gauge theories of the (super-)symmetries of the background (vacuum) superspacetime. These generalizations are the supergravity (SUGRA) theories. Given that the kind of fermions that we can have depends critically on the spacetime dimension, the SUGRA theories that we can construct also depend critically on the spacetime dimension. Furthermore, we can extend the standard bosonic spacetime in different ways by including more than one (N) set of fermionic coordinates. This gives rise to additional supersymmetries relating them and, therefore, to supersymmetric field theories and SUGRA theories with N supersymmetries. The latter are also known as extended SUGRAs (SUEGRAs). The SUEGRAs can be further extended by coupling them to supersymmetric matter (matter fields that fill complete representations of the supersymmetry algebra) and they can be deformed by introducing new couplings among the fields depending on new parameters. The main deformation procedure is that of gauging global symmetries, explained in Section 2.5, but it is not the only one: some supergravities also admit the so-called massive deformations, the main example being Romans' massive N = 2A, d = 10 theory [1083], which is reviewed in Section 22.2. All the possible deformations can, apparently, be taken into account by the embedding tensor formalism, reviewed in Section 2.7.

After the general introduction to extended objects of Chapter 24, in this chapter 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 25.1, in which we will arrive at the diagrams in Figs. 25.5 and 25.6 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 25.1-25.3) 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 25.2) 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 25.3 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 25.4.

In Section 25.5 we will learn how a great deal of information about all these objects is encoded in the spacetime superalgebras of the effective (supergravity) theories. In particular, the superalgebras tell us (up to a point) which extended objects may exist and the amount of unbroken supersymmetry preserved by each of them (always half of the total), as we will check by solving explicitly the Killing spinor equations (Section 25.5.1).

Non-linear σ-models are a common element of many of the actions considered in the main text that contain scalar fields: in the N = 1, d = 4 matter-coupled supergravities studied in Chapter 6 we find σ-models which correspond to Kähler–Hodge manifolds; in the N = 2, d = 4 theories studied in Chapter 7 we find special Kähler and quaternionic-Kähler manifolds, and the latter and real special manifolds naturally arise in the σ-models of the N = 1, d = 5 matter-coupled supergravities studied in Chapter 9. There are many other examples in the text, and that is why generic non-linear σ-models have been included in the generic actions Eqs. (2.147) and (2.178).

In many cases the metrics of these σ-models have isometries and the σ-model action is invariant under an associated global symmetry. Furthermore, there are many instances in which we want to gauge one or several of those symmetries, deforming the action so that it becomes invariant under the local (gauge) version of those symmetries. For example, gauging this kind of symmetry in the above-mentioned supergravities one obtains theories with non-Abelian Yang–Mills fields and symmetries and a scalar potential with many potential uses. Gauged σ-models also arise in the construction of KK-brane effective actions studied on p. 706.

Due to the couplings of the scalars to other fields, the symmetries of a σ-model are not automatically symmetries of the full theory. For general kinds of couplings to certain kinds of bosonic fields (differential forms of various ranks) the situation has been studied in Section 2.6.2. In the above-mentioned supergravity theories the coupling to supergravity (hence, of the σ-model scalars to vectors and spinors) requires the geometries of their σ-models to be Kähler–Hodge, special Kähler, quaternionic-Kähler, or real special. There, the Riemannian structure (the metric) is not the only structure that needs to be preserved for a transformation to be called a symmetry. These geometries and their symmetries have been reviewed in the previous appendices, and here we want to study the gauging of all these symmetries in order of increasing complexity and with an (essentially) homogeneous notation that is used in Chapters 6 and 7 to describe the complete theories.

In spite (or because) of its relentless progress, science is a perpetually unfinished work and so must be a description of any field of research at a given time. The first edition of this book tried to review the foundations and main achievements of the field that we called semiclassical string gravity covering the basics of general relativity, supergravity, and superstring theory aiming to provide a complete and self-consistent introduction to the effective field theory description and the black-hole and black-brane solutions of the latter (ten-dimensional supergravity and some of its compactifications). However, many interesting topics and results had to be omitted then due to lack of space and many others have emerged in the following years and I started feeling quite soon that the book was not complete and the goals I had set forth had not been reached.

Of course, for the aforementioned reasons, it is intrinsically impossible to give a complete and final description of this field in the absolute sense, but I think (the reader will be the judge) that the inclusion of a reasonable number of new topics was necessary and will make the book much more useful. The second edition is the result of trying to cover that necessity while preserving the self-consistency of the book by adding background and complementary material.

The two main gaps I have tried to close are the lack of a complete discussion of the black-hole attractor mechanism and a description of the classification/characterization of the supersymmetric solutions of general (matter-coupled) four-dimensional supergravities. These two subjects are linked by the original discovery of the attractor mechanism in supersymmetric extremal black-hole solutions of N = 2, d = 4 supergravity coupled to vector supermultiplets.

A self-consistent description of these two subjects has required, first, the addition of several new chapters (Chapters 6–8) on matter-coupled N = 1 to N = 8 four-dimensional supergravities, including detailed descriptions of the gaugings of the N = 1 and N = 2 theories.