This chapter offers a survey of ideas and results in the approach to quantum gravity based on supersymmetry, strings, and holography.

Extra spatial dimensions appear naturally in this approach, so to set the stage we begin in Section 12.1 with a discussion of general relativity in more than four spacetime dimensions. In higher dimensions, one encounters a richness of structure with no parallel in 4D. Even in vacuum gravity, this includes black hole solutions with non-spherical horizon topologies, black hole non-uniqueness, and regular multi-horizon black holes. We give an overview of such solutions and their properties, both in the context of Kaluza–Klein theory and for asymptotically flat boundary conditions.

A very interesting extension of general relativity is to include matter in such a way that the action becomes invariant under supersymmetry transformations. Supersymmetry is a remarkable symmetry that relates bosons and fermions. It is the only possible extension of the Poincaré group for a unitary theory with non-trivial scattering processes. Supersymmetry is considered a natural extension of the standard model of particle physics; the study of how supersymmetry is broken at low energies, and its possible experimental consequences, is an important active research area in particle physics. Furthermore, independently of its potential phenomenology, supersymmetry offers strong calculational control and that makes it a tremendously powerful tool for analyzing fundamental properties of quantum field theories.

When supersymmetry and general relativity are combined, the result is supergravity. The metric field is accompanied by a spin-3/2 spinor field and this gives a beautiful and enticing playground for advancing our understanding of quantum gravity. Supergravity theories exist in spacetime dimensions D ≤ 11 and they provide a natural setting for studies of charged black holes. Certain extremal limits of charged black holes in super-gravity are invariant under supersymmetry; such ‘supersymmetric black holes’ are key for understanding the statistical mechanical nature of black hole thermodynamics, specifically the microstates responsible for the Hawking–Bekenstein entropy. An example of a super- symmetric black hole is the extremal Reissner–Nordström solution.

Introduction

The necessity of reconciling general relativity (GR) with quantum physics was recognized by Einstein [1] already in 1916 when he wrote:

Yet, almost a century later, we still do not have a satisfactory reconciliation. Why is the problem so difficult? An obvious response is that this is because there are no observations to guide us. However, this cannot be the entire story because, if there are no observational constraints, one would expect an overabundance of theories, not scarcity!

The viewpoint in approaches discussed in this Chapter is that the primary obstacle is rather that, among fundamental forces of Nature, gravity is special: it is encoded in the very geometry of space-time. This is a central feature of GR, a crystallization of the equivalence principle that lies at the heart of the theory. Therefore, one argues, it should be incorporated at a fundamental level in a viable quantum theory. The perturbative treatments which dominated the field since the 1960s ignored this aspect of gravity. They assumed that the underlying spacetime can be taken to be a continuum, endowed with a smooth background geometry, and the quantum gravitational field can be treated just like any other quantum field on this background. But the resulting quantum GR turned out to be non- renormalizable; the strategy failed by its own criteria. The new strategy is to free oneself of the background spacetime that seemed indispensable for formulating and addressing physical questions; the goal is to lift this anchor and learn to sail the open seas. This task requires novel mathematical techniques and conceptual frameworks.

Introduction

Recent media attention to the centenary of the outbreak of the First World War (WWI) reminds us that it was against this backdrop that Einstein, a Swiss citizen, announced the revolutionary theory of general relativity (GR). The war affected the theory's dissemination. Eddington's report introducing GR to the English-speaking world[1] relied on information from de Sitter in neutral Holland. Inevitably, the theory's adherents were caught up in the conflict, most notably Karl Schwarzschild, who died in 1916 while serving on the Russian front.

In 1915 Einstein was already a decade on from his annus mirabilis of 1905, in which he had announced the theory of special relativity, explained the already well-observed photoelectric effect as due to quantization of light (a vital step towards quantum theory), and explained Brownian motion assuming the reality of atoms, an explanation experimentally confirmed by Perrin in 1908. The second of these three great papers won him the 1921 Nobel prize – and they were not all he published that year! For example, he published the famous E = mc2 equation, which later gave the basis of nuclear fusion and fission (whence Einstein's intervention in the development of atom bombs). Fusion in particular explained how stars could hold themselves up against gravity as long as they do. So Einstein had already triumphed well before 1915.

However, he was aware that his work left an awkwardly unresolved question – the need for a theory of gravity compatible with special relativity that agreed with Newton's theory in an appropriate limit. Here we will not recount Einstein's intellectual development of general relativity, which resolved that problem, nor describe the interactions with friends and colleagues which helped him find the right formulation. Those are covered by some good histories of science, and biographies of Einstein, as well as his own writings.

Introduction

The remarkable advances summarized in the first three parts of this volume refer almost entirely to the well-established realm of classical general relativity (GR). However, Einstein [1] was quite aware of the limitations of his theory. In the context of cosmology he wrote, as early as in 1945,

By now, we know that classical physics cannot always be trusted even in the astronomical world because quantum phenomena are not limited just to tiny, microscopic systems. For example, neutron stars owe their very existence to a quintessentially quantum effect: the Fermi degeneracy pressure. At the nuclear density of ∼ 1015 g/cm3 encountered in neutron stars, this pressure becomes strong enough to counterbalance the mighty gravitational pull and halt the collapse. The Planck density is some eighty orders of magnitude higher! Astonishing as the reach of GR is, it cannot be stretched into the Planck regime; here one needs a grander theory that unifies the principles underlying both general relativity and quantum physics.

Early developments

Serious attempts at constructing such a theory date back to the 1930s with papers on the quantization of the linearized gravitational field by Rosenfeld [2] and Bronstein [3]. Bronstein's papers are particularly prescient in that he gave a formulation in terms of the electric and magnetic parts of the Weyl tensor and his equations have been periodically rediscovered all the way to 2002 [4]! Analysis of interactions between gravitons began only in the 1960s when Feynman extended his calculational tools from QED to general relativity [5]. Soon after, DeWitt completed this analysis by systematically formulating the Feynman rules for calculating the scattering amplitudes among gravitons and between gravitons and matter quanta.

Introduction

Gravitational waves provide an opportunity to observe the universe in a completely new way but also give rise to an enormous challenge to take advantage of this opportunity. When Einstein first found wave solutions in linearized general relativity and derived the quadrupole formula, it became clear that a laboratory experiment to produce and detect gravitational waves was impossible, while it was also clear that any gravitational wave signals produced astronomically were too weak to be detected on earth with the instruments available or thought possible at that time. Nearly 100 years later, we are at the confluence of fundamental science and technology that will soon open this new window.

Several lines of development were required to make the search for gravitational waves realistic. Despite the early recognition by Einstein that linearized gravity had wave solutions, the physical reality of gravitational waves remained in dispute for many decades. The reason for this was the absence of formalisms able to separate physical degrees of freedom in the field equations from coordinate (gauge) effects. A well known, striking example was Einstein's conviction that the Einstein–Rosen cylindrical waves [1] were not physical and furthermore that the character of this exact solution proved that there were no physical gravitational waves in the full theory. While Einstein retrieved the correct interpretation in the nick of time [2], the question remained unsettled until correct, gauge-invariant formulations of the problem were developed. The first of these, from Bondi's group [3-5], used the “news function” to demonstrate that, far from the source, one could quantify the energy carried away by gravitational waves. Further developments in understanding equations of motion, gauge freedom, and other methods to identify gravitational waves in the background spacetime led to approximation methods with greater precision and broader application than the original linear waves [6,7]. In addition, the first half-century of general relativity saw the physically relevant exact solutions of Schwarzschild [8] and, much later, Kerr [9].

Introduction

When general relativity was born 100 years ago, experimental confirmation was almost a side issue. Admittedly, Einstein did calculate observable effects of general relativity, such as the perihelion advance of Mercury, which he knew to be an unsolved problem, and the deflection of light, which was subsequently verified. But compared to the inner consistency and elegance of the theory, he regarded such empirical questions as almost secondary. He famously stated that if the measurements of light deflection disagreed with the theory he would “feel sorry for the dear Lord, for the theory is correct!”.

By contrast, today at the centenary of Einstein's towering theoretical achievement, experimental gravitation is a major component of the field, characterized by continuing efforts to test the theory's predictions, both in the solar system and in the astronomical world, to detect gravitational waves from astronomical sources, and to search for possible gravitational imprints of phenomena originating in the quantum, high-energy or cosmological realms.

The modern history of experimental relativity can be divided roughly into four periods: Genesis, Hibernation, a Golden Era, and the Quest for Strong Gravity. The Genesis (1887–1919) comprises the period of the two great experiments which were the foundation of relativistic physics – the Michelson–Morley experiment and the Eötvös experiment – and the two immediate confirmations of general relativity – the deflection of light and the perihelion advance of Mercury. Following this was a period of Hibernation (1920–1960) during which theoretical work temporarily outstripped technology and experimental possibilities, and, as a consequence, the field stagnated and was relegated to the backwaters of physics and astronomy.

But beginning around 1960, astronomical discoveries (quasars, pulsars, cosmic background radiation) and new experiments pushed general relativity to the forefront. Experimental gravitation experienced a Golden Era (1960–1980) during which a systematic, world-wide effort took place to understand the observable predictions of general relativity, to compare and contrast them with the predictions of alternative theories of gravity, and to perform new experiments to test them.

This chapter aims to provide a broad historical overview of the major developments in General Relativity Theory (‘GR’) after the theory had been developed in its final form. It will not relate the well-documented story of the discovery of the theory by Albert Einstein, but rather will consider the spectacular growth of the subject as it developed into a mainstream branch of physics, high-energy astrophysics, and cosmology. Literally hundreds of exact solutions of the full non-linear field equations are now known, despite their complexity [1]. The most important ones are the Schwarzschild and Kerr solutions, determining the geometry of the solar system and of black holes (Section 1.2), and the Friedmann–Lemaître– Robertson–Walker solutions, which are basic to cosmology (Section 1.4). Perturbations of these solutions make them the key to astrophysical applications.

Rather than tracing a historical story, this chapter is structured in terms of key themes in the study and application of GR:

1. The study of dynamic geometry (Section 1.1) through development of various technical tools, in particular the introduction of global methods, resulting in global existence and uniqueness theorems and singularity theorems.

2. The study of the vacuum Schwarzschild solution and its application to the Solar system (Section 1.2), giving very accurate tests of general relativity, and underlying the crucial role of GR in the accuracy of useful GPS systems.

3. The understanding of gravitational collapse and the nature of Black Holes (Section 1.3), with major applications in astrophysics, in particular as regards quasi-stellar objects and active galactic nuclei.

4. The development of cosmological models (Section 1.4), providing the basis for our understandings of both the origin and evolution of the universe as a whole, and of structure formation within it.

5. The study of gravitational lensing and its astronomical applications, including detection of dark matter (Section 1.5).

6. Theoretical studies of gravitational waves, in particular resulting in major developments in numerical relativity (Section 1.6), and with development of gravitational wave observatories that have the potential to become an essential tool in precision cosmology.

Introduction

Einstein's general relativity is a mathematically beautiful application of geometric ideas to gravitational physics. Motion is determined by geodesics in spacetime, tidal effects between physical bodies can be read directly from the curvature of that spacetime, and the curvature is closely tied to matter and its motion in spacetime. When proposed in 1915, general relativity was a completely new way to think about physical phenomena, based on the geometry of curved spacetimes that was largely unknown to physicists.

While the geometric nature of Einstein's theory is beautiful and conceptually simple, the fundamental working structure of the theory as a system of partial differential equations (PDEs) is much more complex. Einstein's equations are not easily categorized as wave- like or potential-like or heat-like, and they are pervasively nonlinear. Hence, despite the great interest in general relativity, mathematical progress in studying Einstein's equations (beyond the discovery of a small collection of explicit solutions with lots of symmetry) was quite slow for a number of years.

This changed significantly in the 1950s with the appearance of Yvonne Choquet- Bruhat's proof that the Einstein equations can be treated as a well-posed Cauchy problem [1]. The long-term effects of this work have been profound: Mathematically, it has led to the present status of Einstein's equations as one of the most interesting and important systems in PDE theory and in geometrical analysis. Physically, the well-posedness of the Cauchy problem for the Einstein equations has led directly to our present ability to numerically simulate (with remarkable accuracy) solutions of these equations which model a wide range of novel phenomena in the strong-field regime.

The Cauchy formulation of general relativity splits the problem of solving Einstein's equations, and studying the behavior of these solutions, into two equally important tasks: First, one finds an initial data set – a “snapshot” of the gravitational field and its rate of change – which satisfies the Einstein constraint equations, which are essentially four of the ten Einstein field equations.

The discovery of general relativity by Albert Einstein 100 years ago was quickly recognized as a supreme triumph of the human intellect. To paraphrase Hermann Weyl, wider expanses and greater depths were suddenly exposed to the searching eye of knowledge, regions of which there was not even an inkling. For 8 years, Einstein had been consumed by the tension between Newtonian gravity and the spacetime structure of special relativity. At first no one had any appreciation for his passion. Indeed, “as an older friend,” Max Planck advised him against this pursuit, “for, in the first place you will not succeed, and even if you succeed, no one will believe you.” Fortunately Einstein persisted and discovered a theory that represents an unprecedented combination of mathematical elegance, conceptual depth and observational success. For over 25 centuries, spacetime had been a stage on which the dynamics of matter unfolded. Suddenly the stage joined the troupe of actors. As decades passed, new aspects of this revolutionary paradigm continued to emerge. It was found that the entire universe is undergoing an expansion. Spacetime regions can get so warped that even light can be trapped in them. Ripples of spacetime curvature can carry detailed imprints of cosmic explosions in the distant reaches of the universe. A century has now passed since Einstein's discovery and yet every researcher who studies general relativity in a serious manner continues to be enchanted by its magic.

This volume was commissioned by the International Society on General Relativity and Gravitation to celebrate a century of successive triumphs of general relativity as it expanded its scientific reach. Through its 12 Chapters, divided into four Parts, the volume takes us through this voyage, highlighting the advances that have occurred during the last three decades or so, roughly since the publication of the 1979 volumes celebrating the cen- tennial of Einstein's birth.

Introduction

John Friedman

The emphasis of this chapter is on four parts of relativistic astrophysics in which general relativity plays a fundamental role. After briefly reviewing the early history of the subject, we discuss

General relativistic astrophysics encompasses a broader arena, and separate chapters or parts of chapters in this volume are devoted to cosmology, gravitational waves, the inspiral and merger of compact binaries, and black-hole stability.

Relativistic astrophysics began in 1916 on the Russian front, where Karl Schwarzschild wrote two papers, one reporting the solution to the Einstein equation for an incompressible spherical star, the other presenting the celebrated vacuum Schwarzschild spacetime. Schwarzschild was dead within the year, and for the next 47 years his solutions had a twilight existence. In no known stars did general relativity play a significant role, and only a handful of papers in astronomy or astrophysics mentioned the work.

Although sparsely distributed, the exceptions to this neglect were remarkable. In 1931, shortly before Chadwick's discovery of the neutron and shortly after the first paper by Chandrasekhar [2] (following approximate computations by Anderson [3] and Stoner [4] on an upper mass limit of white dwarfs, Landau [5] submitted a paper that independently argued that there was an upper limit on the mass of a collection of degenerate fermions and speculated on the existence of stars with cores of nuclear density.

Introduction

Solutions of the Einstein equations evolve from initial data given on a three-dimensional manifold M. The initial position and velocity of the gravitational field are given by a Riemannian metric g and a symmetric (0, 2) tensor K. The metric g will be the metric induced on M as a spacelike hypersurface in the spacetime S which evolves from the data, and the tensor K will be the second fundamental form of M in S. Thus an initial data set is given by a triple (M, g, K). There is currently interest in higher-dimensional gravity in the physics community, so when convenient we will discuss initial data on an n-dimensional manifold Mn which will evolve to an (n + 1)-dimensional spacetime Sn + 1 (n ≥ 3).

A basic fact of life for the Einstein equations is that the initial data g and K cannot be freely specified, but must satisfy a system of n + 1 nonlinear partial differential equations. These are called the constraint equations, and Section 8.2 deals with recent progress on solving this set of equations. On the one hand the constraint equations present a complication in the study of the initial value problem since it is a difficult (and as yet unsolved) problem to fully analyze their solutions. On the other hand, it is because of the constraint equations that physical notions of energy and momentum can be defined. It is also because of them that geometric and topological restrictions hold in certain cases on the initial manifold M, and for black holes in Σ.

We do not have the space here to give a comprehensive survey of the initial value problem, so instead we have focused on several questions on which there has been recent progress and which are currently active areas of investigation. We have chosen to give brief outlines of the main ideas involved in the study of these specific questions rather than to attempt to touch on all aspects of the field.

Historical perspective

James Clerk Maxwell discovered in 1865 that electromagnetic phenomena satisfied wave equations and found that the velocity of these waves in vacuum was numerically the same as the speed of light [1]. Maxwell was puzzled at this coincidence between the speed of light and his theoretical prediction for the speed of electromagnetic phenomena and proposed that “light is electromagnetic disturbance propagated through the field according to electromagnetic laws” [1].

Because any theory of gravitation consistent with special relativity cannot be an action-at-a-distance theory, in many ways, Maxwell's theory, being the first relativistic physical theory, implied the existence of gravitational waves (GWs) in general relativity (GR). Indeed, years before Einstein derived the wave equation in the linearised version of his field equations and discussed the generation of GWs as one of the first consequences of his new theory of gravity [2,3], Henri Poincaré proposed the existence of les ondes gravifiques purely based on consistency of gravity with special relativity [4]. However, for many years GWs caused much controversy and a lot of doubt was cast on their existence [5-8]. The year 1959 was, in many ways, the turning point – it was the year of publication of a seminal paper by Bondi, Pirani and Robinson [6] on the exact plane wave solution with cylindrical symmetry and the energy carried by the waves [9]. This paper proved that wave solutions exist not just in the weak-field approximation and that GWs in GR carry energy and angular momentum away from their sources. These results cleared the way for Joseph Weber [10] to start pioneering experimental efforts. The discovery of the Hulse–Taylor binary [11], a system of two neutron stars in orbit around each other, led to the first observational evidence for the existence of gravitational radiation [12]. The loss of energy and angular momentum to GWs causes the two stars in this system to slowly spiral in towards each other.

Introduction

David Wands and Roy Maartens

Our Universe provides the grandest arena in which to test General Relativity as a theory of space, time and gravity. It becomes essential to consider both the causal propagation of matter and radiation through space-time and the dynamical evolution of space-time if we are to construct a consistent theoretical framework in which to interpret astronomical observations on the largest observable scales. Einstein himself originally tried to construct a static cosmology but it was soon appreciated that the field equations of General Relativity naturally accommodate dynamical and evolving space-times. Einstein's own static model of a 3-sphere, balancing the gravitational pull of matter against a positive spatial curvature and a cosmological constant was shown to be poised between expansion and collapse and hence unstable to infinitesimal disturbance.

Friedmann and Lemaître showed that Einstein's field equations admit expanding- universe solutions, which have become the basis for modern cosmology, despite Einstein's initial dismissal of the solutions. However persuasive the theoretical models, empirical observations are, of course, necessary to determine the actual dynamics of our observable Universe; the work of Slipher and Hubble in the 1920s [1] persuaded scientists that in fact our Universe is expanding. The logical consequence of this expansion is that either our Universe was hotter and denser in the past (coming ultimately from a Hot Big Bang) or, perhaps, that energy had to be continually created as the universe expanded (the Steady State model). The discovery of the Cosmic Microwave Background (CMB) radiation by Penzias and Wilson in 1965 [2] convinced most astronomers that the Universe did in fact begin at a Hot Big Bang, a finite time in the past. This hot dense plasma, in thermal equilibrium at early times, also provides a setting for the freezing out of the light atomic nuclei as the universe cools below several million degrees Kelvin [3], although heavier elements must be formed later in stars.