Introduction

Consistent theories of quantum gravity in spacetimes that asymptote to anti-de Sitter (AdS) spacetime are equivalent to quantum field theories defined on the conformal boundary of the spacetime [1]. A pedagogical discussion of this “holographic correspondence” may be found in [2, 3] and in Chapter 12 of this volume. While some deeper questions arising from the correspondence remain to be understood from first principles, the conceptual “Gestalt switch” involved in viewing physical processes simultaneously from a gravitational and a field-theoretic perspective has provided an invaluable source of physical intuition as well as computational power. In particular, in the large-N limit of quantum field theories, to be recalled shortly, the gravitational description becomes weakly curved and the tools of general relativity may be harnessed.

This chapter will be concerned with black holes in four-dimensional asymptotically AdS spacetimes. By focusing on charged planar black holes, we will establish an interface with the rich phenomenology of (2+1)-dimensional quantum field theories that has been widely studied in condensed matter physics. Planarity of the horizon will translate into the statement that the dual quantum field theory propagates on a background Minkowski spacetime in 2+1 dimensions. The perhaps more familiar spherical foliation of asymptotically AdS spacetimes would correspond to considering quantum field theories on a spatial sphere; this complicates the field-theoretic physics by introducing a scale, the radius of the sphere, and also does not correspond to a situation of significant interest in condensed matter physics at present.

Introduction

A black ring is a D-dimensional black hole with horizon topology S1 × SD-3. There is a simple heuristic way of understanding why such solutions might exist. Consider a black string, the product of the (D - 1)-dimensional Schwarzschild solution with a flat direction, with horizon topology ℝ × SD-3. Imagine bending the string into a loop, so the topology is now S1 × SD-3. This loop would tend to contract owing to its tension and gravitational self-attraction. However, if the loop rotates then it might be possible to balance these forces with centrifugal repulsion. That this is indeed possible is proved by the existence of an explicit black ring solution of the five-dimensional vacuum Einstein equation [1, 2]. The existence of analogous solutions for D > 5 dimensions (or when a cosmological constant is included) is strongly suggested by the perturbative methods reviewed in Chapter 8.

The discovery of black rings revealed that higher-dimensional black holes can exhibit properties very different from those of four-dimensional holes. First, higherdimensional black holes need not be topologically spherical. Second, higherdimensional black holes are not uniquely parameterized by mass and angular momenta: there exist distinct black ring solutions with the same mass and angular momenta. Furthermore, there exist black ring and Myers–Perry solutions [3] with the same mass and angular momenta. Thus the topology and uniqueness theorems that underpin our understanding of four-dimensional black holes do not extend to higher dimensions in an obvious way.

Black holes are one of the most striking predictions of general relativity. Their main classical properties were understood by the 1970s. In particular, it had been shown that they are very simple objects: stationary vacuum black holes are uniquely characterized by their mass and angular momentum. Astrophysical evidence for black holes has improved dramatically over the past decade and it is now widely accepted that black holes are ubiquitous in our universe.

Naturally enough, the work through the 1970s was focused on black holes in our familiar four spacetime dimensions (D = 4). Recently, there has been an explosion of interest in higher-dimensional black holes. There are at least four reasons for this.

(1)As mentioned above, black holes in D = 4 are special: theymust be spherical, specified by a few parameters, always stable, etc. It is natural to ask whether these properties are characteristic of black holes in general or just a result of four spacetime dimensions.

(2) Recent brane-world ideas suggest that our familiar three spatial dimensions might just be a surface in a higher-dimensional space. In these theories, nongravitational forces are confined to the brane but gravity is higher dimensional. So black holes extend into the extra dimensions.

(3) String theory, one of the most promising approaches to quantum gravity, predicts that spacetime has more than four dimensions. This incorporates older ideas of unification based on the idea that extra dimensions are curled up into a small ball. So, in string theory we must consider higher-dimensional black holes.

Introduction

In this chapter we will study a particular long-wavelength limit of Einstein's equations with a negative cosmological constant in d + 1 dimensions. In such a limit we find that Einstein's equations reduce to the equations of fluid dynamics (relativistic generalizations of the famous Navier–Stokes equations) in d dimensions. While the motivation for our study lies within the AdS/CFT correspondence of string theory, the fluid/gravity correspondence stands on its own and can be viewed as a map between two classic dynamical systems.

Prelude: CFT stress tensor dynamics from gravity

An important consequence of the AdS/CFT correspondence (see Chapter 12) is that the dynamics of the stress(–energy–momentum) tensor in a large class of d dimensional strongly coupled quantum field theories is governed by the dynamics of Einstein's equations with negative cosmological constant in d + 1 dimensions. To begin with, we shall try to provide the reader with some intuition for this statement and argue that searching for a tractable corner of this connection leads one naturally to the fluid/gravity correspondence.

In its most familiar example, the AdS/CFT correspondence asserts that SU (N) N = 4 super Yang–Mills (SYM) theory is dual to type-IIB string theory on AdS5 × S5. It has long been known that in the 't Hooft limit, which involves taking N → ∞ while keeping the coupling λ fixed, the gauge theory becomes effectively classical.

Introduction

Whilst black holes in four dimensions are well mannered, being spherically symmetric or having special algebraic properties which enable them to be found analytically, moving beyond four dimensions many solutions of interest appear to have no manners whatsoever. The problem of finding these unruly black holes becomes that of solving a nonlinear coupled set of partial differential equations (PDEs) for the metric components given by the Einstein equations. In general it is unlikely that closed-form analytic solutions will be found for many of the exotic black holes discussed earlier in this book. If we are to understand their properties then we must turn to numerical techniques to tackle the PDEs that describe them. It is the purpose of this chapter to develop general numerical methods to address the problem of finding static and stationary black holes.

Surely the phrase “the devil is in the detail” could not have a truer application than to numerics. The emphasis of this chapter will be to provide a road map in which we formulate the problem in as unified, elegant and geometric a way as possible. We will also discuss concrete algorithms for solving the resulting formulation, but the extensive details of implementation will not be addressed, probably much to the reader's relief. Such details can be found in the various articles cited in this chapter.

In this chapter we explain the gauge/gravity duality [1–3], which is a motivation for studying black hole solutions in various numbers of dimensions. The gauge/gravity duality is an equality between two theories. On one hand we have a quantum field theory in d spacetime dimensions. On the other hand we have a gravity theory on a (d + 1)-dimensional spacetime that has an asymptotic boundary which is d-dimensional. It is also sometimes called AdS/CFT, because the simplest examples involve anti-de Sitter spaces and conformal field theories. It is often called gauge/string duality, because the gravity theories are string theories and the quantum field theories are gauge theories. It is also referred to as “holography” because one is describing a (d + 1)-dimensional gravity theory in terms of a lower-dimensional system, in a way that is reminiscent of an optical hologram, which stores a three-dimensional image on a two-dimensional photographic plate. This duality is called a “conjecture”, but by now there is considerable evidence that it is correct. In addition, there are some derivations based on physical arguments.

The simplest example involves an anti-de Sitter spacetime. So, let us start by describing this spacetime in some detail. Anti-de Sitter is the simplest solution of Einstein's equations with a negative cosmological constant. It is the Lorentzian analogue of hyperbolic space, which was historically the first example of a non-Euclidean geometry.

Overview

The ultimate fate of black holes subject to the Gregory–Laflamme instability has been an open question for almost two decades. In this chapter we discuss the behavior of an unstable five-dimensional black string and elucidate its final state. Our studies reveal that the instability unfolds in a self-similar fashion, in which the horizon at any given time can be seen as thin strings connected by hyperspherical black holes of different radii. As the evolution proceeds pieces of the string shrink while others give rise to further spherical black holes, and consequently the horizon develops a fractal structure. At this stage its overall topology is still ℝ × S2; the fractal geometry arises along ℝ and has an estimated Hausdorff dimension d ≈ 1.05. However, the ever-thinning string regions eventually shrink to zero size, revealing a (massless) naked singularity. Consequently, this spacetime provides a generic counterexample to the cosmic censorship conjecture, albeit in five dimensions. While we restrict to the five-dimensional case for reasons of computational cost, our observations are intuitively applicable to higher dimensions.

To capture the late-time nonlinear dynamics of the system correctly requires numerical solution of the full Einstein equations. In this chapter, following a brief historical account (section 3.2)we describe details of our numerical implementation (section 3.3) as well as the behavior of the obtained solution (section 3.4). We discuss some additional properties of the solution, including speculation on when quantum corrections are expected to become important, and future directions in section 3.5.

GEOCHEMICAL AND GEOPHYSICAL FOSSILS

In tracing the evolution of the Earth, instead of using physical and chemical methods to reproduce phenomena in the laboratory, research is carried out using the “fossils” of these phenomena. We have already explained how the isotopic ratios recorded in rocks, the oceans, and the atmosphere serve effectively as the “fossils” of ancient chemical differentiation events. Meteorites are still the most important source of information on the origin of the Earth, being “fossils” in which the early state of the Solar System is frozen. We have also seen that the remanent magnetism of rocks can be used as a “fossil” of the Earth’s magnetic field in the past. Studying the evolution of the Earth, therefore, often calls for an ingenious approach, requiring more than simple applications of mathematical, physical, and chemical methods. In this chapter we will discuss a novel approach in this spirit, which may be called “lunar paleopedology” (pedology is the scientific discipline of studying soil). After the monumental achievement of the landing on the Moon in 1969 under the Apollo project, lunar soils sampled by astronauts have been offering a new type of “fossil record” not only for the Moon, but also for the Sun and Earth. We start with a short discussion of the origin of the Earth–Moon system.

ORIGIN OF THE EARTH–MOON SYSTEM

Although the origin of the Moon is still a big mystery, a currently predominant view is that the Moon formed as a result of the impact of a Mars-sized proto-planet with a proto-Earth.[1] An almost identical isotopic composition of oxygen (the most abundant element in planetary objects) between the Moon and Earth suggests that both objects had undergone elemental and isotopic homogenization, most likely through high-temperature vaporization caused by the impact. Numerical calculations of such a giant impact have commonly indicated that the temperature in the impacted material (a proto-Earth) and impactor could have reached more than ten thousand degrees centigrade, enough to vaporize all of the materials involved.[2] The formation age of the Earth–Moon system (or the timing of the giant impact) has been estimated to be about 4.5 billion years ago from an extinct short-lived radioactive isotope 182Hf, whose half-life is 9 million years (see Chapter 2).

PALEOMAGNETISM AND APPARENT POLAR WANDER

The Earth’s magnetic field can be thought of as generated by an enormous magnet placed at the center of the planet. The points that pass through the Earth’s surface when both poles of this enormous magnet are extended are called the north and south magnetic poles. The present north and south magnetic poles differ a little from those of the geographic poles, the difference being about 10° at present. However, if we take an average of the geomagnetic field direction over a few thousand years, the averaged magnetic poles almost perfectly coincide with the geographic poles. As mentioned in Chapter 3, the origin of the geomagnetic field is the fluid motion in the liquid outer core, and it is expected from the geodynamo theory that the magnetic poles align with the rotation axis of the Earth.

When magma erupts on the surface and cools down, it acquires magnetization in the direction of the ambient magnetic field, i.e. the geomagnetic field. The magnetization thus acquired is called thermo-remanent magnetization and has been shown to be extremely stable (cf. Chapter 9). Therefore, the remanent magnetization of volcanic rocks provides a very faithful record of the geomagnetic field at the time of eruption. Accordingly, by measuring the remanent magnetization of volcanic rocks with various ages, we can infer the direction of the geomagnetic field in the past.

Preface to the second edition

A few years ago, Professor David Hilton of the University of California, San Diego mentioned to me that he was still using my book, The Earth: Its Birth and Growth, as suggested reading in his class. The book was published in 1979 by Cambridge University Press. Amazed by its longevity, I became curious about how this seemingly plain small book could have survived in the recent swarm of the media world, in which there are a flood of books on astounding findings in Earth and planetary sciences with colorful pictures and illustrations. I read the book once again, and I was convinced that it was worth revising it by incorporating recent developments.

The new edition has therefore attempted to keep the original style of the first edition: that is, to maintain readability without sacrificing scientific rigor. The concise style of the book is important so that readers can see the big picture without being drowned by a formidable amount of information. Obviously many of the materials in the first edition needed to be updated. Also, given recent developments, I wanted to emphasize in the new edition the importance of integrating a vast range of geophysical and geochemical data to develop a coherent view of Earth’s evolution.