Chapter 1 What is a black hole?

  A black hole is a region of spacewhere the force of gravity is so strong that nothing, not even light, can travel fast enough to escape from its interior. Although they were first conceived in the fertile imaginations of theoretical physicists, black holes have now been identified in the Universe in their hundreds and accounted for in their millions. Although invisible, these objects interact with, and can thus influence, their surroundings in a way that can be highly detectable. Exactly what the nature of that interaction is depends on proximity relative to the black hole: too close and there is no escape, but further afield some dramatic and spectacular phenomena will play out.

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  The term `black hole' was first mentioned in print in an article by Ann Ewing in 1964, reporting on a symposium held in Texas in1963, although she never mentioned who coined the expression.In 1967, American physicist JohnWheeler needed a shorthand for`gravitationally completely collapsed star' and began to popularize the term, though the concept of a collapsed star was developed by fellow Americans Robert Oppenheimer and Hartland Snyder back in 1939. In fact, the mathematical foundations of the modern picture of black holes began rather earlier in 1915, with German physicist Karl Schwarzschild solving some important equations of Einstein's (known as the field equations in his General Theory of Relativity) for the case of an isolated non-rotatingmass in space.

  Two decades later in the UK, a little before Oppenheimer and Snyder's work, Sir Arthur Eddington had worked out some of the relevant mathematics in the context of investigating work by the Indian physicist Subrahmanyan Chandrasekhar on what happens to stars when they die. The physical implications of Eddington's calculations, namely the collapse of massive stars when they have used up all their fuel to form black holes, Eddington himself pronounced to the Royal Astronomical Society in 1935 as being『absurd'. Despite the apparent absurdity of the notion, black holes are very much part of physical reality throughout our Galaxy and across the Universe. Further advances were made in the United States by David Finkelstein in 1958, who established the existence of a one-way surface surrounding a black hole whose significance for what we shall study in the coming chapters is immense. The existence of this surface doesn't allow light itself to break free from the powerful gravitational attraction within and is the reason why a black hole is black. To begin to understand howthis behaviour might arise we need to first understand a profound feature of the physical world: there is a maximum speed at which any particle or any object can travel.

  How fast is fast?

  A lawof the jungle is that if youwant to escape a predator you need to run fast. Unless you have exceptional cunning or camouflage, you will only survive if you are swift. The maximum speed with which a mammal can escape an unpleasant situation depends on complex biochemical relationships between mass,muscle strength, and metabolism. The maximum speed with which the most rapidly travelling entity in the Universe can travel is that exhibited by particles that have no mass at all, such as particles of light (known as photons). This maximum speed can be given very precisely as 299,792,458 metres per second, equivalent to 186,282 miles per second, which is almost approaching a million times faster than the speed of sound in air. If I could travel at the speed of light, I would be able to travel

  from my home in the UK to Australia in one fourteenth of a second, barely time to blink. Light travelling from our nearest star,the Sun, takes just eight minutes to travel to us. From our outermost planet, Neptune, it's a journey time of just a few hours for a photon.We say that the Sun is eight light-minutes away from Earth and that Neptune is a few light-hours away from us. This has the interesting consequence that if the Sun stopped shining or if Neptune suddenly turned purple, no one on Earth could find out about such important information for eight minutes or a few hours respectively.

  Let's now consider how fast light can travel from even more immensely distant points in space back to Earth. The Milky Way,the Galaxy in which our Solar System resides, is a few hundred thousand light-years across. This means that light takes a few hundred thousand years to travel from one side of the Galaxy to the other. The Fornax cluster is the nearest cluster of galaxies to the local group of galaxies (of which the Milky Way is a significant member) and is hundreds of millions of light-years away from us.Thus, an observer on a planet orbiting a star in a galaxy within the Fornax cluster looking back to Earth right now might, if equipped with appropriate instrumentation, see dinosaurs lumbering around on Earth. However, it is only the mind-boggling vastness of the Universe thatmakes the motion of light look sluggish and time-consuming. The role of the speed of light as a mandatory upper limit has an intriguing effect when we start to consider how to launch rockets into space.

  Escape velocity

  Now suppose some disaster occurs with the result that the entire mass of the Earth were shrunk to a point, having no spatial extent whatsoever.We call such an object a singularity. Ithasnowbecome a `pointmass', amassive object that occupies zero volume of space.At a very small distance of only onemetre away fromthis singularity,the escape velocity would be much larger than it was at 1,600 km(and in fact would be about 10% of the speed of light). Closer to the singularity still, just under one centimetre away the escape velocity would be equal to the speed of light. At this distance, light itself would not have sufficient speed to escape this gravitational pull.This is the key idea to understand how black holes work.

  It is worth clarifying use of the word `singularity'.We do not believe that at the end point of a continuing gravitational collapse the matter goes down to a geometric point but rather that our classical theory of gravity breaks down and we enter a quantum regime. From here on, we will use the term singularity to refer to this ultra-dense state.

  The event horizon

  Now imagine you are an astronaut flying a spacecraft and that you are approaching this singularity.While still at some distance from it, you could always throw your engines into reverse and retreat from it. But the closer you get, the harder a dignified retreat becomes. Eventually you reach a distance from which it is impossible to escape, no matter how powerful your onboard engines are. This is because you have reached the event horizon, a mathematically-defined spherical surface, which is defined as being the boundary inside of which the escape velocity would exceed the speed of light. For our thought-experiment about Earth collapsed to a point, this surface would be a sphere of radius only one centimetre with the singularity at its centre, easy enough perhaps for our spacecraft to avoid. However, the event horizon becomes much larger when the black hole is formed from a collapsed star rather than a collapsed planet. The event horizon has an important physical consequence: if you are on that surface or inside it, the laws of physics simply won't allow you to escape because to do so you would need to break the universal speed limit. The event horizon is a mandatory level of demarcation:outside it you have freedom to determine your destiny; inside it,and your future remains unalterably locked within.

  The radius of this spherical surface is named in honour of Karl Schwarzschild, who was mentioned earlier. While a soldier in WorldWar I, Schwarzschild provided the first exact solutions of Einstein's famous field equations that underpin general relativity.The Schwarzschild radius is written as RS=2GM/c2 where Mis the mass of the black hole, G is Newton's gravitational constant,and c is the speed of light. Using this formula, the Schwarzschild radius of the Earth comes out to be just under one centimetre.Similarly, the Schwarzschild radius of the Sun is found to be 3 km,meaning that if the mass of our Sun could all be squashed into a singularity, then at just 3 km away from this point the escape velocity would be equal to the speed of light. A black hole one billion times more massive than the Sun (i.e. having a mass of 109solar masses) would have a Schwarzschild radius one billion times larger (the Schwarzschild radius of a point mass that is not rotating simply scales directly with its mass). As I describe in Chapter 6, such mammoth black holes are believed to be at the centres of many galaxies.

  This description of the event horizon can be reasonably thought of within Newtonian physics. Indeed, physical entities resembling black holes were imagined centuries before Einstein and others profoundly changed our understanding of space and time. The principal thinkers who imagined `dark stars' that resemble black holes were JohnMichell and Pierre-Simon Laplace, starting back in the 18th century, and I will now explain what they did.

  One of the remarkable things about astronomy is how much you can discover about the Universe even when you are stuck on planet Earth. For example, no human being has ever visited the Sun, and yet the presence of helium in the Sun was detected in the late19th century by analysing the spectrum of sunlight. This is particularly remarkable as this constituted the discovery of the element helium itself; it was found on the Sun long before being detected on Earth. Even earlier, in the 18th century, some of the ideas behind black holes were beginning to be formulated, and in particular the idea of what is called a dark star. The person who made the first imaginative leap was verymuch a product of his time.

  John Michell

  The Georgian era was, in England, a time of relative peace. The English CivilWar was long in the past, and England had become a land of relative domestic tranquillity (the rise of Napoleonic France was still some way off ). Like his father before him, the Reverend JohnMichell (Figure 1) received a university education and entered the Church of England. As a rector in Thornhill,West Yorkshire, Michell was able to continue his scientific research,following up his interests in geology, magnetism, gravity, light, and astronomy. In common with other scientists working in England at the time, such as the astronomer WilliamHerschel and the physicist Henry Cavendish (who was a personal friend),Michell was able to ride the wave of the new Newtonian thinking. Sir Isaac Newton had revolutionized the way in which the Universe was perceived, formulating his law of gravitation which explained the orbits of the planets in the Solar System as being due to the same force that caused his famous apple to drop from the tree.

  1. John Michell, polymath.

  Newtonian ideas allowed the Universe to be studied using mathematics, and this fresh breed of scientists was able to deploy this novel world-view into different fields. Michell was particularly concerned to use Newtonian thinking to estimate the distance to nearby stars by using measurements of the light they emitted.He came up with various schemes to do this, by relating a star's brightness to its colour; he also considered binary stars (pairs of stars gravitationally bound to one another) and how their orbital motions could give useful dynamical information.Michell also investigated how stars tend to cluster in particular areas of the sky, testing this against a random distribution and inferring gravitational clustering. None of these ideas was practicable at the time: few binary stars were known (though Herschel was producing some impressive catalogues of various double stars and new objects) and the relationship between a star's brightness and its colour turned out to be not quite asMichell had thought it was.Nevertheless, Michell was straining to do for the wider Universe what Newton had done for the Solar System: allow a scientific,rational, and dynamical analysis of observations to provide new information about the properties, masses, and distances of the heavenly bodies.

  One particular insight that came toMichell followed from the idea that particles of light are, inMichell's words, `attracted in the same manner as all other bodies with which we are acquainted; that is,by forces bearing the same proportion to their vis inertiae [by which he meant mass], of which there can be no reasonable doubt,gravitation being, as far as we know, or have any reason to believe,an universal law of nature'. Such particles emitted from a large star would, he reasoned, be slowed down by the gravitational attraction of the star. Thus the starlight reaching Earth would be slower. Newton had shown that light slows down in glass, and this explained the principle of refraction. If starlight was indeed similarly slowed, Michell argued that it might be possible to detect this slowing by examining starlight through a prism. The experiment was tried, not by Michell, but by the Astronomer Royal, the Reverend Dr NevilMaskelyne, who looked for the diminishing of the refractability of starlight. Cavendish wrote to Michell to tell him that it hadn't worked and that `there is not much likelyhood [sic] of finding any stars whose light is sensibly diminished'.Michell was dismayed, but such astronomical speculations required much guessing of imponderables: was starlight affected by the gravitational attraction of the star from which it is emitted? Michell couldn't be sure. But he was bold enough to make an interesting prediction.

  If a star was sufficiently massive, and gravity really did affect starlight, then the gravitational force could be sufficient to hold back the particles of light completely and prevent them from leaving. Such an object would be a dark star. This little-known cleric writing in his rectory in Yorkshire had thus been the first person to conceive of a black hole. However, so farMichell's own programme of measuring the distances to stars lay in tatters.What was more, his health had been indifferent and this had stopped him using his telescope. Cavendish wrote to him a consoling letter: `if your health does not allow you to go on with[the telescope] I hope it may at least permit the easier and less laborious employment of weighing the world.' This singular example of a joke from Cavendish (who was notoriously buttoned up) refers to another idea thatMichell had conceived. `Weighing the world' meant an experiment in which two large lead spheres at either end of the beam of a torsion balance are attracted by two stationary lead spheres. This allows one to measure the strength of the gravitational force, and thereby infer the weight of the Earth.No one had ever done this before. Michell's idea was brilliant,but he didn't live to complete the project. Instead,Michell's experiment was performed by Cavendish and is now known as Cavendish's experiment. This transfer of credit to Cavendish is more than compensated for by the numerous breakthroughs made by Cavendish which he neglected to publish and were later attributed to subsequent researchers (including `Ohm's' law and`Coulomb's' law).

  Pierre-Simon Laplace

  On the other side of the English Channel, Pierre-Simon Laplace did not enjoy the tranquil idyll afforded by the peaceful period of the English Enlightenment. Laplace lived through the French Revolution, though his career prospered as he influenced the newly founded Institut de France and the 恈ole Polytechnique. He even spent a period asMinister for the Interior under Napoleon, a short-lived appointment the Emperor came to regret. Napoleon realized that Laplace was a first-rate mathematician but as an administrator he was worse than average. Napoleon later wrote of Laplace that `he sought subtleties everywhere, conceived only problems, and finally carried the spirit of "infinitesimals" into the administration'. Napoleon had other administrators to call upon,but the world has had few mathematicians as productive and insightful as Laplace. He made pivotal contributions to geometry,probability, mathematics, celestial mechanics, astronomy, and physics. He worked on topics as diverse as capillary action,comets, inductive reasoning, solar system stability, the speed of sound, differential equations, and spherical harmonics. One of the ideas he considered was dark stars.

  In 1796 Laplace published his Exposition du système du monde.Written for an educated public, this book describes the physical principles on which astronomy is based, the law of gravity and the motion of the planets in the Solar System, and the laws of motion and mechanics. These ideas are applied to various phenomena,including the tides and the precession of the equinoxes, and the book also contains Laplace's speculations on the origin of the Solar System. One particular passage is of special relevance to our story.Laplacemade a calculation of how large an Earth-like body would need to be so that its escape velocity was equal to that of light.He showed, quite correctly, that the gravitational strength on the surface of a star, with density comparable to that of Earth but with a diameter of about 250 times that of the Sun, would be so intense that not even light would be able to escape. Thus, he reasoned, the largest bodies in the Universe would therefore be invisible.Could they still be lurking, undetectable in the dark night sky,while we imagined that the only things `out there' were the bright luminous objects that we can see? The Hungarian astronomer Franz Xaver von Zach requested that Laplace provide the calculations that led to this conclusion, and Laplace obliged,writing this up (in German) for one of the journals that von Zach edited.

  However, Laplace was becoming aware of the wave theory of light.Both Michell's and Laplace's ideas were based in part on the corpuscular theory of light. If light were to consist of tiny particles,then it seemed reasonable that these particles would be affected by a gravitational field and would be bound forever to a star of sufficient size. But the early 19th century saw a number of experiments which seemed to give greater credence to the wave theory of light. If light were instead a wave, then it was harder to see that it should be affected by gravity. Laplace's dark star prediction was quietly omitted from later editions of Exposition du système du monde. After all,Michell and Laplace had been conjecturing and exploring theory, rather than being driven by the need to explain observations and thus, this idea was forgotten for a while. The objects imagined by Michell and Laplace were thus`dark stars', enormous objects in the Universe which by virtue of their mass could sustain planetary systems but by virtue of this same overwhelming bulk could not be observed via the radiation of light. Starlight emitted from the surfaces of Michell's and Laplace's dark stars would be too sluggish to overcome the intense surface gravity.WhatMichell and Laplace could not have guessed was that such gargantuan accumulations of mass would be unstable to collapse. Moreover, in their collapse they would puncture the very fabric of space and time and give rise to a singularity. Thus `black holes' are not `dark stars' and to take the argument forward and begin to meet up with the astronomical discovery of black holes we will first need to understand the nature of spacetime.

  Spacetime

  Our everyday experience leaves us comfortable with the notion that the tangible Universe may be described by one temporal (or time) coordinate t and three spatial coordinates (for example x, y, and z along three mutually perpendicular axes, a construct invented by René Descartes and known as Cartesian coordinates).In 1905, Einstein published his revolutionary paper on Special Relativity, the relativity of motion and stationarity. In 1907,Hermann Minkowski showed how these results could be understood more deeply by considering a four-dimensional spacetime whose points, specified now by the 4-D coordinate (t, x,y, z), correspond to 『events'. An event is something that happens at a particular time (t) and at a particular place (x, y, z). Such 4-D coordinates in what is known asMinkowski spacetime specify exactly where and when an event occurs. Einstein's special theory of relativity could be formulated in terms of Minkowski's spacetime and provides a convenient description of physical processes in different frames of reference that move relative to one another. A `frame of reference' is simply the perspective possessed by a particular observer. Einstein called this theory `special' because it deals only with a particular case, namely reference frames that are non-accelerating (called inertial frames of reference). The special theory can only be applied to uniformly moving, non-accelerated, frames of reference. If you drop a stone,it accelerates towards the ground. The frame of reference attached to the stone is an accelerating frame of reference and cannot be treated by Einstein's special theory.Where you have gravity, you have acceleration.

  This drawback prompted Einstein to formulate a general theory of relativity, which he published a decade after his special theory.What he found was that whereas Cartesian space and Minkowski spacetime were rigid frameworks in which objects `live,move and have their being', spacetime was actually a more responsive entity:it could be curved and otherwise deformed by the presence of mass. Once mass is present in a physical situation then the following inextricably linked behaviour describes reality, neatly summarized by JohnWheeler:

  ·mass acts on spacetime, telling it how to curve！

  ·spacetime acts on mass, telling it how to move

  This behaviour is quantified by Einstein's field equations within General Relativity, which relate the curvature of spacetime to the gravitational field.

  Physicists talk about a gravitational potential well as surrounding a massive object. The cartoon shown in Figure 2 encapsulates how the spacetime is distorted in the vicinity of a couple of black holes,where each region can be regarded as curved in a way which is directly related to its mass and hence to the gravitational force itself. The singularity in spacetime may be regarded as where the curvature in spacetime becomes very high and you go beyond the classical theory of gravity, into the quantum regime. The event horizon surrounding the singularity functions as a one-way

  2. The distortion, i.e. curvature, in spacetime due to the presenceofmasses.

  membrane: particles and photons can enter the black hole from outside but nothing can escape from within the horizon of the black hole out to the external Universe. In fact mass is not the only property that a black hole may possess and be measured by. If the black hole is rotating, that is to say it possesses some spin, then even more extreme behaviour emerges. Before we examine this,we will take a little detour to learn a littlemore about how we may schematically represent spacetime itself.