The Little Book of Black Holes

THE LITTLE BOOK OF BLACK HOLES

Books in the SCIENCE ESSENTIALS series bring cutting-edge science to a

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Memory: The Key to Consciousness,

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The Cosmic Cocktail: Three Parts Dark Matter,

BY KATHERINE FREESE

Life’s Engines: How Microbes Made Earth Habitable,

BY PAUL G. FALKOWSKI

The Little Book of Black Holes,

BY STEVEN S. GUBSER AND FRANS PRETORIUS

the

LITTLE BOOK

of

BLACK HOLES

STEVEN S. GUBSER

& FRANS PRETORIUS

PRINCETON UNIVERSITY PRESS    PRINCETON AND OXFORD

Copyright © 2017 by Steven S. Gubser and Frans Pretorius

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Library of Congress Cataloging-in-Publication Data

Names: Gubser, Steven Scott, 1972– author. | Pretorius, Frans, 1973– author.

Title: The little book of black holes / Steven S. Gubser

and Frans Pretorius.

Description: Princeton, New Jersey ; Oxford :

Princeton University Press, [2017]

Identifiers: LCCN 2017024783| ISBN 9780691163727 (hardback ;

alk. paper) | ISBN 0691163723 (hardback ; alk. paper)

Subjects: LCSH: Black holes (Astronomy)

Classification: LCC QB843.B55 G83 2017 | DDC 523.8/875—dc23

LC record available at https://lccn.loc.gov/2017024783

British Library Cataloging-in-Publication Data is available

This book has been composed in Bembo Std

Printed on acid-free paper. ∞

Printed in the United States of America

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CONTENTS

PREFACE

IT WAS SEPTEMBER 14, 2015, ALMOST EXACTLY 100 YEARS after Albert Einstein formulated the General Theory of Relativity. Two massive detectors, one in Louisiana and one in Washington, were undergoing final preparations for a science run aimed at detecting gravitational waves. Suddenly and unexpectedly, the detectors’ instrumentation recorded a peculiar chirp. If we made this chirp audible, it would sound like a faint, low-pitched thump.

Five months later, after careful scrutiny of the data recorded by these detectors, the Laser Interferometer Gravitational Wave Observatory (LIGO) publicly announced their results. That chirp was exactly the sort of signal they had hoped to detect. It was the distant echo of a pair of black holes caught in the act of merging into a single, larger black hole. The physics community was electrified. It was as if we had lived for all our lives blind to the color red, and at the moment the veil was lifted we saw a rose for the first time.

And what a rose it was! Best estimates from LIGO indicated that the faint thump they recorded was the result of the coalescence more than a billion years ago of two black holes, each of them roughly thirty times the mass of the Sun. During the collision, some three solar masses’ worth of energy from the black holes was vaporized into gravitational radiation.

Black holes and gravitational waves are both consequences of Einstein’s general theory of relativity. General relativity predicts what sort of gravitational waves the LIGO detector should see in the event of a black hole collision, and the chirp recorded on September 14 was beautifully close to expectations. Not only is it a vindication of long-cherished theoretical ideas; this first detection event also augurs a new era of gravitational wave astronomy. The LIGO detectors saw one event of the sort we’ve dreamed of for decades. Now we want to explore a whole new garden of gravitational surprises.

Science seldom involves mathematical certainties, so we should ask, how sure are we that LIGO construed correctly that the little chirp is the distant echo of a billion-year-old black hole merger? Briefly, the answer is, Pretty sure. Everything seems to fit. The signal was seen by both detectors. Nothing else seemed to be happening nearby that would explain the signal. The signal was strong enough to see with the current device, but too weak to be observed by earlier technology. The hypothesis of a black hole merger a billion years ago doesn’t conflict with our general understanding of astrophysics and cosmology. The key point is that we have good hopes of seeing more such events. Indeed, LIGO announced a second confirmed event that occurred on Christmas day 2015, and a third that occurred on January 4, 2017. These events are broadly comparable to the first discovery and should give us significantly more confidence that LIGO is truly observing black hole mergers. Altogether, we believe that we are standing at the dawn of a new age in observational astrophysics—one in which black holes will play a pivotal role.

In this book, we describe black holes, both as astrophysical objects whose existence is now almost beyond doubt, and as theoretical laboratories which allow us to hone our understanding not just of gravity but also of quantum mechanics and thermal physics. Explanations of special relativity and general relativity set the stage, in Chapters 1 and 2. Then in subsequent chapters, we go on to discuss Schwarzschild black holes, rotating black holes, collisions of black holes, gravitational radiation, Hawking radiation, and information loss.

So, just what is a black hole? Essentially, it is a region of spacetime toward which matter is drawn and from which escape is impossible. Let’s focus the discussion on the simplest black holes, known as Schwarzschild black holes (in honor of their discoverer, Karl Schwarzschild). There’s an old saying, What goes up must come down. Inside a Schwarzschild black hole, a stronger statement is true: Nothing can go up. Only down. But we are not sure where down eventually leads. The most straightforward hypothesis, given the mathematics behind Schwarzschild black holes, is that a terrible, infinitely compressed kernel of matter lurks at its core. Colliding with that core is the end of everything. It is even the end of time. This hypothesis is hard to test because no observer who goes into a black hole can ever report back on what he sees.

Before we go on to explore Schwarzschild black holes in more depth, let’s first take a step back and consider gravity in some of its milder forms. From the surface of the Earth, if we impart a sufficiently large upward velocity to an object, then it will keep moving upward forever. The minimal velocity for which this is true is the escape velocity. Neglecting air friction, escape velocity is approximately 11.2 kilometers per second. By way of comparison, it’s hard for a human to throw a ball faster than about 45 meters per second—less than half a percent of escape velocity. The muzzle velocity of a high-powered rifle is roughly 1.2 kilometers per second—so, slightly more than 10% of escape velocity. What we usually mean, then, by What comes up must come down is that Earth’s gravity is strong compared with how forcefully we can propel objects upward using ordinary means.

FIGURE 0.1. Cutaway of a schematic representation of a black hole geometry. Far outside the horizon, spacetime is flat. Moving toward the horizon, it becomes ever more curved but is still time independent, or static. Crossing the horizon however, spacetime becomes dynamical: as time flows, two of the spatial dimensions (having the geometry of a sphere) compress, while the third (not shown) lengthens, until all of space is stretched and squeezed into an infinitely long and thin singularity.

Rocketry is the modern means to conquer Earth’s gravity altogether and send objects into space. To escape Earth’s gravity, it is not strictly necessary for a rocket to go faster than 11.2 kilometers per second (though some rockets do). What can happen instead is that a rocket travels at a slower velocity, but it has enough fuel to keep thrusting upward until it reaches altitudes where Earth’s gravitational field is significantly weaker. The escape velocity from such altitudes is correspondingly smaller. In other words, a rocket designed to take a space probe entirely out of Earth’s gravitational field must be going faster than the escape velocity at the point where the rocket stops firing.

Now we could ask, what if the Earth were much denser? Escape velocity from the surface would be larger because the gravitational field would be more intense. The densest stable form of ordinary matter in the known universe occurs in neutron stars. They pack approximately one and a half times the mass of the Sun into a sphere of only 12 kilometers in radius, though this radius is not very precisely measured. Ordinary matter is completely crushed into the surface by the tremendous gravitational forces, which are something like a hundred billion times stronger than the gravitational field of the Earth. Assuming a 12 kilometer radius, the escape velocity is approximately 60% the speed of light.

But why stop there? As a thought experiment, we could imagine compressing neutron stars even further. If we compress a neutron star to a radius of about four and a half kilometers, then its escape velocity reaches the speed of light. If we go past that point, gravity changes character completely. It’s no longer possible for any form of matter to hold itself up against the pull of gravity. To move forward in time means to move inward in radius. Escape is impossible. This is a black hole.

The central aim of the first few chapters of this book is to make the idea of a black hole more precise. A key concept that we’ll explore is the idea of an event horizon, which is the surface of a black hole. It is a surface in the geometrical sense of being a two-dimensional locus in three-dimensional space. For example, in the simplest example of a Schwarzschild black hole, the horizon is a perfect sphere whose radius is called the Schwarzschild radius. The odd thing about a black hole horizon is that (at least according to conventional understanding), it is not the surface of anything in particular. At the moment you fall through it, you don’t notice anything special. The only problem comes if you try to turn around and get back out. No matter how hard you try—using a rocket, a laser canon, or any other means—and no matter what help you might have from outside, it is impossible to get back outside the horizon, or even to send an SOS signal out to say you’re trapped. Poetically, we might think of a black hole horizon as the lip of a waterfall, beyond which spacetime cascades ineluctably downward into a singularity that destroys all things.

Black holes are more than a thought experiment. They are believed to occur in the universe in at least two situations. One is along the lines of the previous discussion of neutron stars. When large stars run out of nuclear fuel, they collapse in on themselves. This collapse is a messy process in which a great deal of matter is blown off into the surrounding universe in an explosion called a supernova. (In fact, it’s generally thought that supernovae play a crucial role in distributing metals and other moderately heavy elements throughout the universe.) Enough mass can remain behind that it is impossible for a neutron star to form and also remain stable. Instead, this remaining mass collapses to form a black hole whose mass is at least a few times the mass of the Sun. The black holes whose mergers were observed by LIGO are somewhat more massive, but still plausibly produced by stellar collapse.

Much larger black holes are thought to exist at the center of galaxies. Exactly how these black holes formed is more mysterious and may be related to dark matter, the physics of the very early universe, or both. Black holes at the center of galaxies are tremendously massive, containing thousands to billions times as much mass as the Sun. One is thought to be at the center of the Milky Way, containing some 4 million solar masses. We may well ask, how can we be sure that a black hole is present if no signal can escape from a black hole horizon? The answer is that nearby objects respond to the gravitational pull of the black hole. By tracking the motion of stars near the center of the Milky Way, we can be certain that a very massive, very dense object is present there. We cannot prove in this way that it is a black hole. What we can say is that if it isn’t a black hole, then it’s something much stranger. In short, black holes are the simplest possibility, and the modern consensus is that they do exist at the centers of many if not most galaxies.

Black holes are a tremendously useful theoretical laboratory because they are mathematically simple compared to most astrophysical objects. Stars are really very complicated. Nuclear reactions at the core of stars provide their power. The matter inside stars experiences pressures and fluid dynamical motions that we can simulate numerically but certainly do not understand completely. And stars have surface dynamics which are probably as complicated as the Earth’s weather patterns. A black hole, by comparison, is wonderfully simple. In the absence of other matter, black holes must settle down to one of a few definite forms, all of which are explicitly understood as curved geometries which solve Einstein’s equations of general relativity. To be sure, infalling matter complicates the picture, but there is a tolerably good understanding of how ordinary matter behaves as it falls into black holes. Nowadays, there is even a good numerical understanding of what happens when one black hole collides with another, and a central aim of Chapter 6 of this book is to explain how this understanding is achieved and what it means for experiments such as LIGO.

Where things get strange is that black holes aren’t really black. Using quantum mechanics, Stephen Hawking showed that black holes have a definite temperature, related to their surface gravity. In fact, there is a whole field of study known as black hole thermodynamics, in which geometrical properties of black hole solutions are put into precise correspondence with properties familiar from the study of heat: temperature, energy, and entropy. There is even a proposal that black holes in distant parts of