Paths toward understanding black holes - Thesis

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Paths toward understanding black holes

Mayerson, D.R.

Publication date

2015

Document Version

Final published version

Link to publication

Citation for published version (APA):

Mayerson, D. R. (2015). Paths toward understanding black holes.

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Invitation

On 12 June 2015 at 12:00

in the Agnietenkapel at

Oudezijds Voorburgwal

231, in Amsterdam,

I will publicly defend my

thesis entitled

Paths Toward

Understanding

Black Holes

Daniel Mayerson

d.r.mayerson@uva.nl

Paths Toward Understanding

Black Holes

dation for Fundamental Research on Matter (FOM), which is part of the Netherlands Organisation for Scientific Research (NWO).

c

Daniel Mayerson, 2015.

Cover illustration by Anke Schillemans.

All rights reserved. Without limiting the rights under copyright reserved above, no part of this book may be reproduced, stored in or introduced into a retrieval system, or transmitted, in any form or by any means (electronic, mechanical, photocopying, recording or otherwise) without the written permission of both the copyright owner and the author of the book.

Paths Toward Understanding

Black Holes

Academisch Proefschrift

ter verkrijging van de graad van doctor

aan de Universiteit van Amsterdam

op gezag van de Rector Magnificus

prof. dr. D.C. van den Boom

ten overstaan van een door het College voor Promoties ingestelde commissie,

in het openbaar te verdedigen in de Agnietenkapel

op vrijdag 12 juni 2015, te 12.00 uur

door

Daniel Robert Mayerson

Promotor

prof. dr. J. de Boer, Universiteit van Amsterdam

Co-Promotor

prof. dr. M. Shigemori, Kyoto Daigaku

Overige leden

prof. dr. E. P. Verlinde, Universiteit van Amsterdam

prof. dr. C. J. M. Schoutens, Universiteit van Amsterdam

dr. A. Castro Anich, Universiteit van Amsterdam

dr. D. M. Hofman, Universiteit van Amsterdam

prof. dr. E. A. Bergshoeff, Rijksuniversiteit Groningen

prof. dr. F. Larsen, University of Michigan

Publications

This thesis is based on the following publications: [1] M. Baggio, N. Halmagyi, D. R. Mayerson, D. Robbins and B. Wecht

Higher Derivative Corrections and Central Charges from Wrapped M5-branes JHEP 1412, 042 (2014), arXiv:1408.2538 [hep-th].

[2] J. de Boer, D. R. Mayerson and M. Shigemori

Classifying Supersymmetric Solutions in 3D Maximal Supergravity

Class. Quant. Grav. 31, no. 23, 235004 (2014), arXiv:1403.4600 [hep-th]. [3] B. D. Chowdhury, D. R. Mayerson and B. Vercnocke

Phases of non-extremal multi-centered bound states JHEP 1312, 054 (2013), arXiv:1307.5846 [hep-th]. [4] B. D. Chowdhury and D. R. Mayerson

Multi-centered D1-D5 solutions at finite B-moduli JHEP 1402, 043 (2014), arXiv:1305.0831 [hep-th].

Other publications by the author: [5]∗ _{M. Baggio, J. de Boer, J. I. Jottar and D. R. Mayerson}

Conformal Symmetry for Black Holes in Four Dimensions and Irrelevant Deformations

JHEP 1304, 084 (2013), arXiv:1210.7695 [hep-th].

[6]† F. Bigazzi, A. L. Cotrone, J. Mas, D. Mayerson and J. Tarrio

Holographic Duals of Quark Gluon Plasmas with Unquenched Flavors Commun. Theor. Phys. 57, 364 (2012), arXiv:1110.1744 [hep-th]. [7]† F. Bigazzi, A. L. Cotrone, J. Mas, D. Mayerson and J. Tarrio

D3-D7 Quark-Gluon Plasmas at Finite Baryon Density JHEP 1104, 060 (2011), arXiv:1101.3560 [hep-th].

∗_{This publication was already featured in the PhD thesis [8].}

Preface & Thesis Guide

“Begin at the beginning,” the King said, very gravely, “and go on till you come to the end: then stop.”

— Lewis Carroll, Alice in Wonderland

This thesis is an overview of the research work I have performed during my doctoral studies. Very broadly speaking, this work can be summarized as trying to under-stand aspects of black holes, gravity, and geometry, in the context of supergravity and string theory in high-energy theoretical physics.

The two parts of this thesis have been written with entirely different audiences in mind.

The first part consists of chapters 1, 2, 3 and is composed of a self-contained laymen’s introduction to my research. This quest starts with an exposition on quantum mechanics and relativity in chapter 1, and follows the development of these concepts through string theory in chapter 2 to finally end up with discussing my own research in chapter 3. These chapters are aimed at non-physicists (at the level of, say, a popular scientific journal such as Scientific American), although physicists with little or no knowledge of string theory would also hopefully find them entertaining. Equations are non-existent with the one exception of Insert 1.3, and concepts are presented in simplified and intuitive ways. No real prior fa-miliarity with physics is strictly necessary, although some memories of high school physics and mathematics will facilitate understanding. It is important to realize that this means that I have necessarily been incomplete and at times not even en-tirely correct in the wording of the statements presented; this is because accuracy and comprehensiveness are sacrificed in order to present a more convincing and understandable story for the non-physicist. Relevant references can be found in the second part of the thesis, so they are not provided in this part.

The second part of this thesis, consisting of chapters 4, 5, 6, 7, 8 contains the main part of this thesis, and is therefore written in a scientific manner. Chapter 4 introduces black holes in general and the various concepts of the subsequent chapters; it also describes how the research presented in those chapters fits into the bigger picture regarding the study of black holes. Following the introductory chapter 4, chapter 5 discusses my research on the concept of exotic branes; chapter

6 deals with my contributions to the study of higher derivatives in gravitational theories; and chapter 7 details my contributions to multi-centered configurations, especially in the context of fuzzballs. Chapter 8 summarizes the main results of the previous three chapters and provides an outlook on possible interesting future directions of research in these subjects.

People familiar with a few basic concepts in gravity and string theory should be able to read the introductory chapter 4 and certainly understand the big picture; in addition, the outlook chapter 8 can be read to get a further summary of the results and an idea of the future directions of research in these subjects.

After chapter 8 and the bibliography, there are a few final elements in this thesis. Short (mandatory) parts are included that consist of an overview of my contri-butions to my publications used in this thesis, an English summary, and a Dutch summary. Finally, I give acknowledgments to the people who have helped me make this thesis possible.

A quick word on writing styles is probably in order. The first part, meant as a popular scientific writing, is written in quite a colloquial style. The pronoun “I” is used quite extensively, especially when talking about my own research. The second part of this thesis is written in a more conventional scientific way. This includes the almost-exclusive use of the pronoun “we” which is more pervasive in scientific writing, since I also feel this indicates more correctly the nature of the scientific work that I performed, which was always in close collaboration with others.

Quick Guide

Popular-scientific, for laymen; introduction and overview:

,→

At least some familiarity with string theory; introduction and overview:

,→

,→

Chapter 5: Exotic Branes & 3D Supergravity Chapter 6: Higher-Derivative Corrections

Chapter 7: Multi-Centered Solutions & Fuzzballs

Contents

Preface & Thesis Guide vii

I

For Non-Physicists

1

1 From Classical Mechanics to Quantum Gravity 3

1.1 The End of All Physics? . . . 4

1.2 Special Relativity . . . 5

1.3 General Relativity . . . 7

1.4 Quantum Mechanics . . . 10

1.5 Quantum Field Theory & The Standard Model . . . 12

1.6 The Problem of Quantum Gravity . . . 15

1.7 The Big Bang & Cosmology . . . 18

1.8 Black Holes . . . 20

1.9 Summary . . . 22

2 String Theory 25 2.1 Why is Quantum Gravity so Hard? . . . 25

2.2 String Theory: The Idea . . . 28

2.3 Supersymmetry & Other Inconveniences . . . 30

2.4 Holography: A New View On Gravity . . . 31

2.5 String Cosmology . . . 34

2.6 Black Holes, Fuzzballs & Holography . . . 35

2.7 Alternative Approaches to Quantum Gravity . . . 39

2.8 Summary . . . 40

3 My Research 43 3.1 Exotic Branes & (Non-)Geometry . . . 47

3.2 Higher-Derivative Corrections . . . 49

3.3 Multi-Centered Solutions, Fuzzballs & Black Holes . . . 51

II

For Everyone Else

55

4 Introduction 57

4.1 Black Holes in String Theory . . . 57

4.1.1 Holography . . . 60

4.1.2 Fuzzballs From Supertubes . . . 64

4.1.3 Overview . . . 67

4.2 Exotic Branes . . . 68

4.2.1 Seven-branes in Type IIB String Theory . . . 68

4.2.2 Non-geometric Branes From U -duality . . . 70

4.2.3 Exotic Black Hole Microstates . . . 72

4.2.4 What Kinds of Exotic Branes Are There? . . . 73

4.3 Higher-Derivative Corrections . . . 75

4.3.1 Stringy Effects . . . 77

4.3.2 Practical Higher Derivatives . . . 80

4.4 Multi-Centered Solutions & Fuzzballs . . . 81

4.4.1 Multi-Centered Solutions . . . 82

4.4.2 N = 2 Supergravity . . . 83

4.4.3 Fuzzballs in Supergravity . . . 84

4.4.4 N = 4 Supergravity . . . 85

4.4.5 Non-supersymmetric Fuzzballs . . . 86

5 Exotic Branes & 3D Supergravity 89 5.1 Introduction . . . 90

5.2 General Structure of the Theory . . . 93

5.2.1 Scalar cosets and maximal supergravity . . . 93

5.2.2 Maximal supergravity in three dimensions . . . 97

5.3 Structure of SUSY solutions . . . 100

5.3.1 Statement of Results . . . 100

5.3.2 Derivation of Results . . . 103

5.4 Simple Single Center Brane Solutions . . . 111

5.4.1 Ansatz . . . 112 5.4.2 Review of SL(2) Solutions . . . 113 5.4.3 Nilpotent Charges . . . 115 5.4.4 Brane Representatives . . . 116 5.4.5 Semi-simple Charges . . . 119 5.4.6 Other Charges – sl(2)n _{. . . 120}

5.5 More sl(n) Brane Solutions . . . 120

5.5.1 Potential Formalism . . . 121

5.5.2 Various Explicit Examples . . . 125

Preface

5.6 Conclusions . . . 134

5.A Lie Algebra Concepts . . . 136

5.B Mathematical Theorems . . . 141

5.C Construction of e8 . . . 143

5.C.1 Gamma Matrices of so(16) . . . 143

5.C.2 Two Constructions of e8(8) . . . 143

5.C.3 Map Between Constructions . . . 144

5.C.4 Calculations in e8(8) . . . 146 6 Higher-Derivative Corrections 149 6.1 Introduction . . . 150 6.2 Field Theory . . . 152 6.2.1 Four Dimensions . . . 152 6.2.2 Two Dimensions . . . 154

6.3 N = 1 SUGRA Review & Solutions . . . 156

6.3.1 Off-shell Multiplets & Variations . . . 156

6.3.2 Two- and Four-Derivative Actions . . . 158

6.3.3 Supersymmetric Solutions . . . 162

6.4 M5-branes Wrapped On Riemann Surfaces . . . 168

6.4.1 The Scalar Geometry at Two Derivatives . . . 168

6.4.2 The Chern-Simons Terms at Four Derivatives . . . 171

6.4.3 Supersymmetric Completion of the Chern–Simons Terms at Four Derivatives . . . 174

6.4.4 Four-Dimensional Central Charges . . . 177

6.4.5 Two-Dimensional Central Charges . . . 179

6.5 Conclusions . . . 181

6.6 Postlude: Higher-Derivative Corrections and AdS5Black Holes . . 182

6.A SU (2) Conventions & Variations . . . 185

6.A.1 SU (2) & Spinor Conventions . . . 185

6.A.2 SU (2) Structure of SUSY variations . . . 185

6.B Details on N = 1 Superconformal Supergravity . . . 187

6.B.1 Superconformal Action and Variations . . . 188

6.B.2 Gauge-fixing to Poincar´e Supergravity . . . 189

6.C 7D Supergravity Conventions . . . 191

6.C.1 Gauged Maximal Supergravity . . . 192

6.C.2 U (1)2 _{Truncation . . . 193}

6.D Hypercomplex and Quaternionic Geometries . . . 193

6.D.1 Quaternionic-like Manifolds . . . 194

6.D.2 Conformal Symmetry . . . 195

6.D.3 The Map From Quaternionic to Hypercomplex . . . 197

7 Multi-Centered Solutions & Fuzzballs 203

7.1 Introduction . . . 204

7.2 Setup . . . 207

7.2.1 D1-D5-P Black Hole with B-moduli . . . 208

7.2.2 The Supertube Probe . . . 212

7.3 Ergoregions for Supertubes . . . 214

7.4 Extremal Bound States & Moduli . . . 216

7.4.1 The Same Probe at Different Moduli . . . 217

7.4.2 Finding a Supersymmetric Bound Probe at Finite Moduli (Or Not) . . . 220

7.4.3 Qualitative Amount of Breaking . . . 221

7.5 Non-Extremal Bound States . . . 223

7.5.1 Non-Extremal Bound States & Moduli . . . 224

7.5.2 Phase Space of Black Hole-Supertube Bound States . . . . 225

7.6 Conclusions . . . 232

7.A Lifting Analogy . . . 235

7.B Angular Momentum of an Extended Probe . . . 236

7.B.1 Point particle in a magnetic field . . . 237

7.B.2 Extended object in a magnetic field . . . 238

7.C Page Charges . . . 239

7.C.1 D-brane Page Charges . . . 240

7.C.2 F1 Page Charge in IIA . . . 241

7.D Duality Calculations . . . 243

7.E Probe Hamiltonian . . . 244

7.E.1 Calculating the Hamiltonian . . . 244

7.E.2 The Decoupling Limit Hamiltonian . . . 246

7.E.3 Probe angular momentum . . . 248

8 Outlook 249 Bibliography 279 Contributions to Publications 281 Summary 283 Samenvatting 287 Acknowledgments 293

All physical theories, their

mathematical expressions apart, ought to lend themselves to so simple a description that even a child could understand them.

— Albert Einstein, to Louis de Broglie

Part I

1

From Classical Mechanics to

Quantum Gravity

It is often stated that of all the theories proposed in this century, the silliest is quantum theory. In fact, some say that the only thing that quantum theory has going for it is that it is unquestionably correct.

— Michio Kaku, Hyperspace

Before I can comfortably explain my research in sufficiently understandable terms, it helps to first take a journey into the relevant context. For some areas of research, such a context is already fairly explicitly known by laymen: for example, nobody needs to have the context and importance of research into cancer cures explained to them, although perhaps explaining the rationale behind a given specific approach may be less obvious. However, string theory is in a bit of a disadvantaged situation in this respect. Since most people do not deal with the consequences or problems of string theory on a day-to-day basis, they are typically also not familiar with the relevant concepts or the questions that researchers are trying to answer.

This problem is essentially what the first two chapters deal with in this exposition for non-physicists: sketching the context of my research. In the second chapter, I will sketch some basic and important concepts in string theory, which forms the immediate context of my own research. However, to fully appreciate the context in which string theory plays a role, I feel it is important to start our journey even earlier. In this first chapter, we will take a look at the most important conceptual discoveries in twentieth century theoretical physics. I will explain how the highly successful theories of quantum mechanics and general relativity nevertheless nec-essarily lead to internal inconsistencies, which necessitate finding a theory that must encompass both of them at the same time and reconcile their differences – a theory of quantum gravity. People wishing to get a quick flavor of the main points first can try to read the last sections of each of the first two chapters, which are essentially quick summaries of the most important concepts that are discussed. A quick disclaimer is in order. These three chapters are by no means meant to be a

comprehensive review of twentieth century physics or string theory. I have tried to only discuss concepts and issues that contribute to the context sketching that I felt were interesting to understand my own research areas discussed in the final chapter. In doing so, there are naturally many glaring omissions in this story of quantum gravity and string theory that are not immediately necessary for my research – one example of such an omission is the existence of dark matter and dark energy. Nor should this three-chapter laymen’s overview of my research be considered to be an extremely precise and correct description of the subjects covered. A more correct description of the concepts presented here would necessarily take us to a more mathematical discussion, which is exactly what I am trying to avoid in this first part.

Let’s now return to the beginning of the twentieth century and see what the status of theoretical physics was, to understand the context in which general relativity and quantum mechanics were introduced.

1.1

The End of All Physics?

At the beginning of the 20th century, it was broadly thought that our understand-ing of fundamental physics was almost complete. All the fundamental laws of nature were more or less understood: Newton had explained gravity and all of the laws of classical mechanics (then simply called “mechanics”, of course) in his book the Principia, including a successful and elegant theory of gravity. Maxwell had succeeded in writing down equations that completely described electromagnetism in an elegant, unified framework. It seemed that physics was essentially “done” and all that was left to do was to cross some t’s and dot a few i’s.

Of course, this could not have been further removed from the actual truth. Classi-cal mechanics, especially when put together with electromagnetism, had obtrusive inconsistencies buried in it. For one, Maxwell’s theory of electromagnetism seemed to suggest that the speed of light is independent of the observer that measures it. This is totally inconsistent with classical mechanics or our normal intuition: if I am driving at 100 km/hour and you are driving at 150 km/hour in the same direction, I should see you drive by at 50 km/hour. Not so for light – no matter how fast you drive by me, if we look at the same beam of light, we will measure its speed to be exactly the same.

This principle was so weird that people did not really want to believe it at first. Various theories were suggested, such as a theory of the aether, a substance that should be permeating the entire universe and which would carry light waves in the same way that air carries sound waves. If this aether theory were to be true,

1.2. Special Relativity

then depending on the relative velocity of objects with respect to the aether, the speed of light should be measured to be different after all. However, many careful experiments ruled out the existence of such an aether: light was really always travelling at the same speed, no matter what complicated measurement you could cook up to measure it.

The speed of light was surely not the only problem that classical physics had at the turn of the century. Another prominent inconsistency in this framework was the infamous “ultraviolet catastrophe”: classical mechanics predicted that a black body (such as, say: a glowing piece of coal) should emit an infinite amount of radiation per unit area and time in the short wavelength end of the light spectrum. Of course, nobody has ever seen a piece of hot coal emit an infinite amount of energy, so it is clear that something must be very wrong with this calculation. As it turns out, to solve these problems, an entirely new way of thinking about physics was needed. To solve the problem of electromagnetism and the speed of light, we need (special) relativity, which would fundamentally change the way we think about time, space, distances, and velocities. The resolution of the second problem lies in an entirely different area with the introduction of quantum me-chanics, which would also entirely change the way we think about the fundamental building blocks of matter.

1.2

Special Relativity

The laws of electromagnetism clearly state that light should always be seen to travel at the same speed, no matter how fast the observer is moving. On the other hand, classical mechanics (and our intuition) tells us unequivocally that speeds should be relative: if I am moving with respect to you, I should measure a different speed for the same beam of light. With such an obvious contradiction between the predictions of two theories, there is really only way of settling this: by experiment. Since all experiments (as mentioned above) pointed towards the speed of light being constant and unchangeable, clearly we need to throw away our usual notions of relative velocities in classical mechanics. The only question is: What should take its place?

The answer to this question was provided by one of the heroes of theoretical physics of the first half of the twentieth century – Albert Einstein (see also Insert 1.1). His theory of special relativity beautifully explained how the speed of light should always remain constant. But his theory had a price: we had to abandon our intuitive preconceptions about space and time as separate, unmixable entities. Special relativity tells us that space and time only make sense as two aspects of

a combined theory of spacetime. This has profound consequences; for example, if you travel at a large speed compared to me, then special relativity tells us that your clock will seem to me to move at a slower pace than my own. The reason this seems strange and counterintuitive to us is that we simply do not face high enough speeds in our daily life to notice such effects. However, such “time dilation” effects have been measured in a variety of experiments and agree with special relativity’s predictions. In other words, the theory of relativity may seem unnatural, but that’s just tough luck for us: the theory, and not our intuition, is correct. Special relativity places the speed of light on a very particular pedestal. Besides being the same for all observers as described above, the speed of light is also the highest possible speed anything can achieve. It takes more and more energy to accelerate particles to higher and higher velocities, but it would in principle take an infinite amount of energy for any massive object to actually achieve the speed of light exactly.

Insert 1.1: Genius, or Being in the Right Place at the Right Time?

A very typical simplification in popular understanding of the history of sci-ence is to view scientific breakthroughs as produced by lone geniuses, who work away for years in some secluded office or basement, only to emerge tri-umphiantly when they have succeeded in figuring out one of the mysteries of the universe.

This is not a very accurate picture of how most discoveries happen. Usually, scientific breakthroughs happen in a given context; said differently, the time has to be ripe for the discovery to be made. Nor are discoveries typically the product of a lone, secluded genius – often, the best scientists are those that talk to and learn from their colleagues, and combine insights from other people with their own to provide fresh, new views.

One example of a momentous scientific breakthrough was the discovery of the helical structure of DNA. This model is commonly attributed to Watson and Crick, and was based on the latest experimental X-ray diffraction pictures of DNA made by Franklin and Gosling. Earlier models and experiments by Avery, MacLeod, McCarthy, Hershey, Chase, Astbury, and others had started to investigate the structure of DNA but had not quite gotten as far. Watson and Crick’s theoretical model, as well as Franklin and Gosling’s experiments, were the results of fine scientific work and the product of exceptional minds of exceptional people, but it is also clear that the time was ripe for the discovery of helical DNA – if Watson and Crick hadn’t been around, the chances are very high other researchers would have taken their place in making the same

1.3. General Relativity

discovery.

There are a number of exceptions to this general rule, when a scientific discov-ery or breakthrough really does seem to come out of nowhere in a completely unexpected fashion. A recent example in mathematics is Wiles’ proof of Fer-mat’s last theorem; after over seven years of professional seclusion, Andrew Wiles suddenly and unexpectedly announced a proof of this theorem. His proof included over 150 pages of new mathematics, refining existing techniques but also inventing and discovering completely new concepts.

I certainly do not wish to downplay the gifted scientists that get recognized for their breakthroughs, but I do think it is important to emphasize that it is not always simple or clear-cut to precisely determine who is responsible for a particular discovery. I may have mentioned that Einstein revolutionized physics with his discovery of special relativity, but key elements were already in place with (among other things) Maxwell’s discovery of the constancy of the speed of light and Lorentz’ discovery of the so-called Lorentz transforma-tions. The discovery of general relativity was perhaps more of a profound and unexpected departure from the current status quo in physics, which makes it much harder to imagine that the theory would have been discovered at that time and in that form, if it wasn’t for Einstein’s unique and radical insights. Even so, it should be mentioned that Einstein was not working on this theory in isolation – Hilbert and Grossmann were working on similar approaches, to the extent that it is actually not entirely clear if it was Einstein or Hilbert who was first in discovering the correct equations for general relativity. Even though the equations now typically bear Einstein’s name, the gravity action in general relativity (which is a quantity that can be seen as equally important as the actual gravity equations themselves) goes by the name of the Einstein-Hilbert action.

1.3

General Relativity

Einstein wasn’t quite finished with his gifts to theoretical physics after inventing special relativity. There were a number of things that were still quite bothersome in this theory. For one, although it very nicely meshed with the laws of electro-magnetism in explaining the constant speed of light, special relativity did not have anything to say about gravity.

One of the reasons gravity was puzzling is the equality of inertial and gravitational mass. In classical mechanics and Newtonian gravity, two types of mass exist and perform entirely different functions. Inertial mass measures the resistance of a

body to an applied force – encoded in the famous Newton’s second law, F = ma, or force is (inertial) mass times acceleration. If a body has a lot of inertial mass, it means the body has a lot of inertia or “resistance” to an applied force, and it will require a large force to push it into motion. The second type of mass is gravitational mass, and measures how much force gravity exerts on a body: a body with more gravitational mass will have gravity pulling on it more strongly. In classical mechanics, there is no relation whatsoever between the two types of mass; in principle, these could be two separately determined parameters for any given object. However, experiments showed that gravitational and inertial mass were always exactly the same, for any body. Said differently, two objects, regardless of their mass, always fall at the same rate when there is no air resistance; a famous example of this is the Apollo 15 experiment where an astronaut dropped a hammer and a feather on the moon, observing them to fall at exactly the same rate. Why does this happen?

An equivalent but perhaps more thought-provoking way to consider this equality of gravitational and inertial masses is Einstein’s famous thought experiment of the falling elevator. He reasoned that there is no reason that someone, typically trapped in an elevator without windows for this experiment, could tell the differ-ence between being freely falling towards Earth (i.e. being pulled by gravity) and being weightless in empty space. This might seem a bit strange to our intuition, since we are used to “feeling” gravity as something that keeps us on the ground. However, this is because standing on the surface of the Earth is a special situation where the gravitational pull of the Earth is counteracted by the ground pushing us upwards. Next time you are in a freely falling elevator and are coherent enough in this terrifying situation to conduct an experiment, try dropping something. You will see that the object does not fall up or down. From your perspective, it doesn’t “drop” at all but just hangs there, together with you – or, put differently, whatever you drop will still be falling at exactly the same speed as you are after you drop it. Incidentally, the situation is similar for astronauts on a space station in orbit. They are not “weightless” because of the difference in gravity in orbit (which is not that big) but rather because they are “freely falling” – being “in orbit” around a planet means there is no external force keeping you there, as opposed to standing on the ground where the ground continuously pushed you upwards to keep you from falling inwards. In other words, the weightlessness of space is simply because in the absence of external forces, we don’t really “feel” gravity. Why is this? Einstein’s answer to both of these questions (which you may realize are actually the same question) was that the spacetime that he introduced in special relativity is curved. He proposed that the “mass” of an object is simply a measure of how much the object curves spacetime. Moreover, objects simply move around in spacetime in a “natural” fashion: they simply follow the curvature. Note that there is no

1.3. General Relativity

distinction here anymore between gravitational and inertial mass: the beauty of general relativity is that both are automatically equal; it is simply built into the theory. The fact that we don’t “feel” gravity is simply because we don’t “feel” the effects of the curvature of spacetime as we follow it.

The concepts Einstein introduced in special relativity may have seemed unconven-tional, but general relativity is positively mind-blowing. At first, general relativity was viewed with a lot of scepticism by the physics community; moreover, those that believed in its validity thought it might be correct but too unwieldy and com-plicated to ever be able to actually study any physical systems with. Indeed, it is a testament of general relativity’s complexity that today, a hundred years after its introduction, there are still plenty of researchers studying the properties and solutions of the very same, original equations of general relativity that Einstein wrote down at its inception.

General relativity is very well approximated by Newton’s gravitational law. Of course, this must be the case, since Newton’s laws survived for so long and gave many good predictions! In fact, the difference between relativity and Newton’s laws are so tiny that one has to think extremely big, or be able to measure ex-tremely small effects, to be able to see any difference. In the past century, general relativity has successfully predicted galactic phenomena (i.e. thinking big) such as the bending of light by large galactic objects and the large-scale behavior of the universe. It has also predicted certain tiny but measurable effects such as time dilation for GPS satellites and the precession of Mercury’s orbit (which was known for centuries but hitherto unexplainable). At the moment, general relativity is the theory of gravity (see also Insert 1.2).

Insert 1.2: What Does It Mean to Have the “Right” Theory?

It is often said that a theory is “better” than another one or even that a theory is “right”. What do such statements mean?

For a physicist, if theory X is “better” than theory Y, this usually means that theory X gives better agreement with experiments. So for example, Newton’s law of gravity gives us pretty good predictions for how the planets move around the sun, but Einstein’s general relativity gives us slightly better predictions. This was measurable in the precession of Mercury’s orbit, which matched general relativity’s predictions beautifully.

While this is certainly the most important notion of a “better” theory, it is not the only one. One can also refer to theory X as “better” than theory Y if it solves a given inconsistency or problem of theory Y. For example,

quantum mechanics solves the ultraviolet catastrophe of classical mechanics so is better in this sense. Of course, it wouldn’t make sense to introduce a theory Y that solves a given problem of theory X, if many of theory Y’s predictions are now much further away from reality than theory X’s. Hence, usually the two definitions of a “better” theory that I discuss here are used together: theory X can only be better than theory Y if it gives the same or better predictions for experiments as theory Y, and it solves one or many problems or inconsistencies that theory Y has.

Note that, with the above definition, there’s no way of saying which of the theories of general relativity or quantum field theory is “better”. They are both better than each other, depending on what type of experiment you are considering. General relativity reduces to classical mechanics for very small particles, which has the same old problems that quantum mechanics solved; while quantum field theory does not include gravity so must necessarily give wrong answers whenever gravity effects are important, such as for galactic objects or over galactic distances. Clearly, neither theory can be better; rather, if we want a “better” theory, we should have a theory that can give us both the results of quantum field theory and general relativity in their appropriate approximation.

When one speaks of “the” theory, one is making a statement that there is no theory better than it. This is a weighty statement, indeed: this would imply that this theory describes the workings of the universe exactly. This theory, and no other, is the theory of reality. Whether one believes that such a theory really exists or not is more or less outside the realm of physics and rather a matter of philosophy (where those that do believe such a theory exists in principle are typically called Platonists and those that don’t or don’t care are called positivists).

1.4

Quantum Mechanics

With the theory of general relativity, we have resolved some of the puzzles of the beginning of the 20th century: we understand that the speed of light is the same for everyone due to special relativity; moreover, we understand why gravitational and inertial mass are the same due to general relativity. However, around that time there was also an entirely different set of problems, of which the ultraviolet catastrophe mentioned above is one. This set of problems would be tackled and explained by a completely different approach: quantum mechanics.

1.4. Quantum Mechanics

changed at its foundation. Quantum mechanics tells us that, if we measure a particle position very accurately, we have no idea what its momentum (or its ve-locity) is. This seems a bit ludicrous: why can’t I just also measure the particle’s momentum accurately at the same time with a different measurement? Quantum mechanics tells us that, contrary to our intuition, this is simply not possible: if we measure a particle’s position accurately, then by the very act of this measurement, we have ensured that the particle’s velocity or momentum is not determinable. If we then, afterwards, measure the particle’s momentum very carefully, then we have completely destroyed the information we first measured: now we know the particle’s momentum very accurately, but the particle’s position has become, due to the measurement, “smeared out” and thus indeterminable.

Quantum mechanics is full of such odd behaviours. Particles being “smeared out” means they are not really in one given position anymore: one particle can effec-tively be in more than one position at the same time! This weirdness is at the base of many phenomena that we simply cannot grasp with our intuition (which typically relies on classical physics), such as quantum tunneling and quantum en-tanglement. All this strangeness would have resulted in quantum mechanics as a theory being immediately relegated to the trash bin after its inception, if not for one annoying detail: it works. Quantum mechanics was conceived to resolve many of the difficulties and inconsistencies of classical mechanics (such as the ultravio-let catastrophe), and has done so remarkably well. In addition, it has prevailed against innumerable tests and experiments. Many of these tests were devised to see whether we really need such a “weird” theory like quantum mechanics, or whether maybe there is a more satisfying, “normal” theory behind all of the phenomena that may only superficially seem quantum. The results are unequivocal: quantum mechanics is correct, and even more, there can exist no “normal” (classical) theory that gives the same, correct predictions that quantum mechanics does. It’s impor-tant to realize that the basic principles of quantum mechanics, however unusual they might seem, have become part of our daily lives: for example, semi-conductor transistors in computers work only because of phenomena that are purely quantum and unexplainable in classical mechanics, such as quantum tunneling.

Another feature of quantum mechanics is that everything is quantized; in fact, this is so crucial to the theory that it found its way into the name of the theory. Quantization means that everything comes in discrete packets at its most basic level. Even light and radiation come in discrete packets: one can have one or two photons, or particles of light, but never a fraction of a packet. This discrete packet theory, in contrast to the continuous classical mechanics, lies at the basis of how quantum mechanics resolves the ultraviolet catastrophe. This problem arises in classical mechanics because a black body (such as a lump of hot coal) can radiate smaller and smaller packets of higher and higher energy; when we sum

over all of the energies radiated by the black body, we get the infamous impossible infinite energy emitted by the lump of coal. However, in quantum mechanics we are unable to keep making the packets smaller and smaller: at some point we reach one photon and we cannot have anything “smaller” than a single photon. Thus, this infinite energy problem simply does not arise in quantum mechanics.

1.5

Quantum Field Theory & The Standard Model

Quantum mechanics (in its original conception) is essentially the theory of one particle, or at the very most, a given fixed number of particles. It is also a theory that treats space and time as very different concepts: a particle may be very smeared out over space, but never really over time. Said differently, the position (in space) of a particle is dynamic and an output of the theory, while time is fundamentally a parameter, an input.

Having read the above discussion on special relativity, the reader could then begin to worry how this is reconcilable with special relativity, which at its fundaments states that time and space should be fundamentally on the same footing in a unified spacetime. Indeed, quantum mechanics was quite quickly realized to be incompatible with special relativity unless revamped into its current version, called quantum field theory.

Quantum field theory is a theory which inherently obeys both the laws of quantum mechanics and of special relativity. Instead of viewing particles as the fundamen-tal objects of the theory (as in quantum mechanics), now quantum fields are the fundamental objects. These fields permeate everywhere in spacetime; “ripples” of these fields correspond to particles. One particular phenomena that can be explained using the theory of particles as ripples in a field is that of pair-creation: when we pump enough energy into a system, we can create a pair of particles (usu-ally an electron and its antiparticle, a positron) seemingly out of nothing. This is completely incompatible with the original, non-relativistic version of quantum mechanics, which does not allow the number of particles in a system to change in time. However, in quantum field theory, such pair production is very natural: pumping energy into the system is like “pushing” the field, which then creates rip-ples in the field that we interpret as particles. This is analogous to, say, disturbing a calm surface of water, creating ripples in the surface of the water that propagate outward.

Quantum field theory is a very difficult theory to study and to use to calculate quantities of interest. The approach that has proven most successful in quantum field theory calculations is to first assume that all of the fields in the theory do

1.5. Quantum Field Theory & The Standard Model

not interact at all – in other words, the ripples in the fields (particles) simply move through and over each other without seeing the effects of each other: all of the particles in the theory are “free”. Then, we can try to turn on a tiny bit of interaction between the different fields; we let the particles feel a little bit of the effect of each other’s presence. In principle, the effects of turning on these interactions could result in a snowball effect: one interaction between particles could trigger another interaction, and so on. However, in the approximation of small coupling, where we assume the interaction strength between particles is small, we can neglect such snowball effects: the effects of an interaction triggered by an interaction would be much smaller than the effect of the interaction itself, so we can safely ignore these additional secondary effects. Disregarding these additional effects is an approximation that is clearly only valid at small coupling or weak coupling, when the interaction between particles is small. In this case, a whole machinery of calculational methods, pioneered especially by Feynman, exists to calculate quantities of interest using perturbation theory (see Insert 1.3). It is essentially simply a matter of cranking the wheel, following the algorithm invented by Feynman, to calculate the effects of the interactions.

Insert 1.3: Perturbation Theory

To illustrate what it means to do perturbation theory, consider the fraction:

f (x) = 1

1 + x. (1.1)

Clearly, since we know the full expression for this fraction f (x), for any particular numerical value of x, we can simply fill in this number in the fraction and see what the result is. For example f (0) = 1.

A mathematical result tells us that we can approximate this fraction for small values of x by a so-called Taylor series:

f (x) ≈ 1 − x + x2+ · · · , (1.2)

where the dots denote the fact that we are forgetting an infinite number of terms, so the result is not exact. We can consider this perturbation theory as we are perturbing f (x) around the value at x = 0 by small amounts. To illustrate how good this approximation is, a comparison between the true fraction f (x) and this approximate expression is given in the table below for some values of x. We can clearly see that, as x increases, this approximation becomes worse and worse.

second expression (1.2) and didn’t know what the full expression (1.1) looked like? Clearly, we could still calculate what f (x) would be for small values of x, but what we learn from the table below is that we can’t trust expression (1.2) for large values of x. If we had no other way of calculating f (x) for large values of x, we would be stuck!

This is essentially the same situation as we have in a typical quantum field theory. The variable x in the above example is then given by the coupling of the theory, which is why we say that perturbation theory is good for small coupling but cannot give us any meaningful answers for large coupling.

Comparison of f (x) and its approximation for different x

x 0 0.1 0.2 0.3 0.4 0.5 1 10

f (x) 1 0.909 0.833 0.769 0.714 0.667 0.5 0.0909 1 − x + x2 1 0.91 0.84 0.79 0.76 0.75 1 91

This method of calculating quantities in weakly coupled field theories has been highly successful. Perturbation theory calculations in quantum electrodynamics, which is a quantum mechanical version of Maxwell’s electromagnetism theory, have yielded results in agreement with experiment with accuracy up to one part in a trillion (10−12_{). It is hardly possible to exaggerate how preposterously accurate} this is: it is comparable to measuring the distance between the earth and the moon (about 400,000km) to an accuracy of a tenth of a millimeter. No other theory in the history of physics has ever been tested to this precision and still given accurate answers. It is safe to say that this remarkable agreement with experiments gives quantum field theory the undisputed status of most rigorously tested physical theory ever.

However, while the techniques of perturbation theory spawned this success story for quantum electrodynamics (QED), there are other quantum field theories where it is not as successful. Quantum chromodynamics (QCD) is the quantum field theory of strong interactions, which is the fundamental force that keeps protons and neutrons bound tightly together in nuclei of atoms. The fact that protons and neutrons are bound tightly immediately tells us why QCD is so difficult to study: it is a strongly coupled theory, which is thus not amenable to the methods of perturbation theory. It is extremely difficult to perform calculations in QCD and make predictions; the most successful calculations have usually involved doing complicated, heavy numerical simulations on supercomputers. In contrast, basic QED perturbation theory calculations are left as exercises to be done with pen and paper in undergraduate courses on quantum field theory.

Despite these difficulties, quantum field theory has been enormously successful in describing the world of fundamental particles as we know it. Three of the four

fun-1.6. The Problem of Quantum Gravity

damental forces in nature have been described with success using this framework: the electromagnetic force (in quantum electrodynamics and its successor, elec-troweak field theory), the strong force (quantum chromodynamics, as mentioned above), and the weak force (which is responsible for radioactive decay of nuclei). These three forces are described by a combined quantum field theory called the standard model. The only force missing from the list of forces described by the standard model is gravity, which we will get back to later. The standard model has been incredibly successful in predicting the behaviour of the three forces it de-scribes. Most recently, the final piece of the standard model puzzle was resolved: for the entire theory to be consistent, a new particle had to exist with very par-ticular properties – the Higgs boson. This particle was predicted to exist by the standard model decades ago, as its existence was necessary for the consistency of the entire theory. Recently the new particle accelerator LHC at the CERN facility was able to conclusively prove its existence, thus venerating this final ingredient of the standard model once and for all.

1.6

The Problem of Quantum Gravity

In the last few sections, we have described the successes of twentieth century the-oretical physics: Einstein’s general relativity gives us a theory describing gravity, while the three other forces (electromagnetism, weak, and strong) are described with incredible accuracy by the standard model of particle physics using the frame-work of quantum field theory.

Both general relativity and the standard model live in four-dimensional spacetime, where time and space have been put on the same footing and the speed of light is the maximum speed obtainable and is the same for all observers. However, the nature of this spacetime is very different in both theories. In quantum field theory, spacetime is a static background in which the fields and their ripples (particles) move around; spacetime is unchanged and unaffected by the fields that move around in it. However, in general relativity, spacetime is malleable, intimately intertwined with matter, curving due to matter and bending particles’ paths due to its curvature.

In addition, quantum field theory and general relativity treat particles and all their properties with a different philosophy. Quantum field theory, being a descendent of quantum mechanics, has everything quantized – particles, energy, etc. On the other hand, general relativity is philosophically similar to classical mechanics in that everything is simply continuous: there is no minimum quantum of energy or anything else; anything is allowed.

The two pictures painted by general relativity and quantum field theory are very different. So how can it be that these theories have both been so successful in their predictions, if they are so obviously incompatible with each other?

It is important to realize that general relativity and quantum field theory, or the standard model, are really describing two completely different aspects of the universe. Relativity is a theory of gravity, which is a very weak force: a single particle has such a small gravitational pull that it is completely negligible. This means that on short distance scales, where we typically only consider interactions of a few particles, we can typically ignore gravity effects. It is only once we take a very large number of particles together that gravity becomes important. Consider the earth, which certainly contains “a lot” of matter – the gravitational pull of the entire earth can be counteracted by the magnetic field of a simple small magnet holding on to a fridge! Having a large enough number of particles together so that gravity effects are important typically only occurs when we consider very large distances – such as when we consider the movement of planets around stars. On the other hand, the forces involved in electromagnetism and the other forces described by the standard model are much, much larger in magnitude than gravity. These forces remain very important even at short distances. However, these forces tend to disappear when averaging over a large amount of particles. For example, the entire earth is made out of approximately the same amount of electrically positive and negative charged particles, which means the net electromagnetic force that any object feels due to the entire earth is approximately zero.

In other words, we have a theory that describes very large objects (made out of many, many particles) and large distances: relativity; and we have a completely different theory that describes small objects (made out of one or few particles) and short distances: quantum field theory and the standard model. This is the reason that these two very different theories can live side by side in relative harmony: when we do calculations involving gravity, we can typically ignore effects from the other three forces and vice versa (see also the discussion in Insert 1.2).

Even though, as explained above, the theories of quantum field theory and general relativity can coexist relatively peacefully, this is not wholly satisfactory. We would like to have a worldview in which our theories of the very large and very small are mutually consistent. We would like to be able to intuitively see how our theory describing the physics of the very small, when zoomed out, automatically describes the physics of the very large. This would be a theory that unifies the two most important concepts together. General relativity’s dynamical spacetime, curved by the presence of matter, should be combined with quantum field theory’s inherent quantum uncertainties and quantization of all particles and energies. (See also Insert 1.4 for more discussion on unifications of theories.) This would be a

1.6. The Problem of Quantum Gravity

quantized theory of spacetime, a quantum theory of gravity – often simply called quantum gravity.

A good theory of quantum gravity would have to give the same results that quan-tum field theory (specifically, the standard model) yields for very small particles and distances and the same results as general relativity produces for very large amounts of particles and large distances. Perhaps we can even find more than one candidate theory of quantum gravity that satisfies these demands. The question is then: can we think of any particular situation where we would need effects from both theories at the same time? In such a situation, general relativity and quan-tum field theory would give conflicting predictions, so we would need a theory of quantum gravity to tell us what the correct physical prediction in this situation would be. We could then compare and contrast the predictions from different theories of quantum gravity, and choose the one that gives us the right predictions when confronted with experiments.

It turns out it is actually quite difficult to cook up a situation where effects from quantum field theory and general relativity would be necessary at the same time. We typically need to squeeze a lot of matter (to have considerable effects from general relativity) in a very small space (so that quantum field theory effects are important). It turns out there are essentially two situations in the universe in which we find such conditions: on the inside of black holes, and at the very beginning of the universe, at the big bang. Let’s discuss these in more detail.

Insert 1.4: Unification of Theories

Perhaps the first example of a “unified” theory was Newton’s theory of grav-itation: he realized that the force that makes things fall to the ground on the surface of the earth is precisely the same force that makes planets revolve around the sun, or the moon around the earth. Although this may seem ob-vious from our modern point of view, centuries ago these phenomena must have seemed so different that even imagining the force to be the same must have almost seemed preposterous at first! But Newton was right: measuring how things fall to the ground on earth teaches us about the intricacies of how celestial objects more around in the sky.

Another well-known example of a unified theory was Maxwell’s realization that the electric force and the magnetic force are really just two different aspects of the same force – electromagnetism. More recently, the quantum field theory version of electromagnetism – quantum electrodynamics – was unified together with the weak force in electroweak theory.

Finding an underlying, unifying theory of seemingly different phenomena gives us deep insights into physics and strong predictive power. Unification of theories means that less is needed to describe more. For example, all that is required as input in Newton’s gravitational law is a single constant, Newton’s gravitational constant: once the numerical value of this one constant has been determined, all gravitational interactions are completely fixed. Similarly, the theory of electromagnetism fixes the relative strength of the electric and the magnetic force. Before knowledge of electromagnetic unification, in princi-ple one would have to separately measure the strength of the electric and the magnetic force.

The standard model does a fairly successful job of unifying the electromag-netic, weak, and strong forces in one framework. However, it is far from satisfactory: there are a whopping 28 parameters that need to be fixed in the standard model, most of them related to the masses of the many species of fundamental particles (such as quarks and electrons). Ideally, a grand uni-fying theory of everything (sometimes called TOE, or theory of everything) would have only one dimensionful parameter: a measuring stick to use when measuring lengths or masses. This is another attraction of string theory (as will be discussed later on): it is able to encompass a huge variety of theories such as the standard model and contains a single dimensionful parameter – the length of the string. All physics follows without any external input, as would befit a grand unifying theory of everything.

1.7

The Big Bang & Cosmology

The first real piece of evidence that the universe originated from a so-called Big Bang was the (accidental) discovery of the cosmic microwave background (CMB) by Penzias and Wilson. Before this discovery, a rival theory of a steady-state universe, which is a universe which had always existed and will forever continue to exist, was equally popular. The CMB was the first major piece of evidence that this steady-state universe was unlikely and a universe created in a massive explosion was more likely. Since then, massively more evidence has been discovered in favor of a Big Bang, which should have happened approximately 13.7 billion years ago.

By observing how the universe looks now, astronomers and cosmologists can de-duce how the universe must have evolved since the Big Bang and can infer how the universe must have looked like in the past, up to a fraction of a second after the Big Bang. But there the predictions stop. At or right after the Big Bang,

1.7. The Big Bang & Cosmology

all of the matter in the universe must have been squeezed together in a tiny, tiny space – this is exactly the kind of system where general relativity and quantum field theory cannot provide us with answers. We need a theory of quantum gravity to provide us with a solution.

A theory of quantum gravity should be able to tell us what exactly went on in the very first fraction of a second after the Big Bang. It might even be able to tell us something about the Big Bang itself, like how it happened. A theory of quantum gravity might tell us that a Big Bang is always preceded by a big crunch, where the entire universe collapses onto itself – in a sense the opposite of a Big Bang –, and that the universe is doomed to infinitely repeat a cycle of alternatively expanding and shrinking, each time beginning with a Big Bang and ending with a big crunch. Or perhaps quantum gravity will tell us that the Big Bang was a chance process, and that there necessarily exist many parallel universes, universes created by other Big Bangs, perhaps like ours but with slightly different physical parameters.

A theory of quantum gravity should provide us with more than just a satisfying explanation of the Big Bang and what happened a fraction of a second after it. It should also tell us what the consequences are for the universe now, 13 billion years after the Big Bang. It should give predictions for measurements and observations that astronomers can perform in the sky. Such predictions and their confirmation would provide a testing ground for theories of quantum gravities: only the right theory would give us the correct predictions for all such observations.

The problem with measuring such quantum gravity effects at or right after the Big Bang is that these effects are extremely small and difficult to measure. In addition, astronomical observations are obviously limited: we only see the universe that is shown to us as we look upward at our very special place in the universe, on earth. We live on a particular planet which is located at a very particular place in the galaxy and in the universe. It is very difficult to guess what lies beyond the universe that we can see. This all makes it extremely difficult to make strong measurements and statements about the effects of the conditions at and right after the Big Bang. Only recently have advanced satellites such as PLANCK provided us with a possible first glance at such subtle quantum gravity effects – but it is still largely unclear what these experiments will be able to tell us, especially about interesting quantum gravity effects.

1.8

Black Holes

A star is essentially a big ball of burning gas. This gas mainly consists of hydrogen, which at the center of the star is being compressed at such enormous pressures that the hydrogen fuses into heavier elements such as helium. This nuclear fusion reac-tion releases an enormous amount of energy, which counteracts the gravitareac-tional pull of the star and keeps the star from imploding on itself.

This process cannot go on forever, since at a certain point the star’s hydrogen fuel will be depleted and the star must die. Stars can die in various different complicated ways depending on their mass, but one kind of star death has a very interesting result. In this particular type of star, when its fuel is depleted, the star violently sheds its outer layers in a supernova while the rest of the star implodes. With no nuclear fusion pressure left to counteract the gravitational pull, the star’s implosion can only continue to accelerate and the star’s matter will contract into a smaller and smaller volume.

The end result will be a black hole. Due to the concentration of matter at its center, this object has such a high gravitational pull that nothing can escape from travelling too close to it – not even light. The point of no return, past which even light cannot return, is called the event horizon of a black hole.

Schwarzschild was the first to discover that general relativity predicts the existence of black holes and their event horizons. However, even though general relativity provides a good description of what goes on outside of a black hole and what happens around the event horizon, things get more complex as we travel through the horizon to the center of the black hole. At the center of the black hole, general relativity predicts that all of the mass of the black hole is concentrated in a single point, resulting in a point of infinite mass and infinite spacetime curvature – the singularity of the black hole.

These black hole singularities are bad news for general relativity: not only are they clearly a source of various infinities (which always are alarming in physics), but they also lead to general relativity predicting its own limitations. An object falling into a black hole will pass the event horizon and move towards the singularity. After a given time, it must hit the singularity – and then what? Nobody knows what happens at the singularity of a black hole – general relativity cannot give us any predictions for this.

Another problem that arises when unifying quantum mechanics and general rela-tivity for black holes is the so-called information paradox. General relarela-tivity tells us that black holes always look exactly the same on the outside of the black hole. This means that it doesn’t matter if we throw a five-ton elephant and a three-ton

1.8. Black Holes

giraffe together to form a black hole, or just an eight-ton elephant – the result is always the same black hole (with a mass of eight tons). In other words, gen-eral relativity tells us that the black hole loses information about the matter that created the black hole. (See Insert 1.5 for more discussion on the information para-dox.) Such information loss is not consistent with what we would expect from a consistent quantum theory – quantum mechanics (or quantum field theory) never loses information. This may seem a bit strange after the discussion on quantum mechanics above, but nevertheless this is true! A quantum theory of gravity must resolve this paradox and tell us how information is “stored” inside a black hole.

Insert 1.5: The Information Paradox & Hawking Radiation

In the main text, the information paradox is introduced by saying that general relativity predicts that any black hole will look exactly the same and thus not retain any information as to how it was formed. While this is in principle true, perhaps the information paradox can better be thought of by performing a slightly different thought experiment.

Let’s take a black hole, already fully formed – the implosion of the star is already long complete. Now we drop a particle in this black hole. The only thing that will happen is that the black hole has to become slightly larger, as the radius of a black hole is proportional to its mass.

If we wait patiently outside this black hole, we should be able to observe ra-diation coming from the black hole that gives us the information about the particle we dropped in earlier. Indeed, Stephen Hawking did a famous calcu-lation showing that black holes do indeed radiate. This is a purely quantum mechanical phenomenon; in classical general relativity, black holes stay en-tirely black – nothing can escape them. The peculiar thing about this Hawking radiation from black holes, though, is that Hawking’s calculations explicitly tell us it carries absolutely no information at all (in physical terms we say it is thermal radiation). So what’s going on, why isn’t our information coming out? This paradox is a different (more precise) statement of the information paradox.

The crucial point is that Hawking’s calculation was done in the approximation that the black hole itself is unaffected by the radiation. Of course, this is not true in general: if the black hole radiates out a particle, it will grow a little bit smaller. Presumably, if one were able to redo Hawking’s calculation and keep into account how the black hole changes as well, we should be able to see information coming out of the black hole after a while. However, such a calculation is essentially impossible without a quantum theory of gravity, and

even within string theory we don’t really know how to perform this calculation as of now.

Although a good theory of quantum gravity will tell us what happens at the center of black holes, it is also important to note that this will probably not lead to any observable predictions. Although black holes have been found in copious amounts throughout the universe, for obvious reasons it is impossible to observe anything going on inside a black hole (remember that even light cannot escape from inside a black hole!). There are also many reasons to believe that, in fact, the theory of general relativity will suffice plentily to describe the physics of what we can see around a black hole, thus completely obviating the need for quantum gravity in any astronomical observations of black holes in the foreseeable future. Understanding the inside of black holes, as opposed to cosmology, is thus a purely theoretical exercise and curiosity. (If this prompts the reader to wonder why we should even bother thinking about such things, I refer him or her to Insert 1.6.)

1.9

Summary

I have taken you on a quick journey through the some important theoretical physics concepts of the first half of the twentieth century. To accommodate the always constant speed of light, Einstein unified space and time together into spacetime in his special relativity. He later discovered the theory of general relativity, which tells us that this spacetime is curved and that spacetime curvature gives us gravitational forces. In a completely different development, quantum mechanics was invented to solve various problems present in classical mechanics. Quantum mechanics tells us that particles and energy exist only in quantized packets called quanta. Quantum mechanics’ successful union with special relativity, called quantum field theory (in the form of the standard model), has been enormously successful in describing fundamental particles and the world of the very small.

General relativity and quantum field theory are two very successful theories that are mutually incompatible. Somehow a theory that unifies them must exist: a theory of quantum gravity. A good theory of quantum gravity must give us the same predictions of general relativity for large objects and large distances, and the same predictions as quantum field theory for a small amount of particles at short distances. It must also tell us what happens at the point where general relativity and quantum field theory give contradictory answers: when we put a large amount of matter together in a very small space. There are essentially two such systems that we know of that satisfy this criterion: the universe right at and right after the Big Bang; and the deep interior of black holes, where general relativity predicts

1.9. Summary

that the entire mass of the black hole lies at one single point, the singularity of the black hole.

For these reasons, theories of quantum gravity are often studied to see what their predictions would be about the very early universe and the inside of black holes. We will now turn to and discuss what is arguably currently the best candidate quantum gravity theory available (and certainly the most studied), which is string theory.

Insert 1.6: Why Bother?

A question that people in my or related lines of research often get is: What (practical) use does this research have? What practical, everyday applications do you expect?

This question is at worst a sign of ignorance or an expression of quintessential human short-sightedness. Of course I have no idea what practical applications that research into black holes and quantum gravity could possibly have in the future. Anyone claiming anything to the contrary is either a fraud or trying to get money out of you (incidentally, making such claims is not unlike what is required in some applications for research grant money).

Einstein surely did not have the tiniest clue what general relativity could ever be “used” for, if anything. Yet now, a century later, general relativistic ef-fects are a crucial part in the calculations that GPS satellites have to make in orbit in order to give accurate readings of positions. When Maxwell first studied and predicted radio waves as a by-product of his theory of unified elec-tromagnetism, he didn’t have the faintest clue that they could and would be used in the future to transmit information at the speed of light. Quantum mechanics was thought up in the beginning of the twentieth century to solve various theoretical problems that arose from classical mechanics, and in or-der to explain a number of very controlled and contrived (certainly not every-day) experimental results that baffled theorists. It wouldn’t have crossed any of the minds of the founders of quantum mechanics that such an outlandish and fundamentally bizarre theory as quantum mechanics could possibly have every-day applications. Nevertheless, quantum mechanics and the “weird” quantum phenomena of quantum tunneling lie at the very heart of semicon-ductor theory, which is the basis of transistors in every computer in the world. In a lot of research there are clear reasons to conduct the research and obvious every-day applications in mind when embarking on it. These are typically in the realm of “applied” research: nobody needs to be told what the reason is for