Table Of ContentGraduate Texts in Physics
Daniel Grumiller
Mohammad Mehdi Sheikh-Jabbari
Black Hole
Physics
From Collapse to Evaporation
Graduate Texts in Physics
SeriesEditors
KurtH.Becker,NYUPolytechnicSchoolofEngineering,Brooklyn,NY,USA
Jean-MarcDiMeglio,MatièreetSystèmesComplexes,BâtimentCondorcet,Université
ParisDiderot,Paris,France
SadriHassani,DepartmentofPhysics,IllinoisStateUniversity,Normal,IL,USA
MortenHjorth-Jensen,DepartmentofPhysics,Blindern,UniversityofOslo,Oslo,
Norway
BillMunro,NTTBasicResearchLaboratories,Atsugi,Japan
RichardNeeds,CavendishLaboratory,UniversityofCambridge,Cambridge,UK
WilliamT.Rhodes,DepartmentofComputerandElectricalEngineeringandComputer
Science,FloridaAtlanticUniversity,BocaRaton,FL,USA
SusanScott,AustralianNationalUniversity,Acton,Australia
H.EugeneStanley,CenterforPolymerStudies,PhysicsDepartment,Boston
University,Boston,MA,USA
MartinStutzmann,WalterSchottkyInstitute,TechnicalUniversityofMunich,
Garching,Germany
AndreasWipf,InstituteofTheoreticalPhysics,Friedrich-Schiller-UniversityJena,
Jena,Germany
GraduateTextsinPhysicspublishescorelearning/teachingmaterialforgraduate-and
advanced-levelundergraduatecoursesontopicsofcurrentandemergingfieldswithin
physics, both pure and applied. These textbooks serve students at the MS- or
PhD-levelandtheirinstructorsascomprehensivesourcesofprinciples,definitions,
derivations,experimentsandapplications(asrelevant)fortheirmasteryandteaching,
respectively.Internationalinscopeandrelevance,thetextbookscorrespondtocourse
syllabisufficientlytoserveasrequiredreading.Theirdidacticstyle,comprehensive-
nessandcoverageoffundamentalmaterialalsomakethemsuitableasintroductions
orreferencesforscientistsentering,orrequiringtimelyknowledgeof,aresearchfield.
Daniel Grumiller ·
Mohammad Mehdi Sheikh-Jabbari
Black Hole Physics
From Collapse to Evaporation
DanielGrumiller MohammadMehdiSheikh-Jabbari
InstituteforTheoreticalPhysics SchoolofPhysics
TUWien InstituteforResearchinFundamental
Vienna,Austria Sciences
Tehran,Iran
ISSN1868-4513 ISSN1868-4521 (electronic)
GraduateTextsinPhysics
ISBN978-3-031-10342-1 ISBN978-3-031-10343-8 (eBook)
https://doi.org/10.1007/978-3-031-10343-8
©SpringerNatureSwitzerlandAG2022
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MohammadMehdiSheikh-Jabbaridedicates
thisbooktohiswifeYasamanFarzanforall
hersupportandencouragement
DanielGrumillerdedicatesthisbooktohis
parentsIngoGrumiller,WilfriedRaunikar,
andHelgaWeuleforpassingontheir
curiosityaboutLife,theUniverse,and
Everything
Foreword
The study of black holes arguably began in 1783 with John Michell. At the time,
Michell was the rector of St. Michael’s church in the Yorkshire village of Thorn-
hill. Prior to this, he had been educated and become a fellow of Queens’ College
CambridgeandalsohadbeenWoodwardprofessorofgeologyinCambridge.How-
ever, fellows of Cambridge colleges at the time were required to be celibate and
so he migrated on marriage, as many at the time did, to the church. His clerical
duties were sufficiently light that he was able to continue his scientific studies. In
the Proceedings of the Royal Society, he laid out the possibility that there were
starsthatcouldnotbeseen.HisreasoningwasentirelyNewtonian.Hewasableto
calculate the escape velocity of a particle from the surface of a massive body. He
knew that light had large, but finite, velocity as had been shown in the previous
century by Rømer. His conclusion was that if a star was sufficiently condensed,
then it would be invisible. Supposing that the star had mass M, then if its radius
R was such that
2G M
R < N
c2
where G is Newton’s gravitational constant and c is the speed of light, then it
N
wouldbeinvisible.Remarkably,thesameformulaholdsingeneralrelativity.Some
14yearslaterin1796,PierreLaplace,workingindependently,publishedasimilar
result from the other side of the English Channel.
It was not until Einstein proposed the general theory of relativity that mat-
ters progressed. As soon as his theory became known, Karl Schwarzschild found
in 1915 the exact solution of Einstein’s field equations that bears his name. His
solution is a spacetime metric that contains a black hole. For a long time, there
were discussions about whether the horizon was singular or not, and whether the
spacetime singularity at the center of the black hole was unphysical. Despite the
uncertainty about whether the Schwarzschild metric was physically acceptable, in
1939 J. Robert Oppenheimer and Hartman Snyder showed that, under certain cir-
cumstances,astarcouldcollapsetoformablackhole.Nevertheless,considerable
skepticism about the possible existence of black holes remained until the early
1970s.
vii
viii Foreword
A revolution in our understanding of general relativity and black holes, in par-
ticular, started in the late 1950s driven by John Archibald Wheeler in Princeton.
Heseemstohavebeenmotivatedbyattemptstounderstandthefinalstateofstars
and also began to formulate a quantum theory of gravity. This started a period of
great excitement. Two key events were that Roger Penrose proved the first sin-
gularity theorem in 1965. He showed that if a trapped surface formed, then a
spacetime singularity must exist. This result removed one of the possible stum-
bling blocks in understanding black holes. In Oppenheimer–Snyder collapse, one
doesformaspacetimesingularity,butsuchathingmighthavebeenaconsequence
of assuming exact spherical symmetry, something which cannot happen in nature.
Penrose’sresultshowedthatsingularitieswouldformwithoutmakinganyassump-
tions about spatial symmetry. Also in 1965, Roy Kerr found the spacetime metric
that describes the most general stationary black hole in an asymptotically flat
spacetime.Theseresultstriggeredaflurryofkeyresultsinclassicalgeneralrelativ-
ity.Theyledtoanunderstandingoftheastrophysicsofblackholes.Thediscovery
of black hole X-ray sources as stellar remnants, quasars, the black holes in the
centers of galaxies as conclusively proved by the recent results of the event hori-
zontelescope,andtheproductionanddetectionofgravitationalwavesincollisions
involving black holes have removed any uncertainty about their existence.
However,therearestillchallenges.Hawkingshowedthatblackholesemitradi-
ation that appears to be thermal. This was perhaps the first successful result in
quantumgravityandithasgeneratedanenormousfollow-upliterature.Thedeep-
est puzzle is the information paradox. Quantum mechanical information appears
to be destroyed as a black hole evaporates away. Most of us believe that this can-
not happen, largely because of the incredible successes that quantum mechanics
has. Nevertheless, the subject is still open, and there are many that believe that
informationisdestroyed.Therehasbeenalotofrecentprogressandinparticular,
the beginning of our understanding of holography, the discovery of soft hair, and
the application of rigorous quantum information theory is helping an exploration
of this problem.
This book provides an excellent introduction to all of these topics. It is really
thefirsttoattemptsuchasynthesisandprovidesavaluablecompendiumofinfor-
mation on black holes for students of all ages and researchers of all flavors. The
collection of problems at the end of each chapter encourages enthusiasts to gain
a mastery of the subject and undoubtedly will help lead to much-needed further
insights into one of the biggest mysteries in fundamental physics today.
London, England Malcolm Perry
Preface
Black holes are predictions of Albert Einstein’s theory of general relativity and
have been the focus of intense studies for over a century, as they provide a rich
source of theoretical and experimental challenges. Black holes entail some of the
deepestpuzzlesontheroadtowardquantumgravityandengenderedseminaltheo-
reticaldiscoveries,suchasthermodynamicalpropertiesofblackholes,theirrolein
theholographicprinciple,andtherecentmergerofgeometricalandquantuminfor-
mation concepts. Black holes, by their very definition, are hard to detect directly.
Recenttechnologicaladvancementshaveledtonovelmethodsofdetectionthrough
gravitationalwavesandbyobservingtheirshadows.The2020physicsNobelprize
awarded “for the discovery that black hole formation is a robust prediction of the
general theory of relativity” is a testimony of the importance of black holes.
In this book, we elucidate the theoretical concepts of black hole physics and
discusssomeofthemainphenomenologicalsignaturesandprospectsinrecentand
future black hole observations. The scope is narrow in the sense that everything
we present is in one way or another related to black holes; at the same time,
our scope is ambitiously broad, since we cover ground including observational
challenges, geometrical tools, thermodynamics, particle creation, holography, and
quantuminformation.Theprincipalgoalofthisbookistoprovidethereaderwith
an up-to-date status of black hole physics.
This book is mainly intended for three groups:
1. graduate-level physics students who intend to start their research career in the
field of black holes;
2. postdocs who recently changed their research focus toward black holes and
want to get up-to-date on recent and current research topics;
3. advanced researchers who want to teach (or learn) basic and advanced aspects
of black hole physics and the associated mathematical tools.
The main style we pursue is hands-on, which in particular means that there will
be numerous exercises as an integral part of the learning experience conveyed by
our book. We assume the reader is familiar with the basics of general relativity
and with standard undergraduate physics, like special relativity, thermodynamics,
quantum mechanics, and statistical mechanics. Moreover, we assume familiarity
with the basics of quantum field theory on Minkowski space. Apart from these,
ix
x Preface
we have tried to cover the required material in the main text accompanied by
appendices. We mainly focus on established knowledge, but also give a taste of
current trends and issues of dispute. Since the research field of black holes is a
rapidly advancing area on theoretical, numerical, and observational fronts, topics
that are only partially addressed and understood by the time of publication of this
book may well become a part of established physics in just a few years.
Wecoveralltopicsandareasthatweconsideressentialsofblackholephysics.
However,asalwaysthereareseveralinclusionsandexclusionsoftopicsthatsome
readers may find questionable. While our choices naturally are influenced by our
field of research, theoretical aspects of black holes, we aspire to give a broad and
fair overview of all aspects of black hole physics. We have strived to
1. make this a self-contained book on black holes,
2. cover basic, advanced, and expert topics suitable for a 1-year course,
3. provide many exercises and (hints for) solutions, and
4. guide the reader through current research topics.
Whilecoveringawiderangeoftopics,weavoidbeingoverlyencyclopedic;wedo
not intend to give a full literature review on all topics and discuss only the most
essential points. At the end of each chapter, we provide suggestions for further
reading. Finally and importantly, each chapter comes with a set of problems that
are an integral part of the book. There are two types of problems: (1) Exercises
meant to engage the reader with the analysis and discussions of the chapter by
filling in the gaps of calculations and proofs. (2) Problems that are extensions of
the analysis of the chapter. They are interesting and relevant, but we did not find
them crucial enough to be added to the main text.
How to Read and Use This Book
Weprovidenowaroughguidelineforhowtousethisbook.Graduatestudentswho
readthisbookontheirownshouldattempttosolveatleasthalfoftheexercisesat
the end of each chapter, as a cross-check whether they have digested the material
in the main text. If you are such a student and have already some experience with
blackholes,youmayskipreadingsomeoftheearlierchapters,butyoushouldstill
try to complete the exercises just to make sure that you know what you thought
you knew. If you have no experience with black holes whatsoever, then just start
reading Chap. 1 and proceed from there.
Postdocs reading this book may want to skip ahead to their main focus of
interest, which is likely to be in the second half of the book, Chaps. 6–10. Still,
we recommend that you have a look at some of the exercises in the first five
chapters. Essentially, the same is true for more advanced researchers who want to
learn about advanced aspects of black hole physics.
If you plan to use this book as a tool for teaching, then our main recommen-
dation is to drop stuff that you find less relevant and to schedule at least one year
