Foundations of General Relativity: From Geodesics to Black Holes
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Foundations of General Relativity:
From Geodesics to Black Holes
Klaas Landsman
July 31, 2020
Dedicated to (the memory of) Stephen Hawking and Roger Penrose
This cartoon depicts Albert Einstein, carried by the “giants” Isaac Newton, James Clerk Maxwell,
Carl Friedrich Gauss, and Bernhard Riemann, as well as the “dwarfs” Marcel Grossmann,
Gunnar Nordström, Erwin Finlay-Freundlich, and Michele Besso (both groups listed from left
to right). Note that Einstein has difficulty balancing himself, much as village chief Abraracour-
cix does in the original Asterix comic, from which also the dog Idéfix and the violin in the tree, a
reference to the widely ridiculed village musician Assurancetourix. The group to the left consist
of Einstein’s Berlin colleagues Fritz Haber, Walther Nernst, Heinrich Rubens, and Max Planck,
whose interest in general relativity was, as shown here, small. Haber stands for the druid
Panoramix, who is brewing his magic potion, which in this case (World War I was running)
consisted of phosgene and mustard gas. The elderly ladies in the back are Einstein’s second
wife Elsa, with her daughters Ilse and Margot, and his first wife Mileva, with her and Albert’s
sons Hans Albert en Eduard (their illegitimate and “lost” daughter Lieserl is not depicted).1
1 Source: https://www.nature.com/news/history-einstein-was-no-lone-genius-1.18793. Artist:
Laurent Taudin. Idea and copyright: Michel Janssen and Jürgen Renn. Permission pending.
iPreface
This book has grown out of lecture notes for my master’s courses on general relativity (GR)
taught at Radboud University Nijmegen, intended for students with a double bachelor degree in
mathematics and physics who had done a first physics course in relativity and a first mathematics
course in differential geometry. However, as a medium-level (and size) text it is intended for
all students of GR, of any age and orientation, who have a comparable background (although
the subject is developed from scratch, experience shows that the material simply makes no
sense without some prior exposure to it). Compared to most textbooks on GR, this one is
mathematically oriented (and it omits some standard physics material like the classical tests
etc.), but I hope it appeals to mathematicians, physicists, and philosophers (of physics) alike.
My own experience is that a really deep field such as GR (or quantum theory) can only be
learned from a large number of books saying the right things in different ways, as well as by
talking to people working in the field. As such, my first encounter with GR was Einstein’s own
exposition Relativity: The Special and General Theory (Einstein, 1921), which is still in print.
In the Summer of 1981, having just graduated from high-school, this was followed by two books
that were far more difficult, namely Space–Time–Matter by Weyl (1922) and The Mathematical
Theory of Relativity by Eddington (1923), both of which are not only highly mathematical but
also profoundly philosophical in spirit. Weyl makes this point himself quite explicitly:
At the same time it was my wish to present this great subject as an illustration of the
intermingling of philosophical, mathematical, and physical thought, a study which is dear
to my heart. This could only be done by building up the theory systematically from the
foundations and by restricting attention throughout to the principles. But I have not been
able to satisfy these self-imposed requirements: the mathematician predominates at the
expenses of the philosopher. (Weyl, 1918, Vorwort). 2
Indeed, Weyl, as well as Eddington and unquestionably also Einstein himself, was a natural
philosopher in the spirit of the Scientific Revolution, whose mix of physics, mathematics, and
philosophy was the key to its success–it is no accident that Newton was the founder of the
scientific study of gravity and Einstein his successor, working in the same spirit: if any scientific
theory represents the Philosophiae Naturalis Principia Mathematica, it must be GR.
Although I understood little if anything from their books, Weyl and Eddington thus left
an indeledible mark in the way they approached natural science. Still during that same long
Summer vacation between high-school and University, which I regard as one of the high points
of my life, following the advice of an academic friend of my grandfather I bought Gravitation
by Misner, Thorne & Wheeler (1973). For a while I considered this the greatest book written
on any topic whatsoever,3 and when Misner, well in his eighties at the time, came to a talk I
gave in one of the New Directions conferences in Washington DC and even asked a question,
having answered in the positive to my counter-question if he was Charles Misner I petrified.
Nonetheless, I now see its basic flaw: with its xxvi + 1279 pages, it leaves no room for the
reader (except in doing the exercices, which I all duly did), who is overwhelmed and cornered.
2 ‘Zugleich wollte ich an diesem Großem Thema ein Beispiel geben für die gegenseitige Durchdringung
philosophischen, mathematischen und physikalischen Denkens, die mir sehr am Herzen liegt; dies konnte nur
durch einen völlig in sich geschlossenen Aufbau von Grund auf gelingen, der sich durchaus auf das Prinzipielle
beschränkt. Aber ich habe meinen eigenen Forderungen in dieser Hinsicht nicht voll Genüge tun können: der
Mathematiker behielt auf Kosten des Philosophen das Übergewicht.’ Translation: Henry L. Brose (Weyl, 1922).
3 Kaiser (2012) gives an interesting perspective on Gravitation and its history, which confirms its uniqueness.
iiMy next book was The Large Scale Structure of Space-Time by Hawking & Ellis (1973), and
so on, until General Relativity and the Einstein Equations by Choquet-Bruhat (2009). These
are all masterpieces written by the founders of the field, but almost every student or author
on the mathematical side of GR is also indebted to O’Neill (1983) and Wald (1984). Other
influences on this text include Poisson (2004), Plebański & Krasiński (2006), Gourgoulhon
(2012), Malament (2012), Chruściel (2019), and numerous works to be cited in footnotes.
This brings me to the question why an author who wrote only one paper in GR (i.e. Against
the Wheeler–DeWitt equation, published in 1995) is entitled to write a book about the subject,
even if this has been an almost lifelong passion. In one of the Jeeves and Wooster episodes, an
aunt of the latter, an English aristocrat, asks: ‘Bertie, do you actually work?’, upon which, taken
back by the question, Wooster mumbles: ‘Hmm, well, . . . , I know some people who work.’
So yes, I know some people in GR and I talked to them quite a lot. The greatest of these is
Roger Penrose, to whom this book is co-dedicated in honor of his pivotal role in the creation of
mathematical relativity. I first met him during a Seven Pines meeting in Minnesota, where the
organizers had the luminous idea that senior and junior participants share an apartment. I am
not sure to whom this arrangement was more shocking initially, but we got along well, and he
very kindly came to the opening conference of my institute IMAPP in 2005 as a speaker, where
he explained the key ideas of his later book Cycles of Time from 2010. He usually came to my
talks when I was visiting Oxford and joined for lunch or dinner whenever possible.
The other dedicatee, Stephen Hawking, was equally well one of the founders of mathemat-
ical relativity (along with Hilbert, Weyl, Darmois, Lichnerowicz, Choquet-Bruhat, Geroch, and
others, none of whom I have met, although I did visit Hilbert’s grave in Göttingen and heard
Lichnerowicz sing a very lengthy song at one of the Journées Relativistes, a conference series
on GR he founded), but my relationship to him was completely different. I saw him almost on a
daily basis between 1989–1997, when I was a postdoc in DAMTP in Cambridge, but I wasn’t in
his group and I never managed (or dared) to talked to him. I did mingle with his circle though
(my now ex-wife even rented her apartment from Stephen’s second wife, Elaine), and inhaled a
certain culture from this. Stephen was a scientific superstar at the time, but I feel that his best
work predated that kind of stardom, and is especially to be found in his book with Ellis (which
is largely based on his 1966 Adams Prize Essay) and his early work on black holes. It is only
after his death in 2018 that I really came to appreciate his genius and his extraordinary life.4
Thus this book in particular covers the causal theory developed by Hawking and Penrose
and others from the 1960s onwards, including the ensuing singularity theorems (though just the
simplest ones). A complete coverage of this theory is impossible in a work of this size, unde-
sirable for students looking for a first encounter, and unnecessary in view of the recent (Open
Access) encyclopedic treatment by Minguzzi (2019), on which I relied in several ways; though
less exhaustive, the older book by Kriele (1999) also remains useful. Similarly, a complete
description of the PDE approach to GR would require not only a different author, but also much
more space including preliminary material, so that we are fortunate to have Ringström (2009)
for those who want more than the first introduction given here, as well as Klainerman & Nicolò
(2003) and Christodoulou (2008). I have tried to do justice to the modern spirit of mathematical
relativity, which is characterized by a mix of the causal and the PDE theories and culminates
in the cosmic censorship and final state conjectures, described here in an introductory way. See
§1.9 for some brief historical comments on the development of mathematical GR.
4 See Jane Hawking, Music to Move the Stars: A Life With Stephen (MacMillan, 1999) for an unusually honest
account of this life; the later version Travelling to Infinity: My Life with Stephen (Alma Books, 2007) is softer.
iiiResearch on Hawking radiation (which, being a topic in quantum field theory in curved
space-time or in string theory, falls outside the scope of this book, which is purely classical)
requires intimate familiarity with the laws of black hole (thermo)dynamics, as well as with the
black hole uniqueness theorems and conjectures, and so these have been included also, as have
the basic exact black hole solutions (i.e. Schwarzschild, Reissner–Nordström, and Kerr), all at
an introductory level. It seems impossible to me to work on say the “information paradox” for
black holes without being familiar with all these ideas. Otherwise, the scope of the book is
indicated by its subtitle, and in more detail by its table of contents below.
As may be expected more from a work in the humanities than in mathematical physics
(between which the history and philosophy of physics resides), there are many footnotes, placed
where the name suggests. They contain additional information and credits, and may be skipped.
Finally, although this book is still deeply flawed, it would have been even worse if I had
not received help and feedback from a number of very kind students and colleagues, of whom I
would like to mention Jeroen van Dongen, Michel Janssen, Martin Lesourd, John Norton, Jan
Sbierski, . . .
The current version (July 31, 2020) is a draft that is both incomplete and uncorrected.
Sections on the laws of black hole mechanics, black hole uniqueness theorems, constraints and
Lie algebroids, initial value problem via the double null foliation, cosmic censorship, and final
state conjecture will be added later. Comments are welcome at landsman@math.ru.nl.
Please do not spread the file!
ivContents
1 Historical introduction 1
1.1 From physical principles to a mathematical framework . . . . . . . . . . . . . 3
1.2 Riemannian geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6
1.3 Absolute differential calculus and general covariance . . . . . . . . . . . . . . 8
1.4 Towards the gravitational field equations: Entwurf Theorie . . . . . . . . . . . 11
1.5 The Hole Argument . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 13
1.6 Finding the gravitational field equations: November 1915 . . . . . . . . . . . . 14
1.7 Hilbert . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17
1.8 Weyl . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
1.9 Mathematical foundations of GR: Towards the modern era . . . . . . . . . . . 22
1.10 Epilogue: General covariance and general relativity . . . . . . . . . . . . . . . 26
2 General differential geometry 29
2.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.2 Tangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30
2.3 Intermezzo: Dual vector spaces and tensor products . . . . . . . . . . . . . . . 35
2.4 Cotangent bundle . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37
2.5 Other tensor bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.6 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41
3 Metric differential geometry 42
3.1 (Semi) Riemannian metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42
3.2 Lowering and raising indices . . . . . . . . . . . . . . . . . . . . . . . . . . . 43
3.3 Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
3.4 Linear connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47
3.5 General connections on vector bundles . . . . . . . . . . . . . . . . . . . . . . 50
4 Curvature 53
4.1 Curvature tensor for general connections . . . . . . . . . . . . . . . . . . . . . 53
4.2 Riemann tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
4.3 Sectional curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
4.4 Sectional curvature and the Theorema Egregium of Gauß . . . . . . . . . . . . 58
4.5 Spaces of constant curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
4.6 Ricci tensor and Ricci scalar . . . . . . . . . . . . . . . . . . . . . . . . . . . 68
4.7 Submanifolds and hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . 70
4.8 Gauß–Weingarten and Gauß–Codazzi equations . . . . . . . . . . . . . . . . . 71
4.9 Fundamental theorem for hypersurfaces . . . . . . . . . . . . . . . . . . . . . 73
5 Geodesics and causal structure 77
5.1 Geodesic deviation and Jacobi fields . . . . . . . . . . . . . . . . . . . . . . . 77
5.2 The exponential map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
5.3 Basic causal structure in Lorentzian manifolds . . . . . . . . . . . . . . . . . . 83
5.4 Do geodesics extremize length? Local case . . . . . . . . . . . . . . . . . . . 89
5.5 Do geodesics extremize length? Global case . . . . . . . . . . . . . . . . . . . 92
5.6 Existence of geodesics of maximal length . . . . . . . . . . . . . . . . . . . . 94
5.7 Global hyperbolicity and Cauchy surfaces . . . . . . . . . . . . . . . . . . . . 101
v5.8 Global hyperbolicity: AdS as a counterexample . . . . . . . . . . . . . . . . . 109
6 The singularity theorems of Hawking and Penrose 111
6.1 Congruences of geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 113
6.2 Hawking’s singularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . 116
6.3 Null geometry and Penrose diagrams . . . . . . . . . . . . . . . . . . . . . . . 121
6.4 Null congruences and trapped surfaces . . . . . . . . . . . . . . . . . . . . . . 125
6.5 Penrose’s singularity theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
7 The Einstein equations 133
7.1 Integration on manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
7.2 Variation of the Einstein–Hilbert action . . . . . . . . . . . . . . . . . . . . . 136
7.3 The energy-momentum tensor . . . . . . . . . . . . . . . . . . . . . . . . . . 140
7.4 Electromagnetism: gauge invariance and constraints . . . . . . . . . . . . . . . 143
7.5 General relativity: diffeomorphism invariance and constraints . . . . . . . . . . 145
7.6 Existence, uniqueness, and maximality of solutions . . . . . . . . . . . . . . . 149
8 The 3 + 1 split of space-time 154
8.1 Geometric form of the constraints . . . . . . . . . . . . . . . . . . . . . . . . 154
8.2 Lapse and shift . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156
8.3 Beyond Gauß-Codazzi . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
8.4 The 3+1 decomposition of the Einstein equations . . . . . . . . . . . . . . . . 162
8.5 Static and asymptotically flat space-times . . . . . . . . . . . . . . . . . . . . 166
8.6 The origin of diffeomorphism invariance? . . . . . . . . . . . . . . . . . . . . 169
8.7 Hamiltonian formulation of general relativity . . . . . . . . . . . . . . . . . . 175
8.8 Conformal analysis of the constraints: Lichnerowicz equation . . . . . . . . . . 181
9 Black holes 183
9.1 De Sitter space revisited . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183
9.2 The Schwarzschild solution and its geodesics . . . . . . . . . . . . . . . . . . 186
9.3 Extensions of Schwarzschild space-time . . . . . . . . . . . . . . . . . . . . . 191
9.4 The Reissner–Nordström solution . . . . . . . . . . . . . . . . . . . . . . . . 200
9.5 Kerr space-time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206
9.6 Abstract properties of black holes . . . . . . . . . . . . . . . . . . . . . . . . 212
A Lie groups, Lie algebras, and constant curvature 213
A.1 Lie groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 213
A.2 Lie algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 215
A.3 Homogeneous manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 217
A.4 Homogeneous Riemannian and Lorentzian manifolds . . . . . . . . . . . . . . 222
A.5 Symmetric spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
A.6 Proof of the classification of spaces with constant curvature . . . . . . . . . . 224
B Background from formal PDE theory 228
B.1 Distributions and Sobolev spaces on manifolds . . . . . . . . . . . . . . . . . 228
B.2 Linear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 232
B.3 Quasi-linear wave equations . . . . . . . . . . . . . . . . . . . . . . . . . . . 236
viLiterature 239
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