Table Of ContentTopics in the Foundations of General Relativity
and Newtonian Gravitation Theory
(version 4, August 17, 2011)
David B. Malament
Department of Logic and Philosophy of Science
University of California, Irvine
[email protected]
http://www.lps.uci.edu/lps bios/dmalamen
i
To Pen
Contents
Preface iv
1 Differential Geometry 1
1.1 Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Tangent Vectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.3 Vector Fields, Integral Curves, and Flows . . . . . . . . . . . . . . . . . . . . . . . . . . . 16
1.4 Tensors and Tensor Fields on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 22
1.5 The Action of Smooth Maps on Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.6 Lie Derivatives . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
1.7 Derivative Operators and Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44
1.8 Curvature . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 60
1.9 Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65
1.10 Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82
1.11 Volume Elements . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 97
2 Classical Relativity Theory 103
2.1 Relativistic Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103
2.2 Temporal Orientation and “Causal Connectibility” . . . . . . . . . . . . . . . . . . . . . . 111
2.3 Proper Time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118
2.4 Space/Time Decomposition at a Point and Particle Dynamics . . . . . . . . . . . . . . . . 122
2.5 The Energy-Momentum Field T . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 125
ab
2.6 Electromagnetic Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 133
2.7 Einstein’s Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 139
2.8 Fluid Flow . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 146
ii
CONTENTS iii
2.9 Killing Fields and Conserved Quantities . . . . . . . . . . . . . . . . . . . . . . . . . . . . 153
2.10 The Initial Value Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 158
2.11 Friedmann Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
3 Special Topics 170
3.1 G¨odel Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
3.2 Two Criteria of Orbital (Non-)Rotation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191
3.3 A No-Go Theorem about Orbital (Non-)Rotation . . . . . . . . . . . . . . . . . . . . . . . 206
4 Newtonian Gravitation Theory 215
4.1 Classical Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 216
4.2 Geometrized Newtonian Theory — First Version . . . . . . . . . . . . . . . . . . . . . . . 231
4.3 Interpreting the Curvature Conditions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 241
4.4 A Solution to an Old Problem about Newtonian Cosmology . . . . . . . . . . . . . . . . . 249
4.5 Geometrized Newtonian Theory — Second Version . . . . . . . . . . . . . . . . . . . . . . 256
Solutions to Problems 265
Bibliography 292
Preface
Thismanuscriptbeganlifeasasetoflecturenotesforatwo-quarter(20week)courseonthefoundations
of general relativity that I taught at the University of Chicago many years ago. I have repeated the
course quite a few times since then, both there and at the University of California,Irvine, and have over
the years steadily revised the notes and added new material. Maybe now the notes can stand on their
own.
Thecoursewasneverintendedtobeasystematicsurveyofgeneralrelativity. Therearemanystandard
topics that I do not discuss, e.g., the Schwarzschild solution and the “classic tests” of general relativity.
(And I have always recommended that students who have not already taken a more standard course in
thesubjectdosomeadditionalreadingontheirown.) Mygoalsinsteadhavebeento(i)presentthebasic
logical-mathematical structure of the theory with some care, and (ii) consider additional special topics
that seem to me, at least, of particular interest. The topics have varied from year to year, and not all
have found their way into these notes. I will mention in advance three that did.
Thefirstis“geometrizedNewtoniangravitationtheory”,alsoknownas“Newton-Cartantheory”. Itis
nowwellknownthatonecan,afterthefact,reformulateNewtoniangravitationtheorysothatitexhibits
manyofthequalitativefeaturesthatwereoncethoughttobeuniquelycharacteristicofgeneralrelativity.
On reformulation, Newtonian theory too provides an account of four-dimensional spacetime structure in
which (i) gravity emerges as a manifestation of spacetime curvature, and (ii) spacetime structure itself
is “dynamical” in the sense that it participates in the unfolding of physics rather than being a fixed
backdrop against which it unfolds. It has always seemed to me helpful to consider general relativity
and this geometrized reformulation of Newtonian theory side by side. For one thing, one derives a sense
of where Einstein’s equation “comes from”. When one reformulates the empty-space field equation of
Newtonian gravitation theory (i.e., Laplace’s equation 2φ= 0, where φ is the gravitational potential),
∇
one arrives at a constraint on the curvature of spacetime, namely R = 0. The latter is, of course,
ab
just what we otherwise know as (the empty-space version of) Einstein’s equation. And, reciprocally,
this comparison of the two theories side by side provides a certain insight into Newtonian physics. For
example, it yields a satisfying solution (or dissolution) to an old problem about Newtonian cosmology.
Newtonian theoryin a standardtextbook formulationseems to provideno sensible prescriptionfor what
the gravitational field should be like in the presence of a uniform mass-distribution filling all of space.
iv
PREFACE v
(See section 4.4.) But the problem is really just an artifact of the formulation, and it disappears when
one passes to the geometrized version of the theory.
The basic idea of geometrized Newtonian gravitation theory is simple enough. But there are compli-
cations, and I deal with some of them in the present expanded form of the lecture notes. In particular, I
presenttwodifferentversionsofthe theory– whatI callthe “Trautmanversion”andthe “Ku¨nzle-Ehlers
version” – and consider their relation to one another. I also discuss in some detail the geometric signifi-
cance ofvariousconditions onthe RiemanncurvaturefieldRa thatenter intothe formulationofthese
bcd
versions.
A secondspecialtopicthat I consideristhe conceptof“rotation”. It turnsoutto be a ratherdelicate
andinterestingquestion,atleastinsomecases,justwhatitmeans tosaythatabodyisorisnotrotating
within the framework of general relativity. Moreover, the reasons for this — at least the ones I have in
mind — do not have much to do with traditional controversy over “absolute vs. relative (or Machian)”
conceptionsofmotion. Ratherthey concernparticulargeometriccomplexitiesthatarisewhenoneallows
for the possibility of spacetime curvature. The relevant distinction for my purposes is not that between
attributions of “relative” and “absolute” rotation, but rather that between attributions of rotation that
can and cannot be analyzed in terms of motion (in the limit) at a point. It is the latter — ones that
make essential reference to extended regions of spacetime — that can be problematic.
The problemhastwoparts. First,onecaneasilythinkofdifferentcriteriaforwhenanextendedbody
is rotating. (I discuss two examples in section 3.2.) These criteria agree if the background spacetime
structure is sufficiently simple, e.g., if one is working in Minkowski spacetime. But they do not agree in
general. So, at the very least, attributions of rotation in general relativity can be ambiguous. A body
canberotatinginoneperfectlynaturalsensebutnotrotatinginanother,equallynatural,sense. Second,
circumstances can arise in which the different criteria – all of them – lead to determinations of rotation
and non-rotation that seem wildly counterintuitive. (See section 3.3.) The upshot of this discussion is
notthatwecannotcontinuetotalkaboutrotationinthecontextofgeneralrelativity. Notatall. Rather,
we simply have to appreciate that it is a subtle and ambiguous notion that does not, in all cases, fully
answer to our classical intuitions.
A third special topic that I consider is G¨odel spacetime. It is not a live candidate for describing
our universe, but it is of interest because of what it tells us about the possibilities allowed by general
relativity. Itrepresentsapossibleuniversewithremarkableproperties. Foronething,theentirematerial
contentoftheG¨odeluniverseisinastateofuniform,rigidrotation(accordingtoanyreasonablecriterion
of rotation). For another, lightraysand free test particlesin it exhibit a kind of boomerangeffect. Most
striking of all, it admits closed timelike curves that cannot be “unrolled” by passing to a covering space
(because the underlying manifold is simply connected). In section 3.1, I review these basic features of
G¨odel spacetime and, in an appendix to that section, discuss how one can go back and forth between an
intrinsic characterizationof the G¨odel metric and two different coordinate expressions for it.
PREFACE vi
These three specialtopics are treated in chapters 3 and 4. Much of this materialhas been added over
the years. The originalcore of the lecture notes — the review of the basic structure of general relativity
— is to be found in chapter 2.
Chapter 1 offers a preparatory review of basic differential geometry. It has never been my practice to
work through all this material in class. I have limited myself there to “highlights” and general remarks.
ButIhavealwaysdistributedthenotessothatstudentswithsufficientinterestcandofurtherreadingon
their own. On occasion,I have also runa separate “problemsession”and used it for additionalcoaching
on differential geometry. (A number of problems, with solutions, are included in the present version of
the notes.) I suggest that readers make use of chapter 1 as seems best to them — as a text to be read
from the beginning, as a reference workto be consulted when particular topics arise in later chapters,as
something in between, or not at all.
I would like to use this occasion to thank a number of people who have helped me over the years to
learn and better understand general relativity. I could produce a long list, but the ones who come first,
at least, are John Earman, David Garfinkle, Robert Geroch, Clark Glymour, Howard Stein, and Robert
Wald. I am particularly grateful to Bob and Bob for allowing this interloper from the Philosophy
1 2
Department to find a second home in the Chicago Relativity Group. Anyone familiar with their work,
both research and expository writings, will recognize their influence on this set of lecture notes.
Erik Curiel, Sam Fletcher, David Garfinkle, John Manchak, and Jim Weatherall have my thanks, as
well, for the comments and corrections they have given me on earlier drafts of the manuscript.
Matthias Kretschmannwas goodenoughsome years agoto take my handwritten notes on differential
geometry and set them in TEX. I took over after that, but I might not have started without his push.
Finally, Pen Maddy has helped me to believe that this project was worth completing. I shall always
be grateful to her for her support and encouragement.
Chapter 1
Differential Geometry
1.1 Manifolds
We assume familiarity with the basic elements of multivariable calculus and point set topology. The
following notions, in particular, should be familiar.
Rn (forn 1)isthe setofalln-tuples ofrealnumbers x=(x1,...,xn). The Euclideaninner product
≥
(or “dot product”) on Rn is given by x y = x1y1+...+xnyn. It determines a norm x = √x x.
· k k ·
Given a point x Rn and a real number ǫ>0, B (x) is the open ball in Rn centered at x with radius
ǫ
∈
ǫ, i.e., B (x)= y : y x <ǫ . Clearly, x belongs to B (x) for every ǫ>0. A subset S of Rn is open
ǫ ǫ
{ k − k }
if, for allpoints x in S, there is an ǫ>0 suchthat B (x) S. This determines a topology onRn. Given
ǫ
⊆
m,n 1, and a map f:O Rm froman openset O in Rn to Rm, f is smooth (or C∞) if all its mixed
≥ →
partial derivatives (to all orders) exist and are continuous at every point in O.
Asmoothn-dimensionalmanifold(n 1)canbethoughtofasapointsettowhichhasbeenaddedthe
≥
“local smoothness structure” of Rn. Our discussion of differential geometry begins with a more precise
characterization.1
LetM beanon-emptyset. Ann-chartonM isapair(U,ϕ)whereU isasubsetofM and ϕ:U Rn
→
isaninjective(i.e.,one-to-one)mapfromUintoRn withthepropertythatϕ[U] isanopensubsetofRn.
(Hereϕ[U]istheimageset ϕ(p): p U .) Charts,alsocalled“coordinatepatches”,arethemechanism
{ ∈ }
with which one induces local smoothness structure on the set M. To obtain a smooth n-dimensional
manifold, we must lay down sufficiently many n-charts on M to coverthe set, and require that they are,
in an appropriate sense, compatible with one another.
Let (U ,ϕ ) and (U ,ϕ ) be n-charts on M. We say the two are compatible if either the intersection
1 1 2 2
set U =U U is empty, or the following conditions hold:
1 2
∩
1Inthissectionandseveralothers inchapter 1,wefollowthe basiclinesofthepresentation inGeroch[22].
1
CHAPTER 1. DIFFERENTIAL GEOMETRY 2
(1) ϕ [U] and ϕ [U] are both open subsets of Rn;
1 2
(2) ϕ ϕ−1:ϕ [U] Rn and ϕ ϕ−1:ϕ [U] Rn are both smooth.
1◦ 2 2 → 2◦ 1 1 →
(Notice that the second makes sense since ϕ [U] and ϕ [U] are open subsets of Rn and we know what it
1 2
means to say that a map from an open subset of Rn to Rn is smooth. See figure 1.1.1.)
The relation of compatibility between n-charts on a given set is reflexive and symmetric. But it need
not be transitive and, hence, not an equivalence relation. For example, consider the following three
1 charts on R:
−
C =(U ,ϕ ) with U =( 1,1) and ϕ (x)=x
1 1 1 1 1
−
C =(U ,ϕ ) with U = (0,1) and ϕ (x)=x
2 2 2 2 2
C =(U ,ϕ ) with U =( 1,1) and ϕ (x)=x3.
3 3 3 3 3
−
PairsC andC arecompatible,andsoarepairsC andC . ButC andC arenotcompatible,because
1 2 2 3 1 3
the map ϕ ϕ−1:( 1,+1) R is not smooth (or even just differentiable) at x=0.
1◦ 3 − →
Rn
M ϕ
1
U1 ϕ1[U1]
ϕ
2
ϕ [U ]
U 2 2
2
Figure 1.1.1: Two n-charts (U ,ϕ ) and (U ,ϕ ) on M with overlapping domains.
1 1 2 2
We now define a smooth n-dimensional manifold (or, in brief, an n-manifold) (n 1) to be a pair
≥
(M, ) where M is a non-empty set, and is a set of n-charts on M satisfying the following four
C C
conditions.
(M1) Any two n-charts in are compatible.
C
(M2) The (domains of the) n-chartsin coverM, i.e., for every p M, there is an n-chart(U,ϕ) in
C ∈ C
such that p U.
∈
(M3) (Hausdorff condition) Given distinct points p and p in M, there exist n-charts (U ,ϕ ) and
1 2 1 1
(U ,ϕ ) in such that p U for i=1,2, and U U is empty.
2 2 i i 1 2
C ∈ ∩
(M4) is maximal in the sense that any n-chart on M that is compatible with every n-chart in
C C
belongs to .
C
CHAPTER 1. DIFFERENTIAL GEOMETRY 3
(M1) and(M2) are certainly conditions one would expect. (M3) is included, following standardpractice,
simply to rule out pathological examples (though one does, sometimes, encounter discussions of “non-
Hausdorff manifolds”). (M4) builds in the requirement that manifolds don’t have “extra structure” in
the form of distinguished n-charts. (For example, we can think of the point set Rn as carrying a single
(global) n-chart. In the transition from the point set Rn to the n-manifold Rn discussed below, this
“extra structure” is washed out.)
Because of (M4), it might seem a difficult task to specify an n-dimensional manifold. (How is one to
getagriponallthedifferentn-chartsthatmakeupamaximalsetofsuch?) Butthefollowingproposition
shows that the specification need not be difficult. It suffices to come up with a set of n-charts on the
underlying set satisfying (M1), (M2), (M3), and then simply throw in wholesale all other compatible
n-charts.
Proposition 1.1.1. Let M be a non-empty set, let be a set of n-charts on M satisfying conditions
0
C
(M1),(M2),(M3), and let be the set of all n-charts on M compatible with all the n-charts in . Then
0
C C
(M, ) is an n-manifold, i.e., satisfies all four conditions.
C C
Proof. Since satisfies(M1), isasubsetof . Itfollowsimmediatelythat satisfies(M2),(M3),and
0 0
C C C C
(M4). Only (M1) requires some argument. Let C =(U ,ϕ ) and C =(U ,ϕ ) be any two n-charts
1 1 1 2 2 2
compatiblewithalln-chartsin . We showthatthey arecompatiblewithoneanother. We mayassume
0
C
that the intersection U U is non-empty, since otherwise compatibility is automatic.
1 2
∩
First weshow that ϕ [U U ] isopen. (Aparallel argument establishesthat ϕ [U U ] isopen.)
1 1 2 2 1 2
∩ ∩
Consideranarbitrarypointof ϕ [U U ]. Itis ofthe formϕ (p)forsome point p U U . Since
1 1 2 1 1 2 0
∩ ∈ ∩ C
satisfies (M2), there exists an n-chart C =(U,ϕ) in whose domain contains p. So p U U U .
0 1 2
C ∈ ∩ ∩
Since C is compatible with both C and C , ϕ[U U ] and ϕ[U U ] are open sets in Rn, and the
1 2 1 2
∩ ∩
maps
ϕ ϕ−1:ϕ[U U ] Rn, ϕ ϕ−1:ϕ[U U ] Rn,
1 1 2 2
◦ ∩ → ◦ ∩ →
ϕ ϕ−1:ϕ [U U ] Rn, ϕ ϕ−1:ϕ [U U ] Rn,
◦ 1 1 ∩ 1 → ◦ 2 2 ∩ 2 →
are all smooth (and so continuous). Now ϕ[U U U ] is open, since it is the intersection of open
1 2
∩ ∩
sets ϕ[U U ] and ϕ[U U ]. (Here we use the fact that ϕ is injective.) So ϕ [U U U ] is
1 2 1 1 2
∩ ∩ ∩ ∩
open, since it is the pre-image of ϕ[U U U ] under the continuous map ϕ ϕ−1. But, clearly,
∩ 1 ∩ 2 ◦ 1
ϕ (p) ϕ [U U U ], and ϕ [U U U ] is a subset of ϕ [U U ]. So we see that our arbitrary
1 1 1 2 1 1 2 1 1 2
∈ ∩ ∩ ∩ ∩ ∩
point ϕ (p) in ϕ [U U ] is containedin an open subsetof ϕ [U U ]. Thus ϕ [U U ] is open.
1 1 1 2 1 1 2 1 1 2
∩ ∩ ∩
Next we showthatthe map ϕ ϕ−1:ϕ [U U ] Rn is smooth. (Aparallelargumentestablishes
2◦ 1 1 1∩ 2 →
that ϕ ϕ−1:ϕ [U U ] Rn is smooth.) For this it suffices to show that, given our arbitrarypoint
1◦ 2 2 1∩ 2 →
ϕ (p) in ϕ [U U ], the restrictionof ϕ ϕ−1 to some open subset of ϕ [U U ] containing ϕ (p)
1 1 1∩ 2 2◦ 1 1 1∩ 2 1
is smooth. But this follows easily. We know that ϕ [U U U ] is an open subset of ϕ [U U ]
1 1 2 1 1 2
∩ ∩ ∩
containing ϕ (p). And the restriction of ϕ ϕ−1 to ϕ [U U U ] is smooth, since it can be
1 2 ◦ 1 1 ∩ 1 ∩ 2
