S. Lu ˇci ´c
1182056Fibre Bundles in General Relativity
Bachelor’s thesis
Supervised by
Dr. R.I. van der Veen, Dr. J.W. Dalhuisen &
Prof. dr. D. Bouwmeester
July 11, 2016
Mathematical Institute Leiden Institute of Physics
Abstract
Contents
Introduction 1
Notations and Conventions 3
1 Smooth Manifolds 5
1.1 Smooth manifolds . . . 5
1.2 Smooth maps . . . 8
1.2.1 Lie groups . . . 9
1.3 The tangent bundle . . . 9
1.4 The tensor product . . . 12
1.5 Minkowski space andSL(2,C). . . 15
2 Fibre Bundles 19 2.1 Fibre bundles . . . 19
2.1.1 Transition functions . . . 21
2.2 Principal bundles . . . 22
2.2.1 The frame bundle . . . 26
2.3 Associated fibre bundles . . . 27
2.4 Altering the structure group . . . 29
2.4.1 Spin structure . . . 31
2.5 Einstein’s field equations . . . 32
Conclusions 37
Acknowledgments 39
Introduction
In Einstein’s theory of general relativity, the mathematical model of our universe is aspacetime manifoldM, defined as a 4-dimensional smooth manifold which is con-nected, non-compact and space- and time-oriented. Moreover, it has the property that the tangent space at each point of the manifold is isomorphic to Minkowski space, so the metric is represented by the matrixη= diag(1,−1,−1,−1). This is
(more or less) the mathematical expression of Einstein’s postulates that “physics is locally governed by special relativity”, and that gravity is a manifestation of the geometry ofM, more specifically the curvature, which is in turn influenced by the matter which is present in the universe. To quote John A. Wheeler: “Spacetime tells matter how to move; matter tells spacetime how to curve”.
Consider an eventPin spacetime, which is just a pointpinM, and an observer A, which is equipped with a local frame, i.e. a basis for the tangent space at each
point in a some neighbourhood ofp. Suppose for the moment that there is no
gravity, thenMcan be identified with Minkowski space and the observerAcan
actually be equipped with a global frame, i.e. with a basis at each point inM. It is an axiom of physics that any (meaningful) physical theory should be Lorentz covariant, meaning that the equations whichA writes down should be of the
same form for any otheradmissible observer, whose frame is connected to the frame ofAby some restricted Lorentz transformation, i.e. some element of the
restricted Lorentz groupSO↑(1,3). Since we consider all admissible observers,
and since each admissible observer’s frame is connected to that ofAby a unique
Lorentz transformation, we can equivalently say (that is, we have an isomorphism {Admissible observers atp} ∼= SO↑(1,3), but this isomorphism depends on the
chosen observerA) that we consider the whole Lorentz group. We do this at each
will see that this notion generalises and formalises much of what we already know, using the language of fibre bundles and principal bundles. It is however not, simply a generalisation for generalisation’s sake. Much of the standard model, which incorporates the weak, strong, and electromagnetic interaction, is formulated using this framework. But the main reason for studying these objects in relation to general relativity is because using spinors to reformulate problems in general relativity has turned out be very useful. Spinors were first introduced by Paul A.M. Dirac and Wolfgang E. Pauli in quantum mechanics when studying the electron. It was Roger Penrose who primarily introduced and advocated the use of spinors in general relativity [1,2], and two notable results which are still used today are thespin-coefficient formalismsintroduced by Roger Penrose, Ezra T. Newman and Robert Geroch [3,4].
However, there is a natural question which one might ask: under what cir-cumstances spinors can be defined properly on a manifold? To even be able to address this question, one has to properly set up and define the aforementioned language of fibre bundles and principal bundles, and this is what this thesis will be concerned with. We start by developing some manifold theory, and show the relation betweenSL(2,C)and the Lorentz group, where the former comes into
play since it is the group under which spinors transform. After this we will develop some theory on fibre bundles, which allows us to properly define what a spin structure is, which is necessary to have spinors, and we will mention the results on the existence of spin structures on a non-compact manifold. Lastly, we will define what a connection on the tangent bundle of a manifold is, which will enable us to write down the Einstein field equation locally.
Notations and Conventions
The natural numbers are defined asN:= Z≥0, and for anyn∈N≥1, we define
[n] :={1, . . . , n}. For a mapf :A1×. . .×An−→A, whereA1, . . . , AnandA are sets andn∈N≥1, we writef(a1, . . . , an)instead off((a1, . . . , an)), for all (a1, . . . , an)∈A1×. . .×An. Fori∈[n], we defineProji :A1×. . .×An−→Ai, (a1, . . . , an)7−→ ai. A topological space(X,T)is denoted byX, and any non-empty subsetU ofXis equipped with the subspace topology, unless otherwise
stated. A group(G,·, e)is denoted byGand we writeghforg·h, for allg, h∈G.
A right (left) group action of a groupGon a setXis referred to as a right (left)
action ofGon X. Forn, m ∈ N, a mapf : U → V between open subsets U
andV ofRnandRm, respectively, is said to besmoothif it is of classC∞. For K ∈ {R,C}, we defineK∗ :=K\ {0}, and forn∈N≥1, we defineMat(n, K)
to be the set of alln×nmatrices overK, andIn is then×nidentity matrix. The subsetGL(n, K)is the group of all invertiblen×nmatrices overK, and
H(2,C) ={H∈GL(n,C)|H =H†}is the set of all Hermitian matrices, where A†denotes the conjugate transpose ofA∈GL(n,C). Anyg∈GL(n, K)and its
inverseg−1will be written as
g=
g11 · · · g1n
... ...
gn1 · · · gnn
, g
−1 =
g11 · · · gn1
... ...
g1n · · · gnn
,
so thatgij denotes the(i, j)-th entry ofgandgji denotes the(i, j)-th entry of g−1. Throughout this thesis, we will employ theEinstein summation convention,
meaning that in an expression of the formλiei, there is implied a summation over the indexi, whose range will be clear from the context and will usually be the
dimension of the space under consideration. Finally, the Hermitian matrices
σ1=
0 1 1 0
, σ2=
0 −i
i 0
, σ3 =
1 0
0 −1
(1)
Chapter 1
Smooth Manifolds
In this chapter we will introduce the category of smooth manifolds, whose objects (the smooth manifolds) and morphisms (the smooth maps between them) will play an important role throughout this thesis. It provides the natural setting for Einsteins’s theory of general relativity which models spacetime as a 4-dimensional smooth manifold, and underlines the departure from the Newtonian description of gravity as a force in Euclidean space, to Einstein’s description of gravity as a property of spacetime. Furthermore, we will mention some basic properties of the tensor product, and we will discuss Minkowski space and the relation between the restricted Lorentz transformations and the groupSL(2,C), for which we will
borrow some theory on Lie groups.
1.1
Smooth manifolds
Intuitively, a manifold is a space which locally looks ordinary Euclidean space. An example is the earth, which is (ignoring the flattening at the poles) a sphere, and locally looks like a plane. The following definitions will make this precise. Let
n∈N.
Definition 1.1. LetM be a topological space. Ann-dimensional chart forM
is a pair (U, ϕ), where U ⊂ M is open and ϕis a homeomorphism onto an
open subsetϕ(U)ofRn. The (continuous) mapxi:= Proji◦ϕis called thei-th
coordinate function ofϕ, for eachi∈[n], and we refer to the mapsx1, . . . , xnas local coordinates onU.
Definition 1.2. A topological spaceM islocallyn-Euclideanif for eachm∈M
there exists ann-dimensional chart(Um, ϕm)forMwithm∈Um.
Remarks. LetM be a locallyn-Euclidean topological space, letm∈M, and let
is ann-dimensional chart forMatm. For any othern-dimensional chart( ˜Um,ϕ˜m) forM atm, the map
(ϕm◦ϕ˜−m1)|ϕ˜(Um∩U˜m): ˜ϕ(Um∩ ˜
Um)−→ϕ(Um∩U˜m) (1.1) is a homeomorphism between two open subsets ofRn, and is called anoverlap function.
Definition 1.3. LetM be a topological space. Ann-dimensional topological atlas forM is a setA = {(Ui, ϕi)|i ∈ I}, whereI is some indexing set, such that (Ui, ϕi)is ann-dimensional chart forM for eachi∈I, andM =Si∈IUi.
Definition 1.4. A pair (M,A) is an n-dimensional topological manifold ifM
is a topological space which is Hausdorff and second countable, andA is an
n-dimensional topological atlas forM.
Definition 1.5. LetMbe a topological space. Asmoothn-dimensional atlasAfor
Mis ann-dimensional topological atlasAforMsuch that all overlap functions
are smooth. Ann-dimensional chart(U, ϕ)forM isadmissible to a smoothn
-dimensional atlasAforM ifA ∪ {(U, ϕ)}is a smoothn-dimensional atlas for M, andAismaximalif there are non-dimensional charts(U, ϕ)6∈ Awhich are admissible toA. Asmooth structure onM is a maximal smoothn-dimensional
atlasAforM.
It is easy to see that a smoothn-dimensional atlasAfor a topological spaceM
determines a unique maximal smoothn-dimensional atlasMforM; for a proof,
see Proposition 1.17 in [7]. However, twon-dimensional atlasesAandA0forM
need not besmoothly compatible, i.e. there can exist a chart(U, ϕ)∈ Asuch that (U, ϕ)is not admissible toA0. If this is the case, thenAandA0define two different smooth structuresMandM0 onM, and the resulting smoothn-dimensional
manifolds(M,M)and(M,M0)may or may not be “the same”, i.e. there may or
may not exist a diffeomorphism (see Definition 1.9) between them.
This brings up the question of how many “inequivalent” smooth structures can be defined on ann-dimensional topological manifoldM, which has been addressed
by, among others, Simon K. Donaldson, Michael H. Freedman and John W. Milnor (see the discussion on page 40 of [7] and the references mentioned there). In this thesis we will not be concerned with this question, but it is worth mentioning the result by Donaldson on the so-calledfakeR4’s, which states that there is an
uncountable set of4-dimensional smooth manifolds which are all homeomorphic toR4, but pairwise not diffeomorphic to each other1. This result supports the claim
“dimension four is different”, and while it may seem rather far-fetched to look for
1Incidentally, it is nice to note that key ideas in some of the proofs of these and other related
1.1. Smooth manifolds
something physically significant in constructions of this kind, there has been an interest in how these concepts could be used to gain a better understanding of gravity [9–12].
Definition 1.6. A pair(M,M)is ann-dimensional smooth manifold ifM is a
topological space which is Hausdorff and second countable, andMis a maximal smoothn-dimensional atlas forM. They are the objects of the categoryMan∞ of smooth manifolds.
Henceforth, we will refer to ann-dimensional smooth manifold(M,M)as a smooth manifoldM and to ann-dimensional chart forM as a chart forM.
If we say that something holds for each chart forM, we mean that it holds for
each(U, ϕ)∈ M, whereMis the smooth structure onM. From our definition it
follows that every smooth manifold has the property of beingparacompact; see Theorem 1.15 in [7] for a proof.
Examples 1.7. We list some examples of smooth manifolds which we will need later on.
1. The pair(Rn,MRn), whereMRn is thestandard smooth structure onRn
defined byARn ={Rn,I
Rn}, is ann-dimensional smooth manifold.
Iden-tifyingMat(n,R)withRn2, we see thatMat(n,R)is ann2-dimensional
smooth manifold.
2. Then-dimensional sphere(Sn,MSn)is ann-dimensional smooth manifold,
conform Example 1.31 in [7]; this smooth structure onSn is called the standard smooth structure onSn.
3. For any two smooth manifoldsM andM0, the productM×M0(equipped
with the product topology) is clearly an(n+n0)-smooth manifold, whose
smooth structure is defined2by the smooth structures onM andM0.
4. Any non-empty open subsetUof a smooth manifoldMis a smooth manifold
of the same dimension asM, whose smooth structure is the restriction3to U of the smooth structure onM.
5. Ann-dimensional real vector spaceV is ann-dimensional smooth manifold,
conform Example 1.24 in [7].
6. The general linear group GL(n,R) = det−1(R∗) is an open subset of
Mat(n,R)since the determinant function is continuous, soGL(n,R) is
ann2-dimensional smooth manifold. 2IfMandM0are the smooth structures on
M andM0, respectively, then
A×:={(UM ×UM0, ϕM×ϕM0)|(UM, ϕM)∈ AM∧(UM0, ϕM0)∈ AM0}
is an(n+n0)-dimensional smooth atlas forM×M0which defines the smooth structure onM×M0. 3IfM={(U
i, ϕi)|i∈I}is the smooth structure onM, then the restriction ofMtoUis the
1.2
Smooth maps
Now that we know what a manifold is, we want to know if and how we can generalise the concept of a smooth function defined on Euclidean space to a smooth function on a manifold. LetM andM0be smooth manifolds.
Definition 1.8. Letk ∈ N. A map f : M −→ Rkis smoothif for each chart
(U, ϕ)forM the mapf ◦ϕ−1 :ϕ(U) −→ Rkis smooth. The set of all smooth functions fromM toRis denoted byC∞(M), and for any non-empty open subset U ofM, the set of all smooth functions fromU toRis denoted byC∞(M|U).
Remark. The setC∞(M)is naturally a real vector space and a commutative ring,
where the constant map1:M −→R,m−→1is the identity.
Definition 1.9. A continuous mapf :M −→M0 issmoothif the map
(ϕ0◦f◦ϕ−1)|ϕ(U∩f−1(U0)):ϕ(U∩f−1(U0))−→ϕ˜(U0) (1.2)
is smooth for each chart (U, ϕ) for M and for each chart (U0, ϕ0) forM0. A diffeomorphismis a smooth bijective mapf :M −→M0 such thatf−1is smooth,
andM andM0 are calleddiffeomorphicif there exists a diffeomorphism between
them.
Remark. The identity mapIM :M −→M is clearly smooth.
Proposition 1.10. LetM00 be a smooth manifold, and letf : M −→ M0 and g :M0 −→ M00be smooth maps. Then the compositiong◦f :M −→ M00 is
smooth.
Proof. See Proposition 2.10 in [7].
The smooth maps are the morphisms inMan∞, and by the previous remark and proposition, this indeed defines a category.
Definition 1.11. LetU ={Ui|i∈I}be an open cover ofM. Asmooth partition
of unity subordinate toU is a setPU ={pi|i∈I}, where
• eachpi ∈ PU is a smooth map pi : Ui −→ Rsuch that0 ≤ pi(m) ≤ 1 holds for allm∈M,
• for alli ∈ I it holds thatsupp(pi) := {m∈Ui|pi(m)6= 0} ⊂ Ui, and {supp(pi)|i∈I}is locally finite, i.e. for eachm∈M there exist and open subsetU ofM withm∈U such thatU has non-empty intersection with
only finitely many elements of{supp(pi)|i∈I}, and • P
i∈Ipi(m) = 1holds for allm∈M.
Theorem 1.12. For any open coverUofM there exists a smooth partition of unity
PU subordinate toU.
1.3. The tangent bundle
1.2.1 Lie groups
Lie groups come up often in physics, as they are groupsandmanifolds, and can thus properly represent the smooth symmetries so important in physics.
Definition 1.13. A groupGis aLie groupifGis a smooth manifold such that the
multiplicationG×G−→ G,(g, h) 7−→ghand inversionG−→ G,g7−→ g−1
onGare smooth.
Definition 1.14. LetGandG0 be Lie groups. ALie group homomorphismis a
group homomorphismλ:G−→G0which is also smooth.
Examples 1.15. We list some examples of Lie groups which we will encounter later on.
1. The matrix groupGL(n,R)is a Lie group, since matrix multiplication and
inversion (by Cramer’s rule) are both smooth.
2. The circleS1, viewed as a subgroup ofC∗, is a compact Lie group called the
circle group. We will also denote it byU(1).
3. The group of orthogonal matricesO(n,R) = det−1({−1,1})is a closed
subgroup ofGL(n,R)of dimension12n(n−1), as is the indentity component
SO(n,R) = det−1(1). By the closed subgroup theorem (Theorem 20.12
in [7]), these groups are both Lie groups.
4. The special linear groupSL(2,C) ={A∈Mat(2,C)|det(A) = 1}in two
dimensions is a simply connected Lie group of dimension6.
1.3
The tangent bundle
LetM be a smooth manifold, and letm∈M. We assume thatdim(M)∈N≥1.
Definition 1.16. Atangent vector atm is an elementv ∈ HomR(C∞(M),
R)
such thatv(f g) =g(p)v(f) +f(p)v(g)holds for allf, g∈ C∞(M). Thetangent space ofMatmis the real vector space of all tangent vectors atm, and is denoted
byTmM.
Let(U, ϕ)be a chart forM atm. Define for eachi∈[n]the map
∂i|m :C∞(M)−→R
f 7−→Di(f◦ϕ−1)(ϕ(m)),
(1.3)
whereDi(f◦ϕ−1)(ϕ(m))is thei-th partial derivative. By the chain rule this map is a tangent vector atm, for eachi ∈ [n]. As the following proposition shows,
and as makes sense intuitively, the tangent space isn-dimensional and is spanned
Proposition 1.17. The set{∂1|m, . . . , ∂n|m} is basis for TmM, so TmM is of dimensionn.
Proof. See Proposition 3.15 in [7].
Definition 1.18. The cotangent space of M at m is the dual space of TmM, and is denoted byTm∗M. For any chart (U, ϕ) forM atm, the basis dual to
{∂1|m, . . . , ∂n|m}is denoted by{dx1|m, . . . , dxn|m}.
Definition 1.19. Thetangent bundle ofM is the disjoint union T M := a
m∈M
TmM, (1.4)
and thecotangent bundle ofM is the disjoint union
T∗M := a m∈M
Tm∗M (1.5)
There are naturalprojectionsπt:T M −→Mandπc:T∗M −→M.
Note that the fibresπt−1(m)andπc−1(m)are both isomorphic (as real vector spaces) toRn, so consideringT M andT∗Mas sets, they can both be viewed as M×Rn. To however be able to generalise the notion of a smooth vector field
onRnto a smooth vector field onM, we need a way of smoothly assigning to
each point on the manifold an element of the tangent space at that point (i.e. in the fibre ofπover that point). That is, we need a mapV :M −→T M such that V(m0)∈π−t1(m0)holds for allm0 ∈M, and this map should be smooth, so the
tangent bundle should be a smooth manifold. Let(U, ϕ)be a chart forM. Any v ∈ πt−1(U)can be written asv =vi∂i|m0 for some(v1, . . . , vn) ∈ Rn, where
m0 ∈U is such thatπt(v) =m0, so define a map
ψϕ :π−1(U)−→ϕ(U)×Rn
v7−→(ϕ(m), v1, . . . , vn), (1.6)
which is clearly a bijection. Let( ˜U ,ϕ˜)be a chart forM such thatU˜ ∩U 6= ∅. Then
( ˜ψϕ˜◦ψϕ−1)(ϕ(m0), v1, . . . , vn) = ˜ψϕ˜(vi∂i|m0)
= ˜ψϕ˜(viD1i∂˜1|m0+. . .+viDni∂˜n|m0)
= ( ˜ϕ(m0), viD1i, . . . , viDni)
(1.7)
holds by the chain rule for all (ϕ(m0), v1, . . . , vn) ∈ ϕ(U ∩U˜) ×Rn, where
Dij is the (i, j)-th entry of the Jacobian matrix D( ˜ϕ◦ϕ−1)(ϕ(m0)), for each
1.3. The tangent bundle
declaring a subsetV ofT Mto be open ifψϕ(V ∩π−1(U))is open for each chart (U, ϕ) for M, and the smooth structure on T M is determined by the smooth
atlas{(π−1(U), ψϕ)|(U, ϕ) ∈ A}, whereAis a smooth atlas forM. A similar procedure works for the cotangent bundle, and we have the following proposition. Proposition 1.20. The setsT M andT∗Mare2n-dimensional smooth manifolds
such thatπtandπcare smooth.
Proof. See Proposition 3.18 and Proposition 11.9 in [7].
In general, the tangent bundle ofM will not be trivial, i.e. there won’t be
a diffeomorphism4Φ : T M −→ M×
Rn. Note that for any two charts(U, ϕ),
( ˜U ,ϕ˜)forM, we can define a smooth map gUU˜ :U ∩U˜ −→GL(n,R)
m0 7−→D( ˜ϕ◦ϕ−1)(ϕ(m0)). (1.8)
As we will see in the next chapter, these maps actuallydefinethe tangent bundle, and the way in which they do determines how “non-trivial” the tangent bundle is. Note that if we consider a manifold which is covered by a single chart(M, ϕ),
such asRnor some finite-dimensional vector space, thenϕ:M 7−→ϕ(M)is a
diffeomorphism, soψ:T M 7−→ϕ(M)×Rndefines a diffeomorphism between T MandM ×Rn. Now that we have a smooth structure onT M, we can define what a smooth vector field is on a manifold.
Definition 1.21. Asmooth vector field onM is a smooth mapV :M −→T M
such thatπ◦V = IM. The set of all smooth vectorfields onM is denoted by Γ(T M). Asmooth covector field onMis a smooth mapω :M −→T∗Msuch that π◦ω =IM. The set of all smooth covector fields onM is denoted byΓ(T∗M).
The first part of this definition indeed amounts to a smooth assignment of a tangent vector at each pointm0 ∈M, and what is important, at the tangent space Tm0M atm0. As we know from calculus, inRnany vector field can be written as
a linear combination of the vector fields determined by the standard basis, i.e. the smooth functionsEi :Rn7−→ Rn,m 7−→(0, . . . ,1, . . . ,0)for alli∈I, where
the1is in thei-th slot. Similarly, we can define global coordinates in Minkowski
space, i.e. spacetime without gravity, since this is also just a vector space. In general, however, this won’t be possible, which forces us to work locally in a chart (U, ϕ), where we have the coordinate vector fields∂i :U −→T M,m7−→∂i|m.
This leads to the following definition.
Definition 1.22. The tangent bundle ofM isparallelisableif there exist smooth
vector fieldsV1, . . . , Vnsuch that{V1(m0), . . . , Vn(m0)}is a basis forTm0M, for
allm0 ∈M.
4This diffeomorphism should in fact also satisfy some other property, which we will discuss later
When we deal withRn, we are used to taking the standard basis{e1, . . . , en}, which we refer to as right-handed. Since there are many things in physics where some sort of “right-hand rule” comes up, we tend to forget that taking basis is stillonly a choice. To formalise what we mean by this choice, letV be a
finite-dimensional vector space, and letB(V) be the set of all bases forV. We can
define an equivalence relation onB(V), by lettingB, B0∈ B(V)be equivalent if and only ifdet(TBB0)>0, whereTBB0 :V −→V is theR-linear isomorphism
sendingei ∈ B toe0i ∈ B
0 for eachi ∈[dim(V)]. Since for anyB, B0 ∈ B(V)
it holds thatTBB = IV and TBB0 = T−1
B0B, and for any B00 ∈ B(V) it holds
thatTBB00 = TB0B00 ◦TBB0, it follows from the multiplicative property of the
determinant that this is indeed an equivalence relation. The setB(V)/∼clearly has two elements, and an orientation inV is defined as a choice of an element
O ∈ B(V).
We also have the notion of orientability in a smooth manifold, which comes down to a way of consistently choosing an orientation in each tangent space.
Definition 1.23. A smooth manifoldM0 is said to be orientableif there exists
an atlas{(Ui, ϕi)|i∈I}forM0 such that for eachi, j∈IwithUi∩Uj 6=∅, it holds thatdet(D(ϕj◦ϕ−1)(ϕ(m0)))>0for allm0 ∈Ui∩Uj.
The classical example of a non-orientable manifold is theM¨obius strip.
1.4
The tensor product
LetRbe a commutative ring with unity, and letM andN beR-modules5.
Definition 1.24. Thetensor product ofM andN overRis anR-moduleM⊗RN
equipped with anR-bilinear mapT :M×N −→M⊗RN,(m, n)7−→m⊗n
satisfying the universal property
• (Universal property of the tensor product)LetP be anR-module. For each R-bilinear mapB : M ×N −→ P, there exists a unique R-linear map
˜
B:M ⊗RN −→P such that the diagram
M×N M⊗RN
P
T
B ˜
T (1.9)
commutes.
5AnyR-module is assumed to be unital. The dual ofM isM∗
1.4. The tensor product
Proposition 1.25. The tensor productM⊗RN exists and is unique, up to iso-morphism.
Proof. See Proposition 2.12 in [13].
Remarks. TheR-moduleM⊗RN is generated by elements of the formm⊗n withm∈M andn∈N, and it follows from the definition that the equalities
(m+m0)⊗n=m⊗n+m0⊗n, m⊗(n+n0) =m⊗n+m⊗n0,
r(m⊗n) = (rm)⊗n) =m⊗(rn)
(1.10)
hold for allm, m0 ∈M,n, n0 ∈N, andr ∈R.
Proposition 1.26. LetP be anR-module. The maps M⊗RN −→N⊗RM
m⊗n7−→n⊗m,
(M⊗RN)⊗RP −→M⊗RN⊗RP (m⊗n)⊗p7−→m⊗n⊗p, M⊗RN ⊗RP −→M⊗R(N ⊗RP)
m⊗n⊗p7−→m⊗(n⊗p),
(1.11)
areR-module isomorphisms.
Proof. These maps and their inverses are easily constructed using the universal property of the tensor product.
Proposition 1.27. For anyR-moduleP, theR-modulesHomR(M⊗RN, P)and HomR(M,HomR(N, P))are isomorphic. In particular, there is an isomorphism (M ⊗RN)∗∼= HomR(M, N∗).
Proof. See the remarks before Proposition 2.18 in [13].
LetM be a smooth manifold. The real vector spacesΓ(T M)andΓ(T∗M)are naturallyC∞(M)-modules6, and are in fact both reflexiveC∞(M)-modules. To
see this, we will argue thatΓ(T∗M)∼= Γ(T M)∗andΓ(T∗M)∗∼= Γ(T M), from
which the statement then follows immediately. Define a map
f : Γ(T∗M)−→Γ(T M)∗
ω7−→ω,˜ (1.12)
6The multiplications are defined by(f X)(m) :=f(m)X(m)and(f ω)(m) :=f(m)ω(m)
respectively, for allf∈ C∞
where f(ω)(V) = ˜ω(V) := ωV is defined as ωV(m) := ω(m)(V(m)) ∈ R
for allω ∈ Γ(T∗M), V ∈ Γ(T M)and for allm ∈ M. Thenf is well-defined,
since for allω ∈Γ(T∗M), V ∈Γ(T M)and for each chart(U, ϕ)forM, there
are smooth functionsω1, . . . , ωn∈ C∞(M|U)andV1, . . . , Vn∈ C∞(M|U)such that7ω=ω
idxi andV =Vi∂i, and
ωV(m) =ωi(m)dxi|m(Vj(m)∂j|m) =ωi(m)Vj(m)dxi|m(∂j|m) =ωi(m)Vi(m)
(1.13)
holds for allm∈U, soωV ∈ C∞(M). It is clear thatfisC∞(M)-linear and thus
aC∞(M)-module homomorphism. Define a map
f−1: Γ(T M)∗ −→Γ(T∗M)
ϕ7−→ωϕ
(1.14)
and defineωϕ(m)(vm) := ϕ(V)(m) for allϕ ∈ Γ(T M)∗, m ∈ M and for all
vm ∈TmM, whereV ∈Γ(T M)is such thatV(m) = vm, which always exists by Proposition 8.7 in [7]. Thenωϕ(m) ∈ Tm∗M for allϕ∈ Γ(T M) and for all
m ∈ M, and from the proof on pages 265-266 in [14] it follows that this map
is well-defined (i.e. it does not depend on the choice ofV in the definition of ϕm). Since ϕ(V) ∈ C∞(M) for allϕ ∈ Γ(T M)∗ and V ∈ Γ(T M), the map
f−1 indeed maps intoΓ(T∗M). It is easily checked using the definitions that f−1 isC∞(M)-linear and thatf−1 is the inverse off, soΓ(T M)∗ ∼= Γ(T∗M).
Mimicking this proof, we find thatΓ(T∗M)∗ andΓ(T M)are also isomorphic, so Γ(T M)andΓ(T∗M)are both reflexive, enabling us to identify the double dual (Γ(T M)∗)∗(respectively(Γ(T∗M)∗)∗) withΓ(T M)(respectivelyΓ(T∗M)).
Finally, in most physics textbooks, tensors are introduced in a somewhat different manner [15,16], and it’s worth to take a moment to see how it corresponds to the formal definition given here. Letp, q∈Nandm∈M. A(p, q)-tensorT is
aR-multilinear map8T :Tm∗M×p×TmM×q−→R, which descends to a linear
mapT˜:T∗
mM⊗p⊗RTmM
⊗q −→
R, and sinceTmM is finite-dimensional, this corresponds to an element ofTmM⊗p⊗RT
∗
mM⊗q.
7Heredxi
: U −→ T∗M,m 7−→ dxi|mare the coordinate covector fields onU, for each
i∈[n]. Note that the coordinate vector and covector fields constitute a basis forΓ(T M|U)and
Γ(T∗M|U)respectively, whereΓ(T M|U)is just the set of smooth vector fields onU, and similarly
forΓ(T∗M|U).
8This is just the Cartesian productT∗
mM×. . .×Tm∗M×TmM×. . .×TmM, where there
arepcopies ofTm∗Mandqcopies ofTmM, and similarly forTm∗M
⊗p⊗
RTmM⊗q. Of course, all
1.5. Minkowski space andSL(2,C)
1.5
Minkowski space and
SL(2
,
C
)
Minkowski space serves as the model for spacetime in the absence of gravity. One of Einstein’s postulates was that spacetime should “locally look like Minkowski space”, a fact which is mathematically expressed by the fact (as we will see later) that each tangent space to the spacetime manifold is (isomorphic to) Minkowski space. We should define then, what Minkowski space is.
Definition 1.28. The real vector spaceR4 equipped with a non-degenerate
sym-metric bilinear formB :R4×R4 −→ Rof signature(1,3)is calledMinkowski spaceand is denoted byM.
Define the matrix
η=
1 0 0 0
0 −1 0 0
0 0 −1 0
0 0 0 −1
, (1.15)
and letB = {e0, e1, e2, e3}be a basis forM such that9B(v, w) = v·ηw, for
allv, w ∈M. Choose the equivalence class of Bas an orientation inM . The
homogeneous Lorentz groupLis the group consisting of all linear transformations (calledLorentz transformations) ofMwhich preserve the quadratic formQinduced
byB, i.e. all linear mapsΛ :M −→M such thatQ(Λ(v)) =Q(v)holds for all
v∈M. Its matrix representation (with respect to this basis) is the group
O(1,3) :={L∈GL(4,R)|L>ηL=η}. (1.16)
For eachL ∈ O(1,3), it follows fromL>ηL = ηthat(det(L))2 = 1and thus det(L) =±1. Moreover, it follows that(L11)2−(L21)2−(L31)2−(L41)2 = 1, which implies that |L1
1| ≥ 1. This group thus splits up into four connected
components, according to the sign of the determinant and the sign ofL11.
In physics, we often only want to consider Lorentz transformations which reverse neither time nor parity. Mathematically, this means that we have to consider therestricted Lorentz group, which is the subgroup
SO↑(1,3) :={L∈O(1,3)|det(L) = 1∧L11≥1} (1.17) and the identity component ofO(1,3).
We will now show howSL(2,C)andSO↑(1,3)are related. Defineσ0 :=I2,
and note thatH(2,C) =LR{σ0, σ1, σ2, σ3}(theR-linear span of{σ0, σ1, σ2, σ3}).
Indeed, it is clear that any matrix inLR{σ0, σ1, σ2, σ3}is Hermitian. To establish 9Note that such a basis exists by Sylvester’s law of inertia. Also,v·wdenotes the regular inner
the other inclusion, using the Hermitian conditionH=H†for anyH∈ H(2,C)
it is easily shown thatHcan be written as
H = 12Tr(H)σ0+<(H21)σ1+=(H12)σ2+12(H11 −H22)σ3. (1.18)
The following proposition serves as the starting point to establish the relation betweenSO↑(1,3)andSL(2,C).
Proposition 1.29. TheR-linear extensionϕof the assignment
∀i∈ {0,1,2,3}:M 3ei 7−→σi ∈ H(2,C) (1.19)
defines a linear isomorphism betweenM andH(2,C).
Proof. The mapϕis clearly a bijection and thus a linear isomorphism. It is
never-theless useful to write down the inverse, which is given byϕ−1 :H(2,
C)−→M, H→P3
i=0Tr(Hσi)ei. This map is linear because the trace is linear, and because
Tr(σiσj) = 2δij holds for alli, j∈ {0,1,2,3}, it is indeed the inverse ofϕ. The reason why this isomorphism is useful is made clear by the following observation. Letv=viei ∈M. Then
det(ϕ(v)) = det
v0+v3 v1−iv2 v1+iv2 v0−v3
= (v0)2−(v1)2−(v2)2−(v3)2 =Q(v),
(1.20)
so we may equally well work with Hermitian matrices instead of elements ofM. LetS ∈SL(2,C)andv =viei ∈M. For allH ∈ H(2,C), the matrixSHS†is
Hermitian anddet(Sϕ(v)S†) = det(ϕ(v)), soϕ−1(Sϕ(v)S†) =LSv for some
LS∈O(1,3). Note thatLS =L−S. For eachj∈ {0,1,2,3}, we have (LSv)j = (ϕ−1(Sϕ(v)S†))j
= 12viTr(SσiS†σj),
(1.21)
and we find10
2(LS)11=αα+ββ+γγ+δδ, 2(LS)12=αβ+βα+γδ+δγ,
2(LS)21=αγ+γα+βδ+δβ, 2(LS)22=αδ+δα+γβ+βγ,
2(LS)31=i(αγ−γα+βδ−δβ), 2(LS)32=i(αδ−δα+βγ−γβ),
2(LS)41=αα+ββ−γγ−δδ, 2(LS)42=αβ+βα−γδ−δγ,
(1.22)
10We spare the reader the explicit calculations. Hereα, β, γ, δ∈
Care such thatS=
α β γ δ
1.5. Minkowski space andSL(2,C)
2(LS)13=i(−αβ+βα−γδ+δγ), 2(LS)14=αα−ββ+γγ−δδ,
2(LS)23=i(−αδ+δα+βγ−γβ), 2(LS)24=αγ+γα−βδ−δβ,
2(LS)33=αδ+δα−βγ−γβ, 2(LS)34=i(αγ−γα−βδ+δβ),
2(LS)43=i(−αβ+βα+γδ−δγ), 2(LS)44=αα−ββ−γγ+δδ.
(1.23)
It is clear that (LS)ij ∈ R for all i, j ∈ [4], and that (LS)11 > 0 and thus
(LS)11 ≥1holds, and another explicit calculation shows thatdet(LS) = (αδ−
βγ)(αδ−βγ) = 1, soLS ∈SO↑(1,3). We can thus define a map
ρ: SL(2,C)−→SO↑(1,3) S 7−→LS,
(1.24)
which is smooth, as follows from the explicit expressions in (1.22). Sinceρ(σ0) =I4
and
LS1S2(v) =ρ(S1S2)(v) =S1S2ϕ(v)S2†S
†
1
=S1(LS2v)S
†
1
=LS1(LS2(v)),
(1.25)
holds for all S1, S2 ∈ SL(2,C), it is a Lie group homomorphism. Its image ρ(SL(2,C))is connected inSO↑(1,3)sinceSL(2,C)is simply connected, and
ker(ρ) ∼= {−σ0, σ0} =: Z. It is also surjective11, soρdescends to a group
iso-morphismρ : SL/Z 7−→ SO↑(1,3), which is in fact a Lie group isomorphism
by Theorem 21.27 in [7]. This in fact shows thatSL(2,C)is the double cover of
SO↑(1,3). Namely, consider the action ofZ onSL(2,C)defined by matrix
multi-plication. This action is smooth, free and thus proper, as follows from Corollary 21.6 in [7], sinceZis a compact Lie group. Theorem 21.23 then guarantees that the
quotient mapπ: SL(2,C)7−→SL(2,C)/Z is a (smooth) covering map, which is
clearly a double covering asZ∼=Z/2Z. The diagram
SL(2,C)
SL(2,C)/Z SO↑(1,3)
π ρ
ρ
(1.26)
commutes andρis a Lie group isomorphism, soSL(2,C), is the (sinceSL(2,C)is
simply connected) double cover ofSO↑(1,3). Incidentally, this also shows that the
Lorentz group is not simply connected, as it follows thatπ1(SO↑(1,3)) =Z/2Z.
What follows from the above observations is that the Lorentz group is isomor-phic to theprojective special linear groupPSL(2,C) := SL(2,C)/Z(SL(2,C)),
where Z(SL(2,C)) := {λσ0|λ ∈ C : det(λσ0) = 1} = Z is the (normal)
subgroup consisting of all scalar multiples ofσ0 with unit determinant, which
naturally acts on the complex projective lineP1(C). This group can in turn be
identified with the M¨obius group, which is the automorphism groupAut(C∞)
of the Riemann sphereC∞. This observation was key for Penrose to introduce
spinors in general relativity (see chapter 1 in [2]).
We can shortly and informally discuss how 2-spinors arise naturally from the conclusion thatSL(2,C) is the double cover ofSO↑(1,3), since to say of
all this properly, one must really turn to representation theory and study the representations ofSL(2,C)andSO↑(1,3). One can, loosely speaking, define
2-spinors as elements ofC2, which is the representation space of the regular matrix
representation of the special linear group, sinceSL(2,C)acts onC2 by matrix
multiplication. Let{e1, e2}be the standard basis ofC2, and letκ= (ζ, η)∈C2.
The matrix
Hκ: =κκ† =
ζ η
ζ η
=
ζζ ζη ηζ ηη
(1.27)
is clearly Hermitian, and thus defines an elementϕ−1(Hκ) ∈ M. This vector isnull, i.e. Q(ϕ−1) = det(Hκ) = ζζηη−ζηζη= 0. Note also that−κdefines the same matrix, i.e. H−κ = Hκ, and thus the same element ofM. Now let
A ∈ SL(2,C). ThenHAκ = Aκ(Aκ)† = Aκκ†A† = AHA† = LA(ϕ−1(H)), so we can equivalently consider the action ofSL(2,C)on C2 or the action of
the restricted Lorentz group onM, except for the sign-ambiguity which exists sinceκand−κdefine the same element ofM12. This sign ambiguity is then of
course precisely the potential reason why spinors cannot be defined properly on a manifold; more on this can be found in chapter 1 of [2].
12As it is written now, any elementeiθ
κwithθ∈Rdefines the same element ofM. However,
when this is all defined properly, this freedom essentially disappears and we are only left with±κ.
Chapter 2
Fibre Bundles
In this chapter we will introduce the notion of fibre bundles, objects which come up naturally in almost any physical theory that has some group of symmetries associated to1it which encodes the symmetries associated to the specific theory,
called thegauge groupin physics. As mentioned in section 1.5, a group which is of interest in the theory of both special and general relativity is the restricted Lorentz groupSO↑(1,3), the group of “proper” symmetries of Minkowski space.
The demand (which is only there because of physical reasons) that we should have the freedom to consider all observers which are connected to some initially chosen proper frame of reference (i.e. a basis of the tangent space to the spacetime manifold) by a restricted Lorentz transformation can be roughly translated to the mathematical demand that there should be a principalSO↑(1,3)-bundle over
the spacetime manifold. To define this notion properly, and to see how we can extend this to a description of spacetime which allows for spinors, we have to start with the frame bundle of a manifold, which has the bigger groupGL(n,R)as its
symmetry group.
2.1
Fibre bundles
LetM andM0 be manifolds.
Definition 2.1. Asmooth fibre bundle overM is a tripleF = (E, π, F), where
• EandF are manifolds, called thetotal spaceandtypical fibre ofF, respec-tively, and
1Each of the four fundamental forces known in physics has associated to it a group of
sym-metries. For example, the unitary groupU(1)consisting of all complex numbers of norm1is the
symmetry group for electrodynamics, the strong interaction has the special unitary groupSU(3)in 3dimensions, and the weak interaction hasSU(2). The principal bundle approach to incorporating
• π :E −→M is a smooth surjective map, called theprojection ofF, such
that for eachm∈M there exists an open subsetU ofM containingmand
a diffeomorphismϕ:π−1(U)−→U×F, such that the diagram
π−1(U) U ×F
U
ϕ
π|π−1(U)
Proj1
(2.1)
commutes.
Remarks. It follows from the definition thatEm := π−1(m), thefibre ofπ over
m, is diffeomorphic toF, for eachm ∈ M. Any pair(U, ϕ), whereU ⊂M is
open andϕ:π−1(U)−→U ×F is a diffeomorphism, for which (2.1) commutes,
is called alocal trivialisation ofF. Any setC = {(Ui, ϕi)|i ∈ I}, whereI is some indexing set and(Ui, ϕi)is a local trivialisation ofF for alli∈I, such that
M = S
i∈IUi holds, is called atrivialising cover of M. A smooth fibre bundle F = (E, π, F)overM will be referred to as a fibre bundle overM, and will be
written asF −→E −→π Mor simplyE −→π M.
Example 2.2. LetF be a manifold. The triple(M×F,Proj1, F)is a fibre bun-dle overM, called thetrivial bundle over M. A trivialising cover is given by
{(M,Proj1×IF)}.
Definition 2.3. LetF andF0 be fibre bundles overM andM0, respectively. A
bundle map fromF toF0is a pair (Φ, ϕ), whereΦ :E−→E0andϕ:M −→M0
are smooth maps, such that the diagram
E E0
M M0
Φ
π π0
ϕ
(2.2)
commutes.
Remarks. The mapϕin Definition 2.3 is uniquely and completely determined by
Φ, sinceπis surjective; the mapΦis said tocoverϕ. Two fibre bundlesF andF0
overM areequivalent if there exists a bundle map(Φ,IM)fromFtoF0withΦa diffeomorphism; such a bundle map is called abundle equivalencebetweenFand F0, andF istrivialif it is equivalent to(M×F,Proj
1, F). It follows that there
2.1. Fibre bundles
morphism from a fibre bundleFoverM to a fibre bundleF0 overM0 is a bundle
map(Φ, ϕ)fromFtoF0.
2.1.1 Transition functions
Definition 2.4. LetFbe a fibre bundle overM. AsectionofF is a smooth map
s:U →Esuch thatπ◦s=I|U, whereU ⊂Mis open and non-empty. A section is calledlocalifU is a proper subset, andglobalifU =M. For any proper open
subsetU ofMthe set of all local sections is denoted byΓ(E|U), and the set of all global sections is denoted byΓ(P).
LetF −→ E −→π M be a fibre bundle overM andC = {(Ui, ϕi)|i ∈ I} a trivialising cover ofM. Let i, j, k ∈ I be such thatUij := Ui∩Uj 6= ∅and
Uijk:=Ui∩Uj∩Uk6=∅. Then
(ϕi◦ϕ−j1)|Uij×F :Uij×F −→Uij ×F (2.3) is a diffeomorphism, so for eachm∈Uij, the map
ϕi,m◦ϕ−j,m1 :F −→F
f 7−→Proj2,m◦ϕi◦ϕ−j1◦Proj−2,m1
(2.4)
is a diffeomorphism. We can thus define a smooth map2
tij :Uij −→Diff(F)
p7−→ϕi,m◦ϕ−j,m1,
(2.5)
called atransition function. Note that for anym∈Uijand for anye∈π−1(m), the elementsϕi(e) = (m, fi)∈ {p} ×F andϕj(e) = (m, fj)∈ {m} ×F are related as(p, fi) = (p, tij(p)fj). The set{tij :i, j ∈I}of transition functions induced byC is denoted by CC. The transition functions satisfy certain “compatibility
conditions”, as expressed by the following lemma.
Lemma 2.5. LetF −→E −→π M be a fibre bundle overM, letCbe a trivialising
cover ofM, and letCCbe the induced set of transition functions. For alli, j, k∈I,
the conditions
• ∀m∈Ui :tii(m) =IF,
• ∀m∈Uij :tij(m) = (tji(m))−1,
• ∀m∈Uijk :tij(m)◦tjk(m) =tik(m), ( ˇCech cocycle condition)
2The groupDiff(F)is an open submanifoldC∞
(M, M), conform Theorem 7.1 in [18]. What
hold. The setCCis called a cocycle onMassociated to the open covering{Ui:i∈I}
ofM.
Proof. This follows easily from the definition.
We thus see that any fibre bundle with a chosen trivialising cover of the base space determines a set of transition functions which take values inDiff(F). There
is the following converse to this statement.
Theorem 2.6. Let{Ui:i∈I}be an open cover ofM, letF be a manifold, and let {tij :Uij −→Diff(F)}be a set of smooth maps satisfying the conditions of Lemma
2.5. These data determine a unique (up to equivalence) fibre bundle overM with typical fibreF.
Proof. See Theorem 3 in Chapter 16 of [19].
The transition functions determine how the fibreFis “glued” onto the base
manifold, and therefore how ”non-trivial” the fibre bundle is (we have of course seen this before, with the tangent bundle). If all transition functions can be taken to be the identity, then the fibre bundle is clearly equivalent to the trivial bundle. The groupDiff(F)is often be too large to be of interest, but as we will see, all fibre bundles in which we are interested will have a trivialising cover of the base space such that the corresponding transition functions take values in some Lie group. This observation leads us to the concept of a smooth principalG-bundle.
2.2
Principal bundles
LetGandG0 be Lie groups.
Definition 2.7. Asmooth principalG-bundle overM is a tripleP = (P, π, σ),
where(P, π, G)is fibre bundle overM, andσ:P ×G−→P is a smooth right
action ofGonP such that the fibres ofπareG-invariant. In addition, there exists
for eachm∈M alocalG-trivialisationofP, which is a local trivialisation(U, ϕ) ofP such thatϕ(p) = (π(p),ϕ˜(p))for allp ∈ U, whereϕ˜ :π−1(U) −→ Gis G-equivariant.
Remarks. The Lie groupGis called thestructure groupofP; in physics, it is called the gauge group. A smooth principalG-bundle will be referred to as aG-bundle
overM and, if no confusion can arise, will be written asG −→ P −→π M or P −→π M. The action ofGwill be written asp / g, for allp∈Pand for allg∈G.
Each fibre ofπis now diffeomorphic toG, and is thus aG-torsor.
Example 2.8. Consider the trivial fibre bundle overM, and define a right action σJ : (M×G)×G−→GbyσJ((p, g), h) := (p, g)Jh:= (p, gh)for allp∈M
and for allg, h∈G. Then(M×G,Proj1, σJ)is aG-bundle over overM×G,
2.2. Principal bundles
Example 2.9. Define
σ :S3C×U(1)−→S3C
((α1, α2), g)7−→(α1g, α2g),
(2.6)
thenσ is a smooth right action, and since (α1g, α2g) = (α1, α2)implies that g= 1for all(α1, α2)∈S3C, this action is free. SinceU(1)is compact, this is also
a proper action, so3the orbit space
S3C/U(1)is a manifold such that the quotient
map
q:S3C−→S3C/U(1)
(α1, α2)7−→[α1, α2]q
(2.7)
is smooth. Note thatS3/U(1)is diffeomorphic toP1(C), via the map
f :S3C/U(1)−→P1(C)
[α1, α2]q7−→[α1, α2],
(2.8)
and defineπ :=f◦q. Fork∈ {1,2}, defineUk:={[α1, α2]∈P(C2)|αk6= 0} and
ϕk:π−1(Uk)−→Uk×U(1) (α1, α2)7−→
[α1, α2], αk |αk|
. (2.9)
ThenUkis open andϕkis a diffeomorphism, and
ϕk(α1, α2) = (π(α1, α2),ϕ˜k(α1, α2)) (2.10)
holds for all(α1, α2)∈π−1(Uk), where the map
˜
ϕk:π−1(Uk)−→U(1) (α1, α2)7−→
αk |αk|
(2.11)
isU(1)-equivariant. It follows thatC:={(U1, ϕ1),(U2, ϕ2)}is a trivialising cover
ofP1(C), and thus that
U(1)−→S3
C−→P
1(
C) (2.12)
is aU(1)-bundle overS3C, called thecomplex Hopf bundle.
complex Hopf bundle is sometimes also written asS1 −→S3−→S2, to emphasize
the involvement of the spheres.
Lemma 2.10. Let P be a G-bundle over M. The action of Gon P is free, and transitive on the fibres ofπ.
Proof. Letp∈P, and suppose there exists someg∈Gsuch thatp=p / g. Let
(U, ϕ)be a localG-trivialisation ofP such thatπ(p)∈U. Then
(π(p),ϕ˜(p)) = (π(p / g),ϕ˜(p / g))
= (π(p),ϕ˜(p)g)) (2.13) and thusg=e, so the action is free. Letm∈M, letp, q∈Pm, and let(U, ϕ)be a localG-trivialisation ofP such thatm∈U. Thenϕ˜(p),ϕ˜(q)∈G, so there exists
someg∈Gsuch thatϕ˜(q) = ˜ϕ(p)g= ˜ϕ(p / g), and
ϕ(q) = (π(q),ϕ˜(q)) = (π(p / g),ϕ˜(p / g)) =ϕ(p / g),
(2.14)
so sinceϕis injective, it follows thatq=p / g, so the action is transitive on the
fibres ofπ.
LetP be aG-bundle overM, and letCbe a trivialising cover ofM consisting
of localG-trivialisations. The transition functions for aG-bundleP overM are
readily recovered fromC. Let(Ui, ϕi),(Uj, ϕj) ∈ Cbe such thatUi∩Uj 6= ∅. Letm ∈ Ui ∩Uj andp, p0 ∈ π−1(m). Then p0 = p / g for some g ∈ G, so
ϕi(p0)(ϕj(p0))−1=ϕi(p)gg−1(ϕj(p))−1=ϕi(p)(ϕj(p))−1, and we can a define a mapgij : Uij −→ G,m 7−→ ϕi(p)(ϕj(p))−1, wherepis any element in the fibre ofπoverm, andCC ={gij :Uij −→G|i, j∈I}.
Definition 2.11. LetP be aG-bundle overM, and letP0 be aG0-bundle over
M0. Aprincipal bundle map fromP toP0 is a triple(Φ, ϕ, λ), where(Φ, ϕ)is a bundle map fromP toP0 andλ:G−→G0is a Lie group homomorphism, such
that the diagram
P×G P0×G0
P P0
Φ×λ
σ σ0
Φ
(2.15)
2.2. Principal bundles
Remarks. TwoG-bundlesP andP0overM areequivalentif there exists a bundle
equivalence(Φ,IM)betweenP andP0such that(Φ,IM,IG)is a principal bundle map fromPtoP0, called aG-bundle equivalence betweenPandP0, andP istrivial
if it is equivalent to(M ×G,Proj1, σ). There is thus a categoryP-Bun, where
any object is anH-bundlePoverN, for some Lie groupHand some manifoldN,
and a morphism from anP to aG-bundleP0 overM is a principal bundle map
(Φ, ϕ, λ)fromP toP0. If we fix the Lie groupGand the manifoldM, we get a
subcategoryPG-Bun(M). As the following lemma shows, this last category is
quite restrictive.
Lemma 2.12. LetP,P0 ∈PG-Bun(M). If(Φ,
IM,IG)is a principal bundle map
fromP toP0, thenΦis a diffeomorphism.
Proof. Letp, q∈P be such thatΦ(p) = Φ(q). Thenpandqare elements of the
same fibre ofπ, sinceπ(p) = π0(Φ(p)) = π0(Φ(q)) = π(q), so there exists a
uniqueg∈Gsuch thatq =p / g. Then
Φ(q) = Φ(p / g)
= Φ(q)/0g, (2.16)
which implies thatg=eand thus thatp=q, soΦis injective. Letp0∈P0, and
letp∈π−1(π0(p0)). ThenΦ(p)andp0are elements of the same fibre ofπ0, since π0(Φ(p)) =π(p) =π0(p0), so there exists a uniqueg∈Gsuch thatp0= Φ(p)/0g.
Then
Φ(p / g) = Φ(p)/0g
=p0, (2.17)
soΦis bijective. The inverse is given by
Φ−1 :P0 −→P p0 7−→p / gpp0,
(2.18)
where p ∈ π−1(π0(p0)), andgpp0 ∈ Gis the unique group element such that
p0 = Φ(p)/0g
pp0 holds. ThenΦ−1is a smooth map which preserves the fibres of
π0such thatΦ−1(p0/0g) = Φ−1(p0)/ gholds for allp0 ∈P and for allg∈G, so
F−1is a principal bundle map.
The following lemma illustrates another important property of principal bun-dles.
Proof. Suppose thatPis trivial, and let(Φ,IG)be a principal bundle map. Define
s:M −→P
m7−→Φ−1(m, e), (2.19)
thensis smooth andπ◦s=IM, sos∈Γ(P). Suppose thatΓ(P)is non-empty and lets∈Γ(P). Define
Φs:M×G−→P
(m, g)7−→s(m)/ g, (2.20)
then(Φs,IM)is a bundle map fromM ×GtoP, and
Φs((m, g)Jh) = Φs(m, gh) =s(m)/ gh
= (s(m)/ g)/ h
= Φs(m, g)/ h
(2.21)
holds for allm ∈M and for allg, h ∈ G, so(Φs,IM,IG)is a principal bundle map fromM ×GtoP. By Lemma 2.12 this map is an equivalence fromM×G
toP, soP is trivial.
2.2.1 The frame bundle
Definition 2.14. Letm∈M. Aframe atmis an ordered basisem= (e1, . . . , en) forTmM. The set of all frames atmis denoted byLmM, andLM := ˙Sm∈MLmM is the set of all frames at all points inM.
Since any element ofLM is a frame at some point in the manifoldM, there
is a natural projectionπLM :LM −→ M sending eachem ∈LM to the point
m∈M at whichem is a frame.
Lemma 2.15. The setLMis an(n+n2)-dimensional manifold such that
πLM :LM −→M
em 7−→m
(2.22)
is a smooth map.
Proof. See section 3.3 in [20].
Letem= (e1, . . . , en)∈LmM, and letg= (gij)∈GL(n,R). Define
2.3. Associated fibre bundles
thene / gis again a frame atm. Now writeeas a column vector, i.e. as(e1· · ·en); thene / gcan be viewed as the column vector
(eigi1· · ·eigin) = (e1· · ·en)
g11 · · · g1n
... ...
gn1 · · · gnn
(2.24)
which makes it clear that(e / g)/ h=e /(gh)holds for allh∈GL(n,R), so that
σLM :LM×GL(n,R)−→LM
(e, g)7−→e / g (2.25)
is a right action ofGL(n,R)onLM. It is clear that this action is free, and that it
is transitive when restricted toLmM, for eachm∈M.
Lemma 2.16. The tripleF M:= (LM, πLM, σLM)is a principalGL(n,R)-bundle overM.
Proof. See section 3.3 in [20].
Remark. TheGL(n,R)-bundleF Mis called theframe bundle ofM.
Lemma 2.17. The tangent bundle is parallelisable if and only if the frame bundle F MoverM is trivial.
Proof. The existence ofnlinearly independent sections of the tangent bundle is
clearly equivalent to the existence of a global section ofF M. By Lemma (2.13), the result follows.
2.3
Associated fibre bundles
LetF be a manifold equipped with a smooth left action4τ :F×G−→F, letP be aG-bundle overM, and consider the action
∆ : (P×F)×G−→P ×F
((p, f), g)7−→(p / g, g−1. f). (2.26)
The action∆is smooth sinceσandτ are, andPF := (P ×F)/Gis a topological space equipped with the quotient topology. Denote by[p, f]the equivalence class inPF of(p, f)∈P×F. Since
˜
π :P×F −→M
(p, f)7−→π(p) (2.27)
4In analogy to the right action defined onP, this action will be written asg . f, for allg∈G
is aG-equivariant map with respect to∆, it descends to a continuous map
πF :PF −→M
[p, f]7−→π(p). (2.28)
Lemma 2.18. The triplePF := (PF, πF, F)is a fibre bundle overM.
Proof. See Theorem 6.87 in [21].
The bundlePF is called thefibre bundle associated toP viaτ, or simply an
associated fibre bundle ofP. There is a slight abuse of notation here, since the associated fibre bundle depends on the specific actionτ and the notation does not
reflect this. However, there will be no possibility for confusion due to this. As we will see now, many fibres bundles which come up naturally, are associated to the frame bundle.
Example 2.19. Consider the frame bundle ofM, and the left actionτ1ofGL(n,R)
onRndefined by5matrix multiplication. Note thatτ1is smooth. Then
˜
Φ :LM ×Rn−→T M (e, f)7−→fiei
(2.29)
is a well-definedGL(n,R)-equivariant map, since
(g−1. f)i(e / g)i=gjifjekgki =gkigjifjek =δjkfjek =fjej
(2.30)
holds for allg∈GL(n,R)and for all(e, f)∈LM×Rn, so it descends to a map
Φ :LMRn −→T M such that the diagram
LMRn T M
M M
Φ
πRn πt
IM
(2.31)
commutes; it follows that(Φ,IM)is a bundle map fromLMRn to the tangent
bundleTM := (T M, πt,Rn). In fact, it is a bundle equivalence. Letm ∈ M,
letX∈TmM, and let(U, ϕ)be a chart forM aroundmwith local coordinates
5The elements of
2.4. Altering the structure group
x1, . . . , xnonU. TheneUm := (∂1|m, . . . , ∂n|m)is a frame atm, soV =fi∂i|m for somefmU := (f1, . . . , fn) ∈ Rn, and the element[eU
m, fmU] ∈LMRn is such
thatΦ([e, f]) = X, so Φis surjective. Let[e, f],[˜e,f˜] ∈ LMRn be such that Φ([e, f]) = Φ([˜e,f˜]). Then πLM(e) = πLM(˜e), so there exists a unique g ∈ GL(n,R)such that˜e=e / g. Thenfiei= ˜fi(e / g)iimplies thatf˜=g−1. f, so Φis injective. The inverse map is given by
Φ−1 :T M −→LMRn
X7−→[eUm, fmU], (2.32)
wherem ∈ M is such thatX ∈TmM. Note thatΦ−1 is well-defined, since if ( ˜U ,ϕ˜)is another chart forM aroundm, theneU˜ =eU/ gandfU˜ =g−1. fU, wheregis the Jacobian atϕ˜(m)of the overlap functionϕ◦ϕ˜−1. We thus see that the tangent bundle can be viewed as an associated fibre bundle ofF M.
The above description of the tangent bundle reveals how the formal definition of tangent vectors and vector fields corresponds to the way in which they are usually presented in the physics literature, namely as a set of components (or component functions) with respect to some basis, such that if the basis transforms by a basis transformationsΛ, then the components transform with the inverse transformationΛ−1.
Example 2.20. Consider again the frame bundle of M, and the action τ of
GL(n,R)on6(Rn)∗ defined by(g . f)i = fj(g−1)ji for allf ∈ (Rn)∗and for allg∈GL(n,R). From the previous example, it is clear that the associated fibre
bundleF M(Rn)∗is equivalent to the cotangent bundleTM∗ := (T∗M, πc,Rn).
To close this section, we make an informal remark which brings up the point made about representations in section 1.5. Namely, any linear representationRof
the Lie groupGdefines a smooth left action ofGon the representation space of R, and thus also an associated fibre bundle, and most associate bundles which are
important in physics come from a representation of a Lie group.
2.4
Altering the structure group
As mentioned before, the relevant group in general relativity is the restricted Lorentz groupSO↑(1,3), so the frame bundle, with its structure groupGL(n,R),
is not the structure we need, as it would allow physically for many non-admissible observers. We would therefore like toreducethe structure group. The first most obvious choice is to reduceGL(n,R) to the subgroupO(n,R), which means
that we are left with only orthormal frames. Eventually, we need to restrict to the Lorentz groupSO↑(1,3). Once we have done that, we can consider the
6The elements of(
question whether we can “lift” the structure group7toSL(2,C), since having this
group is essentially what allows us to define spinors on a spacetime. To define what orthogonality means, we need an inner product in each tangent space: a Riemannian metric. We first need a definition. Define the set
T∗M⊗T∗M := a m∈M
Tm∗M⊗Tm∗M. (2.33)
This set inherits in an analogous way as for the tangent and cotangent bundle a smooth structure fromM. Since the dimension of the tensor product of two vector
spaces is the product of their respective dimensions, it is a(n+n2)-dimensional
smooth manifold.
Definition 2.21. ARiemannian metricis an elementg∈Γ(T∗M⊗T∗M)such thatg(m)∈Tm∗M⊗Tm∗M is symmetric and positive-definite for eachm∈M.
Ifgis a Riemannian metric onM, we say that(M, g)is a Riemannian manifold. Lemma 2.22. A Riemannian metric exists.
Proof. Let{(Uα, ϕα)|α ∈ I}be an atlas forM, and letC = {pα|α ∈ J}be a smooth partition of unity subordinate the covering{Uα|α ∈ I}ofM. Let (Uα, ϕα)be a chart with local coordinatesx1α, . . . , xnαonUα, and define
gα:U −→T∗M⊗T∗M m7−→
n
X
j=1
dxjα|m⊗dxjα|m
(2.34)
Thengαis symmetric and positive-definite, and the map
g:M −→T∗M⊗T∗M m7−→X
α∈I
pα(m)gα(m) (2.35)
defines a Riemannian metric onM.
If(M, g)is a Riemannian manifold, then a construction completely analogous to the construction of the frame bundle allows us to construct the orthonormal frame bundle OM = (OM, πOM, σOM) over M, which has O(n,R) has its
structure group. Similarly, if the manifold is orientable, we may construct the oriented orthonormal frame bundle, whose structure group isSO(n,R)8. We see
that refining the group is in a way equivalent to introducing more structure to the manifold, and to get to the Lorentz group, we need to have not a Riemannian metric, but a Lorentzian metric.
7We say lift becauseSL(2,
C)is the double cover ofSO↑(1,3).
2.4. Altering the structure group
Definition 2.23. ALorentzian metric onM is an elementgL∈Γ(T∗M⊗ T∗M) such thatgL(m) is symmetric of signature(1, n−1)for eachm ∈ M. IfM is equipped with a Lorentzian metric gL, we say that(M, gL) is a Lorentzian
manifold.
From now on, we will restrict to our attention to connected, non-compact and 4-dimensional smooth manifoldM, and assume that there is a Lorentzian manifoldgLdefined onM. This assumption if of course physically motivated, as compact manifolds have certain unphysical properties; see section 1.5 in [2] and references therein, and because it is a postulate of general relativity that we have a Lorentzian metric.
The Lorentzian metric gL allows us to classify the tangent vectors to M as follows. For anym ∈ Mand Vm ∈ TmM, we say thatVm isspacelike if
gL(m)(Vm, Vm) < 0,timelike ifgL(m)(Vm, Vm) = 0, and null if it holds that
gL(m)(Vm, Vm) = 0. Then we say that(M, gL)istime orientedif there exists an everywhere non-vanishing smooth vector field such thatgL(m)(V(m), V(m))> 0for allm ∈ M. If(M, g)is time oriented also oriented, then(M, g)is called
spacetime oriented. For the same reasons as discussed above, we will assume that Mis oriented and time oriented9. From this it follows that we may further reduce the group toSO↑(1,3), as follows from corollary 1 in [22] and the discussion
on page 171 in [20], and we thus get aSO↑(1,3)-bundle overM, which will be
denoted bySO↑(M).
2.4.1 Spin structure
We can now define what a spin structure onMis. Definition 2.24. Aspin structure onMconsists of
• a principalSL(2,C)-bundleSL(2,C)−→S(M)−→ MoverM, and
• a smooth mapΦ : S(M)7−→SO↑(M)such that the diagram
S(M)×SL(2,C) SO↑(M)×SO↑(1,3)
S(M) SO↑(M)
M
Φ×ρ
σs σl
Φ
πs πl
(2.36)
commutes, whereπs : S(M) 7−→ Mandπl : SO↑(M) 7−→ Mare the bundle projections, andσsandσlare the actions of respectivelySL(2,C)
andSO↑(1,3)onM.
SinceSL(2,C)is not a subgroup ofSO↑(1,3)but its double cover, we also
say that a spin structure is a lift of the structure groupSO↑(1,3)toSL(2,C).
A spin structure does not always exist. A result by Robert Geroch [23] states that a spin structure exists onMif and only if there exist four smooth vector fieldse1, e2, e3 ande4 onM, such that{e1(m), e2(m), e3(m), e4(m)}forms a
basis forTmMfor which it holds that the value ofgL(m)(ei(m), ej(m))is1if
i = j = 1,−1 ifi = j ∈ {2,3,4}, and0 ifi, j ∈ [4] and i 6= j. Another
way to express the obstruction to having a spin structure is the following. There is a topological invariant called the second Stiefel-Whitney class, which is the elementw2(M) ∈ H2(M,Z/2Z) in the second ˇCech cohomology groupofM
with coefficients inZ/2Z(see section 11.6 in [24]).
2.5
Einstein’s field eq uations
In this section, we will introduce the concept of a connection on the tangent bundle of a manifold, which is necessary in order to write Einstein equation. One specific type of connection, namely the Levi-Civita connection, will turn out to be the appropriate choice connection in general relativity. Having done this, it becomes a fairly straightforward matter to write down the field equations in a local chart for the manifold. LetMbe a smooth manifold.
Definition 2.25. A connection onTM is aR-linear map
D: Γ(T M)−→Γ(T M) ⊗
C∞(M)Γ(T ∗M)
(2.37)
such thatD(f·V) = f·D(V) +V ⊗df holds for allV ∈Γ(T M)and for all
f ∈ C∞(M).
Remark. Since for anyV ∈ Γ(T M)it holds thatD(V) = Pn
i=1fiVi ⊗αi for
somen∈N≥1, wherefi ∈ C∞(M),Vi∈Γ(T M)andαi∈T∗M for alli∈[n], we can define
D(V)(U) := n
X
i=1
αi(U)fiVi ∈Γ(T M) (2.38) for allU ∈Γ(T M).
2.5. Einstein’s field eq uations
us to consider the change of one vector field on a manifold with respect to another vector field. LetDbe a connection onTM. This motivates the following definition.
Definition 2.26. Thecovariant derivative induced byDis theR-bilinear map
C: Γ(T M)×Γ(T M)−→Γ(T M) (U, V)7−→DUV,
(2.39)
whereDUV :=D(V)(U)for allU, V ∈Γ(T M).
Remarks. Note thatDf UV = f DUV andDU(f V) = f DUV +U(f)V holds for allf ∈ C∞(M)and for allU, V ∈Γ(T M), as follows immediately from the
definition ofD. It is clear that for any openU ⊂M, the mapCrestricts to a map C|U : Γ(T M|U)×Γ(T M|U)−→Γ(T M|U).
Fix a chart (U, ϕ) for M with local coordinates x1, . . . , xn on U (for the
remainder of this section; whenever we write locally, we mean that we are working in the chart(U, ϕ).). Since the image of two vector fields underCis again a vector field, we can express the result in terms of the basis local frame{∂1, . . . , ∂n}on
U. So for anyi, j ∈[n], there are smooth functionsΓ1ij, . . . ,Γnij ∈ C∞(M|U) such that
D∂i∂j = Γ
k
ij∂k. (2.40)
The elements{Γk
ij :i, j, k ∈ [n]}are called theChristoffel symbolsassociated toD, and are in the phyisics literature often introduced axiomatically in order
to define a “new” kind of derivative, which is supposed to replace the ordinary derivative. Here we see how that works formally.
Definition 2.27. Thetorsion ofDis theR-bilinear map T : Γ(T M)×Γ(T M)−→Γ(T M)
(U, V)7−→DUV −DVU−[U, V]
(2.41)
Here [U, V]is theLie bracket of the vector fields U andV, defined for all f ∈ C∞(M)by[U, V]f =U(V(f))−V(U(f)).
Lemma 2.28. The torsion isC∞(M)-bilinear.
Proof. LetU, V ∈Γ(T M)andf ∈ C∞(M). Then it holds thatDf UV =f DUV andDV(f U) =f DVU+V(f)U, and
[f U, V]g= (f U)(V(g))−V(f U(g))
=f(U(V(g))−V(f)U(g)−f V(U(g)) =f[U, V]g−V(f)U(g)
for allg∈ C∞(M), soDf UV −DV(f U)−[f U, V] =f(DUV −DVU −[U, V] and thusT(f U, V) = f T(U, V). The torsion is clearly antisymmetric, soT is
C∞(M)-bilinear
By the previous lemma, the torsion mapT descends to aC∞(M)-linear map
ˆ
T : Γ(T M)⊗Γ(T M) 7−→ Γ(T M). The image of two vector fields under this
mapTˆcan locally be expressed in terms of the Christoffel symbols: using equation 2.40 and the definition, we find that
T(∂i, ∂j) =D∂i∂j−D∂j∂i−[∂i, ∂j] = Γkij∂k−Γkji∂k,
(2.43)
since the partial derivates commute when acting on smooth functions. What is maybe the most important in object in general relativity (besides the metric), is the curvature ofD.
Definition 2.29. Thecurvature ofDis theR-trilinear map F : Γ(T M)×3 −→Γ(T M)
(U, V, W)7−→DU(DVW)−DV(DUW)−D[U,V]W
(2.44)
Lemma 2.30. The curvature isC∞(M)-trilinear.
Proof. LetU, V, W ∈Γ(T M)andf ∈ C∞(M). Then
Df U(DVW) =f DU(DVW),
DV(Df UW) =DV(f DUW)
=f DV(DUW) +V(f)DUW,
D[f U,V]W =Df[U,V]−V(f)UW
=f D[U,V]W −V(f)DUW,
(2.45)
so
F(f U, V, W) =f F(U, V, W) (2.46) SinceF(U, V, W) =−F(V, U, W), it follows thatF(U, f V, W) =f F(U, V, W).
Finally, it holds that
DU(DV(f W)) =DU(f DVW +V(f)W)
=f DU(DVW) +U(f)DVW +V(f)DUW+
U(V(f))W,
DV(DU(f W)) =f DV(DUW) +V(f)DUW +U(f)DVW+
V(U(f))W,
D[U,V](f W) =f D[U,V]W + [U, V](f)W,
2.5. Einstein’s field eq uations
and thusF(U, V, f W) =f F(U, V, W).
From Lemma 2.30 it follows thatFdescends to aC∞(M)-linear map
ˆ
F : Γ(T M)⊗Γ(T M)⊗Γ(T M)−→Γ(T M), (2.48)
and thus thatF corresponds to aC∞(M)-linear map
R: Γ(T∗M)⊗Γ(T M)⊗Γ(T M)⊗Γ(T M)−→ C∞(M), (2.49)
called theRiemann curvature tensor. Again, we can locally express the components of the mapFˆin terms of the Christoffel symbols. A quick calculation shows that
ˆ
F(∂i, ∂j, ∂k) = (∂iΓljl)∂l−(∂jΓlil)∂l+ ΓljkΓl
0
il∂l0−ΓlikΓl 0
jl∂l0. (2.50)
For physical reasons, it turns out be interesting to consider a very specific type of connection, namely the Levi-Civita connection, which always exists, and is unique.
Lemma 2.31. There exists a unique connectionDon(M, g), called theLevi-Civita connection, such thatDis torsion-free, i.e. TD = 0, andg-compatible, i.e. for all
U, V, W ∈Γ(T M)it holds thatU(g(V, W)) =g(DUV, W) +g(V, DUW).
Proof. See Theorem 13.9 in [21].
Now let(M, gL)be aspacetime manifold, i.e. a non-compact and connected Lorentzian manifold, and letDbe the Levi-Civita connection associated togL. Since the modulesΓ(TM|U)andΓ(T∗M|U)are now free of rank4with bases given by {∂1, ∂2, ∂3, ∂4}and{dx1, dx2, dx3, dx4}respectively, we have the isomorphism
Hom(Γ(T∗M|U)⊗Γ(TM|U)⊗3)= Γ(∼ TM|U)⊗Γ(T∗M|U)⊗3, (2.51) soRcorresponds to an elementR ∈ Γ(TM|U)⊗Γ(T∗M|U)⊗3. This means that there exist smooth functions{Rα
βγδ ∈ C
∞(U) :α, β, γ, δ∈[4]}such that
R=Rαβγδ∂α⊗dxβ⊗dxγ⊗dxδ, which is the local representation of the Riemann curvature tensor. TheRicci tensorRicis defined as the contraction
Ric =Rαβαδdxα(∂α)dxβ⊗dxγ
=Rαβαδdxβ⊗dxδ, (2.52)
where for each α0 ∈ [4],dxα0(∂α0) : M 7−→ C∞(M) sends anym ∈ U to10
dxα0|m(∂α0|m) = 1, and is thus the constant function with value1 on U, so
