B2r(0) for all t ∈ (α, ˜˜ β). In particular f(β˜) > f(α˜). From the mean value theorem, there is a t?∈(α, ˜˜ β)such that
f˙(t?) = f(β˜)−f(α˜) β˜−α˜
>0.
Now sincet?is an element of ˜I, one also hasγ(t?)∈B2r(0)and hence ˙f(t?) =hX(γ(t?)),γ(t?)i<
µ0<0 which is a contradiction. HenceBr(0)is an attracting neighbourhood of 0.
The proof for the existence of a repelling neighbourhood follows from the fact, that it is
attracting for−X.
Lemma1.1.19. Let X∈X(M)be a smooth vector field on M and let p∈ M be an attractor or repeller of X. Letγ:(α,β)→M be a maximal integral curve of X repelled or attracted by p, i.e. eitherγ(t)→ p orγ(−t)→p for t→∞. Then eitherγis a constant curve or it is not closed, i.e. for all t16=t2∈(α,β) one also hasγ(t1)6=γ(t2).
Proof: Assume there aret1,t2∈(α,β)such thatγ(t1) =γ(t2) =:qand denote∆:=t2−t1. Then
˜
γ(t):=γ(∆+t)fulfils ˜γ(t1) =qand ˙˜γ(t1) =Xγ(t˜ 1)=Xγ(t2) =Xγ(t1). Henceγand ˜γcoincide by uniqueness of integral curves. The limitst→∞andt→ −∞exist if and only ifγis constant,
in particularγ≡ p.
1.2 causal structure of lorentzian manifolds
This section will focus on pseudo-Riemannian manifolds(M,g) of dimensionn and signature (1,n−1). Such a signature is also called Lorentzian signature. Basically this section follows [O’N83] and provides basic definition and concepts concerning the causal structure in Lorentzian geometry.
Definition First the causal character of a vector is defined. Considerp ∈ M, a non-vanishing tangent vectorX∈TpMis called
timelike ⇔ g(X,X)<0 null ⇔ g(X,X) =0 spacelike ⇔ g(X,X)>0 causal ⇔ g(X,X)≤0.
A embedded submanifold N of (M,g) may not have a induced metric that is degenerate.
In that case, the null vector that defines the degenerate direction of gp in TpN also is called isotropic vector. Any non-trivialtotally isotropic vector subspace3V ⊂TpMin Lorentzian signature is of dimension1. An orthonormal frame{ei}ofTpMwill always refer to a basis with indices running from 0 ton−1 and for which gp ei,ej
=ηijwhereηis the Minkowski metric. Hence the timelike direction is given bye0
Remark1.2.1. Let beei an orthonormal frame inTpMwith g(e0,e0) =−1. Then the frame{ni}, defined by
ni =e0+ei fori>0 (1.59)
n0= n−1 n−2e0−
√n−1
n−2 (e1+· · ·+en−1). (1.60) is a basis of null vectors with the propertyg(ni,nj) = δij−1, where δij is the Kronecker delta.
Such a basis will be callednull basis.
3 A subspaceV ⊂TpMwill be called totally isotropic ifgp(X,Y) =0 for allX,Y∈ V.
22 Chapter 1: differential geometry
Regarding to the causal character of a vector one defines the following subsets of the tangent space.
Definition Let(M,g)be a Lorentzian manifold,p∈ M. Then
TpM:= nX∈TpM | kXk2:=g(X,X)<0o
(1.61) CpM:= nX∈TpM | kXk2:=g(X,X) =0,o
(1.62) KpM:=TpM∪CpM. (1.63) The subset of all timelike vectorsTpMis calledtime coneat p, while the subset of all null vectors CpMis callednull coneat p. To distinguish them from a similar definition that is given later, they will be referred to as tangent time cone and tangent null cone. The union KpMof both sets is called(tangent) causal cone.
For keeping notations short, g(X,X) is shorten to kXk2 if there is no confusion about the metric in use 4. Otherwise it will be made explicit by writing kXk2g. The causal character of curves in(M,g)is specified in a similar way.
Definition A smooth curveγ:I→M, with I⊂Rbeing an interval, is called timelike ⇔ kγ˙(t)k2<0 ∀t∈I
null ⇔ kγ˙(t)k2=0 ∀t∈I spacelike ⇔ kγ˙(t)k2>0 ∀t∈I causal ⇔ kγ˙(t)k2≤0 ∀t∈I
if in addition it has nowhere vanishing tangent vector ˙γ(t). Assumeγto be smooth only piece-wise and V : I → TM a not necessarily continuous map with V(t) ∈ Tγ(t)Mbeing a timelike vector for allt∈ I. Thenγ usually is called timelike (null, causal), if it is timelike (null, causal) on its smooth parts and if in addition for allt∈ I
gγ(t) γ˙(t−),V(t)·gγ(t) γ˙(t+),V(t)>0.
Here ˙γ(t+):=lims&tγ˙(s)and ˙γ(t−):=lims%tγ˙(s).
Definition A Lorentzian manifold(M,g)is said to betime-orientableif it admits a non-vanishing timelike vector fieldOT ∈X(M). Thetime-orientationwith respect to that vector field then is the decomposition
TpM=T↑pM∪T↓pM (1.64) at each point p ∈ M, with T↑pM := {X ∈ TpM|g(X,OT) < 0} being the set offuture-directed timelike vectors atp, whileT↓pM:={X∈TpM|g(X,OT)>0}is the set ofpast-directedtimelike vectors atp. A similar notation is used for the null and the causal cone.
CpM\ {0}=C↑pM∪C↓pM. (1.65) KpM\ {0}=K↑pM∪K↓pM. (1.66) Consequently, if(M,g)is time-oriented with respect to OT ∈ X(M), then a causal vector X ∈ TpMor a causal curveγ:I→Mwith tangent vector ˙γ(t)is called
future-directed ⇔ g(γ,˙ OT)<0 ∀t∈I org(X,OT)<0 past-directed ⇔ g(γ,˙ OT)>0 ∀t∈I
org(X,OT)>0
Definition [causality relation]Let(M,g)be an time-oriented Lorentzian manifold,U⊂ Mand x,y∈ M. Then the following notation is used
4 Recall that despite the notation, this is not a norm.
1.2 causal structure of lorentzian manifolds 23
x≪Uy ⇔
∃γ:[0,e6=0]→U smooth on(0,e) γ(0) =x,γ(e) =y
γfuture-directed, timelike curve x<Uy ⇔
∃γ:[0,e6=0]→U smooth on(0,e) γ(0) =x,γ(e) =y
γfuture-directed, causal curve x≤Uy ⇔ x<U yorx=y.
IfU=Mthe subscriptUis omitted and the notation isx≪y,x <yand x≤y. Following the notation in [HE73] special subsets characterising the causal structure of a Lorentzian manifold will now be defined. To this end let beS,T⊂ M. Thechronological future/past ofSrelative toT is
I+(S,T):={x ∈T| ∃y∈S:y≪T x} (1.67) I−(S,T):={x ∈T| ∃y∈S:y≫T x}. (1.68) IfS = {p}is a single point, the chronological sets will be denoted I±({p},T) =: I±(p,T)and forT being the whole manifold, it will be written as I±(S,M) =: I±(S). Thecausal future/past ofSrelative toTis
J+(S,T):= (S∩T)∪ {x∈T| ∃y∈S:y≤Tx} (1.69) J−(S,T):= (S∩T)∪ {x∈T| ∃y∈S:y≥Tx}. (1.70) Again short notations will be J±(p,T)for single points andJ±(S)if applied to the whole mani-fold. Now the set of sets
[
p∈M
I+(p),I−(p),J+(p),J−(p)
is called thecausal structureof(M,g). Thefuture/past horismos ofSrelative toTis
E+(S,T):=J+(S,T)\I+(S,T) (1.71) E−(S,T):=J−(S,T)\I−(S,T) (1.72) Short notations areE±(p,T),E±(S).
Definition LetV=V(p,U)be one of the causal sets defined above. TheclosureV(p,U),interior V(◦ p,U)andboundary∂V(p,U)are the topological quantities taken with respect to the open set U.
ConsiderU⊂Mto be open, then the setsI±(S,U)are open, since ify∈Ucan be reached by a future- or past-directed timelike curve fromS, then it has a sufficiently small neighbourhood that can be reached by a small variation of that curve, without changing its causal character.
Moreover one finds for the closure, interior and boundary of the sets defined above
I±(p,U) =J±(p,U) I◦±(p,U) =J◦±(p,U) ∂I±(p,U) =∂J±(p,U). (1.73) Furthermore for all points x ∈ E±(S,U) there exists a future- or past-directed null geodesic γ:I→Mwithγ(0)∈Sandγ(e) =xfor some e≥0. IfUis a convex normal neighbourhood of p ∈ M, thenE±(p,U) is the union of null geodesics starting in p. As a result one gets for a convex normal neighbourhoodU:
I±(p,U) =J◦±(p,U) E±(p,U) =∂J±(p,U). (1.74) Definition Let (M,g)be a time-oriented Lorentzian manifold, U ⊂ Mand p ∈ M. Then the geodesic future/past null coneinprelative toUis
Cp±(U) ={p} ∪
x∈U
∃γ:[0,e]→Unull geodesic : γ(0) =p, γ(e) =x
. (1.75)
24 Chapter 1: differential geometry
In particular, ifUis a convex normal neighbourhood of p, then the geodesic null coneCp±(U) inpcoincides with the horismosE±(p,U)of p.
Definition Let(N,h)be a submanifold of a Lorentzian manifold(M,g)with dim(M)>2 and hbeing the bilinear form induced onNbyg. Then(N,h)is called
spacelike submanifold ⇐⇒ his a positive definite metric.
timelike submanifold ⇐⇒ his a Lorentzian metric.
null submanifold ⇐⇒ hdegenerates.
Consider N to be a codim = 1 manifold. Then the above definitions are equivalent to the existence of a vectornp∈TpMfor eachp∈ Nsuch that the tangent spaceTpNis orthogonal to np, i.e. g(np,V) =0∀V∈TpN.(N,h)is
spacelike ⇐⇒ knpk2<0 ∀p∈ N.
timelike ⇐⇒ knpk2>0 ∀p∈ N.
null ⇐⇒ knpk2=0 ∀p∈ N.
If(N,h)is a null hypersurface, thennpidentifies theisotropic directiononTpN.
From here we will implicitly assume that the considered Lorentzian manifolds are time-oriented.
1.2.1. Geodesics on Lorentzian Manifolds
A curveγ : I → M is ageodesic, if∇γ˙γ˙ = 0 alongγ. It is amaximal geodesic, if its domain I is inextendible. An often used fact is that for eachp∈ MandX∈TpMthere is a unique maximal geodesicγX : I(X)→Msuch thatγX(0) = pand ˙γ(0) = X. The intervalI(X)depends on the vectorX. Moreover the set
Dp:=X∈TpM|1∈ I(X) (1.76) is open and star-shaped with respect to the origin 0∈ TpM.
Definition Letα,β∈R+\ {0}. A geodesic segmentγ:[t0,t0+α]→Misclosedif γ(t0+α) =γ(t0) γ˙(t0+α) =βγ˙(t0). It is calledperiodic, if β=1.
Assume β<0 and consider the geodesicη :[t0,t0−βα] →M defined byη(t) :=γ(α−tt0 + t0+1
βt). Thenη(t0) =γ(t0+α) =γ(t0)and ˙η(t0) = 1
βγ˙(t0+α) = γ˙(t0). Thereforeη = γby uniqueness of geodesics. In particular the choice ˜t:=t0−1−β
βα∈[t0,t0−βα]∩[t0,t0+α]leads to 1βγ˙(t˜) =γ˙(t˜), which is a contradiction, since βwas assumed to be negative. In particular this shows the impossibility of valuesβ<0.
As a matter of fact due to [O’N83, Proposition 7.13], the maximal geodesic extension of a closed segment is maximal if and only ifβ =1. Moreover following the proof therein one has the following corollary.
Corollary1.2.2. Letγ:I→M be a maximal null geodesic extension of a closed geodesic segment such thatγ(t0) =γ(t0+α)andγ˙(t0+α) =βγ˙(t0), then
I=
−∞,t0+αβ−1β
forβ>1 R forβ=1 t0−α1−ββ,∞
forβ∈(0, 1)
(1.77)
1.2 causal structure of lorentzian manifolds 25
In some situations it is beneficial to have a definition for curves that are geodesics up to reparametrisation.
Definition Let (M,g) be a semi-Riemannian manifold, I ⊂ R an interval. A smooth curve γ: I → Mis apregeodesicif it has nowhere vanishing differentialdγand there exists a smooth functionc:I→Rsuch that
(∇γ˙γ˙) (t) =c(t)γ˙(t). (1.78) In most situation it is much simpler to get pregeodesics instead of geodesics. That often it is sufficient to work with the former is provided by the following lemma.
Lemma1.2.3. Letγ : I → M be a pregeodesic satisfying Equation (1.78) for some smooth function c.
Then there exists a reparametrisation5h:I0→I such thatγ˜ :=γ◦h is a geodesic in M.
An outline to the proof is given in [O’N83] of which a detailed elaboration is provided in the appendix. Some of its calculations will be used later. For time- or spacelike pregeodesics the reparametrisation is done by normalising to constant length. The reparametrised curve ˜γthen satisfies dsdg(γ˜0, ˜γ0) =0.
1.2.2. Exponential Map
Consider p ∈ M. The maximal domain DpM ⊂ TpM defined in Equation (1.76) by maximal geodesics originating inpcan be used to define theexponential mapexppin p. It is
expp: DpM → M X 7→ γX(1),
where γX is the maximal geodesic with initial tangent vector X ∈ DpM. The exponential map is a local diffeomorphism in a neighbourhood of the origin 0 ∈ TpM, i.e. there exists a star-shaped neighbourhoodU⊂ Dp of 0 such that expp:U→expp(U)is a diffeomorphism [O’N83, Proposition3.30]. The set expp(U)is called anormal neighbourhoodof p. As a consequence there is a normal neighbourhood for eachp∈ M. An open setU ⊂Mis calledconvexprovided it is a normal neighbourhood of each of its points. A simple important consequence of the existence of a convex neighbourhood for each point inM (see for example [O’N83, Proposition5.7]) is that geodesics do not “end” within the manifold. The fact will be made a lemma, as it is used later on.
Lemma1.2.4. Let U ⊂ M be a convex open set and γ : [0,t0) → U a geodesic such that the limit limt→t0γ(t)∈ U exists. Then t0<∞andγadmits a geodesic extension beyond t0.
Another important property is that locally the exponential map expp is a radial isometry (Gauß lemma), i.e. gp(X,W) = gq
hdexppi
X(X),h dexppi
X(W) for all X,W ∈ TpM with the identification TX TpM
' TpM and with the notation q = expp(X) ∈ Up. In case where a geodesic starting at a point p is explicitly written in terms of the exponential map expp, a consequence of Corollary1.2.2can be formulated as follows.
Lemma1.2.5. Let p∈ M be a point and X∈ TpM. Consider the geodesic defined byγ(t) =expp(tX). Ifγis a closed geodesic withγ(t0) =γ(t0+α)andγ˙(t0+α) =βγ˙(t0))forα,β∈R+\ {0}then
(i) t0<α1−ββ forβ∈(0, 1)and there is no restriction to t0ifβ≥1.
(ii) ∃t∈[t0,t0+α]withγ(t) =p.
The restriction to positive values ofβis reasonable since by Corollary1.2.2the set of geodesics with β ≤ 0 is empty. The proof is left to the appendix just as the proof for the following proposition.
5 A reparametrisation is a smooth, surjective map with nowhere vanishing tangent vector
26 Chapter 1: differential geometry
Proposition1.2.6. Consider (M,g) to be a time-oriented Lorentzian manifold, p ∈ M, U a normal neighbourhood of p andCp(U)the geodesic null cone in p. Furthermore letN ∈X(U)be a vector field on U with the following properties:
N |U \{p}6=0 kN k2
Cp(U)=0.
Nx∈TxCp(U) for x∈ Cp(U)
In particularN defines the isotropic direction on the tangent space ofCp(U)\ {p}. Let T∈ T(p,0)M be a tensor which is annihilated byN along the null coneCp(U)\ {p}, i.e.
T(N,·, . . . ,·) =|C
p(U)0.
Then T vanishes at p
Tp=0.
1.2.3. Jacobi Fields and Causality Theorem
Null, time and causal cone of a point locally can be related to the image of the exponential map at that point. The main theorems dealing with this issue will be summarised in this section. A more extended analysis can be found in [O’N83] and [HE73].
First some notations will be fixed. Let N ⊂ M be some submanifold of (M,g) with non-degenerated induced metrich. The generic projections of the tangent space or normal space at a point p∈ Nare denotedπNp :TpM→TpNand π⊥p :TpM→TpN⊥. A geodesicγ:I → Mis said to benormaltoNifγ(0)∈ Nand ˙γ(0)∈TN⊥.
Definition Thenormal connection∇⊥:X(N)⊥→Γ T∗N⊗TN⊥
of Nacting on sections of the normal bundle is denoted by
∇⊥XY:=π⊥∇gXY, (1.79) whereX∈ X(N)andY∈X(N)⊥. The difference of the Levi-Civita and the normal connection is denoted by
˜II(X,Y):=∇gXY− ∇⊥XY. (1.80) Definition Letγbe a geodesic on(M,g). A vector field J ∈Xγ(M)onγis anJacobi fieldif the Jacobi equation
∇γ˙∇γ˙J=Rg(J, ˙γ)γ˙ (1.81) is satisfied. Providedγis normal to N, a vector field J∈Xγ(M)onγis aN-Jacobi fieldif
(i) J(0)∈Tγ(0)N
(ii) πN(∇γ˙J) (0) =II˜(J(0), ˙γ(0)) (iii) Jis Jacobi field onγ.
As a matter of fact for eachX,Y∈TpMthere is a unique Jacobi fieldJonγsatisfyingJ(0) =X and(∇γ˙J) (0) =Y[O’N83, Lemma8.5]. Furthermore in case of the geodesicγbeing normal to the submanifoldNthen a Jacobi field Jis the variation vector field of a variation of γthrough normal geodesics if and only if it is anN-Jacobi field onγ[O’N83, Proposition10.28]. Avariation ofγ:I= [t0,t1]→Mis a family of curves
δγ:I×(−e,e)→M (1.82) such that∀t ∈ I : γ(t) = δγ(t, 0). It is a geodesic variation, if δγ(·,s)are geodesics for all fixed s∈(−e,e). If the derivative with respect to the second parameter is indicated by an prime then the vector fieldδγ0(t, 0)onγis thevariation vector fieldofδγ.