Defining a Time-Orientation

Now that we have shown that our adjoined manifoldS

FMi has well-defined

Lorentzian metric ˜g, we can discuss time-orientations. Again, it is not the case in general that every collectionτiof time-orientations can induce a time-

orientation onS

FMi. However, if we assert that theτiare pairwise compat-

ible, then we obtain the following result.

Lemma 6.4.If the fij preserve time-orientations, then there is a time-

orientationτ˜ onS

FMi such that theφi preserve time-orientations.

Proof. Recall that fij preserves time-orientations whenever dfij(τi(p)) =

τj(fij(p)) for every pin Aij. We define the time orientation ˜τ by ˜τ([p, i]) =

dφi(τi(p)). Observe first that this is well-defined, since whenever [fij(p), j] =

[p, i], we have that

We now check that ˜τ is smooth, i.e. we check that for every point inS

FMi

there is an open neighbourhood and a locally-defined vector field that agrees with ˜τ. So, let [p, i] be a point inS

FMi, and considerpinMi. By assumption

the time-orientationτionMi is smooth, so there is some open neighbourhood

U ofpand a vector fieldXU onUsuch that for eachp′inU, it is the case that

XU(p′)∈τi(p). Consider now the subsetφi(U), which is an open neighbour-

hood of [p, i] (since φi is an open map). Since φi is a smooth embedding, in

particular it is a diffeomorphism onto φi(Mi) and thus by Lemma A.9 it is a

diffeomorphism once restricted toU. Thus we can pushforward the vector field XU alongφi to obtain the vector field (φi)∗XU. Recall that the pushforward

of a vector field along a diffeomorphism is defined to be:

(φi)∗XU([p′, i]) :=dφi◦XU(φ−i 1([p

′

, i])) =dφi◦XU(p′).

We now show that this vector field will suffice to witness the smoothness of ˜τ at [p, i]. Consider some element [q, j] of φi(U). Since φi(U) ⊆ φi(Mi),

it must be the case that [q, j] = [fij(p′), j] = [p′, i] for some p′ ∈ Aij ∩

U. Then ˜τ([p′_{, i]) = dφ}

i(τi(p′)). It follows from the definition of (φi)∗XU

that (φi)∗XU([p′, i]) = dφi◦XU(p′). By assumption XU(p′) ∈ τi(p′), thus

dφi◦XU(p′)∈dφi(τi(p′)) = ˜τ([p′, i]), and we may conclude that ˜τ is indeed

smooth. Observe that the canonical mapsφipreserve time-orientations follows

immediately from the construction of ˜τ. ⊓⊔

We suggested in Section 5.3.3 that when a spacetime is non-Hausdorff, it may not be the case that the two definitions of a time-orientation coincide. As such, we cannot conclude from the above result that the existence of a time-oreintation ˜τ entails the existence of a globally-defined timelike vector field. As a small aside, the following is a sufficient condition to guarantee the existence of globally-defined timelike vector field on the adjoined manifold S

FMi.

Lemma 6.5.Let{Xi}be a collection of globally-defined, timelike vector fields

for the Mi such that the diagram

T Aij T Aji Aij Aji dfij Xi fij Xj

commutes for each i and j. Then (S

FMi, g) has a globally-defined timelike

vector field.

Proof. By assumption, we can use Lemma 5.13 to define a section ˆX of the bundleS

T(S

FMi). We define the vector field ˜X : SFMi → T(SFMi) to be ˜X :=

Ψ◦Xˆ. This means that ˜

X([p, i]) =Ψ([(p, Xi), i]) = ([p, i],(dφi)p(Xi(p))).

It is not hard to see that this is indeed a globally-defined section ofT(S

FMi)

– the argument is similar to that of Thm.6.3. To see that ˜X is timelike, suppose we have some [p, i] ∈ S

FMi, and consider ˜X([p, i]) ∈ T[p,i]SFMi.

Then ˜X([p, i]) = ([p, i],(dφi)p(Xi(p))). By definition of the Lorentzian metric

˜

g, we have that ˜

g[p,i]((dφi)p(Xi(p)),(dφi)p(Xi(p))) = (gi)p(Xi(p), Xi(p)).

which is timelike since we assumed thatXi is timelike onMi. ⊓⊔

The next theorem is a summary of the results thus far.

Theorem 6.6.Let {(Mi, gi)} be a countable collection of Hausdorff space-

times, and F= (X,A,f)an adjunction system in which: 1.Xconsists of the spaces Mi,

2. eachAij is an open Lorentzian submanifold ofMi, and

3. each fij : Aij → Mj is an isometric embedding that preserves time-

orientation.

Then the adjunction space S

FMi possesses a Lorentzian metric that makes

the canonical maps φi act as open, isometric embeddings that preserve time-

orientation.

Proof. We can use Theorem 6.3 to conclude that S

FMi has a Lorentzian

metric, and we can use Lemma 6.4 to define a time-orientation ˜τ onS

FMi

that makes everyφi preserve time-orientations. ⊓⊔

We will refer to adjunction spaces satisfying the conditions of Theorem 6.6 as adjoined spacetimes. We complete this section by showing that the adjoined spacetime (S

FMi,g) possesses a certain universal property.˜

Lemma 6.7.Let F be as in Theorem 6.6. Suppose that there are smooth isometric embeddings ψi :Mi →N that commute on overlaps. Then there is

a unique isometric embedding from S

FMi toN.

Proof. We know from Lemma 5.11 that there is at least a unique smooth embeddingξ:S

FMi→N where [p, i]7→ψi(p). To see thatξis an isometry,

we need to show that ˜g[p,i](v, w) = gNξ([p,i])((dξ)[p,i](v),(dξ)[p,i](w)), however

this follows from our assumption that each ψi is an isometric embedding.

˜

g[p,i](v, w) = (gi)p((dφ−i 1)[p,i](v),(dφ−i1)[p,i](w))

= (gN₎

ψ(p)((dψi)p◦(dφ−i1)[p,i](v),(dψi)p◦(dφ−i1)[p,i](w))

= (gN₎

ψ(p)(d(ψi◦φ−i1)[p,i](v), d(ψi◦φ−i 1)[p,i](w))

= (gN)ξ([p,i])((dξ)[p,i](v),(dξ)[p,i](w))

Thusξis an isometric embedding fromS

FMi intoN. ⊓⊔

6.1.3 Causal Properties of Adjoined Spacetimes