As it was mentioned in the introduction, we try to point out a possible way to construct a new causal boundary for M, called the light boundary or L–boundary, introduced by R. Low in [47]. It will be a conformal extension of the conformal manifold and its construction requires the use of geometrical structures of the space of light rays. A first study of this new boundary is done in [7] including some initial results and examples comparing it with the c-boundary developed in [27]. Later, a more complete article [8] was release.
Preliminary: the causal boundary
In [27], the c–boundary or causal boundary is introduced to improve others already existing boundaries (Geroch’s [26], Schmidt’s [64], etc.) It is conformally invariant and defined intrinsically adding endpoints, as subsets in the spacetime M, to causal inextensible curves.
Definition 12
We will say that \(W\subset M\) is an indecomposable past set or IP if W is open, non–empty and a past set, that is \(I^{-}(W)=W\), which can not be expressed as the union of two proper subsets satisfying the same previous properties.
If there exists \(p\in M\) such that \(I^{-}(p)=W\), we will say that W is a proper IP or PIP. We will name W as a terminal IP or TIP whenever such \(p\in M\) does not exist.
The future causal boundary or future c–boundary is defined by the set of all TIP contained in M.
In an analogous way, the past c–boundary is defined from indecomposable future sets, PIFs and TIFs.
The following proposition characterizes the TIPs as chronological past of future inextendible timelike curves and states that points in the c–boundary can also be defined by causal curves. Analogue results can be stated for TIFs. See [32, Prop. 6.8.1] and [23, Prop. 3.32(i)] for proofs.
Proposition 11
Let M be a strongly causal spacetime.
-
1.
A set \(W\subset M\) is a TIP if and only if there exists a future inextendible timelike curve \(\lambda \subset M\) such that \(I^{-}(\lambda )=W\).
-
2.
If \(\lambda \subset M\) is a future inextendible causal curve, then \(I^{-}(\lambda )\) is a TIP.
An immediate consequence of Proposition 11 is the following corollary.
Corollary 4
Let M be a strongly causal spacetime. \(I^{-}(\gamma )\) is a TIP for any light ray \(\gamma \in {\mathcal {N}}\).
Observe that the points \(p\in M\) can be identified by its corresponding PIPs \(I^{-}(p)\) or PIFs \(I^{+}(p)\) and the gaps and points at infinity of M are identified with TIPs \(I^{-}(\lambda )\) or TIFs \(I^{+}(\lambda )\) for some future inextensible timelike curve \(\lambda \) (Fig. 9). Therefore, by Corollary 4, light rays also define points in the c–boundary.
Remark 9
As Proposition 11 suggests in its statements, not every TIP can be defined by light rays because such ideal points may be accessible only by timelike curves. For example, let us consider \(M=\{(t,x)\in \mathbb {M}^2 : t<-2\vert x \vert \}\) as submanifold of the 2–dimensional Minkowski spacetime with the standard metric \({\mathbf {g}}=-dt\otimes dt + dx \otimes dx\). The ideal point (0, 0) is not accessible by light rays contained in M. In fact, the TIP defined by the curve \(\lambda (s)=(s,0)\) for \(s\in (-\epsilon , 0)\) is the whole spacetime M, that is \((0,0)\sim I^{-}(\lambda )=M\) (see Fig. 10). This example can be easily generalized to higher dimensions.
The identification \(p\sim I^{-}(p)\sim I^{+}(p)\) presents some inconsistencies affecting even to the topology of the spacetime’s completion and many authors have tried to solve this problem (Budic et al. [12], Rácz [62], Szabados [66, 67], Harris [29, 30], Marolf et al. [49, 50] among others) but finally, Flores et al. [23] stated a consistent definition of the c–boundary. Anyway, this matter is beyond the scope of this review, so, for our purposes, we will only need the classical definition of the causal boundary above. See [24, 63] and [23] for an overview on this subject.
Low’s idea
As indicated in [47], Low introduces a new idea for the construction of a causal and conformal boundary in M. It consists in the addition of skies at infinity, in virtue of the equivalence between events and skies by the sky map (see Sect. 3).
It is expected that a sky at infinity will be the limit of the skies of the points of a fixed light ray when the parameter tends to the future/past endpoint of its domain of definition. Low wonders if this is a new way to obtain the c–boundary using the geometry of the space of light rays.
In order to build these boundary points, we will take some future–directed inextensible null geodesic \(\gamma :\left( a,b\right) \rightarrow M\), then we will consider the curve \({\widetilde{\gamma }}:\left( a,b\right) \rightarrow \mathrm {Gr}^{m-2}\left( {\mathcal {H}}_{\gamma }\right) \) defined by
$$\begin{aligned} {\widetilde{\gamma }}\left( s\right) = T_{\gamma }S\left( \gamma \left( s\right) \right) \end{aligned}$$
where \(S(\gamma (s))\in \varSigma \) is the sky of \(\gamma (s)\in M\). Since each S(p) is diffeomorphic to the sphere \(\mathbb {S}^{m-2}\), then \(T_{\gamma }S\left( \gamma \left( s\right) \right) \) is contained in the Grassmannian manifold \(\mathrm {Gr}^{m-2}\left( {\mathcal {H}}_{\gamma }\right) \) of \(\left( m-2\right) \)–dimensional subspaces of \({\mathcal {H}}_{\gamma }\subset T_{\gamma }{\mathcal {N}}\). Then, we can define endpoints of the curve \({\widetilde{\gamma }}\) by
$$\begin{aligned} \begin{array}{l} \ominus _{\gamma } = \lim _{s\mapsto a^{+}}{\widetilde{\gamma }}\left( s\right) \in \mathrm {Gr}^{m-2}\left( {\mathcal {H}}_{\gamma }\right) , \\ \oplus _{\gamma } = \lim _{s\mapsto b^{-}}{\widetilde{\gamma }}\left( s\right) \in \mathrm {Gr}^{m-2}\left( {\mathcal {H}}_{\gamma }\right) , \end{array} \end{aligned}$$
(5.1)
when the limits exist. In general, the existence of \(\ominus _{\gamma }\) and \(\oplus _{\gamma }\) is not clear, although the compactness of \(\mathrm {Gr}^{m-2}\left( {\mathcal {H}}_{\gamma }\right) \) assures the existence of accumulation points when \(s\mapsto a^{+},b^{-}\). Moreover, even in case of the existence of the limits, we wonder if \(\ominus ,\oplus :{\mathcal {N}}\rightarrow \mathrm {Gr}^{m-2}\left( {\mathcal {H}}\right) \) are smooth distributions in \({\mathcal {N}}\). Low defines the future/past endpoint of the light ray \(\gamma \subset M\) for this new boundary of M as the integral manifold of the distributions \(\oplus / \ominus \), which will comprise all light rays arriving at/emerging from the same point at infinity than \(\gamma \), see [47].
Recall that, at the end of Sect. 2.5, we have seen that \(\left( {\mathcal {H}}_{\gamma },\left. \omega \right| _{{\mathcal {H}}_{\gamma }}\right) \) is a symplectic vector space for any \(\gamma \in {\mathcal {N}}\), where \(\left. \omega \right| _{{\mathcal {H}}_{\gamma }}\) satisfies the expression (2.11). Then, it is easy to show that for any sky \(X\in \varSigma \) such that \(\gamma \in X\), then \(T_{\gamma }X\) is its own symplectic orthogonal vector space, that is,
$$\begin{aligned} T_{\gamma }X=\left( T_{\gamma }X\right) ^{\perp }\equiv \{ \langle J \rangle \in {\mathcal {H}}_{\gamma }:\left. \omega \right| _{{\mathcal {H}}_{\gamma }}\left( \langle J \rangle ,\langle K \rangle \right) =0 \text { for all } \langle K \rangle \in T_{\gamma }X \} \end{aligned}$$
therefore \(T_{\gamma }X\) is a lagrangian subspace of \({\mathcal {H}}_{\gamma }\). So, if we denote by \({\mathscr {L}}\left( {\mathcal {H}}\right) \subset \mathrm {Gr}^{m-2}\left( {\mathcal {H}}\right) \) the manifold of Lagrange grassmannian subspaces in \({\mathcal {H}}\) and by \({\mathscr {L}}\left( {\mathcal {H}}_{\gamma }\right) \subset \mathrm {Gr}^{m-2}\left( {\mathcal {H}}_{\gamma }\right) \) the submanifold of Lagrange grassmannian subspaces in \({\mathcal {H}}_{\gamma }\), then \(\varLambda \in {\mathscr {L}}\left( {\mathcal {H}}_{\gamma }\right) \) if and only if \(\dim \varLambda = m-2\) and \(\left. \omega \right| _{\varLambda }=0\). So, the image of the maps \(\ominus ,\oplus \) can be restricted to \({\mathscr {L}}\left( {\mathcal {H}}\right) \) by
$$\begin{aligned} \ominus ,\oplus :{\mathcal {N}}\rightarrow {\mathscr {L}}\left( {\mathcal {H}}\right) , \end{aligned}$$
as well as \({\widetilde{\gamma }}\subset {\mathscr {L}}\left( {\mathcal {H}}_{\gamma }\right) \) holds.
The distributions \(\oplus \) and \(\ominus \) are independent of each other and they permit to build the future and past boundaries respectively. Therefore, the construction of one of these boundaries is also independent of that of the other. Thus, we will describe the construction of the future boundary from the distribution \(\oplus \), taking into account that the process to obtain the past boundary from \(\ominus \) is analogous.
We propose the following hypotheses for the general case \(\dim M \ge 3\):
- H1:
-
\(\left( M,{\mathcal {C}}\right) \) is strongly causal, null–pseudo convex, light non–conjugate and sky–separating.
- H2:
-
The distribution \(\oplus : {\mathcal {N}} \rightarrow {\mathscr {L}}\left( {\mathcal {H}}\right) \), defined by \(\oplus _{\gamma }=\lim _{s\mapsto b^{-}}T_{\gamma }S\left( \gamma \left( s\right) \right) \) for any maximally and future–directed parametrized light ray \(\gamma :\left( a,b\right) \rightarrow M\), is differentiable and regular.
Definition 13
A Lorentz conformal manifold M satisfying conditions H1 and H2 is said to be a L–spacetime.
Notice that H1 is required for \({\mathcal {N}}\) having good topological and differentiable properties and H2 are basic conditions on \(\oplus \) so that the L–boundary could be built.
For the sake of simplicity, we do not use the labels future and past in the definition of L–spacetime as the property H2 is verified for \(\oplus \) or \(\ominus \) respectively. We understand that this may be a bit ambiguous, but in this way we will avoid too many adjectives in later definitions. Therefore, as in what follows, we will build only the future boundary, L–spacetime should be understood as future L–spacetime.
Hypotheses for the 3–dimensional case
As a first approximation to the general case, we will study the L–boundary for 3–dimensional conformal manifolds following the original Low’s idea. (Fig. 11) Observe that in such case \({\mathcal {N}}\) is also 3–dimensional and the Lagrangian grassmannian manifold \({\mathscr {L}}\left( {\mathcal {H}}\right) \) becomes \(\mathbb {P}\left( {\mathcal {H}}\right) \) (in fact, \(\mathrm {Gr}^{1}\left( {\mathcal {H}}\right) ={\mathscr {L}}\left( {\mathcal {H}}\right) =\mathbb {P}\left( {\mathcal {H}}\right) \)). So the curve \({\widetilde{\gamma }}\) is contained in \(\mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \simeq \mathbb {S}^1\).
Notice that, if M is light non–conjugate, then the curve \({\widetilde{\gamma }}\left( s\right) =T_{\gamma }S\left( \gamma \left( s\right) \right) \in \mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \simeq \mathbb {S}^{1}\) is injective and therefore the continuity of \({\widetilde{\gamma }}\) would imply that the limits \(\oplus _{\gamma }\) and \(\ominus _{\gamma }\) exist consisting in lines in \(\mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \) (Fig. 11).
So, when M is 3–dimensional, the hypothesis H2 becomes
- H2:
-
The distribution \(\oplus : {\mathcal {N}} \rightarrow \mathbb {P}\left( {\mathcal {H}}\right) \), defined by \(\oplus _{\gamma }=\lim _{s\mapsto b^{-}}T_{\gamma }S\left( \gamma \left( s\right) \right) \) for any maximally and future–directed parametrized light ray \(\gamma :\left( a,b\right) \rightarrow M\), is differentiable and regular.
The rest of this Sect. 5 is devoted to the construction of the future L–boundary for \(m=3\), but we also believe that a similar way can be travelled in order to get the L–boundary for any dimension \(m\ge 3\).
The space \(\widetilde{{\mathcal {N}}}\) of tangent spaces to skies
First, let us consider the following natural map
$$\begin{aligned} \begin{array}{rrcl} \sigma : &{} {\mathbb {P}}{\mathbb {N}} &{} \rightarrow &{} \mathbb {P}\left( {\mathcal {H}}\right) \\ &{} \left[ u\right] &{} \mapsto &{} T_{\gamma _{\left[ u\right] }}S\left( \pi ^{{\mathbb {P}}{\mathbb {N}}}_{M}\left( \left[ u\right] \right) \right) \end{array} \end{aligned}$$
(5.2)
where \(\pi ^{{\mathbb {P}}{\mathbb {N}}}_{M}:{\mathbb {P}}{\mathbb {N}}\rightarrow M\) is the canonical projection. The assumption of M being light non–conjugate, by [7, Lem. 2.5], gives us the injectivity of \(\sigma \). But, moreover, this map permits to embed M (by its bundle \({\mathbb {P}}{\mathbb {N}}\) of null directions) into the geometry of \({\mathcal {N}}\) (by the bundle \(\mathbb {P}\left( {\mathcal {H}}\right) \) of lines in the contact structure \({\mathcal {H}}\)).
Proposition 12
The map \(\sigma :{\mathbb {P}}{\mathbb {N}}\rightarrow \mathbb {P}\left( {\mathcal {H}}\right) \) defined in (5.2) is a diffeomorphism onto its image. Moreover, \(\widetilde{{\mathcal {N}}}=\mathrm {Im}\left( \sigma \right) \) is an open submanifold of \(\mathbb {P}\left( {\mathcal {H}}\right) \).
(Sketch of the proof)
This proposition in proven in [8, Prop. 5.1] in two steps. First, the differentiability of \(\sigma \) is shown [8, Lem. 5.1] by the construction of \(\sigma \left( \left[ u\right] \right) \in \mathbb {P}\left( {\mathcal {H}}\right) \) for \(\left[ u\right] \in {\mathbb {P}}{\mathbb {N}}\) by differentiable composition of maps, using null geodesics variations which fix the point \(\pi ^{{\mathbb {P}}{\mathbb {N}}}_{M}([u])\in M\). Second, the differential \(d\sigma _{\left[ u\right] }\) is an isomorphism. In this second step, it is necessary to prove that the curve \({\widetilde{\gamma }}\left( t\right) =\sigma \left( \left[ \gamma '\left( t\right) \right] \right) \) is regular whenever the parameter t is affine [8, Lem. 5.2]. Finally, since both \({\mathbb {P}}{\mathbb {N}}\) and \(\mathbb {P}\left( {\mathcal {H}}\right) \) are 4–dimensional then the image \(\sigma \left( {\mathbb {P}}{\mathbb {N}}\right) =\widetilde{{\mathcal {N}}}\) is open in \(\mathbb {P}\left( {\mathcal {H}}\right) \). \(\square \)
Observe that, by construction, the manifold \(\widetilde{{\mathcal {N}}}\) is the space of all tangent spaces to skies of points of M. In [8, Sec. 3.1], \(\widetilde{{\mathcal {N}}}\) is named as the blow up of M.
Notice that, if \(\gamma =\gamma \left( t\right) \) is a null geodesic, then \(\sigma \left( \left[ \gamma '\left( t\right) \right] \right) =T_{\gamma }S\left( \gamma \left( t\right) \right) \in \mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \). So, the endpoints of the curve
$$\begin{aligned} {\widetilde{\gamma }}\left( t\right) \equiv \sigma \left( \left[ \gamma '\left( t\right) \right] \right) = T_{\gamma }S\left( \gamma \left( t\right) \right) \end{aligned}$$
(5.3)
define the distributions \(\oplus \) and \(\ominus \). Assuming the hypotheses H1, we have the following implications
$$\begin{aligned} {\widetilde{\gamma }}\left( t_1\right) ={\widetilde{\gamma }}\left( t_2\right)&\Rightarrow T_{\gamma }S\left( \gamma \left( t_1\right) \right) = T_{\gamma }S\left( \gamma \left( t_2\right) \right)&\text { (by definition)} \\&\Rightarrow S\left( \gamma \left( t_1\right) \right) = S\left( \gamma \left( t_2\right) \right)&\text { (by light non--conjugate)} \\&\Rightarrow \gamma \left( t_1\right) = \gamma \left( t_2\right)&\text { (by injectiveness of }S) \\&\Rightarrow t_1 = t_2&\text { (by injectiveness of }\gamma ) \end{aligned}$$
then any \({\widetilde{\gamma }}\) is also an injective curve.
In [8, Prop. 6.1], it is shown that \(\widetilde{{\mathcal {N}}}\subset \mathbb {P}\left( {\mathcal {H}}\right) \) is a submanifold with boundary under the hypotheses H1, H2 for both \(\oplus \) and \(\ominus \) with \(\oplus _{\gamma }\ne \ominus _{\gamma }\) for all \(\gamma \in {\mathcal {N}}\) . This last hypothesis is not critical, it is just a technical condition to simplify the construction. It is possible to show a more general statement only under conditions H1 and H2 with the same procedure used in [8], which is sufficient for the construction of the future boundary. Notice that, if \({\mathcal {N}}_{U}\) is the open set of all light rays passing through the globally hyperbolic, normal and causally convex open \(U\subset M\) according to remark 1, we can consider two smooth spacelike Cauchy surfaces \(C,C_{-}\subset U\) such that \(C_{-}\subset I^{-}(C)\). Observe that if \({\mathcal {H}}_U=\bigcup _{\gamma \in {\mathcal {N}}_U}{\mathcal {H}}_{\gamma }\) then
$$\begin{aligned} \mathbb {P}\left( {\mathcal {H}}_U\right) \simeq {\mathcal {N}}_{U} \times \mathbb {S}^{1} \simeq C \times \mathbb {S}^{1}\times \mathbb {S}^{1} . \end{aligned}$$
So, we can choose a coordinate system \({\widetilde{\psi }}=(x,y,\theta , \phi )\) where \((x,y,\theta )\) are coordinates for \(C\times \mathbb {S}^{1}\) and \(\phi \in \left[ 0,2\pi \right) \) is a coordinate for the fibres \(\mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \simeq \mathbb {S}^{1}\) that can be built from the initial values of the Jacobi fields at the Cauchy surface \(C\subset U\). If \(P\in \mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \) is a line such that \(P=\mathrm {span}\{\langle J \rangle \}\), then we can choose a representative J such that \(J(0),J'(0)\in T_{\gamma (0)}C\). Since \({\mathbf {g}}\left( J(0),\gamma '(0)\right) =0\) and \({\mathbf {g}}\left( J'(0),\gamma '(0)\right) =0\), then it is possible to write
$$\begin{aligned} J\left( 0\right) =w \cdot {\mathbf {e}}, \quad \text { and } \quad J'\left( 0\right) =v \cdot {\mathbf {e}} \end{aligned}$$
where \(T_{\gamma (0)}C \cap \left\{ \gamma '\left( 0\right) \right\} ^{\perp }=\mathrm {span}\{{\mathbf {e}}\}\) with \(\left\{ \gamma '\left( 0\right) \right\} ^{\perp } = \left\{ u\in T_{\gamma (0)}M: {\mathbf {g}}\left( \gamma '\left( 0\right) , u \right) = 0 \right\} \). Since for any \(0\ne \alpha \in \mathbb {R}\) we have \(P=\mathrm {span}\{\langle \alpha J \rangle \}=\mathrm {span}\{\langle J \rangle \}\) then the homogeneous coordinate
$$\begin{aligned} \phi =\left[ w:v\right] \end{aligned}$$
(5.4)
or, even the polar coordinate \(\phi =\arctan (w/v)\), determines the line \(P\in \mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \).
For any \(\gamma \in {\mathcal {N}}_U\) there exists \(s_{\gamma }\in \mathbb {R}\) smoothly depending on \(\gamma \) such that \(\gamma \left( s_{\gamma }\right) \in C_{-}\)Footnote 3. Moreover, since for all \(\gamma \in {\mathcal {N}}\), every curve \(\sigma \left( [\gamma '(s)] \right) ={\widetilde{\gamma }}(s)\) is injective, then we can assume that
$$\begin{aligned} 0<\phi \left( \sigma \left( [\gamma '(s_{\gamma })] \right) \right)< \phi \left( \sigma \left( [\gamma '(s)] \right) \right)< \phi \left( \oplus _{\gamma }\right) < 2\pi \end{aligned}$$
for all \(\gamma \in {\mathcal {N}}_U\) and \(s>s_{\gamma }\) in the domain of \(\gamma \). Due to \(\oplus \) is a smooth distribution then the function \(\phi _{\oplus }=\phi \circ \oplus :{\mathcal {N}}_U\rightarrow \left[ 0,2\pi \right) \) is smooth, therefore the future boundary of \(\widetilde{{\mathcal {N}}}_U=\mathbb {P}\left( {\mathcal {H}}_U\right) \cap \widetilde{{\mathcal {N}}}\) corresponding to \(\oplus \) can be locally written by
$$\begin{aligned} \partial ^{+} \widetilde{{\mathcal {N}}}_U =\left\{ \phi = \phi _{\oplus } \right\} \end{aligned}$$
which can be seen as the graph of \(\oplus \). Then \(\oplus :{\mathcal {N}}\rightarrow \mathbb {P}\left( {\mathcal {H}}\right) \) is a diffeomorphism onto its image and the future boundary of \(\widetilde{{\mathcal {N}}}\) is
$$\begin{aligned} \partial ^{+}\widetilde{{\mathcal {N}}} = \oplus \left( {\mathcal {N}}\right) . \end{aligned}$$
The past boundary \(\partial ^{-} \widetilde{{\mathcal {N}}}\) can be shown to be smooth analogously. It is summarized in the following proposition.
Proposition 13
Let M be a 3–dimensional conformal manifold under the hypotheses H1 and H2. Then \(\oplus :{\mathcal {N}}\rightarrow \partial ^{+}\widetilde{{\mathcal {N}}}\) is a diffeomorphism and \(\partial ^{+}\widetilde{{\mathcal {N}}}=\oplus \left( {\mathcal {N}}\right) \) is a smooth manifold embedded in the boundary of \(\widetilde{{\mathcal {N}}}\).
If both \(\oplus \) and \(\ominus \) satisfy the condition H2 with \(\oplus _{\gamma }\ne \ominus _{\gamma }\) for all \(\gamma \in {\mathcal {N}}\), a trivial corollary can be stated.
Corollary 5
Let M be a 3–dimensional conformal manifold under the hypotheses H1 and H2 for both \(\oplus \) and \(\ominus \) and \(\oplus _{\gamma }\ne \ominus _{\gamma }\) for all \(\gamma \in {\mathcal {N}}\). Then the closure of \(\widetilde{{\mathcal {N}}}\) is a smooth manifold with boundary \(\partial \widetilde{{\mathcal {N}}}= \partial ^{+} \widetilde{{\mathcal {N}}} \cup \partial ^{-} \widetilde{{\mathcal {N}}}\) where \(\partial ^{+} \widetilde{{\mathcal {N}}}=\oplus \left( {\mathcal {N}}\right) \) and \(\partial ^{-} \widetilde{{\mathcal {N}}}=\ominus \left( {\mathcal {N}}\right) \) are embedded in \(\mathbb {P}\left( {\mathcal {H}}\right) \) .
It is known that a projectivity from \(\mathbb {S}^{1}\) to \(\mathbb {R}\cup \{\infty \}\) can be defined by choosing three points \(s_1,s_2,s_3\in \mathbb {S}^{1}\) and assigning them the corresponding value in \(\mathbb {R}\cup \{\infty \}\). Then we can define, in a smooth way, a projectivity in each fibre \(\mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) \simeq \mathbb {S}^{1}\). If \(U\subset M\) is a globally hyperbolic, causally convex and normal open set and \(C,C_{-}\subset U\) are smooth spacelike Cauchy surfaces such that \(C_{-}\subset I^{-}(C)\), we can observe that \(\oplus \) is a section of the bundle \(\pi ^{\mathbb {P}\left( {\mathcal {H}}\right) }_{{\mathcal {N}}}:\mathbb {P}\left( {\mathcal {H}}_U\right) \rightarrow {\mathcal {N}}_U\) and we can consider two disjoint sections \(\sigma \left( {\mathbb {P}}{\mathbb {N}}(C)\right) \in \mathbb {P}\left( {\mathcal {H}}_U\right) \) and \(\sigma \left( {\mathbb {P}}{\mathbb {N}}(C_{-})\right) \in \mathbb {P}\left( {\mathcal {H}}_U\right) \), then they permit to define a projectivity \({\mathbf {t}}\) in such a way that
$$\begin{aligned} {\mathbf {t}}\left( \sigma \left( {\mathbb {P}}{\mathbb {N}}\left( C_{-}\right) \right) \right) =-1, \qquad {\mathbf {t}}\left( \sigma \left( {\mathbb {P}}{\mathbb {N}}\left( C\right) \right) \right) =0, \qquad {\mathbf {t}}\left( \partial ^{+} \widetilde{{\mathcal {N}}}_U \right) =1 . \end{aligned}$$
We can see \({\mathbf {t}}\) as a new coordinate for the fibres of \(\mathbb {P}\left( {\mathcal {H}}_{U}\right) \). It is easy to check that \({\mathbf {t}}\) is related to \(\phi \) by
$$\begin{aligned} {\mathbf {t}}\left( P_{\gamma }\right) =\frac{\left( \phi ^{\vee }-\phi ^{\wedge }\right) \left( \phi \left( P_{\gamma }\right) -\phi ^0\right) }{\left( 2\phi ^0 -\left( \phi ^{\wedge }+\phi ^{\vee }\right) \right) \phi \left( P_{\gamma }\right) +\left( 2\phi ^{\wedge }\phi ^{\vee }-\phi ^0\left( \phi ^{\wedge }+\phi ^{\vee }\right) \right) } \end{aligned}$$
(5.5)
for \(P_{\gamma }\in \mathbb {P}\left( {\mathcal {H}}_{\gamma }\right) -{\widetilde{\infty }}_{\gamma }\) where \({\widetilde{\infty }}\) is the section corresponding to the infinite of \({\mathbf {t}}\) which verifies \({\widetilde{\infty }}\cap \overline{\widetilde{{\mathcal {N}}}_U} = \varnothing \) and where we have denoted \(\phi ^{\vee }=\phi \circ \sigma \left( [\gamma '(s_{\gamma })]\right) \), \(\phi ^{\wedge }=\phi \circ \oplus (\gamma )\) and \(\phi ^0=\phi \circ \sigma \circ \xi ^{-1}(\gamma )\) with \(\xi :{\mathbb {P}}{\mathbb {N}}(C)\rightarrow {\mathcal {N}}_U\) the diffeomorphism of diagram (2.3).
So, we can express the trivialization \( \mathbb {P}\left( {\mathcal {H}}_U\right) \simeq {\mathcal {N}}_{U} \times \mathbb {S}^{1}\) by a map
$$\begin{aligned} \begin{array}{rrcl} \varepsilon : &{} {\mathcal {N}}_{U}\times \mathbb {R} &{} \rightarrow &{} \mathbb {P}\left( {\mathcal {H}}_{U}\right) -{\widetilde{\infty }} \\ &{} \left( \gamma ,{\mathbf {t}}\right) &{} \mapsto &{} {\widetilde{\gamma }}\left( {\mathbf {t}}\right) \end{array} \end{aligned}$$
(5.6)
which is a diffeomorphism because its expression in coordinates is just \(\left( (x,y,\theta ),{\mathbf {t}}\right) \mapsto (x,y,\theta ,{\mathbf {t}})\) and \({\mathbf {t}}\) is said to be a projective parameter. Now it is clear that \(\varepsilon \left( {\mathcal {N}}_{U}\times (-1,1)\right) \subset \widetilde{{\mathcal {N}}}_U \) with
$$\begin{aligned} \sigma \left( {\mathbb {P}}{\mathbb {N}}\left( C_{-}\right) \right)= & {} \varepsilon \left( {\mathcal {N}}_{U}\times \{-1\}\right) , \qquad \sigma \left( {\mathbb {P}}{\mathbb {N}}\left( C\right) \right) =\varepsilon \left( {\mathcal {N}}_{U}\times \{0\}\right) , \qquad \\&\qquad \qquad \partial ^{+} \widetilde{{\mathcal {N}}}_U =\varepsilon \left( {\mathcal {N}}_{U}\times \{1\}\right) \end{aligned}$$
and, for any light ray \(\gamma \in {\mathcal {N}}_U\), the curve \(\sigma \left( \left[ \gamma '({\mathbf {t}})\right] \right) ={\widetilde{\gamma }}\left( {\mathbf {t}}\right) =\varepsilon \left( \gamma , {\mathbf {t}}\right) \) is defined for \({\mathbf {t}}\in \left( -1,1\right) \), but it is naturally extended to any \({\mathbf {t}}\in \mathbb {R}\) by \({\widetilde{\gamma }}\left( {\mathbf {t}}\right) = \varepsilon \left( \gamma ,{\mathbf {t}}\right) \). Moreover, the tangent vector to \({\widetilde{\gamma }}\) satisfies
$$\begin{aligned} {\widetilde{\gamma }}'\left( {\mathbf {t}}\right) =\left( \frac{\partial }{\partial {\mathbf {t}}} \right) _{{\widetilde{\gamma }}\left( {\mathbf {t}}\right) } . \end{aligned}$$
(5.7)
The map \(\varepsilon \) defined by choosing Cauchy surfaces \(C,C_{-}\subset U\) is called a local projective synchronization and it will permit us to build the future L–boundary. The construction of the past boundary can be done by reversing the time. Notice that, if \(\oplus _{\gamma }\ne \ominus _{\gamma }\) for all \(\gamma \in {\mathcal {N}}\), the section \(\sigma \left( {\mathbb {P}}{\mathbb {N}}(C_{-})\right) \in \mathbb {P}\left( {\mathcal {H}}_U\right) \) can be replaced by \(\partial ^{-}\widetilde{{\mathcal {N}}}_U\) in order to define the projective parameter. In this case, we have that \(\varepsilon :{\mathcal {N}}_U\times (-1,1)\rightarrow \widetilde{{\mathcal {N}}}_U\) is a diffeomorphism (see [8, Prop. 5.2]).
Remark 10
If M is globally hyperbolic conformal manifold with Cauchy surface \(C\subset M\) diffeomorphic to \(\mathbb {R}^{2}\), then \({\mathcal {N}}\simeq C \times \mathbb {S}^{1}\) and therefore the projective parameter \({\mathbf {t}}\) can be defined for \(\mathbb {P}\left( {\mathcal {H}}\right) \) since \(\mathbb {P}\left( {\mathcal {H}}\right) \simeq {\mathcal {N}}\times \mathbb {R}\cup \{\infty \}\simeq C\times \mathbb {S}^{1}\times \mathbb {R}\cup \{\infty \}\) where C is a global Cauchy surface. In this case we will call universal projective parameter to the parameter \({\mathbf {t}}\in \mathbb {R}\).
Remark 11
The bijectivity of \(\varepsilon \) implies the injectivity of the curves \({\widetilde{\gamma }}\). This generalizes the condition of light non–conjugation of M, because \({\widetilde{\gamma }}({\mathbf {t}})=\varepsilon (\gamma ,{\mathbf {t}})\) extends \({\widetilde{\gamma }}({\mathbf {t}})=\sigma ([\gamma '({\mathbf {t}})])\) outside of \(\widetilde{{\mathcal {N}}}\).
Distributions in \(\widetilde{{\mathcal {N}}}\)
In order to simplify, we will work only with the boundary \(\partial ^{+}\widetilde{{\mathcal {N}}}\) because all the construction for \(\partial ^{-}\widetilde{{\mathcal {N}}}\) can be done analogously.
Now, we will define two distributions in \(\overline{\widetilde{{\mathcal {N}}}}=\widetilde{{\mathcal {N}}}\cup \partial ^{+}\widetilde{{\mathcal {N}}}\): the former will be \({\mathcal {D}}^{\sim }\) in \(\widetilde{{\mathcal {N}}}\) and the latter \(\partial ^{+}{\mathcal {D}}^{\sim }\) in \(\partial ^{+}\widetilde{{\mathcal {N}}}\). We will determine the conditions so that the union \(\overline{{\mathcal {D}}^{\sim }}={\mathcal {D}}^{\sim }\cup \partial ^{+}{\mathcal {D}}^{\sim }\) is a smooth distribution in \(\overline{\widetilde{{\mathcal {N}}}}\). The orbits of the distribution \(\partial ^{+}{\mathcal {D}}^{\sim }\) will corresponds to the points of the future boundary.
First, let us call \({\mathcal {P}}\) to the regular distribution in \({\mathbb {P}}{\mathbb {N}}\) defined by the fibres \({\mathbb {P}}{\mathbb {N}}_q\) with \(q\in M\) then, trivially, the map \(\zeta :M\rightarrow {\mathbb {P}}{\mathbb {N}}/{\mathcal {P}}\) defined by \(\zeta \left( q\right) ={\mathbb {P}}{\mathbb {N}}_q\) is a diffeomorphism. Then, passing the distribution \({\mathcal {P}}\) to \(\widetilde{{\mathcal {N}}}\) by the diffeomorphism \(\sigma \), we obtain the distribution \({\mathcal {D}}^{\sim }\).
Observe that the orbits of \({\mathcal {D}}^{\sim }\) are
$$\begin{aligned} \sigma \left( {\mathbb {P}}{\mathbb {N}}_{q}\right) =\{ \sigma \left( \left[ v\right] \right) \in \widetilde{{\mathcal {N}}}:\left[ v\right] \in {\mathbb {P}}{\mathbb {N}}_{q} \} \end{aligned}$$
being 1–dimensional compact submanifolds when \(\mathrm {dim}~M=3\), then \({\mathcal {D}}^{\sim }\) is a regular distribution and \(\widetilde{{\mathcal {N}}}/{\mathcal {D}}^{\sim }\) is a differentiable manifold. Moreover the quotient map \({\widetilde{\pi }}:\widetilde{{\mathcal {N}}} \rightarrow \widetilde{{\mathcal {N}}}/{\mathcal {D}}^{\sim }\) is a submersion.
Now, we have the following diagram
where \({\widetilde{\sigma }}\) is the map defined by \({\widetilde{\sigma }}\left( {\mathbb {P}}{\mathbb {N}}_q\right) =\sigma \left( {\mathbb {P}}{\mathbb {N}}_q\right) \in \widetilde{{\mathcal {N}}}/{\mathcal {D}}^{\sim }\), the maps \(\kappa \) and \({\widetilde{\pi }}\) are the corresponding submersions, and \(\sigma \), \(\zeta \) and \({\widetilde{\sigma }}\) are diffeomorphisms. Therefore, we can observe that
$$\begin{aligned} {\widetilde{S}}={\widetilde{\sigma }}\circ \zeta : M \rightarrow \widetilde{{\mathcal {N}}}/{\mathcal {D}}^{\sim } \end{aligned}$$
(5.9)
is a diffeomorphism. There is a different proof of this result in [7, Prop. 2.6].
The second distribution to be defined is \(\partial ^{+} {\mathcal {D}}^{\sim }\). Recall that, by hypotheses, \(\oplus \) is a 1–dimensional regular distribution in \({\mathcal {N}}\), then \(\oplus \) is integrable and the orbits of \(\oplus \) define a regular foliation. By Proposition 13, \(\oplus : {\mathcal {N}}\rightarrow \partial ^{+}\widetilde{{\mathcal {N}}}\) is a diffeomorphism, then the images of the orbits of \(\oplus \) define a regular foliation in \(\partial ^{+}\widetilde{{\mathcal {N}}}\) corresponding to the distribution \(\partial ^{+} {\mathcal {D}}^{\sim }\). Of course, the distribution \(\partial ^{-}{\mathcal {D}}^{\sim }\) in \(\partial ^{-}\widetilde{{\mathcal {N}}}\) needed to build the past boundary can be defined by \(\ominus \).
The next step is to describe the distribution \(\overline{{\mathcal {D}}^{\sim }}={\mathcal {D}}^{\sim }\cup \partial ^{+} {\mathcal {D}}^{\sim }\) in order to study its smoothness. For this purpose, we will construct explicitly the orbits of \({\mathcal {D}}^{\sim }\).
Fix some auxiliary metric \({\mathbf {g}}\in {\mathcal {C}}\) and some globally hyperbolic, normal and causally convex open \(U\subset M\) with \(C\subset U\) a smooth spacelike Cauchy surface in U as in Remark 1. We denote by \({\mathcal {N}}_{U}\subset {\mathcal {N}}\) the open set of all light rays passing by U and hence \({\mathcal {N}}_{U}\) is diffeomorphic to \(C\times \mathbb {S}^1\) and then, we can consider all light rays \(\gamma \in {\mathcal {N}}_{U}\) parametrized such that \(\gamma '\left( 0\right) \in \varOmega \left( C\right) =\{ u\in \mathbb {N}^{+}\left( U\right) : {\mathbf {g}}\left( u,T\right) =-1 \}\) for some future–directed timelike vector field \(T\in {\mathfrak {X}}\left( M\right) \). Since M is strongly causal, by [32, Prop. 6.4.7], we can assume without any lack of generality, there is no imprisoned light ray in the closure \({\overline{U}}\) where U is assumed to be relatively compact, so U can be chosen such that \(\gamma \cap U\) has only one connected component for all \(\gamma \in {\mathcal {N}}_{U}\).
Let us consider an orthonormal frame \(\{ E_1, E_2, E_3 \}\) on the local Cauchy surface C such that \(E_2, E_3\) are tangent to C and \(E_1\) is future–directed timelike related to the conformal structure \(\left( M,{\mathcal {C}}\right) \). For a light ray \(\gamma \in {\mathcal {N}}_{U}\) such that \(\gamma \simeq \left( c,\theta \right) \in C\times \mathbb {S}^{1}\), we define \(\left\{ {\mathbf {E}}_i\left( \gamma ,{\mathbf {t}}\right) \right\} _{i=1,2,3}\) the extension of the frame \(\left\{ E_i\left( c\right) \right\} _{i=1,2,3}\) by parallel transport to \(\gamma \left( {\mathbf {t}}\right) \) along \(\gamma \) related to the metric \({\mathbf {g}}\), where \({\mathbf {t}}\) is the projective parameter defined in Sect. 5.4 by a local projective synchronization \(\varepsilon \).
The regular dependence on parameters of the solutions of initial value problems of ODEs [31, Ch. 5] assures the smooth dependence of the frames \(\left\{ {\mathbf {E}}_i\left( \gamma ,{\mathbf {t}}\right) \right\} _{i=1,2,3}\) on \(\left( \gamma ,{\mathbf {t}}\right) \).
If \(\theta \in \left[ 0,2\pi \right) \simeq \mathbb {S}^{1}\), then we define the lightlike vector
$$\begin{aligned} V\left( \gamma ,{\mathbf {t}},s\right) = {\mathbf {E}}_1\left( \gamma ,{\mathbf {t}}\right) + \cos \left( \theta + s \right) {\mathbf {E}}_2\left( \gamma ,{\mathbf {t}}\right) +\sin \left( \theta + s \right) {\mathbf {E}}_3\left( \gamma ,{\mathbf {t}}\right) \in \mathbb {N} \end{aligned}$$
which depends smoothly on \(\left( \gamma ,{\mathbf {t}}\right) \) and defines the line
$$\begin{aligned} \varLambda \left( \gamma ,{\mathbf {t}},s\right) =\left[ V\left( \gamma ,{\mathbf {t}},s\right) \right] = \mathrm {span}\{V\left( \gamma ,{\mathbf {t}},s\right) \}\in {\mathbb {P}}{\mathbb {N}} . \end{aligned}$$
By means of the diffeomorphisms \(\sigma \) and \(\varepsilon \) of Sect. 5.4, and the canonical projections \(p_1:{\mathcal {N}}\times \left( -1,1\right) \rightarrow {\mathcal {N}}\) and \(p_2:{\mathcal {N}}\times \left( -1,1\right) \rightarrow \left( -1,1\right) \), we define the following differentiable maps
$$\begin{aligned} \begin{array}{l} {\widetilde{X}}\left( \gamma ,{\mathbf {t}},s\right) =\sigma \left( \varLambda \left( \gamma ,{\mathbf {t}},s\right) \right) \in \widetilde{{\mathcal {N}}} \\ X\left( \gamma ,{\mathbf {t}},s\right) = p_1 \circ \varepsilon ^{-1}\left( {\widetilde{X}}\left( \gamma ,{\mathbf {t}},s\right) \right) \in {\mathcal {N}} \\ \tau \left( \gamma ,{\mathbf {t}},s\right) =p_2 \circ \varepsilon ^{-1}\left( {\widetilde{X}}\left( \gamma ,{\mathbf {t}},s\right) \right) \in \left( -1,1\right) . \end{array} \end{aligned}$$
(5.10)
Observe that, for fixed \(\left( \gamma ,{\mathbf {t}}\right) \in {\mathcal {N}}_{U}\times \left( -1,1\right) \), the curve \(X_{\left( \gamma ,{\mathbf {t}}\right) }\left( s\right) =X\left( \gamma ,{\mathbf {t}},s\right) \) is a parametrization of the 1–dimensional submanifold \(S\left( \gamma \left( {\mathbf {t}}\right) \right) \cap {\mathcal {N}}_U\). We will denote
$$\begin{aligned} \gamma _{\left( {\mathbf {t}},s\right) }= X\left( \gamma ,{\mathbf {t}},s\right) \in {\mathcal {N}} \end{aligned}$$
when we will need to use a parameter for the light ray. Also, the function \(\tau _{\left( \gamma ,{\mathbf {t}}\right) }\left( s\right) =\tau \left( \gamma ,{\mathbf {t}},s\right) \) is the value of the parameter of \(\gamma _{\left( {\mathbf {t}},s\right) }\) at the point \(\gamma \left( {\mathbf {t}}\right) \) from C, then the identity
$$\begin{aligned} \gamma _{\left( {\mathbf {t}},s\right) }\left( \tau \left( \gamma ,{\mathbf {t}},s\right) \right) =\gamma \left( {\mathbf {t}}\right) \end{aligned}$$
(5.11)
holds. Moreover, \({\widetilde{X}}_{\left( \gamma ,{\mathbf {t}}\right) }\left( s\right) ={\widetilde{X}}\left( \gamma ,{\mathbf {t}},s\right) \) is the curve of lines of classes of Jacobi fields tangent to the light ray \(X_{\left( \gamma ,{\mathbf {t}}\right) }\left( s\right) \) at the point \(\gamma \left( {\mathbf {t}}\right) \).
Recall that the map \(\xi :{\mathbb {P}}{\mathbb {N}}(C) \rightarrow {\mathcal {N}}_U\) of diagram (2.3) is a diffeomorphism, so the curve defined by
$$\begin{aligned} c_{\left( \gamma ,{\mathbf {t}}\right) }\left( s\right) =\pi ^{{\mathbb {P}}{\mathbb {N}}(C)}_{C}\circ \xi ^{-1}\left( \gamma _{({\mathbf {t}},s)}\right) \end{aligned}$$
(5.12)
is smooth, but it can also be written by
$$\begin{aligned} c_{\left( \gamma ,{\mathbf {t}}\right) }\left( s\right) =\gamma _{({\mathbf {t}},s)}\cap C =\gamma _{\left( {\mathbf {t}},s\right) }\left( 0\right) \in C. \end{aligned}$$
Now, we replace the parameter s for the arc–length parameter. Fix some auxiliary metric \({\mathbf {g}}\in {\mathcal {C}}\) in M, since the Cauchy surface C is differentiable and spacelike, the restriction \(\left. {\mathbf {g}}\right| _{TC\times TC}\) is a Riemannian metric on C. If consider any \(\langle J_{\left( \gamma ,{\mathbf {t}},s\right) }\rangle \in {\widetilde{X}}\left( \gamma ,{\mathbf {t}},s\right) \), since M is assumed to be light non–conjugate, then for any \({\mathbf {t}}>0\) we have that any representative \(J_{\left( \gamma ,{\mathbf {t}},s\right) }\in \langle J_{\left( \gamma ,{\mathbf {t}},s\right) }\rangle \) satisfies
$$\begin{aligned} J_{\left( \gamma ,{\mathbf {t}},s\right) }\left( 0\right) \ne 0~(\mathrm {mod} \gamma '_{\left( {\mathbf {t}},s\right) }\left( 0\right) ) \end{aligned}$$
and by Lemma 1,
$$\begin{aligned} c'_{\left( \gamma ,{\mathbf {t}}\right) }\left( s\right) \ne 0 \end{aligned}$$
therefore we can parametrize the curves \(c_{\left( \gamma ,{\mathbf {t}}\right) }\) with the arc–length parameter \({\mathbf {s}}\) defined in C by the restriction of \({\mathbf {g}}\).
Lemma 5
For every \(\gamma _0\in {\mathcal {N}}\) and every \(\delta \in (0,1)\) there exists \(\epsilon >0\) and a neighbourhood \({\mathcal {N}}_U^{\epsilon }\subset {\mathcal {N}}\) of \(\gamma _0\) such that the curves \(c_{\left( \gamma ,{\mathbf {t}}\right) }\left( {\mathbf {s}}\right) \in C\) can be defined for \(\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \in {\mathcal {N}}_{U}^{\epsilon }\times (1-\delta ,1) \times (-\epsilon ,\epsilon )\), where \({\mathbf {s}}\) is the arc–length parameter.
Proof
Fix some auxiliary \({\mathbf {g}}\in {\mathcal {C}}\). Let us consider a neighbourhood \({\mathcal {N}}_U\) of \(\gamma _0\) such that \(U\subset M\) is relatively compact and globally hyperbolic with Cauchy surface \(C\in U\). For \(p\in C\) and \(r>0\), we will denote by \(B_r(p)\) the ball in C of radius r centered at p determined by the restriction of the metric \({\mathbf {g}}\) to C. Now consider some \(\epsilon > 0\) such that the ball \(B_{2\epsilon }(\gamma _0\cap C)\) is fully contained in C. So, let us call \({\mathcal {N}}_U^{\epsilon }=\{ \gamma \in {\mathcal {N}}_U: \gamma \cap C\in B_{\epsilon }(\gamma _0 \cap C)\}\subset {\mathcal {N}}_U\).
For any \((\gamma ,{\mathbf {t}})\in {\mathcal {N}}_U^{\epsilon } \times \left( 1-\delta ,1\right) \) there exist \(a_{(\gamma ,{\mathbf {t}})},b_{(\gamma ,{\mathbf {t}})}>0\) such that the maximal domain of definition of the segment of the curve \(c_{(\gamma ,{\mathbf {t}})}\) contained in \(B_{2\epsilon }(\gamma _0\cap C)\) is the interval \(I_{(\gamma ,{\mathbf {t}})}=\left( -a_{(\gamma ,{\mathbf {t}})},b_{(\gamma ,{\mathbf {t}})} \right) \subset \mathbb {R}\).
Let us assume that \(b_{(\gamma ,{\mathbf {t}})}<\epsilon \). If we take a sequence \(\{{\mathbf {s}}_n\}\subset I_{(\gamma ,{\mathbf {t}})}\) such that \({\mathbf {s}}_n\mapsto b_{(\gamma ,{\mathbf {t}})}\), since \({\mathbb {P}}{\mathbb {N}}(C)\) is relatively compact, then the sequence
$$\begin{aligned} \left[ u_n \right] = \left[ \gamma '_{({\mathbf {t}},{\mathbf {s}}_n)}(0) \right] \in {\mathbb {P}}{\mathbb {N}}(C) \end{aligned}$$
has a convergent subsequence. Assuming that this subsequence is \(\{[u_n]\}\) itself, then
$$\begin{aligned} \left[ u_n \right] \mapsto \left[ u \right] \in {\mathbb {P}}{\mathbb {N}}(C) . \end{aligned}$$
Moreover, for any \(\left[ u_n \right] \) there exists \(\left[ v_n \right] \in {\mathbb {P}}{\mathbb {N}}_{\gamma ({\mathbf {t}})}\) such that \(\gamma _{\left[ v_n \right] }=\gamma _{\left[ u_n \right] }\in {\mathcal {N}}\). Since \({\mathbb {P}}{\mathbb {N}}_{\gamma ({\mathbf {t}})}\) is compact, then we can consider that \(\left[ v_n \right] \mapsto \left[ v \right] \in {\mathbb {P}}{\mathbb {N}}_{\gamma ({\mathbf {t}})}\). Now, because \({\mathcal {N}}\) is Hausdorff, then \(\gamma _{\left[ v \right] }=\gamma _{\left[ u \right] }\in {\mathcal {N}}\) and hence we have \(\gamma _{\left[ u \right] }\in S\left( \gamma ({\mathbf {t}})\right) \). This implies that \(c_{(\gamma ,{\mathbf {t}})}({\mathbf {s}})\) exists for \({\mathbf {s}}=b_{(\gamma ,{\mathbf {t}})}\) and the interval \(I_{(\gamma ,{\mathbf {t}})}\) is not maximal. This fact contradicts the assumption of \(b_{(\gamma ,{\mathbf {t}})}<\epsilon \).
Therefore, \(b_{(\gamma ,{\mathbf {t}})}\ge \epsilon \) for all \((\gamma ,{\mathbf {t}})\in {\mathcal {N}}_U^{\epsilon } \times \left( 1-\delta ,1\right) \). The case \(a_{(\gamma ,{\mathbf {t}})}\ge \epsilon \) can be shown analogously. Then \(c(\gamma ,{\mathbf {t}},{\mathbf {s}})=c_{(\gamma ,{\mathbf {t}})}({\mathbf {s}})\) can be defined by
$$\begin{aligned} c:{\mathcal {N}}_U^{\epsilon } \times \left( 1-\delta ,1\right) \times \left( -\epsilon ,\epsilon \right) \longrightarrow B_{2\epsilon }(\gamma _0\cap C)\subset C \end{aligned}$$
\(\square \)
The arc–length parameter \({\mathbf {s}}\in (-\epsilon ,\epsilon )\) of the curves \(c_{(\gamma ,{\mathbf {t}})}\) can replace the previous variable s in the maps \({\widetilde{X}}\), X and \(\tau \). We will denote again \({\widetilde{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \), \(X\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \), \(\tau \left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \) the corresponding maps of (5.10) with the new variable \({\mathbf {s}}\) and defined as
$$\begin{aligned} \left\{ \begin{array}{l} {\widetilde{X}}:{\mathcal {N}}^{\epsilon }_{U}\times \left( 1-\delta ,1\right) \times \left( -\epsilon ,\epsilon \right) \longrightarrow \widetilde{{\mathcal {N}}}_{U} \\ X: {\mathcal {N}}^{\epsilon }_{U}\times \left( 1-\delta ,1\right) \times \left( -\epsilon ,\epsilon \right) \longrightarrow {\mathcal {N}}_{U} \\ \tau : {\mathcal {N}}^{\epsilon }_{U}\times \left( 1-\delta ,1\right) \times \left( -\epsilon ,\epsilon \right) \longrightarrow \left( -1,1\right) \end{array} \right. \end{aligned}$$
(5.13)
Observe that (Fig. 12), by construction, we have
$$\begin{aligned} {\widetilde{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) = \varepsilon \left( X\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) , \tau \left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right) \end{aligned}$$
(5.14)
and moreover
$$\begin{aligned} \left\{ \begin{array}{l} X\left( \gamma ,{\mathbf {t}},0\right) =\gamma _{\left( {\mathbf {t}},0\right) }=\gamma \\ \tau \left( \gamma ,{\mathbf {t}},0\right) ={\mathbf {t}} \end{array} \right. \end{aligned}$$
(5.15)
holds for all \({\mathbf {t}}\in \left( 1-\delta ,1\right) \).
It is important to remark that the curve \({\widetilde{X}}_{\left( \gamma ,{\mathbf {t}}\right) }({\mathbf {s}})={\widetilde{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \) is a parametrization of the orbit of the distribution \({\mathcal {D}}^{\sim }\) passing through \({\widetilde{\gamma }}\left( {\mathbf {t}}\right) \), that is the submanifold \(\widetilde{S\left( \gamma \left( {\mathbf {t}}\right) \right) } \subset \widetilde{{\mathcal {N}}}\). This implies that \({\mathcal {D}}^{\sim }\) is generated by the tangent vectors \(\frac{\partial {\widetilde{X}}}{\partial s}\left( \gamma ,{\mathbf {t}},0\right) \in T\widetilde{{\mathcal {N}}}\) so, by (5.14), we have
$$\begin{aligned} \frac{\partial {\widetilde{X}}}{\partial s}\left( \gamma ,{\mathbf {t}},0\right) =\left( d\varepsilon \right) _{\left( \gamma ,{\mathbf {t}}\right) }\left( \frac{\partial X}{\partial s}\left( \gamma ,{\mathbf {t}},0\right) , \frac{\partial \tau }{\partial s}\left( \gamma ,{\mathbf {t}},0\right) \right) \end{aligned}$$
(5.16)
for all \(\left( \gamma ,{\mathbf {t}}\right) \in {\mathcal {N}}^{\epsilon }_{U}\times \left( 1-\delta ,1\right) \).
Smoothness of the distribution \(\overline{{\mathcal {D}}^{\sim }}=\partial ^{+}{\mathcal {D}}^{\sim }\cup {\mathcal {D}}^{\sim }\): current status
We have defined the distributions \({\mathcal {D}}^{\sim }\) and \(\partial ^+ {\mathcal {D}}^{\sim }\) separately in \(\widetilde{{\mathcal {N}}}\) and in \(\partial ^+ \widetilde{{\mathcal {N}}}\) respectively, so we have that \(\overline{{\mathcal {D}}^{\sim }}=\partial ^{+}{\mathcal {D}}^{\sim }\cup {\mathcal {D}}^{\sim }\) is a distribution that, a priori, could even be non–continuous. In this section, we will study the conditions under which the distribution \(\overline{{\mathcal {D}}^{\sim }}\) is smooth.
The following Theorem 5.1 clarifies a loophole in [8, Thm. 7.1] and allows to establish equivalent conditions to the differentiability of \(\overline{{\mathcal {D}}^{\sim }}\).
Recall that when we say that a smooth map \(f:A\times (a,b) \rightarrow B\) can be smoothly (or differentiably) extended to \(A\times (a,b]\) we mean that there exists \(\epsilon >0\) and a smooth map \({\overline{f}}:A\times (a,b+\epsilon ) \rightarrow B\) such that \({\overline{f}}=f\) in \(A\times (a,b)\).
Theorem 5.1
Under the hypotheses H1 and H2 the following conditions are equivalent:
-
1.
\(\overline{{\mathcal {D}}^{\sim }}\) is smooth in \(\overline{\widetilde{{\mathcal {N}}}_U}\).
-
2.
\({\widetilde{X}}\) can be smoothly extended to \({\mathbf {t}}=1\).
-
3.
X can be smoothly extended to \({\mathbf {t}}=1\).
-
4.
\(\tau \) can be smoothly extended to \({\mathbf {t}}=1\).
-
5.
\(h(\gamma , {\mathbf {t}})=\frac{\partial \tau }{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \) can be smoothly extended to \({\mathbf {t}}=1\).
Proof
1) \(\Rightarrow \) 2)
Assuming \(\overline{{\mathcal {D}}^{\sim }}\) is smooth, since it is 1–dimensional, then there exists a non–zero vector field \({\widetilde{\varPhi }}\in {\mathfrak {X}}\left( \overline{\widetilde{{\mathcal {N}}}}\right) \) such that \(\overline{{\mathcal {D}}^{\sim }}=\mathrm {span}\{{\widetilde{\varPhi }}\}\). For any \({\widetilde{\gamma }}(1)\in \partial ^{+}\widetilde{{\mathcal {N}}}\) there exists a flow box of \({\widetilde{\varPhi }}\), that is, a smooth map \({\widetilde{F}}:\widetilde{{\mathcal {U}}}\times (-{\overline{\epsilon }},{\overline{\epsilon }}) \rightarrow \overline{\widetilde{{\mathcal {N}}}}\) such that \(u\mapsto {\widetilde{F}}_{{\widetilde{\gamma }}({\mathbf {t}})}(u)={\widetilde{F}}\left( {\widetilde{\gamma }}({\mathbf {t}}),u\right) \) is an integral curve of \({\widetilde{\varPhi }}\). Making smaller the neighbourhood of definition of \({\widetilde{F}}\) and since \(\varepsilon \) is a diffeomorphism, we can define the map
$$\begin{aligned} \begin{array}{rccl} \overline{{\widetilde{X}}}: &{} {\mathcal {U}} \times \left( 1-\delta ,1\right] \times \left( -{\overline{\epsilon }},{\overline{\epsilon }}\right) &{} \longrightarrow &{} \overline{\widetilde{{\mathcal {N}}}} \\ &{} \left( \gamma ,{\mathbf {t}},u\right) &{} \longmapsto &{} \overline{{\widetilde{X}}}\left( \gamma ,{\mathbf {t}},u\right) ={\widetilde{F}}\left( \varepsilon (\gamma ,{\mathbf {t}}),u\right) . \end{array} \end{aligned}$$
with \({\mathcal {U}}\subset {\mathcal {N}}\). Then we have that
$$\begin{aligned} \left( {\overline{X}}\left( \gamma ,{\mathbf {t}},u\right) , {\overline{\tau }}\left( \gamma ,{\mathbf {t}},u\right) \right) =\varepsilon ^{-1}\left( \overline{{\widetilde{X}}}\left( \gamma ,{\mathbf {t}},u\right) \right) \in {\mathcal {U}} \times \left( 1-\delta ,1\right] \end{aligned}$$
where \({\overline{X}}\) and \({\overline{\tau }}\) are differentiable maps for \({\mathbf {t}}\le 1\).
By Lemma 5, we can reparametrize the maps \(\overline{{\widetilde{X}}}\), \({\overline{X}}\) and \({\overline{\tau }}\) by arc–length of the curves \(z_{(\gamma ,{\mathbf {t}})}(u)={\overline{X}}\left( \gamma ,{\mathbf {t}},u\right) \cap C\) and we obtain the maps (called in the same way)
$$\begin{aligned} \left\{ \begin{array}{l} \overline{{\widetilde{X}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \in \overline{\widetilde{{\mathcal {N}}}} \\ {\overline{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \in {\mathcal {N}}_{U}^{\epsilon } \\ {\overline{\tau }}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \in \left( 1-\delta ,1\right] \end{array} \right. \end{aligned}$$
for \({\mathbf {s}}\in (-\epsilon ,\epsilon )\). Since the curves \({\mathbf {s}}\mapsto \overline{{\widetilde{X}}}(\gamma ,{\mathbf {t}},{\mathbf {s}})\) describe the orbits of \(\overline{{\mathcal {D}}^{\sim }}\) and they are parametrized by the same arc–length parameter then
$$\begin{aligned} \overline{{\widetilde{X}}}={\widetilde{X}} \quad , \qquad {\overline{X}}=X \quad , \qquad {\overline{\tau }}=\tau \end{aligned}$$
for \({\mathbf {t}}<1\). Therefore \(\overline{{\widetilde{X}}}\) is a smooth extension of \({\widetilde{X}}\).
2) \(\Rightarrow \) 3)
Trivially, since \(\varepsilon \) is a diffeomorphism, then if \(\overline{{\widetilde{X}}}\) is a smooth extension of \(\overline{{\widetilde{X}}}\) to \({\mathbf {t}} = 1\) we have
$$\begin{aligned}&\overline{{\widetilde{X}}}(\gamma ,{\mathbf {t}},{\mathbf {s}})= \varepsilon \left( {\overline{X}}(\gamma ,{\mathbf {t}},{\mathbf {s}}),{\overline{\tau }}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right) \\&\quad \Longleftrightarrow \quad \varepsilon ^{-1}\circ \overline{{\widetilde{X}}}(\gamma ,{\mathbf {t}},{\mathbf {s}})= \left( {\overline{X}}(\gamma ,{\mathbf {t}},{\mathbf {s}}),{\overline{\tau }}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right) \end{aligned}$$
for \({\mathbf {t}} \le 1\) and therefore \({\overline{X}}\) (and also \({\overline{\tau }}\)) is a smooth extension to \({\mathbf {t}}=1\).
3) \(\Rightarrow \) 4)
Let us assume that \({\overline{X}}(\gamma ,{\mathbf {t}},{\mathbf {s}})\) is a smooth extension of \(X(\gamma ,{\mathbf {t}},{\mathbf {s}})\) to \({\mathbf {t}}=1\), then we have that
$$\begin{aligned} \frac{\partial {\overline{X}}}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \in T_{\gamma _{({\mathbf {t}},{\mathbf {s}})}}S\left( \gamma _{({\mathbf {t}},{\mathbf {s}})}\left( \tau (\gamma ,{\mathbf {t}},{\mathbf {s}})\right) \right) \subset {\mathcal {H}}_{\gamma _{\left( {\mathbf {t}},{\mathbf {s}}\right) }} \end{aligned}$$
for \({\mathbf {t}}<1\).
Since the curves \(c_{\left( \gamma ,{\mathbf {t}}\right) }({\mathbf {s}})={\overline{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \cap C\) are smooth and parametrized by arc–length as in Eq. (5.12), by continuity we have that
$$\begin{aligned} \vert c'_{\left( \gamma ,1\right) }({\mathbf {s}})\vert =\lim _{{\mathbf {t}}\mapsto 1} \vert c'_{\left( \gamma ,{\mathbf {t}}\right) }({\mathbf {s}})\vert =1 \end{aligned}$$
where \(\vert \cdot \vert \) denotes the norm related to the restriction of the metric \({\mathbf {g}}\in {\mathcal {C}}\) to the local Cauchy surface \(C\subset U\), hence we have that \(c'_{\left( \gamma ,1\right) }({\mathbf {s}})\ne 0\). Then \(\frac{\partial {\overline{X}}}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \ne 0\) for all \({\mathbf {t}}\le 1\) and so we obtain
$$\begin{aligned} \left[ \frac{\partial {\overline{X}}}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right] =\mathrm {span}\left\{ \frac{\partial {\overline{X}}}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right\} \in \mathbb {P}\left( {\mathcal {H}}_{\gamma _{\left( {\mathbf {t}},{\mathbf {s}}\right) }}\right) . \end{aligned}$$
Notice that, since \(X\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \!=\!{\overline{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \) for \({\mathbf {t}}\!<\!1\) and \({\widetilde{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \!=\!\mathrm {span}\left\{ \frac{\partial X}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right\} \) for \({\mathbf {t}}<1\), then we have \({\widetilde{X}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) = \mathrm {span}\left\{ \frac{\partial {\overline{X}}}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right\} \) for \({\mathbf {t}}<1\).
By the diffeomorphism \(\varepsilon \), we have that
$$\begin{aligned} \left( {\overline{X}}(\gamma ,{\mathbf {t}},{\mathbf {s}}), {\overline{\tau }}(\gamma ,{\mathbf {t}},{\mathbf {s}}) \right) =\varepsilon ^{-1}\left( \left[ \frac{\partial {\overline{X}}}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},{\mathbf {s}}\right) \right] \right) \end{aligned}$$
therefore \({\overline{\tau }}\) is a smooth extension of \(\tau \).
4) \(\Rightarrow \) 5)
Let \({\overline{\tau }}\) be a smooth extension of \(\tau \) to \({\mathbf {t}}=1\). Trivially, \({\overline{h}}(\gamma ,{\mathbf {t}})=\frac{\partial {\overline{\tau }}}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \) is an smooth extension of \(h(\gamma ,{\mathbf {t}})=\frac{\partial \tau }{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \) to \({\mathbf {t}}=1\).
5) \(\Rightarrow \) 1)
First, let us show that \({\overline{h}}(\gamma ,1)=\frac{\partial {\overline{\tau }}}{\partial {\mathbf {s}}}\left( \gamma ,1,0\right) =0\). Observe that, by Eq. (5.15), \(\tau (\gamma ,{\mathbf {t}},0)={\mathbf {t}}\) and since \({\overline{\tau }}\) is continuous, we have \({\overline{\tau }}(\gamma ,1,0)=1\). Moreover, since \({\overline{\tau }}(\gamma ,{\mathbf {t}},{\mathbf {s}})< 1\) for all \((\gamma ,{\mathbf {s}})\) and \({\mathbf {t}}<1\), then \({\overline{\tau }}(\gamma ,1,{\mathbf {s}})\le 1\). So, since for every \(\gamma \), the function \(f({\mathbf {s}})={\overline{\tau }}(\gamma ,1,{\mathbf {s}})\) reaches its maximum at \({\mathbf {s}}=0\), then, the smoothness of \({\overline{\tau }}\) brings \(\frac{\partial {\overline{\tau }}}{\partial {\mathbf {s}}}\left( \gamma ,1,0\right) =f'(0)=0\).
Now, let us show that \(\varPhi \left( \gamma ,{\mathbf {t}}\right) =\frac{\partial X}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \) can be smoothly extended to \({\mathcal {N}}^{\epsilon }_{U} \times \left( 1-\delta ,1\right] \). Notice that in (5.10) and (5.13), we have defined \(X(\gamma ,{\mathbf {t}},{\mathbf {s}})=\gamma _{({\mathbf {t}},{\mathbf {s}})}\) for \({\mathbf {t}}<1\) such that \(c_{(\gamma ,{\mathbf {t}})}({\mathbf {s}})=\gamma _{({\mathbf {t}},{\mathbf {s}})}(0)\in C\) is arc–length parametrized and, by lemma 1, the tangent vector \(\langle J_{(\gamma ,{\mathbf {t}})} \rangle =\frac{\partial X}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \) of the variation of light rays \(X(\gamma ,{\mathbf {t}},{\mathbf {s}})\) at \({\mathbf {s}}=0\) can be chosen such that \(J_{(\gamma ,{\mathbf {t}})}(0)=c'_{(\gamma ,{\mathbf {t}})}(0)\).
We can consider the fibre bundle \(\pi :{\mathcal {H}}\rightarrow \mathbb {P}({\mathcal {H}})\) and any smooth non–zero local section \({\overline{\omega }}:\widetilde{{\mathcal {U}}}\subset \mathbb {P}\left( {\mathcal {H}}\right) \rightarrow {\mathcal {H}}\) in some neighbourhood \(\widetilde{{\mathcal {U}}}\) of some \({\widetilde{\gamma }}_0\left( 1\right) \in \partial ^{+}\widetilde{{\mathcal {N}}}\). Without any lack of generality, we can assume that \(\widetilde{{\mathcal {U}}}=\varepsilon \left( {\mathcal {N}}^{\epsilon }_{U}\times (1-\delta ,1+\delta )\right) \).
Since \(\pi \left( {\overline{\omega }}({\widetilde{\gamma }}({\mathbf {t}}))\right) ={\widetilde{\gamma }}({\mathbf {t}})\), then we have
$$\begin{aligned} {\overline{\omega }}({\widetilde{\gamma }}({\mathbf {t}}))\in {\widetilde{\gamma }}({\mathbf {t}}) . \end{aligned}$$
We denote by \(\langle Z_{(\gamma ,{\mathbf {t}})} \rangle \) the class of Jacobi fields along \(\gamma \in {\mathcal {N}}^{\epsilon }_{U}\) defined by \({\overline{\omega }}({\widetilde{\gamma }}({\mathbf {t}}))\), that is
$$\begin{aligned} {\overline{\omega }}({\widetilde{\gamma }}({\mathbf {t}}))=\langle Z_{(\gamma ,{\mathbf {t}})} \rangle \in {\mathcal {H}}_{\gamma } . \end{aligned}$$
For any \({\mathbf {t}}\ne 0\), any representative \(Z_{(\gamma ,{\mathbf {t}})}\in \langle Z_{(\gamma ,{\mathbf {t}})} \rangle \) verifies that
$$\begin{aligned} Z_{(\gamma ,{\mathbf {t}})}(0) \ne 0~(\mathrm {mod}~\gamma '(0)) \end{aligned}$$
because, in other case, we will have \(\langle Z_{(\gamma ,{\mathbf {t}})}\rangle \in {\widetilde{\gamma }}(0)\cap {\widetilde{\gamma }}({\mathbf {t}})\). But this is not possible since \({\widetilde{\gamma }}\) is an injective curve for all \({\mathbf {t}}\in \mathbb {R}\), as noted in Remark 11.
In fact, by locality of \({\overline{\omega }}\), we can assume that \(Z_{(\gamma ,{\mathbf {t}})}(0)\) is far from 0 because \(Z_{(\gamma _0,1)}(0)\ne 0~(\mathrm {mod}~\gamma _0'(0))\), it means that for all \((\gamma ,{\mathbf {t}})\in {\mathcal {N}}^{\epsilon }_{U}\times (1-\delta ,1+\delta )\) we have
$$\begin{aligned} \vert Z_{(\gamma ,{\mathbf {t}})}(0) \vert ^2={\mathbf {g}}\left( Z_{(\gamma ,{\mathbf {t}})}(0), Z_{(\gamma ,{\mathbf {t}})}(0)\right) \ge \epsilon _0 > 0 \end{aligned}$$
for some \(\epsilon _0 > 0\). We can call \(f(\gamma ,{\mathbf {t}})=\vert Z_{(\gamma ,{\mathbf {t}})}(0) \vert \) the smooth function which does not annihilate for all \((\gamma ,{\mathbf {t}})\in {\mathcal {N}}^{\epsilon }_{U}\times (1-\delta ,1+\delta )\).
If we define
$$\begin{aligned} Y_{(\gamma ,{\mathbf {t}})}= \frac{1}{f(\gamma ,{\mathbf {t}})} \cdot Z_{(\gamma ,{\mathbf {t}})} \end{aligned}$$
we have that
$$\begin{aligned} \omega \left( {\widetilde{\gamma }}({\mathbf {t}})\right) =\frac{1}{f(\gamma ,{\mathbf {t}})} \cdot {\overline{\omega }}\left( {\widetilde{\gamma }}({\mathbf {t}})\right) =\langle Y_{(\gamma ,{\mathbf {t}})} \rangle \in {\widetilde{\gamma }}({\mathbf {t}}) \end{aligned}$$
is another smooth non–zero local section defined for all \((\gamma ,{\mathbf {t}})\in {\mathcal {N}}^{\epsilon }_{U}\times (1-\delta ,1+\delta )\) verifying
$$\begin{aligned} {\mathbf {g}}\left( Y_{(\gamma ,{\mathbf {t}})}(0), Y_{(\gamma ,{\mathbf {t}})}(0)\right) =1 . \end{aligned}$$
Take into account that, since \({\widetilde{\gamma }}({\mathbf {t}})=T_{\gamma }S\left( \gamma \left( {\mathbf {t}}\right) \right) \) is 1–dimensional, then the initial vectors \(Y_{\left( \gamma ,{\mathbf {t}}\right) }\left( 0\right) \) determine the value of the section \(\omega \). In fact, if \({\overline{Y}}\) is a another Jacobi field along \(\gamma \) such that \(\langle {\overline{Y}} \rangle \in {\widetilde{\gamma }}({\mathbf {t}})\) with the same initial vector \({\overline{Y}}\left( 0\right) =Y_{(\gamma ,{\mathbf {t}})}(0)\), then \(K=Y_{(\gamma ,{\mathbf {t}})}-{\overline{Y}}\) is also a Jacobi field along \(\gamma \) verifying \(K\left( 0\right) =0~\left( \mathrm {mod}~\gamma '(0)\right) \) and so, \(K\in {\widetilde{\gamma }}\left( 0\right) \cap {\widetilde{\gamma }}\left( {\mathbf {t}}\right) \). Since every curve \({\widetilde{\gamma }}\) is injective, then \(K=0\) and therefore \(\langle {\overline{Y}} \rangle =\langle Y_{(\gamma ,{\mathbf {t}})} \rangle \).
Recall that the curves \(c_{(\gamma ,{\mathbf {t}})}\) have been parametrized by arc–length, so
$$\begin{aligned} {\mathbf {g}}\left( c'_{(\gamma ,{\mathbf {t}})}(0), c'_{(\gamma ,{\mathbf {t}})}(0)\right) =1 \end{aligned}$$
and we can take representatives \(Y_{(\gamma ,{\mathbf {t}})}\) such that \(Y_{(\gamma ,{\mathbf {t}})}(0)=c'_{(\gamma ,{\mathbf {t}})}(0)\), therefore by construction of \(X(\gamma ,{\mathbf {t}},{\mathbf {s}})\) and \(\omega \), then
$$\begin{aligned} \omega \left( {\widetilde{\gamma }}\left( {\mathbf {t}}\right) \right) =\langle Y_{\left( \gamma ,{\mathbf {t}}\right) }\rangle = \langle J_{\left( \gamma ,{\mathbf {t}}\right) }\rangle =\frac{\partial X}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \end{aligned}$$
(5.17)
holds for \({\mathbf {t}}<1\).
Therefore \({\overline{\varPhi }}\left( \gamma ,{\mathbf {t}}\right) =\omega \left( \varepsilon (\gamma ,{\mathbf {t}})\right) =\omega \left( {\widetilde{\gamma }}\left( {\mathbf {t}}\right) \right) \) is a smooth extension of \(\varPhi \left( \gamma ,{\mathbf {t}}\right) =\frac{\partial X}{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \) defined in \({\mathcal {N}}^{\epsilon }_{U} \times \left( 1-\delta , 1+\delta \right) \).
Now, the expression of Eq. (5.16) can be extended as
$$\begin{aligned} \overline{{\widetilde{\varPhi }}}\left( \gamma ,{\mathbf {t}}\right) =\left( d\varepsilon \right) _{\left( \gamma ,{\mathbf {t}}\right) }\left( {\overline{\varPhi }}\left( \gamma ,{\mathbf {t}}\right) , {\overline{h}}\left( \gamma ,{\mathbf {t}}\right) \right) \end{aligned}$$
for all \(\left( \gamma ,{\mathbf {t}}\right) \in {\mathcal {N}}^{\epsilon }_{U}\times \left( 1-\delta ,1\right] \) such that
$$\begin{aligned} \frac{\partial {\widetilde{X}}}{\partial s}\left( \gamma ,{\mathbf {t}},0\right) =\overline{{\widetilde{\varPhi }}}\left( \gamma ,{\mathbf {t}}\right) \end{aligned}$$
because \({\overline{\varPhi }}\left( \gamma ,{\mathbf {t}}\right) =\frac{\partial X}{\partial s}\left( \gamma ,{\mathbf {t}},0\right) \) and \({\overline{h}}\left( \gamma ,{\mathbf {t}}\right) =\frac{\partial \tau }{\partial s}\left( \gamma ,{\mathbf {t}},0\right) \) for \({\mathbf {t}}<1\).
Notice that a curve \(\varGamma \left( s\right) \in {\mathcal {N}}\) is an integral curve of \(\oplus :{\mathcal {N}}\rightarrow \mathbb {P}\left( {\mathcal {H}}\right) \) if \(\varGamma '\left( s\right) \in \oplus _{\varGamma \left( s\right) }\). So, the curve \({\widetilde{\varGamma }}\left( s\right) =\varepsilon \left( \varGamma \left( s\right) ,1\right) \) is a leaf of the distribution \(\partial ^{+}{\mathcal {D}}^{\sim }\) if \(\varGamma '\left( s\right) \in \oplus _{\varGamma \left( s\right) }\), that is
$$\begin{aligned} {\widetilde{\varGamma }}'\left( s\right) =\left( d\varepsilon \right) _{\left( \varGamma \left( s\right) ,1\right) }\left( \varGamma '\left( s\right) , 0 \right) \in \partial ^{+}{\mathcal {D}}^{\sim } \Longleftrightarrow \varGamma '\left( s\right) \in \oplus _{\varGamma \left( s\right) } . \end{aligned}$$
(5.18)
Now, since \(\frac{\partial {\overline{X}}}{\partial s}\left( \gamma ,{\mathbf {t}},0\right) \in {\widetilde{\gamma }}({\mathbf {t}})\) for \(1-\delta<{\mathbf {t}}<1\), by continuity we have
$$\begin{aligned} \frac{\partial {\overline{X}}}{\partial s}\left( \gamma ,1,0\right) \in {\widetilde{\gamma }}(1)=\oplus _{\gamma } \end{aligned}$$
and also
$$\begin{aligned} \frac{\partial \overline{{\widetilde{X}}}}{\partial s}\left( \gamma ,1,0\right)&=\left( d\varepsilon \right) _{\left( \gamma ,1\right) }\left( \frac{\partial {\overline{X}}}{\partial s}\left( \gamma ,1,0\right) , \frac{\partial {\overline{\tau }}}{\partial s}\left( \gamma ,1,0\right) \right) = \\&=\left( d\varepsilon \right) _{\left( \gamma ,1\right) }\left( \frac{\partial {\overline{X}}}{\partial s}\left( \gamma ,1,0\right) , 0 \right) . \end{aligned}$$
Then, by (5.18), \(\frac{\partial \overline{{\widetilde{X}}}}{\partial s}\left( \gamma ,1,0\right) \in \partial ^{+}{\mathcal {D}}^{\sim }\).
Therefore, \(\overline{{\widetilde{\varPhi }}}\left( \gamma ,{\mathbf {t}}\right) =\frac{\partial \overline{{\widetilde{X}}}}{\partial s}\left( \gamma ,{\mathbf {t}},0\right) \) is a smooth vector field defining \(\overline{{\mathcal {D}}^{\sim }}={\mathcal {D}}^{\sim } \cup \partial ^{+}{\mathcal {D}}^{\sim }\) for all \(1-\delta <{\mathbf {t}}\le 1\), so \(\overline{{\mathcal {D}}^{\sim }}\) is a differentiable distribution (see Fig. 13). \(\square \)
As an immediate consequence we have the following result.
Corollary 6
The distribution \(\overline{{\mathcal {D}}^{\sim }}\) in \(\overline{\widetilde{{\mathcal {N}}}}\) is smooth if for each light ray \(\gamma _0\in {\mathcal {N}}\) there exist \(U\subset M\) such that \(\gamma _0\in {\mathcal {N}}_U=\{ \gamma \in {\mathcal {N}} : \gamma \cap U \ne \varnothing \}\) satisfying the hypotheses of Theorem 5.1.
At this point, it has not yet been possible to prove whether the hypotheses H1 and H2 are sufficient conditions to establish the existence of the extension of the function \(h(\gamma ,{\mathbf {t}})=\frac{\partial \tau }{\partial {\mathbf {s}}}\left( \gamma ,{\mathbf {t}},0\right) \) to \({\mathbf {t}}=1\) or whether there is some counterexample of this. So, this is an open question that should be studied in future work.
The canonical future extension of M
Let us assume that \(\overline{{\mathcal {D}}^{\sim }}\) is a smooth distribution. Notice that the leaves of \({\mathcal {D}}^{\sim }\) are always compact but the ones of \(\partial ^{+}{\mathcal {D}}^{\sim }\) could be not, making of \(\overline{{\mathcal {D}}^{\sim }}\) a non–regular distribution. When, moreover, \(\overline{{\mathcal {D}}^{\sim }}\) becomes a regular distribution then the quotient
$$\begin{aligned} \overline{\widetilde{{\mathcal {N}}}} / \overline{{\mathcal {D}}^{\sim }} = \widetilde{{\mathcal {N}}} / {\mathcal {D}}^{\sim } \cup \partial ^{+}\widetilde{{\mathcal {N}}} / \partial ^{+}{\mathcal {D}}^{\sim } \end{aligned}$$
is a differentiable manifold. In this case, by the diffeomorphism \({\widetilde{S}}:M\rightarrow \widetilde{{\mathcal {N}}} / {\mathcal {D}}^{\sim }\) of Eq. (5.9), and because \(\partial ^{+}\widetilde{{\mathcal {N}}}\) is the boundary of \(\overline{\widetilde{{\mathcal {N}}}}\), then \( \partial ^{+}\widetilde{{\mathcal {N}}} / \partial ^{+}{\mathcal {D}}^{\sim }\) is the boundary of \(\overline{\widetilde{{\mathcal {N}}}} / \overline{{\mathcal {D}}^{\sim }}\). Then we can extend \({\widetilde{S}}\) by
$$\begin{aligned} {\widetilde{S}}:{\overline{M}}\rightarrow \overline{\widetilde{{\mathcal {N}}}} / \overline{{\mathcal {D}}^{\sim }} \end{aligned}$$
where \({\overline{M}} = M \cup \partial ^{+} M \) with
$$\begin{aligned} \partial ^{+} M = \partial ^{+}\widetilde{{\mathcal {N}}} / \partial ^{+}{\mathcal {D}}^{\sim } \end{aligned}$$
in such a way that \(\left. {\widetilde{S}}\right| _{\partial ^{+} M}\) is the identity map. Then there exists a differentiable structure in \({\overline{M}}\), compatible with the one in M, such that the extension \({\widetilde{S}}\) is a diffeomorphism inducing in \(\partial ^{+} M\) a differentiable structure.
This is summarized in the following corollary.
Corollary 7
If \(\overline{{\mathcal {D}}^{\sim }}\) is a smooth and regular distribution then the quotient \(\overline{\widetilde{{\mathcal {N}}}} / \overline{{\mathcal {D}}^{\sim }} \) is a differentiable manifold with boundary \( \partial ^{+} M= \partial ^{+}\widetilde{{\mathcal {N}}} / \partial ^{+}{\mathcal {D}}^{\sim }\). Moreover, there exists an extension
$$\begin{aligned} {\widetilde{S}}:{\overline{M}}=M \cup \partial M\rightarrow \overline{\widetilde{{\mathcal {N}}}} / \overline{{\mathcal {D}}^{\sim }} \end{aligned}$$
of the diffeomorphism \({\widetilde{S}}:M\rightarrow \widetilde{{\mathcal {N}}} / {\mathcal {D}}^{\sim }\) of Eq. (5.9) such that \(\left. {\widetilde{S}}\right| _{\partial ^{+} M}\) is the identity map and it induces in \(\partial ^{+} M\) a differentiable structure such that the extended map \({\widetilde{S}}\) is a diffeomorphism.
Definition 14
The extension \({\overline{M}}\) constructed in the Corollary 7 is called the canonical future extension of \(\left( M,{\mathcal {C}}\right) \) and the boundary \(\partial ^{+} M\) is the future L–boundary.
Again, we only focus on future boundary, but the construction of the canonical past extension \(M\cup \partial ^{-}M\) and the past boundary \(\partial ^{-} M\) can be done using the distribution \(\ominus \) analogously.
In virtue of maps (5.6) and (5.9), we obtain a double fibration extended to the boundaries which is analogous to the double fibration (1.1) of twistor theory. So, we have
where \(\pi ^{\mathbb {P}\left( {\mathcal {H}}\right) }_{{\mathcal {N}}}\) is the canonical projection that can be expressed by \(\pi ^{\mathbb {P}\left( {\mathcal {H}}\right) }_{{\mathcal {N}}}=p_1\circ \varepsilon ^{-1}\) as in Eq. (5.10) and \(\rho \) is the extension of the submersion given by \(\rho ={\widetilde{S}}^{-1}\circ {\widetilde{\pi }}\) where \({\widetilde{\pi }}\) is the quotient map given in diagram (5.8).
Remark 12
The regularity of \(\overline{{\mathcal {D}}^{\sim }}\) is achieved, for example when the leaves of \(\partial ^{+}{\mathcal {D}}^{\sim }\) are compact, because then, all the leaves of \(\overline{{\mathcal {D}}^{\sim }}\) are compact. This holds when M is globally hyperbolic with compact Cauchy surface C. In this case we have that \({\mathcal {N}}\simeq {\mathbb {P}}{\mathbb {N}}(C)\) is compact and since, by hypothesis, \(\oplus \) is a regular distribution, then the leaves of \(\partial ^{+}{\mathcal {D}}^{\sim }\) must be compact, therefore \(\overline{{\mathcal {D}}^{\sim }}\) is regular and the canonical extension \({\overline{M}}\) is a differentiable manifold. This is the case of de Sitter \(d{\mathcal {S}}^{m}\) spacetimes and Robertson-Walker models without initial or final singularity.
If the leaves of \(\partial ^{+}{\mathcal {D}}^{\sim }\) are not compact, the canonical extension can still exist, as for Minkowski \(\mathbb {M}^m\) spacetimes.
In Example 7, the canonical extension of Minkowski is built for \(\mathrm {dim}(M)=3\). For higher dimension it can be computed integrating the distribution \(\oplus \) in \({\mathcal {N}}\) as done in [7, Sec. IV.B]. In this reference, the L–boundary of \(\mathbb {M}^3\) is constructed by restriction of the one of \(\mathbb {M}^4\). This can be done because there exists an embedding \(\mathbb {M}^3\hookrightarrow \mathbb {M}^4\) such that the maximal null geodesic in \(\mathbb {M}^3\) are maximal null geodesics in \(\mathbb {M}^4\). Moreover, any Cauchy surface \({\overline{C}}\subset \mathbb {M}^4\) defines a Cauchy surface \(C={\overline{C}}\cap \mathbb {M}^3\) in \(\mathbb {M}^3\) and, since we have an embedding \(T\mathbb {M}^3\hookrightarrow T\mathbb {M}^4\), then \({\mathbb {P}}{\mathbb {N}}(C)\hookrightarrow {\mathbb {P}}{\mathbb {N}}({\overline{C}})\) is an embedding. Therefore, \({\mathcal {N}}_{\mathbb {M}^3}\hookrightarrow {\mathcal {N}}_{\mathbb {M}^4}\) is an embedding and this implies that \(T{\mathcal {N}}_{\mathbb {M}^3}\hookrightarrow T{\mathcal {N}}_{\mathbb {M}^4}\) is another embedding. Then, denoting \(\overline{{\mathcal {H}}}\) and \({\overline{\oplus }}\) the contact structure and the future limit distribution related to \(\mathbb {M}^4\) and \({\mathcal {H}}\) and \(\oplus \) the same geometric objects related to \(\mathbb {M}^3\), then it is easy to see that
$$\begin{aligned} {\mathcal {H}}_{\gamma }= \overline{{\mathcal {H}}}\cap T_{\gamma }{\mathcal {N}}_{\mathbb {M}^3}, \qquad \oplus _{\gamma }= {\overline{\oplus }}\cap T_{\gamma }{\mathcal {N}}_{\mathbb {M}^3} \qquad \text { for } \gamma \in {\mathcal {N}}_{\mathbb {M}^3}\subset {\mathcal {N}}_{\mathbb {M}^4} . \end{aligned}$$
We will use this procedure in Example 8 for the 3–dimensional de Sitter spacetime which is embedded in \(\mathbb {M}^4\).
Example 7
Consider the 3–dimensional Minkowski spacetime block
$$\begin{aligned} \mathbb {M}^3_{(a,b)} = \left\{ (t,x,y)\in \mathbb {M}^3: a<t<b \right\} . \end{aligned}$$
To simplify, we will only compute the future L–boundary for \(-\infty<a<0\) to avoid \(\oplus = \ominus \) when \(b=\infty \) and to keep \(C\equiv \{t=0\}\) as Cauchy surface for initial values of the null geodesics. First, observe that, by (2.13), the homogeneous coordinate \(\phi \) in Eq. (5.4) is
$$\begin{aligned} \phi =[-s:1]=\left[ 1:\frac{1}{-s}\right] \end{aligned}$$
when \(s\ne 0\) then, by Eq. (5.5), we can consider the projective parameter \({\mathbf {t}}(s)=\frac{(b-a)s}{(b+a)s-2ab}\) and whence \(s({\mathbf {t}})=\frac{2ab{\mathbf {t}}}{(b+a){\mathbf {t}}-(b-a)}\). In Example 5, we have seen that \(\mu (\theta ,s,\tau )\in C\) whenever \(\tau =-s\) and this value does not depend on \(\theta \in \left[ 0,2\pi \right) \). Moreover, in virtue of Remark 6 and Examples 4 and 5, the tangent space of the sky \(S\left( \gamma \left( {\mathbf {t}}\right) \right) \in \varSigma \) at \(\gamma \) can be written as
$$\begin{aligned} T_{\gamma }S\left( \gamma \left( {\mathbf {t}}\right) \right) = \mathrm {span}\left\{ \textstyle { s({\mathbf {t}})\left( \sin \theta _0 \left( \frac{\partial }{\partial x} \right) _{\gamma } - \cos \theta _0 \left( \frac{\partial }{\partial y} \right) _{\gamma } \right) + \left( \frac{\partial }{\partial \theta } \right) _{\gamma } } \right\} . \end{aligned}$$
(5.20)
Therefore, the orbit of the distribution \(\overline{{\mathcal {D}}^{\sim }}\) in \(\overline{\widetilde{{\mathcal {N}}}}\) passing through \({\widetilde{\gamma }}({\mathbf {t}})\simeq (x_0,y_0,\theta _0,{\mathbf {t}}_0)\) corresponds to the integral curve \(c\left( r\right) =\left( x\left( r\right) ,y\left( r\right) ,\theta \left( r\right) ,{\mathbf {t}}\left( r\right) \right) \) of the vector field
$$\begin{aligned} {\widetilde{\varPhi }}=s({\mathbf {t}})\left( \sin \theta \frac{\partial }{\partial x} - \cos \theta \frac{\partial }{\partial y}\right) +\frac{\partial }{\partial \theta } \in {\mathfrak {X}}\left( \widetilde{{\mathcal {N}}}\right) . \end{aligned}$$
So, after integration, it can be written by
$$\begin{aligned} c\left( r\right)&=\textstyle {\left( x_0 + s({\mathbf {t}}_0) \left[ \cos \theta _0 -\cos \left( \theta _0+r\right) \right] \, , \right. } \\&\textstyle { \left. y_0 + s({\mathbf {t}}_0) \left[ \sin \theta _0 -\sin \left( \theta _0+r\right) \right] \, , \, \theta _0+r \, , \, {\mathbf {t}}_0\right) } \end{aligned}$$
and verifies
$$\begin{aligned} \left\{ \begin{array}{l} \left( x-x_0 - s({\mathbf {t}}_0) \cos \theta _0\right) ^2+ \left( y-y_0 - s({\mathbf {t}}_0) \sin \theta _0\right) ^2=s^2({\mathbf {t}}_0)\\ \theta = \theta _0+r \\ {\mathbf {t}}={\mathbf {t}}_0 \end{array} \right. . \end{aligned}$$
(5.21)
Since \(\lim _{\mathbf {t_0}\mapsto 1^{-}}s({\mathbf {t}}_0)=b\) then, whenever \(b<\infty \), the light rays in \(\mathbb {M}^{3}_{(a,b)}\) of the orbit of \(\partial ^{+}{\mathcal {D}}^{\sim }\) passing through \({\widetilde{\gamma }}(1)\simeq (x_0,y_0,\theta _0, 1)\) corresponds to the sky in \(\mathbb {M}^{3}\) of the point \(p=\left( b, x_0 + b \cos \theta _0 , y_0 + b \sin \theta _0\right) \), see Fig. 14a. Then the future boundary \(\partial ^{+} M\) can be identified with the topological boundary of \(\mathbb {M}^{3}_{(a,b)}\) as a set in \(\mathbb {M}^{3}\), that is
$$\begin{aligned} \partial ^{+} M = \{ (t,x,y)\in \mathbb {M}^3 : t=b\} . \end{aligned}$$
In case of \(b=\infty \), we can develop the squares of the first equation in (5.21) and divide by \(s({\mathbf {t}}_0)\) to obtain
$$\begin{aligned} \left\{ \begin{array}{l} \frac{1}{s({\mathbf {t}}_0)}\left( x-x_0\right) ^2 - 2\left( x-x_0\right) \cos \theta _0 + \frac{1}{s({\mathbf {t}}_0)}\left( y-y_0\right) ^2 - 2\left( y-y_0\right) \sin \theta _0=0\\ \theta = \theta _0+r \\ {\mathbf {t}}={\mathbf {t}}_0 \end{array} \right. \end{aligned}$$
then, taking the limit \(\lim _{\mathbf {t_0}\mapsto 1^{-}}s({\mathbf {t}}_0)=+\infty \), we have that the orbit of \(\partial ^{+}{\mathcal {D}}^{\sim }\) passing through the point with coordinates \((x_0,y_0,\theta _0,1)\in \overline{\widetilde{{\mathcal {N}}}}\) verifies
$$\begin{aligned} \left\{ \begin{array}{l} \left( x-x_0\right) \cos \theta _0 + \left( y-y_0\right) \sin \theta _0=0\\ \theta = \theta _0 \\ {\mathbf {t}}=1 \end{array} \right. . \end{aligned}$$
Therefore, as it is illustrated in Fig. 14b, the orbit consists of the light rays with tangent vector \(v=\left( 1, \cos \theta _0, \sin \theta _0 \right) \) intersecting the Cauchy surface \(C\equiv \{t=0\}\) at the points of the straight line
$$\begin{aligned} \left\{ \begin{array}{l} \left( x-x_0\right) \cos \theta _0 + \left( y-y_0\right) \sin \theta _0 =0 \\ t= 0 . \end{array} \right. \end{aligned}$$
It is trivial to see that for any \(\beta \in {\mathcal {N}}\) in the above orbit of \(\oplus \), the chronological past of \(\beta \) is
$$\begin{aligned} I^{-}\left( \beta \right) = \left\{ \left( t,x,y\right) \in \mathbb {M}^3: t< \left( x-x_0\right) \cos \theta _0 + \left( y-y_0\right) \sin \theta _0 \right\} \end{aligned}$$
then, any point in the future L-boundary corresponds to a point in the future c-boundary.
In both cases, the L–boundary coincides with the part of c–boundary of \(\mathbb {M}^{3}_{(a,b)}\) accessible by light rays, but if \(b=\infty \), it is not possible for the L–boundary to obtain the points of the timelike c–boundary because there exist inextensible timelike curves with chronological past which is not the chronological past of any light ray (see Remark 9).
Example 8
We consider the 3–dimensional de Sitter spacetime \(d{\mathcal {S}}^{3}\) embedded in \(\mathbb {M}^4\) as the set verifying
$$\begin{aligned} -t^{2}+x^{2}+y^{2}+z^{2}=1 \, . \end{aligned}$$
(5.22)
As seen in Example 4, a null geodesic in \(\mathbb {M}^4\) can be written by
$$\begin{aligned} \gamma (s)=\left( s,x_0+s \cos \theta _0\sin \phi _0,y_0+s \sin \theta _0\sin \phi _0,z_0+s \cos \phi _0 \right) \in \mathbb {M}^{4} \end{aligned}$$
where \(\gamma (0)\in {\overline{C}}\equiv \{t=0\}\). Then \(\gamma \) is a null geodesic in \(d{\mathcal {S}}^{3}\) if Eq. (5.22) is satisfied, so
$$\begin{aligned} -s^2 + \left( x_0+s\cos \theta _0 \sin \phi _0 \right) ^{2} + \left( y_0 +s\sin \theta _0 \sin \phi _0\right) ^{2} + \left( z_0+s\cos \phi _0\right) ^{2} = 1 \end{aligned}$$
and simplifying we get
$$\begin{aligned}&\left( x_0\cos \theta _0 +y_0\sin \theta _0\right) \sin \phi _0+z_0\cos \phi _0=0\nonumber \\&\quad \Rightarrow \quad \cot \phi _0=-\frac{x_0 \cos \theta _0+y_0 \sin \theta _0}{z_0} . \end{aligned}$$
(5.23)
Taking \(t=0\) in Eq. (5.22), we obtain a Cauchy surface \(C\subset d{\mathcal {S}}^{3}\), in fact it is 2–sphere, restriction of the Cauchy surface \({\overline{C}}\equiv \{t=0\}\subset \mathbb {M}^4\) to \(d{\mathcal {S}}^{3}\). We can parametrize C by
$$\begin{aligned} \left\{ \begin{array}{l} x=\cos u \sin w \\ y=\sin u \sin w \\ z=\cos w \end{array} \right. \end{aligned}$$
(5.24)
where \((x_0,y_0,z_0)=(\cos u_0\sin w_0,\sin u_0 \sin w_0,\cos w_0)\).
By Remark 6, the tangent space to the sky of \(\gamma (s)\) related to \(\mathbb {M}^4\) is
$$\begin{aligned} T_{\gamma }{\overline{S}}\left( \gamma (s)\right)&= \textstyle { \mathrm {span}\left\{ s \left( \sin \theta _0 \sin \phi _0\left( \frac{\partial }{\partial x} \right) _{\gamma } - \cos \theta _0 \sin \phi _0\left( \frac{\partial }{\partial y} \right) _{\gamma }\right) + \left( \frac{\partial }{\partial \theta } \right) _{\gamma }, \right. } \\&\textstyle { \left. s\left( -\cos \theta _0 \cos \phi _0 \left( \frac{\partial }{\partial x} \right) _{\gamma } - \sin \theta _0 \cos \phi _0 \left( \frac{\partial }{\partial y} \right) _{\gamma } + \sin \phi _0 \left( \frac{\partial }{\partial z} \right) _{\gamma }\right) +\left( \frac{\partial }{\partial \theta } \right) _{\gamma } \right\} } \end{aligned}$$
and then
$$\begin{aligned} {\overline{\oplus }}_{\gamma }&=\lim _{s\mapsto \infty } T_{\gamma }S\left( \gamma (s)\right) = \\&=\textstyle { \mathrm {span}\left\{ \sin \theta _0 \sin \phi _0\left( \frac{\partial }{\partial x} \right) _{\gamma } - \cos \theta _0 \sin \phi _0\left( \frac{\partial }{\partial y} \right) _{\gamma }, \right. } \\&\textstyle { \left. -\cos \theta _0 \cos \phi _0 \left( \frac{\partial }{\partial x} \right) _{\gamma } - \sin \theta _0 \cos \phi _0 \left( \frac{\partial }{\partial y} \right) _{\gamma } + \sin \phi _0 \left( \frac{\partial }{\partial z} \right) _{\gamma } \right\} } . \end{aligned}$$
Integrating \({\overline{\oplus }}\), we obtain that its orbit passing through \((x_0,y_0,z_0,\theta _0,\phi _0)\in {\mathcal {N}}_{\mathbb {M}^4}\) is defined by
$$\begin{aligned} \left\{ \begin{array}{l} x(\tau ,\eta ) = x_0+\tau \sin \theta _0 \sin \phi _0 - \eta \cos \theta _0 \cos \phi _0 \\ y(\tau ,\eta ) = y_0-\tau \cos \theta _0 \sin \phi _0 - \eta \sin \theta _0 \cos \phi _0 \\ z(\tau ,\eta ) = z_0+ \eta \sin \phi _0 \\ \theta (\tau ,\eta ) = \theta _0 \\ \phi (\tau ,\eta ) = \phi _0 \end{array} \right. \end{aligned}$$
which verifies the equation
$$\begin{aligned} \left\{ \begin{array}{l} \cos \theta _0 \sin \phi _0 \cdot \left( x-x_0\right) + \sin \theta _0 \sin \phi _0 \cdot \left( y-y_0\right) + \cos \phi _0 \cdot \left( z-z_0\right) =0 \\ \theta = \theta _0 \\ \phi = \phi _0 . \end{array} \right. \end{aligned}$$
(5.25)
Substituting (5.23) and (5.24) in (5.25), we obtain the expression of the orbit of the field \(\oplus \) in \({\mathcal {N}}_{d{\mathcal {S}}^3}\) as the restriction of the orbit of \({\overline{\oplus }}\) to \(d{\mathcal {S}}^3\). So, we have
$$\begin{aligned}&\cos \theta _0 \sin \phi _0 \cos u \sin w + \sin \theta _0 \sin \phi _0 \sin u \sin w+ \cos \phi _0 \cos w=0 \\ \Rightarrow&\tan w \left( \cos \theta _0 \cos u + \sin \theta _0 \sin u \right) =-\cot \phi _0 \\ \Rightarrow&\tan w \cos \left( u -\theta _0 \right) =-\cot \phi _0 \\ \text { (by }(5.23))\Rightarrow&\tan w \cos \left( u -\theta _0 \right) =\tan w_0 \cos \left( u_0 -\theta _0 \right) \end{aligned}$$
and therefore, the orbits of \(\oplus \) must satisfy the equation
$$\begin{aligned} \tan w \cos \left( u -\theta _0 \right) =\tan w_0 \cos \left( u_0 -\theta _0 \right) . \end{aligned}$$
(5.26)
For any \(\gamma _0\in {\mathcal {N}}_{d{\mathcal {S}}^3}\) with coordinates \((u_0,w_0,\theta _0)\simeq \gamma _0\), it is straightforward to check that (u, w) in Eq. (5.26) corresponds with a maximal circumference on the Cauchy surface \(C=\mathbb {S}^2\). This circumference is the limiting curve of the intersection of the lights rays in \(S(\gamma _0(s))\) with C whenever \(s\mapsto \infty \) (see Fig. 15a). For fixed \(\theta =\theta _0\) we obtain all maximal circumferences passing through the points \((u,w)=\left( \theta _0\pm \frac{\pi }{2},\frac{\pi }{2}\right) \) (see Fig. 15b). Then, moving \(\theta _0\) in \(\left[ 0,2\pi \right) \), we get all maximal circumferences in the sphere C. Observe that each maximal circumference can be obtained twice as solutions of Eq. (5.26) for two different light rays with the same values of \((u_0,w_0)\) and the antipodal values \(\theta _0\) and \(\theta _0+\pi \). So, although the space of maximal circumferences in C is diffeomorphic to the projective space \(\mathbb {P}(\mathbb {R}^3)\), the orbits of \(\oplus \) do not coincide for antipodal values of \(\theta _0\). Therefore, we have that the space of orbits of \(\oplus \), that is, the future L–boundary is \(\partial ^{+}(d{\mathcal {S}}^3)\simeq \mathbb {S}^2\). Notice that c–boundary and L–boundary of \(d{\mathcal {S}}^3\) coincide.
Definition 15
A L–spacetime is said to be proper if the total distribution \(\overline{{\mathcal {D}}^\sim }\) is smooth and regular with Hausdorff space of leaves.
If M is a proper L–spacetime, the future L–boundary \(\partial ^{+} M = \partial ^{+}\widetilde{{\mathcal {N}}} / \partial ^{+}{\mathcal {D}}^{\sim }\) built in Corollary 7 is a smooth boundary for the manifold \({\overline{M}} = M \cup \partial ^{+} M\). Moreover, the hausdorffness of the quotient space \(\overline{\widetilde{{\mathcal {N}}}}\) over the regular distribution \(\overline{{\mathcal {D}}^\sim }\) assures that \({\overline{M}}\) is Hausdorff. This can be summarized in the following result.
Corollary 8
If M is a proper L–spacetime then the canonical extension \({\overline{M}}\) of \(\left( M,{\mathcal {C}}\right) \) exists.