Uniqueness of maximal spacetime boundaries

Given an extendible spacetime one may ask how much, if any, uniqueness can in general be expected of the extension. Locally, this question was considered and comprehensively answered in a recent paper of Sbierski, where he obtains local uniqueness results for anchored spacetime extensions of similar character to earlier work for conformal boundaries by Chru\'sciel. Globally, it is known that non-uniqueness can arise from timelike geodesics behaving pathologically in the sense that there exist points along two distinct timelike geodesics which become arbitrarily close to each other interspersed with points which do not approach each other. We show that this is in some sense the only obstruction to uniqueness of maximal future boundaries: Working with extensions that are manifolds with boundary we prove that, under suitable assumptions on the regularity of the considered extensions and excluding the existence of such ''intertwined timelike geodesics'', extendible spacetimes admit a unique maximal future boundary extension. This is analogous to results of Chru\'sciel for the conformal boundary.


Introduction
Questions of (low-regularity) spacetime (in-)extendibility have a long history within mathematical general relativity and are closely related to several important physical problems such as the nature of the incompleteness predicted from the singularity theorems and strong cosmic censorship. The former has lead people to consider various ways of defining a boundary of spacetime (and attaching such boundaries to spacetime). As we will see, some of these old constructions are now providing useful inspirations, tools and reality checks in investigating uniqueness questions. The latter has of course been crucial motivation in studying low-regularity (in-)extendibility theory from the beginning in the hopes that the usually very general results developed in this field might provide useful additions to more PDE based approaches.
In this general framework the usual procedure for determining whether a concrete spacetime or concrete class of spacetimes is extendible admits an extension ι : (M, g) → (M ext , g ext ) (with (M ext , g ext ) being a spacetime and ι an isometric embedding) or not is to follow one of two paths: either an explicit extension of the spacetime is found/constructed or it is shown that the spacetime satisfies some criteria that are known to be general obstructions to extendibility within a certain class of extensions. For instance, blow up of any curvature scalar (e.g., the scalar curvature or the Kretschmann scalar) is an immediate obstruction to C 1,1 -extendibility, that is there cannot exist a proper extension with g ext ∈ C 1,1 . However, different strategies are required in order to explore the inextendibility of a spacetime in a lower regularity class (e.g C 0 -or C 0,1 -regularity).
Here a lot of new tools and techniques have been developed in the last six years, leading to several nice results. For example, the question of C 0 -inextendibility was first tackled by Sbierski [19], who proved that the Minkowski and the maximally extended Schwarzschild spacetime are C 0inextendible. We now have a collection of low regularity inextendibility criteria foremost amongst them timelike geodesic completeness: In the first place, in [19] it was proven that if no timelike curve intersects the boundary of M in the extension, ∂ι(M ), then the spacetime is inextendible. This result already pointed to the idea that, under certain additional assumptions, timelike (geodesic) completeness would yield the inextendibility of a spacetime (in a low regularity class). Indeed, in [5] it was proven that a smooth globally hyperbolic and timelike geodesically complete spacetime is C 0 -inextendible. 1 More importantly for us, [4] also showed that if the past boundary, ∂ − ι(M ), is empty, then the future boundary, ∂ + ι(M ), has to be an achronal topological hypersurface. This is a bit more generally applicable as often the behaviour to the past (or future) is better understood and there are several spacetimes, especially when looking towards cosmological models, that are future or past timelike geodesically complete but not both. Together with a structure result on the existence of certain nice coordinates around any boundary point by Sbierski (cf. Proposition 14, this leads one to suspect that if M is extendible but the past boundary is empty, ι(M ) ∪ ∂ + ι(M ) should be a topological manifold with boundary and, as we will discuss in Section 3 indeed this is the case).
Surprisingly, in case (M, g) is an arbitrary extendible spacetime, the general (i.e., without imposing additional symmetry, field equations or any strong regularity) question of uniqueness of extensions appears to have only recently come up, despite it being a very natural one.
Sbierski [21] proved the local uniqueness of C 0,1 loc -extensions up to (and including) the boundary in the following sense: Let (M, g) be a globally hyperbolic spacetime and consider two C 0,1 locextensions ι 1 and ι 2 satisfying that there exists a future directed timelike curve γ : [0, 1) → M (also called the anchoring curve) such that ι 1 • γ has a limit point p 1 ∈ ∂ι 1 (M ) and ι 2 • γ a limit point p 2 ∈ ∂ι 2 (M ) as t → 1. Then, there exist suitable open subsets U 1 of ι 1 (M ) and U 2 of ι 2 (M ) containing ι 1 • γ and ι 2 • γ such that the restriction of the identification map id := ι 1 • ι −1 2 to these subsets extends to a C 1,1 loc -isometric diffeomorphism id : U 1 ∪ (∂ι 1 (M ) ∩ ∂U 1 ) → U 2 ∪ (∂ι 2 (M ) ∩ ∂U 2 ). Hence, this implies local uniqueness of C 0,1 loc extensions that 'extend through the same region'. These statements are nicely analogous to earlier local uniqueness results for conformal boundaries by Chruściel [2], albeit the details of the proofs clearly differ due to the different setting and the lower regularities Sbierski considers. Sbierski also provides explicit examples that this local uniqueness fails if one allows extensions which are no longer C 0,1 loc . Once one has local uniqueness, the next natural question is if there is a sensible notion of 'maximal extension' and whether such maximal extensions may be globally unique in some sense.
In this paper we aim to answer these questions. However our setup is (out of necessity for our methods but also because of general considerations, cf. the discussion in Remark 10) a bit different from the classical spacetime extensions as we really focus on the boundary and on future directed timelike geodesics. This leads us to consider a different type of extensions of M having the following properties: (i) First, we consider a class of extensions in which the 'extended' manifold is a topological manifold with boundary.
(ii) Secondly, the 'extended' manifolds we work with can be seen as the result of 'attaching' to the original spacetime M the limit points of inextendible incomplete (in M ) timelike geodesics. That is, every point in the boundary should be the endpoint of a future directed timelike geodesic. Further, we need to keep tight control on the topology of the extension at the boundary points. This is achieved by demanding that the manifold topology of the extension can be reconstructed in a very precise way from the timelike geodesics of the original spacetime. This description of a topology via so-called 'timelike thickenings' (see Definition 7) is reminiscent of the old g-boundary construction by Geroch (see [6]) and further motivated by an analogous use of 'null thickenings' in Chruściel's [2] work on maximal conformal boundaries.
(iii) Third, sets of the form ι(M )∪∂ + ι(M ) should furnish examples of these new "future boundary extensions" -at least for well behaved spacetime extensions (M ext , g ext ). We show that this is indeed the case if (M, g) is globally hyperbolic, the past boundary of (M ext , g ext ) is empty and g ext is C 2 in Section 3. In particular, whether ι(M ) ∪ ∂ + ι(M ) satisfies point two appears to be closely tied to the regularity of g ext : It should still work for g ∈ C 1,1 , but becomes quite doubtful below that threshold. One may thus interpret (ii) as a regularity condition.
We call these types of extensions regular future g-boundary extensions and refer to Definition 9 for the exact definitions. We will further motivate this definition in Section 2. Our main goal will be to construct a unique maximal regular future g-boundary extension (provided any such extension exists in the first place), where uniqueness is in the sense of the equivalence in Definition 20, i.e., the composition of the associated embeddings extends to a homeomorphism of topological manifolds with boundary. Note that our regular future g-boundary extensions do not come with a concept of extension of the metric to the boundary, so at this point our uniqueness really is topological in nature and we in particular don't claim anything about uniqueness of the metric on the boundary. This also means that we cannot use the metric at the boundary for our proofs, contrary to our main inspirations of [2,21]. However, in case there were a way of extending the metric to the boundary one might be able to combine our result with techniques from Sbierski's local results to obtain uniqueness of the metric on the boundary as well, but this would have to be explored in some future work. Another avenue for further exploration is that, except for the compatibility results in Section 3, we at this point do not investigate under which criteria given spacetimes possess a regular future g-boundary extension. This question would lead back to the general question of spacetime boundary constructions based on attaching endpoints to incomplete geodesics which generally are rather ill behaved topologically even when excluding the obvious potential offender of 'intertwined' timelike geodesics, that is roughly geodesics which never separate nor remain arbitrarily close as their affine parameter approaches the limit of their interval of existence (Definition 34), as an old example in [7] shows.
Outline of the paper We start by motivating and giving the definition for a regular future g-boundary extension in Section 2 and discussing its relation with the usual concept of spacetime extensions in Section 3. Our procedure to construct a unique maximal regular future g-boundary extension, assuming that at least one regular future g-boundary extension exists and that the original spacetime (M, g) does not contain any intertwined timelike geodesics, is as then follows: First (Section 4), we define an ordering relation via embeddings and then essentially 'glue' together an ordered collection of regular future g-boundary extensions by taking the disjoint union and then identifying all points which are related by the ordering. This makes it straightforward to verify that the resulting object is still a regular future g-boundary extension. Since here the family we are gluing is assumed to be ordered, we can still allow (M, g) to have intertwined timelike geodesics in principle (but the ordering via embeddings implicitly guarantees that these intertwined geodesics would not acquire endpoints in the considered family). This gives us maximal extensions in a set-theoretic sense by a standard Zorn's Lemma type argument, inspired by Choquet-Bruhat and Geroch's [1] proof of the uniqueness of the maximal Cauchy development (see also Ringström's [16] detailed presentation of this proof), cf. Corollary 33. We would like to point out at this point that the extra conditions required on the topology in the definition of a regular future g-boundary extension (beyond being a topological manifold with boundary for which the interior is homeomorphic to M ) are necessary in our proof. The rough reason for this is that these conditions fix a preferred topology on the extension based on timelike thickenings (Definition 7) and provide a (very useful) neighborhood basis for points on the boundary. This allows us to control the topology as we pass to the quotient.
To obtain uniqueness in Section 5 an extra obstruction has to be taken into account: the (possible) existence of intertwined timelike geodesics, can lead to the existence of inequivalent maximal extensions. This problem is already known from the study of the Taub-NUT spacetime (or the simpler example of Misner [14]), which has two inequivalent maximal conformal boundary extensions (see e.g [2], Section 5.7 for a discussion). This leads us to the main result (cf. Theorem 46) of our paper: Theorem 2. Let (M, g) be a strongly causal C 2 spacetime. If (M, g) is regular future g-boundary extendible and does not contain any intertwined future directed timelike geodesics, then there exists a unique maximal regular future g-boundary extension in the sense of Definition 36.
The proof here is rather similar to the above: We do an analogous 'take the disjoint union and then identify' quotient construction for two arbitrary regular future g-boundary extensions but now use that we excluded intertwined timelike geodesics (instead of the ordering) to show that the quotient space is again a regular future g-boundary extension. This then implies that any two set-theoretic maximal elements have to coincide.
Finally we note that while our proofs are based on Zorn's Lemma, our second main Theorem, Theorem 2, can also be obtained more constructively without invoking Zorn's Lemma, cf. the discussion in Remark 47.
Acknowledgements This article originally started from work on MvdBS' Masters thesis written at the University of Tübingen. We would like to thank Carla Cederbaum for her support and bringing this collaboration together. We would further like to thank Eric Ling for bringing some of these problems to our attention and stimulating discussions. MG acknowledges the support of the German Research Foundation through the SPP2026 "Geometry at Infinity" and the excellence cluster EXC 2121 "Quantum Universe" -390833306. MvdBS thanks Carla Cederbaum for her financial support during the development of this research project, the Studienstiftung des deutschen Volkes for granting him a scholarship during his Master studies and Felix Finster for his support.

Future boundary extensions
In the first place, we consider a C k spacetime as a connected time-oriented Lorentzian manifold (M, g) without boundary with a C k -regular metric g. Furthermore, timelike curves are smooth curves whose tangent vector is timelike everywhere. Note that, comparing with our main sources, this convention coincides with the one in [21], but differs from the one in [4], where they use piecewise smooth timelike curves. However this does not make a difference for the resulting timelike relations. The following basic concepts play an important role in our study.
Definition 3 (C l spacetime extension). Fix k ≥ 0 and let 0 ≤ l ≤ k. Let (M, g) be a C k spacetime with dimension d. A C l spacetime extension of (M, g) is a proper isometric embedding ι ι : (M, g) → (M ext , g ext ) where (M ext , g ext ) is C l spacetime of dimension d. If such an embedding exists, then (M, g) is said to be C l extendible. The topological boundary of M within M ext is ∂ι(M ) ⊂ M ext . By a slight abuse of notation we will sometimes also call (M ext , g ext ) the extension of (M, g), dropping the embedding ι.
Note that it does in general not hold that ∂ι [20]). One of the advantages of working with ∂ + ι(M ) and ∂ − ι(M ) is that, as we mentioned in the introduction, if one of them is empty, the other becomes particularly nice.
As advertised in the introduction our main extension concept will not be the spacetime extensions of Definition 3 but rather certain 'future boundary extensions', a concept which we will develop now. Of course all our constructions (with all their caveats) should work analogously for a past boundary. Definition 6 (Candidate for a future boundary extension). Let (M, g) be a spacetime with an at least C 2 -metric and let (N, τ ) be a topological space. If there exists a topological embedding ι : M → N such that ι(M ) is open and ι(M ) = N , then we say that ((N, τ ), ι) is a candidate for a future boundary extension of (M, g). We may suppress both τ and ι notationally if they are clear from context. We denote by π TM : T M → M the natural projection map from the tangent bundle to M . We also fix a complete Riemannian background metric h T M on T M and throughout this section all distances in T M will be measured with respect to this background metric. 2 We denote by T t M the set of timelike tangent vectors, i.e., Before we can proceed we need to do some preparatory work defining certain sets based around timelike geodesics of M which will play an important role in describing regularity of extensions at the boundary via topological properties. Given a fixed X ∈ T M and r > 0, let B r (X) denote the open ball in T M around X. Moreover, for any X ∈ T M , let γ X : (a X , b X ) → M be the unique inextendible geodesic in M with initial data γ X (0) = π T M (X),γ X (0) = X. Note that X → a X is upper semi-continuous and X → b X is lower semi-continuous.
Definition 7 (Timelike thickening). Let (M, g) and ((N, τ ), ι) as above. For X ∈ T t M and r > 0 the timelike thickening of radius r generated from X is where the timelike boundary thickening O ∂ X,r and the timelike interior thickening O int X,r are defined as follows: and These are natural analogues of the thickenings of null geodesics considered in [2].
Remark 8. Note that O X,r , while indexed by objects intrinsic to (M, g), also depends on (N, τ ) and the embedding ι : M → N . In all our applications (M, g) will be fixed, however, we will sometimes need to consider different N . Whenever there is any chance of confusion we will indicate in which N we are considering the timelike thickening by writing O N X,r instead of merely O X,r . Now we are ready to define our concept of (regular) future (g-)boundary extensions: Definition 9 (Regular future g-boundary extension). Let (M, g) be a C 2 -spacetime. We say that a topological manifold with boundary N is a future boundary extension of (M, g) if there exists a homeomorphism ι : M → int(N ) and for any p ∈ ∂N there exists a future directed timelike curve γ : [0, 1) → M with p = lim t→1 − ι(γ(t)). If further 1. for any p ∈ ∂N there exists a future directed timelike geodesic γ : 2. and all timelike thickenings O N X,r are open and for any p ∈ ∂N and any future directed timelike geodesic γ : : n, m ∈ N} is a neighborhood basis of p, then we say that N is a regular future g-boundary extension.
Let us first note that in Section 3 we show that for globally hyperbolic (M, g) any C 0 -spacetime extension (M ext , g ext ) in the sense of Definition 3 with empty past boundary gives rise to a future boundary extension N := ∂ + ι(M ) ∪ ι(M ) ⊂ M ext . If (M ext , g ext ) is a C 2 -extension with empty past boundary, then N will be a regular future g-boundary extension. This suggests viewing conditions (1) and (2) in Definition 9 as hidden regularity assumptions and is the reason we introduced the name of regular future g-boundary extensions. The "g" refers to "geodesic" as we demand that all points in the boundary are reached by timelike geodesics and also refers back to old constructions of a "geodesic boundary" by Geroch and others, see [6] and [7], highlighting some similarities in spirit to our approach. The idea of Geroch's g-boundary is the following: given a geodesically incomplete spacetime M one considers the set of incomplete geodesics. This set can be endowed with an equivalence relation which, intuitively, considers as equivalent incomplete geodesics that become arbitrarily close (as they approach the singularities of M ). This set of equivalence classes is called the g-boundary. Note that the resulting object of attaching this g-boundary to the original spacetime M is only a topological space: i.e. in general it is not a manifold anymore and issues with non-Hausdorffness may appear. However, it was more recently shown that it is possible to find a finer topology on the topological space that arises from 'attaching' the g-boundary to the original spacetime M such that this space becomes Hausdorff in the new topology ( [3]). It remains to be seen whether this could be used in actually constructing regular future g-boundaries or proving regular future g-boundary extendibility. Remark 10. Our main reason for switching to work with topological manifolds with boundary instead of the classical concept of spacetime extensions from Definition 3, where the extension is itself again a spacetime without boundary, is that a uniqueness result for a maximal extension (with the "standard" ordering defined via the existence of a global embedding) is clearly impossible when going beyond the boundary as one can freely modify the topology of M ext \ ι(M ) as well as the extended metric g ext on M ext \ ι(M ). However, recent results of (Sbierski, [21]) show that there is a strong local uniqueness up to and including the boundary. We tried adapting the definition of an ordering relation to only demand the existence of an embedding of some open neighborhood of the boundary (cf. Remark 23), however for such modified orderings it is not readily apparent that set theoretic maximal elements even have to exist: The problem here appears to be that when trying to construct set theoretic upper bounds via taking unions over the elements in an infinite totally ordered set of extensions (and identifying appropriately) one quickly runs into the issue that -in order to ensure that the resulting object is a manifold -we would need a common neighborhood of the boundary into which all other neighborhoods progressively embed, however such a common neighborhood need not exist, as the considered neighborhoods could contract to just the boundary itself. Indeed we expect that this process would generally only produce a manifold with boundary. Working with topological manifolds with boundary from the beginning avoids these issues.

Preliminary topological considerations
As we already remarked in the introduction, condition (2) in Definition 9 will be necessary to control the topology of our upcoming quotient space constructions. In this preliminary section we will give a first example on how (2) controls the topology by showing that it guarantees second countability, even if (N, τ ) is not assumed to be a manifold with boundary already.
For this we now define timelike thickenings in M itself (in analogy of timelike thickenings in candidates ((N, τ ), ι) for future boundary extensions) by for X ∈ T t M and r > 0. We are interested in the interplay between the topologies of M and N and properties of the sets O M X,r and O N X,r . Remark 11. Clearly for any candidate for a future boundary extension ((N, τ ), ι) of (M, g) we So there is, as expected, a quite strong relationship between O M X,r and O N X,r . On the other hand, the O N X,r are a priori relatively independent of the topology on N (except for O N X,r ∩ ι(M ) having to be open) and demanding "regularity" is exactly forcing a stronger relation between the O N X,r and the topology on N . We define Definition 12 (Candidate for a regular future g-boundary extension). Let (M, g) be a spacetime with C 2 -metric. We say that a candidate ((N, τ ), ι) for a future boundary extension is a candidate for a regular future g-boundary extension if all timelike thickenings O N X,r are open and for any p ∈ N \ι(M ) there exists a future directed timelike geodesic γ : and for any such geodesic γ the collection {O Ṅ γ(1− 1 n ), 1 m : n, m ∈ N} is a neighborhood basis for p.
We will next prove that if N is a candidate for a regular future g-boundary extension of M , then the topology on N is always second countable and can be described entirely by the family of timelike thickenings in N and the topology on M .
Lemma 13. Let (M, g) be a spacetime with an at least C 2 -metric and let ((N, τ ), ι) be a candidate for a regular future g-boundary extension of (M, g). Then for any countable dense subset {X i } i∈N of T t M and any countable basis {U i } i∈N for the manifold topology of M the collection Hence establishing that a candidate for a regular future g-boundary extension is indeed a regular future g-boundary extension boils down to finding homeomorphisms from open neighborhoods of "boundary points" p ∈ N \ ι(M ) to open subsets in the half space [0, ∞) × R d−1 (clearly, ι induces a manifold structure on the "interior" ι(M ) and ι being a homeomorphism between M and the open set ι(M ) ⊂ N takes care of compatibility of charts) and showing Hausdorffness while second countability then follows automatically.

Compatibility with other extension concepts
As a further preliminary step, let us -as promised -investigate under which conditions we can strip down a spacetime extension ι : M → M ext in the sense of Definition 3 to just ι(M ) ∪ ∂ + ι(M ) while retaining a sensible structure, namely that of a topological manifold with boundary or even of a regular future g-boundary extension, on the resulting space. The following result in [20] plays an important role in investigating this.
The previous Proposition implies that points beneath the graph of the Lipschitz function f are in the inside of the "original" spacetime ι(M ). An easy way to ensure that points above the graph of f are in M ext \ ι(M ) (as, in general, it cannot be ruled out that some of these points are in ι(M ) or ∂ι(M ), cf. the comments in [21]) is to assume that the past boundary is empty. Under this assumption, we immediately have the following: Proof. Let γ : [0, b) → M ext be a suitable timelike curve. The set {t : γ(t) ∈ ∂ + ι(M )} is non-empty by assumption and we set s := inf{t : γ(t) ∈ ∂ + ι(M )}. By openness of ι(M ) and continuity of γ we have s > 0 and γ([0, s)) ⊂ ι(M ). Clearly s ≤ b 0 by definition, so s ∈ (0, b). It remains to Achronality of ∂ + ι(M ), which follows from (the time reversed version of) Theorem 5, implies that . We now proceed as before: Since the first part of the lemma in particular applies to vertical coordinate lines in the chart ϕ, it is clear that points below the graph of f are inside ι(M ) while points above the graph of f are outside of ι(M ) ∪ ∂ + ι(M ). Therefore, given a globally hyperbolic spacetime and considering low regularity extensions with a disjoint future and past boundary, it follows that ι(M ) ∪ ∂ + ι(M ) is a topological manifold with boundary: taking around every point p ∈ ∂ + ι(M ) a future boundary chart ϕ : V → R ε 0 ,ε 1 and defining the homeomorphism φ : is a topological manifold but not a smooth manifold 3 . We have thus shown: Lemma 16. Let (M, g) be globally hyperbolic and (M ext , g ext ) a C 0 extension with empty past boundary, then N := ι(M ) ∪ ∂ + ι(M ) with the subspace topology induced from M ext is a topological manifold with boundary and a future boundary extension of (M, g).
The following Lemma establishes that, given a regular enough extension of a globally hyperbolic spacetime, every point of its future boundary is intersected by a timelike geodesic.
To establish that given a regular enough extension of a globally hyperbolic spacetime for which the past boundary is empty N := ι(M ) ∪ ∂ + ι(M ) is a regular g-boundary extension it only remains to show that all sets O N X,r are open (in the subspace topology τ s induced on N from N ⊂ M ext ) and for any p ∈ ∂ + ι(M ) and any future directed timelike geodesic γ : Lemma 18. Let (M, g) be a globally hyperbolic C 2 spacetime and (M ext , g ext ) a C 2 spacetime extension with empty past boundary.
Proof. We first show that any O N X,r is τ s -open. Take an arbitrary X ∈ T t M, r > 0. Then we define So the problem reduces to arguing that the sets O ext,ε Collecting results we have shown Proposition 19. Let (M, g) be a globally hyperbolic C 2 spacetime and (M ext , g ext ) a C 2 spacetime extension with empty past boundary, then N := ι(M ) ∪ ∂ + ι(M ) with the subspace topology induced from M ext is a topological manifold with boundary and a regular future g-boundary extension of (M, g). 4 Ordering relation and existence of maximal elements 4.1 Partial ordering and equivalence of regular future g-boundary extensions In this short section we introduce an equivalence relation on the collection of regular future gboundary extensions.
Definition 20. Let (M, g) be a C 2 spacetime and (N 1 , ι 1 ), (N 2 , ι 2 ) be two regular future g-boundary extensions of M . We say (N 1 , ι 1 ) ∼ = (N 2 , ι 2 ) if there exists a homeomorphism (of topological manifolds with boundary) ψ 12 : N 1 → N 2 that is compatible with the homeomorphisms ι 1 : M → int(N 1 ) and ι 2 : M → int(N 2 ), i.e., such that is the identity map for (M, g). In other words, we demand that 4 Assuming the Riemannian background metric h ext on T Mext is chosen to satisfy B h ext r (X ) = (T ι)(B h r (X)), but for given X, r this can always be achieved. Else one could also use different radii to obtain appropriate subset relations.
Hence ≤ indeed defines a partial ordering on the set I of all equivalence classes of regular future g-boundary extensions.
Remark 23. For spacetime extensions M ext we had considered the following definition in the second author's Master thesis (see [22]). Let ι 1 : M → M ext,1 and ι 2 : M → M ext,2 be spacetime extensions (i.e. such as in Definition 3) of (M, g). In this Remark, no assumption on the regularity class nor on the causal properties (e.g. global hyperbolicity) of (M, g) or the extensions is made. We define the following relations: • We say that (M ext,1 , ι 1 ) = ∂ (M ext,2 , ι 2 ) provided there exist open neighborhoods U 1 and U 2 satisfying that ∂ι 1 (M ) ⊂ U 1 , ∂ι 2 (M ) ⊂ U 2 and ι −1 , and an embedding ψ 12 : U 1 → U 2 whose restriction ψ 12 : ι 1 (M ) ∩ U 1 → ι 2 (M ) ∩ U 2 is surjective (and thus a homeomorphism) and which is compatible with the extensions, i.e. such that In [22] the second author showed (Lemma 60 in [22]) that this defines an equivalence relation. We label the family of equivalence classes by I ext .
These proofs are relatively straightforward but tedious to write down precisely (as one has to constantly change the neighborhoods one is working on). This changing of neighborhoods becomes an issue when considering an (uncountable) totally ordered subfamily of equivalence classes of extensions {[ι α ]} α∈A and trying to construct an upper bound for this subfamily by 'gluing' together all U α ∪ ι α (M ) and identifying points appropriately, as we already discussed in the paragraph above Section 2.1. Let us remark that the relations "= ∂ " and "≤ ∂ " are compatible with Definitions 20 and 22 above: If (M ext,1 , ι 1 ), (M ext,2 , ι 2 ) have empty past boundary and M is globally hyperbolic, then (M ext,1 , ι 1 ) = ∂ (M ext,2 , ι 2 ), resp. ≤ ∂ , then the corresponding future boundary extensions N 1 and N 2 satisfy N 1 ∼ = N 2 , resp. N 1 N 2 (note that even if the future boundary extensions are not regular, the relations ∼ = and are well defined).

Existence of set theoretic maximal elements
In this section we will show that the set of equivalence classes of regular future g-boundary extensions I of a given spacetime (M, g) contains at least one set-theoretic maximal element proceeding via a standard Zorn's lemma proof. In other words, we show that there exist upper bounds for any arbitrary totally ordered subset J = {[N α ]} α∈A ⊂ I of regular future g-boundary extensions. This is organized as follows. First, a candidate for a representative of an upper bound, N upp , for any such totally ordered J is constructed by gluing together all the N α 's: We take a disjoint union, identify points via the embeddings ψ αβ from the ordering relation and take N upp to be the quotient space (with the quotient topology) and define a natural map ι upp : M → N upp via ι upp = π • ι α for any ι α .
Next, we need to show that this quotient space belongs to I, i.e. is itself a regular future g-boundary extension for M . As quotient topologies are in general quite badly behaved, especially with respect to separation axioms and potentially second countability (if {[N α ]} α∈A is not countable), some care is necessary. The order relation straightforwardly gives us that π is an open map (cf. Lemma 24) which implies that N upp is indeed Hausdorff (cf. Lemma 29) and that ((N upp , τ q ), ι upp ) is a candidate for a future boundary extension. For second countability we first establish the "regularity" part of Definition 9, i.e., we show that ((N upp , τ q ), ι upp ) is a candidate for a regular future g-boundary extension (cf. Lemma 27). Once this is done, second countability follows from Lemma 13. Lastly, openness of π straightforwardly allows us to project charts for N α onto the quotient N upp showing that it is indeed a topological manifold with boundary (cf. Proposition 30).
where the equivalence relation ∼ is defined as follows for two arbitrary points p ∈ N α ⊂ N and q ∈ N β ⊂ N : Note that this is indeed an equivalence relation and implies p ∼ q ⇐⇒ p = q if p, q ∈ N α . We will denote the quotient map by π, i.e., π : N → N upp , p → π(p). We endow N upp with the quotient topology τ q , i.e., where α ∈ A can be chosen arbitrarily as π(ι α (p)) = π(ι β (p)) for any α, β ∈ A since ψ αβ = ι β • ι −1 α on ι α (M ) by definition of . Note that a priori both N and N upp may depend on the choices of representatives, but this doesn't bother us for the moment as we only aim to show the existence but not necessarily uniqueness of a regular future g-boundary extension N upp for which [N upp ] is an upper bound for the totally ordered set J . However, we will see in Remark 31 that the equivalence class [(N upp , ι upp )] obtained from this process is in fact independent of the chosen representatives.
Continuity and openness of π together with injectivity of π Nα now immediately imply that ι upp = π • ι α = π Nα • ι α (for any α ∈ A) is a topological embedding onto the open set ι upp (M ) = π(ι α (M )). Further, τq by continuity of π. So ((N upp , τ q ), ι upp ) is a candidate for a future boundary extension of (M, g) as in Definition 6 and we may define a family of timelike thickenings for N upp as in Definition 7. Our next crucial step is to show that ((N upp , τ q ), ι upp ) is a candidate for a regular future g-boundary extension of (M, g) as in Definition 12. We start with the following Lemma Lemma 25. Let X ∈ T t M, r > 0. We have where O Nα X,r denotes the timelike thickening corresponding to X, r in for all α ∈ A: Let p ∈ O Nα X,r . Either p = ι α (γ Y (t)) ∈ int(N α ) for some Y ∈ B r (X) ⊂ T M and t ∈ (0, b Y ). Then π(p) = ι upp (γ Y (t)) by definition of ι upp and hence π(p) ∈ O Nupp X,r . Or p = lim Nα t→b − Y ι α (γ Y (t)). Then, by continuity of π, π(p) = lim

To show the other inclusion let
Then we must have q ∈ N α for some α ∈ A. For any open neighborhood U of q in N α we have that π(U ) = π Nα (U ) is a τ q -open (because π is an open map, see Lemma 24) neighborhood of p in N upp , so ι upp (γ Y (t)) must be contained in π Nα (U ) for all t sufficiently close to b Y . This means exists and equals q. Thus q ∈ O Nα X,r , i.e., Figure 1: While the boundary of N upp may be "larger" than the boundary of any of the N α 's, we can lift any point p ∈ N upp \ ι upp (M ) to some N α and since N α is a manifold with boundary there exists a suitable relatively compact neighbourhood U α such that any O Nα X,r ⊂ U α will satisfy π(O Nα X,r ) = O Nupp X,r .
Because N α is a topological manifold (with boundary) we can find an open U α ⊂ N α such that q ∈ U α and the closure of U α in N α is compact and contained in π −1 (V )∩N α . Set T n :=γ(1 − 1 n ), r m = 1 m and let n, m ∈ N be such that O Nα Tn,rm ⊂ U α (such an n, m exists because N α is a regular future g-boundary extension), cf. Figure 1.
By continuity of π and definition of ι upp , we would have π(y) = x.
Since we assumed x ∈ O Nupp Tn,rm \ π(O Nα Tn,rm ), we would have y ∈ N α \ O Nα Tn,rm . However, by definition of O Nα Tn,rm , whenever lim Nα t→t − 0 (ι α • γ Y )(t) exists in N α for any 0 < t 0 ≤ b Y , then this limit will belong to O Nα Tn,rm . So lim Nα t→t − x (ι α • γ Y )(t) cannot exist. Hence there must exist a different sequence t k for which (ι α • γ Y )(t k ) → y = y in N α (the diverging case can again be excluded by relative compactness). By continuity of π we must again have x = π(y ) but π Nα is injective, so y = y , giving a contradiction. Remark 26. Note that the proof only used injectivity of π Nα : N α → N upp , compatibility of the embedding ι upp with π and ι α for all α ∈ A, i.e., that ι upp = π • ι α for any α ∈ A, and that π is an open map. Importantly we did neither use that the family {[N α ]} α∈A was totally ordered nor any further details on the definition of the equivalence relation. Hence we will be allowed to use this fact (and any results deriving directly from it) in the construction in next section, Section 5, as well.
It remains to show that all timelike thickenings O Nupp X,r are open and that for any future directed timelike geodesic γ : : n, m ∈ N} is a neighborhood basis for p. This follows immediately from the previous Lemma 25.
Remark 28. Again, the proof only uses injectivity of the π Nα , compatibility of the embedding ι upp with π and ι α for all α ∈ A and that π is an open map. Hence we will be allowed to use this fact in the construction in next section, Section 5, as well and will in fact do so to obtain Lemma 43.
Let us now turn towards topology. As already pointed out Lemma 13 immediately gives second countability. We next show Hausdorffness.
We are now ready to equip N upp with suitable charts turning it into a topological manifold with boundary and put everything together.
Proposition 30. Let (M, g) be a C 2 spacetime and J a totally ordered set of of regular future g-boundary extensions. Then N upp is a regular future g-boundary extension of (M, g).
Proof. Thanks to Lemmas 27 and 29, it only remains to show that (N upp , τ q ) carries the structure of a topological manifold with boundary, i.e., that there exist suitable charts. The idea is to construct charts on N upp using the charts on N α (for each α ∈ A) and composing them with the quotient map π. Take a point p ∈ N upp , take α ∈ A and p α ∈ N α such that p = π(p α ) and a coordinate chart (U α , x α ) around p α in N α . Note that if p α ∈ ι α (M ) = int(N α ), x α is a homeomorphism onto an open subset of R d , while if p α ∈ N α \ ι α (M ), x α is a homeomorphism onto an open set in the half space [0, ∞) × R d−1 . As π is an open map, π(U α ) is an open neighborhood of p in N upp . Then, on N upp we define the map x : Nα (p)), noting that π Nα : U α → π(U α ) is a bijection. In the following it will be proven that (π(U α ), x) is a coordinate chart for N upp .
We show that the map x is bicontinuous. By definition of x it holds that is open (since x α is continuous) and π is an open map. Hence, x is continuous. In order to see that x −1 is continuous, simply note that x −1 = π • x −1 α is the composition of continuous maps. Since we only need a topological manifold, there is no further compatibility between charts we'd have to check.
Remark 31. Let us at this point remark that while (N upp , ι upp ) might depend on the chosen representatives (N α , ι α ) of [(N α , ι α )], its equivalence class [(N upp , ι upp )], which is now well-defined as we just established that N upp is a regular future g-boundary extension, does not: For every α let (N α , ι α ) and (N α , ι α ) be two regular future g-boundary extensions with [(N α , ι α )] = [(N α , ι α )] and consider ψ : defined by ψ(p α ) := π (ψ αα (p α )), where ψ αα is the homeomorphism arising from the equivalence relation (N α , ι α ) ∼ = (N α , ι α ) (with π the projection p α → [p α ] ∈ N upp for N upp ) for p α ∈ N α ⊂ α N α . Clearly this is well defined, surjective and satisfies ψ(p α ) = ψ(p β ) for p α , p β with [p α ] = [p β ] (noting that for (N α , ι α ) (N β , ι β ) also (N α , ι α ) (N β , ι β ) and ψ ββ | ψ αβ (Nα) = ψ α β • ψ αα • ψ −1 αβ since all ψ ij are uniquely determined from ι j • ι −1 i by extending continuously). So by the universal property of the quotient space there exists a well-defined continuous and surjective map Analogously, just switching the roles of N upp and N upp , we obtain a continuous and surjective map By construction (using that ψ −1 Let us observe that this immediately gives the following Corollary. Proof. This follows directly from the existence of upper bounds for every totally ordered subset J and Zorn's Lemma. Of course, such set theoretic maximal elements are expected to be non-unique. For instance, we believe the two inequivalent extensions of the Misner and Taub-NUT spacetimes (described in [11] and [2], cf. in particular Prop. 5.16, respectively) to both be maximal elements. However a proof of this in our case is less immediate than the corresponding proof of [2,Prop. 5.16] due to the non-constructive nature of Zorn's lemma.

Existence of a unique maximal regular future g-boundary extension
The example of Misner spacetime given in [14] suggests that uniqueness necessitates an additional condition to be imposed on (M, g). Together with the work by Chruściel on uniqueness of conformal boundaries, [2], it seems natural to consider the following additional condition Definition 34 (Intertwined timelike geodesics). Let γ 1 : [0, b Y 1 ) → M , Y 1 :=γ 1 (0), and γ 2 : , be two future directed future inextendible timelike geodesics in a C 2 spacetime (M, g). Then, we say that γ 1 and γ 2 are not intertwined provided one of the following conditions holds: (i) For any radii r > 0, ρ > 0 there exist If neither of these conditions hold, then we say that γ 1 and γ 2 are intertwined.
Heuristically, two curves γ 1 and γ 2 are not intertwined if they merge, i.e. they approach each other and remain arbitrarily close (case (i) of the previous definition), or part, i.e. there exists a fixed distance at which these curves will, as long as defined, never be (case (ii) of the previous definition). In other words, it is not possible that these geodesics come arbitrarily close to each other without remaining close afterwards. Intertwined geodesics, as pointed out by Chruściel [2] and Sbierski [21], appear for example in the Taub-NUT or Misner spacetime and lead to the existence of distinct extensions of the original spacetime. In particular, a "common" extension of two arbitrary (i.e., non-ordered) regular future g-boundary extensions N α and N β might fail to be Hausdorff if there exist intertwined timelike geodesics in M .
Before proceeding let us remark that condition (i) in Definition 34 could be rewritten using Proof. Fix r > 0, n ∈ N and set T n :=γ(b − 1 n ). Chooseρ(r, n) > 0 such that {γ Y (b − 1 n ) : Y ∈ Bρ (r,n) (γ(0))} ⊂ B r (T n ), noting that such aρ(r, n) exists by continuous dependence of tangents to geodesics on the initial data. Then Note that the latter set is compact. Now if γ ([s , b )) ⊂ O M Tn,r for any s ∈ (0, b ), but by point (i) in Definition 34 there existss So by the above γ (s k ) is contained in a compact set for all k. This shows that γ is an inextendible timelike curve partially imprisoned in a compact set, contradicting strong causality of (M, g) (see e.g. [12] Prop. 2.5) In the remainder of this section we show that, indeed, (M, g) not containing any intertwined future directed timelike geodesics is a sufficient condition for the existence of a maximal regular future g-boundary extension (provided that (M, g) is regular future g-boundary extendible at all).
Definition 36. A regular future g-boundary extension (N max , ι max ) of (M, g) is said to be a maximal regular future g-boundary extension if any other regular future g-boundary extension (N, ι) Remark 37.
1. By this definition any maximal regular future g-boundary extension automatically has to be unique in the following sense: If (N max , ι max ) and (N max ,ι max ) are two maximal regular future g-boundary extensions, then [N max ] = [N max ], i.e., there exists a homeomorphism between them which, when pulled back by the embeddings ι max resp.ι max , gives the identity on M .
2. Clearly the equivalence class of any maximal regular future g-boundary extension has to be a maximal element for the partially ordered set However, a set theoretic maximal element of I need not satisfy that its representatives are maximal regular future g-boundary extensions in the sense of the above Definition 36.
3. If any two set theoretic maximal elements [N ] max and [N ] max for I are equal, then any representative for their equivalence class is a maximal regular future g-boundary extension.
Definition 38. Let (M, g) be a C 2 spacetime. It is called regular future g-boundary extendible if the set the set of regular future g-boundary extensions of (M, g) is non-empty.
For instance, a sufficient condition for a C 2 globally hyperbolic (M, g) to be regular future gboundary extendible is that there exists a C 2 spacetime extension (in the usual sense, cf. Definition 3) with empty past boundary (cf. Section 3).
As mentioned, our goal of this section is to show that there exists a maximal regular future g-boundary extension if (M, g) does not contain any intertwined future directed timelike geodesics. The strategy of the proof proceeds as follows: We first show that if (M, g) does not contain any intertwined timelike geodesics we can, essentially, do the same construction as in the previous section for any two regular future g-boundary extensions N α and N β . That is, if we define N := N α N β / ∼ for an appropriate equivalence relation, then N naturally becomes a regular future g-boundary extension. Our strategy essentially follows the one in Section 4.2, and we are even are able to make direct use of some of the results from that section, such as Lemmas 25 and 27 (cf. Remarks 26 and 28). However showing openness of the quotient map π and Hausdorffness of the quotient topology (cf. Lemma 44) becomes much more involved (and for both our proofs rely on not having intertwined timelike geodesics in M ).
Once we have established this, we may choose N α and N β to be representatives of set theoretic maximal elements [N ] max and [N ] max to conclude that any two set theoretic maximal elements are equal (cf. Theorem 46), which establishes that N max is indeed a maximal regular future g-boundary extension in the sense of Definition 36.
1. If p and q both lie in N α or both lie in N β , then p ∼ q iff p = q.

That this is indeed an equivalence relation (transitivity is not immediately obvious as we only
demand the existence of a suitable Y and this Y may a priori depend on both p and q) will follow from Lemma 41.
3. If one has N α N β and w.l.o.g. p ∈ N α , q ∈ N β , then p ∼ q according to Definition 39 if and only if q = ψ αβ (p), i.e., if p ∼ q according to the definition in (10): If q = ψ αβ (p), then for any future directed timelike geodesic γ : Lemma 41. Let (N α , ι α ) and (N β , ι β ) be two regular future g-boundary extensions of a C 2 spacetime Proof. Fix p and q and Y 0 ∈ T t M such that lim Nα

t) exists and equals q:
Since N β is a regular future g-boundary extension, the Tn,rm } n,m∈N , where T n :=γ Y 0 (1 − 1 n ) and r m = 1 m , is a neighborhood basis of q. Since N α is a regular future g-boundary extension as well, {O Nα Tn,rm } n,m∈N is a neighborhood basis for p. Since {O Nα Tn,rm } n,m∈N is a neighborhood basis for p and ι α • γ Y → p by assumption, for any n ∈ N we can find 0 < t n,m < b Y such that ι α • γ Y (t) ∈ O Nα Tn,rm for all t ∈ (t n,m , b Y ). By the definitions of O Nα Tn,rm and O M Tn,rm and Remark 11, this implies γ Y (t) ∈ O M Tn,rm for all t ∈ (t n,m , b Y ). Hence, again appealing to the definitions and Remark 11, we obtain ι β • γ Y (t) ∈ O N β Tn,rm for all t ∈ (t n,m , b Y ). Since this works for any n, m and {O N β Tn,rm } n,m∈N is a neighborhood basis for q we get lim Note that it was essential for the above proof that we could choose the same T n =γ Y 0 (1− 1 n ), r m for the neighborhood bases in N α and in N β by the second condition in Definition 9 because we already had one geodesic γ Y 0 with the right limiting behavior in N α and N β . We will encounter this again when showing that π is an open map.
So, ∼ from Definition 39 is indeed an equivalence relation and we may define the quotient spacẽ As in Section 4.2 we equipÑ with the quotient topology τ q . We proceed by showing that also in this case the quotient mapπ : N α N β →Ñ is open provided (M, g) does not contain any intertwined timelike geodesics.
Lemma 42. Let (N α , ι α ) and (N β , ι β ) be two regular future g-boundary extensions of a strongly causal C 2 spacetime (M, g). If no two timelike geodesics γ 1 , γ 2 : [0, 1) → M with ι α • γ 1 converging to a p 1 ∈ N α \ ι α (N α ) and ι β • γ 2 converging to a p 2 ∈ N β \ ι β (N β ) are intertwined, then the projection mapπ : The more interesting case is q / ∈ ι β (M ). This impliesπ(q) =π(p) for a unique p ∈ U \ ι α (M ) and that there exists Let us again denote T n :=γ Y (1 − 1 n ) and r m := 1 m . We will show that there exist n 0 , m 0 ∈ N for which we may take Tn,rm } n,m∈N is a neighborhood basis at p (remembering that N α is a regular future g-boundary extension and p = lim t→1 − ι α • γ Y (t)) and U is an open neighborhood of p, we must have O Nα Tn 0 ,rm 0 ⊂ U for some n 0 , m 0 . Since N α is a topological manifold, we can further w.l.o.g. assume that O Nα Tn 0 ,rm 0 is compact and also contained in U . Fix these n 0 , m 0 ∈ N and assumẽ We now distinguish two cases: Either q 0 is contained in ι β (M ) or q 0 / ∈ ι β (M ). In the first case , then any future directed timelike geodesic c in M with ι α •c terminating in p 0 would be intertwined with the original γ Y 0 .
It thus remains to show that this limit exists. By relative compactness of O Nα Tn 0 ,rm 0 there always exists a sequence t k → 1 such that ι α •γ Y 0 (t k ) converges. Let's denote this limit by p 0 . We will exploit the fact that (M, g) does not contain any intertwined timelike geodesics to argue that actually lim t→1 − (ι α • γ Y 0 )(t) = p 0 . Since p 0 ∈ N α \ ι α (M ), there must exist some timelike geodesic c : [0, 1) → M such that p 0 = lim t→1 − (ι α • c)(t). Since c and γ Y 0 cannot be intertwined by assumption they either satisfy point (i) in Definition 34, in which case Lemma 35 applies and we can conclude that for any n, m ∈ N there exists s ≡ s (n, m) such that ( there exist s 1 , s 2 ∈ (0, 1) and r, ρ > 0 such that O Ṁ γ Y 0 (s 1 ),r ∩ O Ṁ c(s 2 ),ρ = ∅. But this is impossible because for any s 1 , s 2 ∈ (0, 1) and r, ρ > 0 we have ,ρ for all large enough k: On the one hand, for any s 1 ∈ (0, 1) there clearly exists K such that t k ≥ s 1 for all k ≥ K and then γ Y 0 (t k ) ∈ O Ṁ γ Y 0 (s 1 ),r for all r > 0. On the other hand, for any s 2 ∈ (0, 1), ρ > 0 the set O Nα is an open set (as N α is a regular future g-boundary extension), contains p 0 = lim t→1 (ι α • c)(t) and ,ρ for all k ≥ K.
This map is well-defined and, sinceπ is an open map, a (topological) embedding onto the open is a candidate for future boundary extension of (M, g). We may now appeal to Lemma 27 and Remark 28 to conclude that in fact Lemma 43. Under the assumptions of Lemma 42 ((Ñ , τ q ),ι) is a candidate for a regular future g-boundary extension of (M, g).
Next it is shown that, provided there are no intertwined timelike geodesics in M , (Ñ , τ q ) is Hausdorff.
Proof. Consider two distinct points p, q ∈Ñ . We will separate two cases: Either p, q ∈π(N a ) for some a ∈ {α, β} or p ∈π(N a ) \π(N b ) and q ∈π(N b ) \π(N a ) for some a, b ∈ {α, β} with a = b. So, let p, q ∈π(N a ) and let p a , q a ∈ N a be the unique points such that p =π(p a ) and q =π(q a ) (noting thatπ Na is injective). Hausdorffness of N a implies that there exist disjoint neighborhoods U, V ⊂ N a of p a and q a respectively and hence, by openness ofπ and injectivity ofπ| Na ,π(U ) and π(V ) are disjoint open neighborhoods of p and q respectively.
Lastly, coordinate charts can be defined onÑ . This process is again analogous to Section 4.2. Proof. Let p ∈Ñ , w.l.o.g. p ∈π(N α ), and choose p α ∈ N α such that p =π(p α ). Let (U α , x α ) be a coordinate chart around p α in N α . Asπ is an open map,π(U α ) is an open neighborhood of p inÑ . Then, onÑ we define the map x : Remark 47. Sections 4.2 and 5 are actually more independent of each other than they might seem at first reading. This is important as Corollary 33 relies heavily on Zorn's Lemma while Theorem 46 doesn't. Moreover, if we assumed that (M, g) contains no intertwined timelike geodesics from the beginning, our proof would not need to resort to Zorn's Lemma (and, in particular, we would be able to not only prove the existence of a unique maximal extension, but also to construct it). In the following, let N α and N β be two arbitrary regular future g-boundary extensions of a strongly causal spacetime (M, g): • In the proof of Theorem 46 we actually show that if (M, g) contains no intertwined timelike geodesics, gluing together N α and N β (and identifying points appropriately as in Definition 39) yields a 'larger' regular future g-boundary extensionÑ .
• However, the previous conclusion can also be generalized to an arbitrarily large number of regular future g-boundary extension, assuming again that (M, g) has no intertwined timelike geodesics. Let I be the set of regular future g-boundary extensions of (M, g). Then,Ñ := ( α∈A N α )/ ∼ is a regular future g-boundary extension (by the proof of Theorem 46 and using thatÑ is second countable even for uncountable unions as it is a candidate for a regular future g-boundary extension). As any N α ∈ I can be embedded inÑ , it is clear that this is the unique maximal regular future g-boundary extension.
• The 'dezornified' version of Theorem 46 is similar to Sbierski's dezornification [18] of the proof of the existence of a unique maximal globally hyperbolic development of a given initial data set by Choquet-Bruhat and Geroch. In the first place, the statement thatÑ = (N α N β )/ ∼ is a regular future g-boundary extension is similar to Theorem 2.7 in [18] 5 . Secondly, the strategy of using the previous result to glue all regular future g-boundary extensions together in order to construct the maximal regular future g-boundary extension is very similar to Theorem 2.8 in [18], which states the existence of a unique maximal globally hyperbolic development 6 .
Let us remark that we can also formulate an "if and only if" version of Theorem 46 as follows: A regular future g-boundary extendible strongly causal C 2 spacetime (M, g) has a maximal future g-boundary extension if and only if (M, g) does not admit any intertwined timelike geodesics γ 1 and γ 2 such that ι 1 • γ 1 acquires an endpoint in some regular future g-boundary extension (N 1 , ι 1 ) and ι 2 •γ 2 acquires an endpoint in some other regular future g-boundary extension (N 2 , ι 2 ). Clearly this is sufficient for the proof of Theorem 46 to go through. For the "only if" part we note first that if N max is a maximal regular future g-boundary extension, then there cannot exist any intertwined timelike geodesics γ 1 and γ 2 in M such that ι max • γ 1 and ι max • γ 2 have future endpoints in N max : Assume that γ 1 and γ 2 are two geodesics in M such that ι max •γ 1 and ι max •γ 2 have future endpoints p 1 respectively p 2 in N max , then either • p 1 = p 2 , in which case γ 1 and γ 2 will satisfy condition (i) in Definition 34 because O Nmaẋ γ 1 (0),r is an open neighborhood of p 2 and vice versa. . This immediately implies that γ 1 and γ 2 will satisfy condition (ii) in Definition 34.
So in both cases γ 1 and γ 2 are not intertwined. Since any regular future g-boundary extension embeds into N max the existence of a maximal regular future g-boundary extension further implies that M cannot have any intertwined timelike geodesics γ 1 and γ 2 in M such that ι 1 • γ 1 acquires an endpoint in some regular future g-boundary extension (N 1 , ι 1 ) and ι 2 • γ 2 acquires an endpoint in some other regular future g-boundary extension (N 2 , ι 2 ). All of this is in line with the corresponding converse statement for conformal boundary extension in [2,Thm. 4.5]. However, it is at this point unclear to us if this would already imply somehow that (M, g) cannot contain any intertwined future directed timelike geodesics at all.

Discussion
We already discussed some of the limitations of our approach (such as only getting regular future gboundary extensions and not necessarily spacetime extensions and requiring quite a bit of 'hidden' regularity), so we would like to end with several possibilities and open questions for extending our work and potential applications thereof. Most of these have been mentioned throughout the paper already but we will collect them here and give a little more detail.
First, let us note that even though our results are only about the existence of a maximal future g-boundary extension this may have some consequences for C 2 spacetime extensions as well because of the compatibility results in Section 3. For instance, if for a given C 2 spacetime which is globally hyperbolic, past timelike geodesically complete and contains no intertwined timelike geodesics one could show that the maximal future g-boundary extension has non-compact and connected boundary, then maximality and invariance of domain imply that no C 2 spacetime extension can have compact ∂ + ι(M ).
Since C 2 spacetime extensions are anyways rather nice and well understood an important follow up question would be how far one can lower the regularity of a spacetime extension (M ext , g ext ) in Section 3 while retaining its conclusions. That g ext ∈ C 1,1 is sufficient should be very straightforward to check. If g ext ∈ C 1 , openness of all O N X,r is still expected to be unproblematic and, with some more work, also the neighborhood property of Similarly, one may ask if it is really necessary to assume global hyperbolicity of M or that ∂ − ι(M ) = ∅ for compatibility. Regarding global hyperbolicity we note that if we keep ∂ − ι(M ) = ∅ even without global hyperbolicity we still get a future boundary extension: Since [4, Thm. 2.6] establishes that then ∂ + ι(M ) is always an achronal topological hypersurface and a standard argument then produces suitable charts near this hypersurface [15,Prop. 14.25]. The arguments in Lemma 18 should also go through without global hyperbolicity. However global hyperbolicity cannot be dropped from Lemma 17 (i.e. [21,Lem. 3.1]), cf. [21,Rem. 3.4,3.], so we will not obtain a regular future g-boundary extension without it. While this solves some issues (like no longer needing to prove significant parts of Lemma 18), it invariably introduces others (Hausdorffness, manifold structure, etc.). This is again reminiscent of similar issues in picking a suitable topology in the various boundary constructions such as of course the g-boundary of Geroch [6] itself, but also the bundle (or b-) boundary construction of Schmidt [17] or the causal (or c-) boundary introduced by Geroch, Kronheimer and Penrose [8] (although there also idealized endpoints of complete timelike geodesics are attached), cf. e.g. [3].
Turning towards comparing our results with [2] we note that, while our proof of Theorem 46 largely follows the same overall strategy as [2, Thm. 4.5] of constructing a larger extension from at least two given ones by taking unions and identifying appropriately, [2] does this in the null bundle of M whereas we work directly in the topological manifolds with boundary. In contrast to our arguments, [2] can work with geodesics up to and including the boundary and in particular has local uniqueness as in [2,Prop. 3.5]. Having analogous tools in our setting would simplify parts of the proofs, e.g., in Lemma 42 one could argue with geodesic uniqueness instead of using the "no intertwined geodesics" assumption. On the other hand we do not have to show that our charts at the boundary are compatible nor have to construct a conformal metric that extends to the boundary. There is an interesting reformulation of this result in [2,Thm. 5.3], namely the existence of a unique future conformal boundary extension with strongly causal boundary which is maximal in the class of future conformal boundary extensions with strongly causal boundaries. It would be interesting to see if, for some sensible definition of strong causality for our boundaries, an analogous result remains available. Certainly Lemma 42 seems amenable to a strong causality/non-imprisoning argument.
Of course the bigger open questions are more conceptual. In Section 3 we have discussed associating a (regular) future (g-)boundary extension to a given spacetime extension. Conversely one could ask if, given a regular future g-boundary extension, there is any hope of characterizing (intrinsically) when one can extend this further to a spacetime extension. Similarly, it is open if one could develop any (in-)extendibility criteria ensuring the (non-)existence of future g-boundary extensions that do not come from spacetime extendibility and compatibility. In this sense the present article can be considered a first starting point proposing a concept of regular future gboundary extensions which, excluding pathological behavior of timelike geodesics in the original spacetime, naturally admits unique maximal elements, and many open questions remain to be explored.