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A rigidity result on the ideal boundary structure of smooth space-times

Following a survey of the abstract boundary definition of Scott and Szekeres, a rigidity result is proved for the smooth case, showing that the topological structure of the regular part of this boundary in invariantly defined.


Introduction
The most intuitively natural definition of a boundary to a space-time (M, g) would seem to be obtained by topologically embedding M in a larger manifold M of the same dimension by means of a map φ : M → M , followed by taking the topological boundary ∂ φ M def = ∂(φ(M)) of the image φ(M) in M . (Where φ is of differentiability class C ℓ , ℓ 1 we shall refer to this as a C ℓ envelopment, or simply a smooth envelopment if the class C ℓ is understood.) The problem with this is that, unless further conditions are imposed, ∂ φ M depends critically on the choice of φ, the different possible ∂ φ M for different φ being so variable in structure that they contain no invariant information.
Because of this difficulty, this approach was abandoned early on the history of relativity, in favour of boundary constructions linked more closely to the detailed geometry of M. If, however, we were to restrict ourselve to regular envelopments, meaning envelopments where φ was a metric isometry into a "regular" (in some sense) ( M ,ĝ), then we might hope for more rigidity (that is, constancy of topological structure) in ∂ φ M. To provide an overall context for this and related constructions Scott and Szekeres [1] have developed the abstract boundary construction which incorporates all possible boundaries ∂ φ M with appropriate identifications being made between corresponding subsets of the boundaries (boundary sets). It is already known that if the metric is used certain topological properties of boundary sets are preserved under equivalence, even allowing for quite unpleasant properties of the metric [2]. This work cited can be said to serve as an interesting example of the relative ease with which certain simple results about the abstract boundary can be obtained, and also the crucial rôle played by our "regularity/extendibility" assumptions (q.v.); however, the work is not used here.
In this paper we review the abstract boundary construction, we survey possible definitions of the notion of "regular", and we show within this framework that we can achieve an appropriate rigidity of structure for a boudary constructed from regular envelopments. More precisely, we show that if we consider only boundaries that are regular and satisfy a Lipschitz condition, then all representatives of an equivalence class of boundary sets, in the sense of the abstract boundary, are homeomorphic.
In what follows, by a pseudo-Riemannian manifold (M, g) we will mean a Hausdorff, paracompact topological manifold M, without a boundary in the usual sense, that is equipped with a C 1 atlas and a metric g. ¶ If, further, M is connected, n = dim M 2, and the metric g is of Lorentz signature (−+ · · · +), then we shall refer to (M, g) as a space-time. If g is C ℓ , some ℓ 1, the covariant differential with respect to the unique torsion-free metric-compatible connection on M will be denoted ∇, or g ∇ if we wish to emphasise the rôle of the metric. In fact, we mostly use the notation of [3]. For example, X p ∈ T p (M) will denote the value of the vector field X at the point p ∈ M.
In referring to envelopments a boundary set is a subset B ⊂ ∂ φ . (We will normally have primed objects belonging to an envelopment (M, M ′ , φ ′ ), with corresponding unprimed objects belonging to an envelopment (M, M , φ).)

Regular abstract boundaries
The equivalence relation used in the definition of the abstract boundary is as follows. Suppose that we are given boundary sets B, B ′ of two envelopments (M, M , φ), It is easily seen that B ✄ B ′ if and only if "one cannot approach B ′ from within M without also approaching B". In a sense, then, this means that B is "bigger" than B ′ , and this is the reason for the notation. As an example, let φ ′ (r, Θ) = (r + 1, Θ) ∈ R n \ {0}.
(5) ¶ Note that we require all manifolds to have differentiable structure, but make no demands on g at the moment, beyond continuity; note also that saying that (M, g) is C k means that g is C k , and the atlas of M is at least C k+1 .
One says that boundary sets B and B ′ are equivalent, B ∼ B ′ , if they cover each other: B✄B ′ and B✁B ′ . The equivalence class containing the boundary set B is denoted [B], and is called an abstract boundary set. If [B] contains a singleton boundary set {p}, then [B] may also be denoted by [p], and is called an abstract boundary point.
The collection of all abstract boundary points constitutes the abstract boundary BM of M; that is, According to this definition, boundary points and abstract boundary points admit quite a rich further classification [1], which need not concern us here. Note that the abstract boundary construction is not as developed (or, perhaps, useful!) as such constructions as the bundle boundary (b-boundary) of Schmidt [4,5,6,7], the conformal boundary (c-boundary) of Geroch et al. [8,9,5], or even the "A-boundary" of Clarke [5]. The latter turns out to be very closely related to a certain ("Lipschitz") regular abstract boundary.
We now introduce the rôle of a metric. Let (M, M , φ) be an envelopment of a C k , k 1 pseudo-Riemannian manifold (M, g). A point p ∈ ∂ φ , which has a neighbourhood U p such that M p def = φ(M)∪U p can be endowed with a C ℓ (some 1 ℓ k) pseudo-Riemannian metricĝ extending g, i.e., g = ψ * ĝ , is called C ℓ -regular. The envelopment may then be referred to as (M, g, M,ĝ, φ).
For regular points we will use both the original setting of the abstract boundary [1] just described, and also a definition from earlier, unpublished work of C. J. S. Clarke and S. M. Scott on defining topologies on the abstract boundary. (Actually, the authors of [1] mention that their classification requires merely an affine connection, not necessarily the Levi-Cività connection of a pseudo-Riemannian metric.) Definition 2.1 Let E be a collection of envelopments and and define a relation ∼ on X E in the following way: This is an equivalence relation, and we set M E = X E /∼. Let π be the projection and write ∂ E M (or ∂ E ) for M E \ M. We will use the notation φ : M → M φ for envelopments, and for later use will also assume, as is entirely reasonable, that each M φ is assumed to have an atlas A M φ for which the "pullback" φ * A M φ is equivalent to the atlas A M on M. (We will impose a further condition on atlases in §4.) The key use of the above formalism is that it allows us to vary construction of the completion M E as we vary our chosen set of envelopments E. The usual abstract boundary is ∂ E 0 M, where E 0 is the collection of all smooth envelopments φ of M into some smooth manifold M φ . (In order to be absolutely correct mathematically, "all" needs to be qualified in such a way as to ensure that E is a set and not a proper class-for instance by working in a category of concretely defined manifolds.) In the next section we will survey in detail possible candidates for "regular" maps, metrics etc. in terms of a class Λ of maps between Euclidean spaces. For a given choice of Λ we will denote by Λ the corresponding category of class Λ maps between manifolds (e.g.Λ = C k ), and by Λ(M.N) the set of morphisms of Λ from manifold M to manifold N.
By, e.g., the categorical statement"Λ ⊂ C 1 " we mean that Λ(M, Armed with a suitable category of maps, then, we make an important set of definitions, the first of which generalises our earlier notion of "regularity" for Λ C 1 , but is identical to the earlier notion for Λ = C 1 .
Definition 2.2 Let Λ be a category of maps as above, and denotes the set of sections B → T ) be a given pseudo-Riemannian metric of signature (n, n − ν), i.e., a non-degenerate, symmetric bilinear form at every point of M. We shall say that g is Λ-extendible about p, where p ∈ ∂ φ for some envelopment φ ∈ E 0 , if the following holds: there is a neighbourhood U p of p (in M φ ) and a pseudo-Riemannian metricĝ p on U p ,ĝ p ∈ Λ(U p , PsR n ν (U p )), for which φ * ĝ p = g. Here PsR n ν (M) denotes the subbundle of the symmetric tensor product Symm(T * M ⊗ T * M) with fibres consisting of non-degenerate bilinear forms of the appropriate signature. Now let We call this a regular abstract boundary.
Note that R[Λ] can be identified, loosely at least, with the abstract boundary ∂ E[Λ] M associated with the collection of all envelopments with "everywhere Λ-regular" boundaries.
One would expect these to be useful abstract boundaries in relativity theories, where we would like to seek properties of all possible metric extensions (of some class) through a "boundary point", without using or requiring detailed knowledge of the analytic form of the metric involved.

Various regularity classes
We mention a few possible choices for our extension classes Λ( · ), although we will only consider the first in this paper. For clarity we assume that we are speaking of Lorentzian metrics.
"Smooth regularity", Λ = C k , k 1. Such regular abstract boundaries, the first to be considered, are the subject of unpublished work by C. J. S. Clarke and S. M. Scott.
"Existence theorem regularity." There are several possibilities that suggest themselves immediately here, relying on the existence theorems of [10]. These rely on conditions such as the difference between the metric that we consider and a fixed "asymptotically flat" or "background" metric lying in the Sobolev space H 2.5+ε (R n−1 ), ε > 0 (the metrics checked are the restrictions of a full space-time metric to hypersurfaces). We could allow extensions such that through any point there is a spacelike hypersurface on which induced data satisfies the conditions; or such that through any point there is a set of space-like hypersurfaces whose normals at the point form an open set and which satisfy the conditions, or such that there is a (local or global) foliation satisfying the conditions.
"Non-quantum physics regularity." It could be argued that a fairly robust criterion for pair production to become significant within a 3-dimensional region of "size" L where the components of the Riemann tensor in a "reasonable" frame are larger than R is that the argument being either merely on dimensional grounds, or by requiring that virtual pairs acquire sufficient energy within the region during the Heisenberg uncertainty time to become non-virtual. We would, then, require Λ to consist of metrics for which this condition did not hold, i.e., those metrics where we might expect Einstein's theory to describe "reality" accurately.
"Distributional (Cauchy-Schwarz or Colombeau) regularity." Here we would, technically, go outside our formalism and allow extensions-and presumably, although not necessarily, interior metrics also-that were not functions at all, but either Cauchy-Schwarz (linear) distributions, or Colombeau's "generalised functions" (non-linear distributions). To cope with such generality, we would probably have to impose further cnditions on the Levi-Cività connection components of these extended metrics, e.g., square-integrability in the first case and integrability in the second. Physically, these would describe, respectively, impulsive gravitational waves in their most obvious form, and "stringy" space-times admitting conical singularities.
"Geroch-Traschen regularity" [11], Λ = C 0 ∩ W 1,2 loc (intersection of categories of maps being defined by the intersection of function classes) We call these metrics of Geroch-Traschen type. (and the second.....) deleted Physically, these appear to encompass gravitational wave space-times, though the metric must be transformed so that it is continuous. This "Rosen form" transformation typically destroys at least the C 1 structure of the manifold in question. We will propose a way to deal with this in a forthcoming publication PSfrag replacements (in which we will also allow degenerate metrics, though still requiring squareintegrability of Levi-Cività connection components).
In this paper we consider only the first of these items.

Lipschitz boundaries
In this section we recall some material from [5], in which certain proscriptions on the nature of an atlas entail that the A-boundary can be constructed. This turns out to be a rather useful boundary construction for later work of Clarke and colleagues on singularity theorems, appearing in the same reference. It amounts to the requirement that the boundary "not wiggle about too much". . . Let A = {(W α , p α )} α∈A be an atlas for an envelopment M φ of M, and write U α = p α (W α ∩ φ(M)) ⊂ p α (W α ) ⊂ R n . For brevity, we will write U * α for that part of the boundary of the open set U α , which, speaking somewhat loosely, corresponds to parts of "the boundary of M" which "have two sides" and are "regular" with respect to our chosen g and Λ: See Figure 1. Note that we exclude points of intŪ α from the definition of U * α . This is because portions of a boundary which are not "Lipschitz hypersurfaces" cannot make up the "Lipschitz boundaries" that we define, following [5], in a moment. (In particular, this excludes higher codimension potential boundaries like the z-axis for M = U = R 3 \ {z-axis}, φ = inclusion.) Finally, we impose a regularity condition on our chosen points U α .
Definition 4.1 We say that the envelopment M φ has a Lipschitz boundary if the following three conditions hold, for each of the charts W α (the last condition, depending on pairs of charts, is for the benefit of the A-boundary construction and will not concern us for the remainder of this work): There is a vector field k α : R n → R n (assumed to be as smooth as the atlases A, A φ ), such that for each x ∈ U * α , we have |k α (x)| = 1 and the existence of some Note that the choice φ = identity, A = A φ leads to a case of the A-boundary construction, where the subsets U * i of ∂U i are taken to be the maximal such subsets through which g is Λ-extendible. (The union of M with the A-boundaryM \ M constructed in [5] can be exhibited as a (C 0 ) manifold-with-boundaryM def =M (A) (the details are in the quoted reference).) We shall call this the maximally regular (Clarke) A-boundary construction.
LetK be a Lipschitz constant forf x ; it follows that is the interior of a cone. We give these cones z ± we call a proper inward cone and that with "+" rather than "−" a proper outward cone. Remark. In the reference quoted there is no reference to (what we call) proper outward cones-but if condition (B α ) holds, there must be a proper outward cone as well as a proper inward cone.
The existence of proper inward cones will be used in the next section. First we make a new definition.

The bundle metric-classical (C 1 ) case
If Λ ⊂ C 1 and g ∈ C 1 then there is a (continuous) connection on (M, g) and the frame bundle LM (structure group GL(n, R)) admits a topological metric d, induced by a C 0 Riemannian structure as in [5,Chap. 3], [7] or [6].
The same construction gives a topological metricd p on L(φ(M) ∪ U p ) (p ∈ ∂ φ ), resulting from the (continuous) Levi-Cività connection ofĝ p . However, there are at least two potential problems with this.
Firstly, the metric in the frame bundle depends on the boundary point:d p depends on the point p. This, however, will not prove to be a problem when we come to the crux of the matter, Theorem 5.2, but we must keep the subscript p in mind! Secondly, we have the following matter which rather complicates our procedings. For points x, y ∈ φ(M), we certainly haved p (x,ỹ) d(x,ỹ) (tildes denoting points of fibres as usual)-loosely,d p d-but we may haved p (x,ỹ) < d(x,ỹ). This is evident from Figure 2 (takeĝ p to be flat). Unfortunately, in a crucial argument below (Theorem 5.2) we will want to haved p (

Using Lipschitz boundaries to imbed the Cauchy completion LM in L M
It is quite sufficient for our needs, however, to have a mere bound "φ * d constant×d p ", in a small neighbourhood of each point of the boundary. We will call into play the Lipschitz condition imposed on the boundary in the last section. But first, a technical (but straightforward) lemma. A sketched proof only will be given-the reader unsatisfied with this is invited to turn to the more rigourous, and general, treatment of a forthcoming paper. For now, though, note that we require that Λ ⊂ C 1 .
Proof sketch. Given y, we can choose U so small that sup x∈U,i,j |g ij ((x) − η ij | and sup x∈U,i,j,k |g ij,k ((x)| are as small as desired (i.e., g is close to a pseudo-Euclidean metric in the C 1 strong Stiefel-Whitney topology [12]), so the Christoffel symbols of the metric g are as small as desired. We use the usual continuous dependance of the solutions of a (C 1 ) differential equation on its parameters (see, e.g., Theorem IV.2.1 of [13]), here specifying our curve by means of two extra parameters, namely its endpoints. The solution parallelly transported vector field depends continuously on the (small) Christoffel symbols of g, so the solution is close to that for a flat metric, so for a short curve γ, where c depends only on ω γ(0) . (Of course it depends on the metric g, but this is fixed once and for all.) This "closeness" is uniform for "endpoint parameters" in some neighbourhood U ⊂ R n × R n of the origin, by the continuous dependence cited. We get the result.
By choosing a subsequence if necessary, we can assume that φ(x i ) →x ∈ M , and hence that φ(y i ) →x. From Proposition 4.1, if necessary by excluding a finite initial part of the sequence and applying a fixed homothety, we can ensure that there are inward cones of the form where in the coordinate chart (U α , p α ) thus fixed we write w i = p α (φ(x i )) and z i = p α (φ(y i )). Recall that K is the Lipshitz constant in the definition of • C. Let Π : R n → R n−1 be the projection onto the first n − 1 coordinates, and let e denote Euclidean distance in the coordinate system now chosen. Set Then p α (x), z i , w i are on the same side of the hyperplane Σ i , P i is a ball of radius (1+1/K)e i and centre w * i def = Π(w i ), while Q i is a ball of radius ((K +1)e i +z n i −w n i )/K e i and centre z * i def = Π(z i ). Since |w * i − z * i | e i , these balls have a non-empty intersection containng some point But from Lemma 5.1 again, the far right hand side tends to zero, and so the result is proved.
Another, perhaps more satisfying, way of stating this result is as an "imbedding theorem". (We use "imbedding" rather than "embedding" as a mnemonic for "injection"-there is no topological content to the stament.)

Corollary 5.3 Let φ ∈ E[Λ] have a Lipschitz boundary. For any boundary point
is one-one, i.e., it is an imbedding.

Application to regular abstract boundaries
We see that we can neither "coalesce" two regular boundary points into one without destroying the Lipschitz nature of the boundary near the points; similarly we cannot "tear" one Lipschitz regular boundary point into two: , of more than one point cannot be covered by a single (Lipschitz regular ) point q ∈ ∂ ψ , ψ ∈ LE[Λ].
Proof. Let p i , i = 1, 2, be distinct points of B, assumed to be both covered by a single boundary point q ∈ ∂ ψ . We first deal with the first of the 'potential problems' mentioned just before §5.1. Of course we can construct a single (C 1 ) coordinate chart (U, τ ) containing the two points, using the tubular neighbourhood theorem and a C 1 curve connecting the two points (the latter existing by connectedness of M). Since both these points are C 1 regular boundary points, then we may assume that U p 1 ∩ U p 2 = ∅ (c.f. Definition 2.2), and thus that there is a simultaneous extension of g to M p 1 ,p 2 def = φ(M) ∪ U p 1 ∪ U p 2 . We may then construct the bundle metricd p 1 ,p 2 as we did earlier when considering extensions about a single point. (The astute reader will have noted that a version of Theorem 5.2 using this new bundle metric will not be possible in general, for we cannot now shrink our neighbourhood to make "g ij ≈ η ij ". Fortunately, we only need to use Theorem 5.2 about the single (regular) point q. As a consequence of this we do not need the boundary ∂ φ to be Lipschitz, but only the boundary ∂ ψ about the single point q.) Note that, as yet, we have said nothing about the fibre elements E k,j , k = 1, 2. Now ψ(x i,j ), i = 1, 2, both converge to the covering point q. We can choose the fibre components of thex i,j so that these sequences also converge to the same point q ∈ L M ψ . This means thatd q -distance between the relevant points tends to zero, so by Theorem 5.2 we must have Butd p 1 ,p 2 (p 1 ,p 2 ) = 2R > 0, sod p 1 ,p 2 (x 1,j ,x 2,j ) > R for large j. Since "d p 1 ,p 2 d", for all large enough j, in contradiction.
We arrive at a result sitting well with intuition. Corollary 6.2 Let Λ ⊂ C 1 and g ∈ C 1 . Lipschitz regularity of boundary sets means that they "cannot be blown up further" without destroying regularity. To be more precise, consider two regular boundary sets Assume that the first envelopment φ has a Lipschitz boundary. Then no point of B 1 can cover more than one point of B 2 .
If both envelopments have Lipschitz boundaries and B 1 ∼ B 2 , then each point of B 1 is equivalent to precisely one point of B 2 .
Proof. Let p ∈ B 1 , and let Q Then Q is either empty, in which case p covers no subset of B 2 , or is the maximal subset of B 2 covered by p. In the latter case we apply Theorem 6.1, to get that Q must be a singleton. Thus no point of B 1 can cover more than one point of B 2 .
For the second part, note that by symmetry, the reverse of this statement is true, and the result follows.
The following lemma, although perhaps rather useless in other situations, does not per se depend on the Lipschitz, regular nature of the boundary sets to which we will apply it. Lemma 6.3 If B 1 ∼ B 2 and the function p → Q of Corollary 6.2 is singleton-valued (in which case we say p → q, if Q = {q}), and indeed maps each p to an equivalent q, then B 1 → B 2 : p → q is continuous.
Proof. If not, then there is an open neighbourhood V of q ∈ B 2 such that there is no neighbourhood U of p all of whose boundary points correspond to points of V . If there are small enough neighbourhoods U intersecting B 1 only at p, then we get a trivial contradiction. So there is a sequence of boundary points p i ∈ B 1 , equivalent to points outside V , which converge to p.
But q covers p, so there is a sub-neighbourhood , with respective images under φ converging in j to the respective p i ∈ B 1 , must have images under ψ which converge to points outside of V . However, the "diagonal" sequence By symmetry, we have the following. Note that this means that every representative of a Lipschitz C 1 -regular abstract boundary point [p] ∈ LR[M, g, Λ] is itself a point. Of course, the nature of a Lipschitz boundary forces the original point p to be a point of an (n − 1)-dimensional boundary hypersurface. As was alluded to earlier, it says that any envelopment of the space-time in which our original point is "blown up" to be represented by anything other than a single point destroys C 1 regularity of at least part of the new representative, whether this new boundary is Lipschitz or not.
This gives us a rather complete characterisation of those sections of boundaries of (e.g.) space-times through which the metric can be extended in a smooth (C 1 ) fashion. We are on firm mathematical, if not physical, ground when we suggest that each point of a Lipschitz of boundary of a space-time through which the metric can be extended smoothly, has something to distinguish it from all others.
It is a natural question to ask whether the same (or a similar) characterisation is valid when we relax the smoothness assumption somewhat, allowing, e.g., gravitational shock waves. After all, we would certainly hope that the presence of gravitationl waves in the space-time and extension thereof cannot "distort" the boundary so that the topological type of a portion of "boundary hypersurface", through which extension is possible, could depend on the precise way in which the extension is performed.
Since boundaries are of physical interest precisely in situations of collapse, where strong gravitational fields are liable to generate shock-and impulse-waves, it is important to be able to relax as far as possible the smoothness assumptions in this work. It would be a problem indeed for our regular abstract boundaries if it turned out that "nonsmooth perturbations" of a smooth metric could result in a space-time where, unlike the smooth case, the topological type of a portion of "regular boundary hypersurface" could depend on the precise way in which the extension is performed. In a forthcoming paper we will extend our results in this direction.