The boundary of the chronology violating set

A sufficiently general definition for the future and past boundaries of the chronology violating region is given. In comparison to previous studies, this work does not assume that the complement of the chronology violating set is globally hyperbolic. The boundary of the chronology violating set is studied and several propositions are obtained which confirm the reasonability of the definition. Some singularity theorems related to chronology violation are considered. Among the other results we prove that compactly generated horizons are compactly constructed.


Introduction
This work is devoted to the study of the boundary of the chronology violating region of spacetime. This boundary shares some features that are reminiscent of the Cauchy horizon H(S) of an achronal hypersurface. In fact, in some cases in which the chronology violation can be removed by passing to a suitable covering (that 'counts' the numbers of times a timelike curve crosses a hypersurface S), this equivalence can be made manifest [17]. Unfortunately, in the general case there is no such correspondence and the boundary of the chronology violating set must be studied for its own sake.
As today there are few results concerning the boundary of the chronology violating set C , and indeed the very definition of future and past boundary seems to be missing in the literature. In some cases, as done by Thorne [30] and Hawking [9], the character and properties of the boundary ∂C are obtained by assuming a globally hyperbolic complement M\C, and by identifying the future boundary of the chronology violating set with the past Cauchy horizon of such chronological complement. Of course, this definition is not completely satisfactory as it is well posed only for spacetimes which admit a globally hyperbolic chronological region.
The development of closed timelike curves in spacetimes which admit non-compact partial Cauchy hypersurfaces is fairly well understood. Hawking argued [9] that if the closed timelike curves originated by the actions of an advanced civilization, then the generators of the Cauchy horizon, followed in the past direction, would enter the space region were those actions took place. This fact should be expected since the generators themselves represent the flow of the information which signals the fact that a causality violation took place. Without a breaking of the spacetime continuum those generators would have to enter a compact region, namely the future Cauchy horizon would have to be compactly generated. However, Hawking showed that compactly generated horizons cannot form and some well known gaps in the proof, connected with a tacit differentiability assumption on the horizon, have been recently solved [24], thus confirming the validity of chronology protection at the classical level. Similar issues connected with Tipler's analysis [31] have also been clarified [24].
It has been shown that Hawking's no go theorem on the formation of timelike curves can be circumvented either by relaxing the assumption on the compact generation of the horizon or by admitting violation of the null energy condition, see [6,8,[25][26][27][28] (we shall say more on this in Sect. 3). The reader is warned that on this topic some imprecise or misleading statements can be repeatedly found in the literature; the most relevant example is given by Hawking's claims [9,Sect. 3] that (a) 'absence of closed null geodesics' on compact Cauchy horizons would be unstable, that is, the least perturbation of the metric would cause the horizon to contain closed null geodesics; (b) 'presence of closed null geodesics' would be stable. At present there is no convincing proof for these claims, and some studies seem to suggest different conclusions [5,22].
In this work we study the boundary of the chronology violating set without making restrictive assumptions and we eventually obtain a definition of its future and past parts. As it happens for the concept of Cauchy horizon, the results of this work could prove useful for the study of singularities under chronology violation. Indeed, we shall argue that a deeper understanding of this boundary could clarify the mutual relationship between chronology violation and geodesic incompleteness (i.e. singularities).
Let us recall that a spacetime (M, g) is a connected, time-oriented Lorentzian manifold of arbitrary dimension n + 1 ≥ 2, where g ∈ C k , k ≥ 3, has signature (−, +, . . . , +). As a matter of notation, the boundary of a set is denoted with a dot. In some cases in which this notation could be ambiguous the dot is replaced by the symbol ∂. The subset symbol ⊂ is reflexive, i.e. X ⊂ X. A set is achronal if no timelike curve joins two of its points. If S is a closed achronal set, the Cauchy development D + (S) is the set of those p ∈ M such that every past inextendible causal curve ending at p intersects S. The Cauchy horizon is H + (S) = D + (S)\I − (D + (S)).
Let us also recall that a future lightlike ray is a future inextendible achronal causal curve, in particular it is a lightlike geodesic. Past lightlike rays are defined analogously. A lightlike line is an achronal inextendible causal curve, hence a lightlike geodesic without conjugate points. In this work, unless otherwise specified, all the curves will be future directed, thus, for instance, a past lightlike ray ends at its endpoint.
The condition of absence of lightlike lines is is implied under the null genericity and the null convergence conditions by null completeness (as these three conditions together imply the existence of conjugate points on any null geodesic [1,10]). Therefore, in the study of singularity theorems it is often a good strategy to assume the absence of lightlike lines and to look for contradictions.
The chronology violating region C := {x : x ≪ x} is the set formed by those points through which passes at least one closed timelike curve. The relation x ∼ y if x ≪ y and y ≪ x is an equivalence relation in C and, as it is well known since the work by Carter, it splits the chronology violating region into (open) equivalence classes denoted in square bracket, [x] = I + (x) ∩ I − (x). Two points belonging to the same class have the same chronological future and the same chronological past.

The boundary of a chronology violating class
In this work we are going to study the boundary of a generic chronology violating class since the boundary of the chronology violating region can be recovered from those. In this respect the following result [18,Theor. 4.5] is worth recalling. Proof. It is a consequence of the fact that a sequence of future lightlike rays σ n of starting points x n → x has as limit curve a future lightlike ray of starting point x [18], and analogously in the past case. Clearly, by lemma 2.2, Fig. 1). A set F is a said to be a future set if I + (F ) ⊂ F . A future set is open iff I + (F ) = F . If F is future then J + (F ) ⊂F which implies that the closureF is future. Analogous definitions and results hold for past sets, in particular F is a future set iff M\F is a past set. The boundary of a future set is an achronal boundary [1].
The achronal boundary ∂I − ([r]) will be particulary important in what follows. The proof of the next result is rather standard. Proof. Let σ n be a timelike curve connecting p n ∈ I − ([r]) to r, with p n → p. By the limit curve theorem [18] either there is a continuous causal curve connecting p to r, which is impossible because p / ∈ I − ([r]) or there is a future inextendible continuous causal curve σ contained in I − ([r]). No point of this curve can be contained in If the causal curve γ connects x to y then between x and y no point of it can belong to I − ([r]) otherwise x ∈ I − ([r]), a contradiction. Let z ∈ γ\{y}, and take z ′ ≪ z, then z ′ ≪ y and since I + is open and y ∈ ∂I − ([r]) we have z ′ ∈ I − ([r]). Taking the limit z ′ → z we obtain z ∈ I − ([r]) thus z ∈ ∂I − ([r]). The causal curve obtained by joining γ with the lightlike ray starting from y must be achronal as it is contained in ∂I − ([r]) and thus it is a lightlike ray.  We do not impose that the generator be a maximally extended lightlike ray contained in A. In other words, as a matter of terminology, if σ : [0, b) → M is a generator of A then σ : [a, b) → M, 0 ≤ a < b, is also a generator of A. Let us come to the proof of achronality. Assume by contradiction that there is a timelike curve σ : [0, 1] → M whose endpoints p = σ(0) and q = σ(1) belong to B f ([r]). There cannot be a value of t ∈ (0, 1) such that σ(t) ∈[ r] otherwise as I + is open, and p, q ∈[r], we would have r ≪ σ(t) ≪ r, that is σ(t) ∈ [r], in contradiction with σ(t) ∈[r]. Thus either σ((0, 1)) is contained in [r] or it is contained in M\[r]. The former case would imply p ∈ I − ([r]), a contradiction. In the latter case it is possible to find z ∈ σ((0, 1)) ∩ M\[r], and as p ≪ z ≪ q and I + is open, r ≪ z ≪ r, a contradiction. Proof. Let us prove the former inclusion, the latter being analogous.
Let us come to the last statement. As p ∈ R p ([r]) there is a past lightlike ray η contained in[r] ending at p. As p ∈ I + ([r]) ∩[ r] ⊂ R f ([r]), there is a future lightlike ray σ passing through p and contained in [ r]. This ray is the continuation of the past lightlike ray η. Indeed, assume that they do not join smoothly at p. Take a point x ∈ I + (r) ∩ η\{p} (recall that I + is open), so that, because of the corner at p, σ\{p} ⊂ I + (x). Again, since I + (x) is open and σ ⊂[r] we have x ≪ r, thus since r ≪ x, we conclude x ≪ x which is impossible as x ∈ η ⊂[r]. We have therefore obtained a lightlike geodesic γ = σ • η passing through p entirely contained in[r].  The previous and the next results will prove the reasonability and the good behavior of these definitions. In a direction, ). In the other direction, assume  Definition 2.14. Given an achronal set S the edge of S, edge(S), is the set of points q ∈S such that for every open set U ∋ q there are p ∈ I − (q, U), r ∈ I + (q, U), necessarily not belonging to S, such that there is a timelike curve in U connecting p to r which does not intersect S.
It is useful to recall that edge(S) is closed andS\S ⊂ edge(S) ⊂S.  ). However, we can redefine x by slightly shortening σ so that we can assume σ ⊂ M\[r]. It remains to prove that q ∈ B p ([r]), from which it follows, as Since the previous analysis can be repeated for every U ∋ q, we can find a sequence x n / ∈ [r], x n → q, x n ≪ q. As I + (r) is open we can assume x n ≫ r, but since x n ≪ q and q ∈[r], we have also x n ≪ r, thus x n ∈ [r], a contradiction. We conclude that edge(B f ([r])) ⊂ edge(B p ([r])) and the other inclusion is proved similarly.
From the previous proposition it follows that edge , however, the reverse inclusion does not hold in general (see figure 1). Contrary to what happens with Cauchy horizons the generators of the boundary do not need to reach its edge.
. An analogous past version also holds.
Proof. The first identity follows from the chain of equalities, and let q ≪ p sufficiently close to p that q ∈ I + ([r]). Since q ≪ p, we have q ∈ I − ([r]) thus q ∈ [r] and taking the limit q → p we obtain p ∈ [r], but p / Let us come to the last statement. Since ). An analogous past version also holds.
By contradiction assume not, then for every causally convex neighborhood U ∋ q and x, y ∈ U, x ≪ q ≪ y, we consider the neighborhood of q, I + (x) ∩ I − (y). By assumption this neighborhood contains some point z ∈ B\B f ([r]). The timelike curve η ⊂ U joining x to z and then z to y does not intersect ) between x and z because, as z ∈ B, and B f ([r]) ⊂ B it would imply that B is not achronal. Analogously, η cannot intersect B f ([r]) between z and y because, as z ∈ B, and B f ([r]) ⊂ B it would imply that B is not achronal. Since every point admits arbitrarily small causally convex neighborhoods we have proved q ∈ edge(B f ([r])) a contradiction.
The answer is affirmative and shows in particular that no point of edge In the next theorem Int ∂I − ([r]) denotes the interior with respect to the topology induced on the achronal boundary ∂I − ([r]). and hold. Analogous past versions also hold.
be a sequence such that p n → p. The timelike vertical curve σ passing through p n intersects ∂I − ([r]) at some point q n different from p n because p n ∈ I − (r). It cannot be q n ≪ p n otherwise q n ∈ I − ([r]) while q n ∈ ∂I − ([r]), a contradiction. Thus we have just p n ≪ q n . Since q n ∈ ∂I − ([r]) and I + is open for every U ∋ q n there is some point and since U is arbitrary q n ∈ṙ. Furthermore, we have q n ∈ I + (p n ) = I + ([r]), q n ∈ṙ ∩ I + ([r]), and the continuity of the graphing function x 0 (x) of the achronal boundary implies q n → p, Coming to the last identity, the inclusion  The next example proves that edge(B f ([r])) is not necessarily acausal and that in fact edge(B f ([r])) could be generated by inextendible lightlike lines (see figure 2).
The metric can be written in the Kaluza-Klein reduction form If we focus on sets that are rotationally invariant the causal sets corresponding to those are obtained just considering the metric in square brackets rather than the full metric. This is a general feature of spacelike dimensional reduction, and rests on the fact that the horizontal lift of a causal curve on the base is a causal curve in the full spacetime and the projection of a causal curve of the full spacetime is a causal curve on the base. Furthermore, for what concerns causality the metric in square brackets can be multiplied by a conformal factor so that in the end the casuality is determined by the metric −dt 2 + sin 2αdr 2 .
The idea is to consider the disk S = {x : t(x) = 0, r(x) ≤ 1}, represented in the reduced spacetime by the segment [0, 1] and define C ± = {y : t(y) = ±k} ∩ D ± (S). For reasons of symmetry C ± is a, possibly empty, disk but for k sufficiently small C ± has non-vanishing radius. The fact that the causality can be reduced to that of a 2dimensional spacetime, and the fact that  Proof. Let x ≤ y ≤ z with x, z ∈ [r]. We know that r ≪ x and z ≪ r thus r ≪ y ≪ r, that is y ∈ [r], which proves that [r] is causally convex.
, which proves that [r] is chronologically convex.
Let us come to the last statement. Let x ≤ y ≤ z with x, z ∈ [r]. If x ≪ y or y ≪ z then it is easy to construct a timelike curve connecting x to z which passes arbitrarily close to y. Since this timelike curve is necessarily contained in [r] (because x, z ∈ [r] and I + is open) we get y ∈ [r]. We can therefore assume that x is connected to y by an achronal lightlike geodesic and analogously for the pair y, z. If the two geodesic segments do not join smoothly it is possible again to construct, using the smoothing of the corner argument, a timelike curve which connects x to z which passes arbitrarily close to y. We can therefore consider the case in which x and z are connected by a lightlike geodesic segment γ passing through y.
Let The set B f ([r]) ∩ B p ([r]) is not necessarily causally convex, see Figure 3. If we follow a generator of B f ([r]) in the past direction we may suspect that as long as the geodesic stays in[r] its points belong to B f ([r]). This is false as Figure 1 shows, however, if the geodesic does not enter B p ([r])\B f ([r]) then it is true as the next proposition proves.   Consider 1+2 Minkowski spacetime of coordinates (t, x, y) and identify the hyperplanes t = −2 and t = 2, so that the spacetime N between the two slices becomes totally vicious. Next remove from t = −1 an ellipse (including the interior) whose minor axis is 2 and whose major axis is 4. Do the same on the slice t = 1 but let the new ellipse be rotated of π/2 radians with respect to the former. A point belonging to both  The differentiability of topological hypersurfaces generated by past inextendible lightlike geodesics has been studied in [2][3][4]24]. This analysis was carried out having in mind Cauchy horizons but, as it is clarified with [3,Theor. 2.3], the results hold in general. Points at which the generators leave the hypersurface in the future direction are called future endpoints. The quoted works prove that at non-future endpoints the hypersurface is C 1 , at future endpoints at which ends only one generator the hypersurface is still C 1 and at future endpoints at which ends more than one generator the hypersurface is non differentiable. Therefore these results hold unchanged for Furthermore, Chruściel and Galloway [4] have given an example of Cauchy horizon which is non-differentiable in a dense set. They first constructed [4, Theor. 1.1] a compact set C = R 2 \K ⊂ R 2 having a connected Lipschitz boundary such that on the spacetime M = (−1, 1) × R 2 , endowed with the usual Minkowski metric, E + ({0} × C) was non-differentiable on a dense set.
We construct an example of spacetime in which[r] is non-differentiable on a dense set as follows. We remove from the just constructed spacetime the sets {0} × C and {1/2} × C and we identify the interior of the upper-side of {0} × C with the interior of lower-side of {1/2} × C. This operation introduces closed timelike curves and the boundary of the chronology violating region is a subset of what, before the removal of the sets, was E + ({0} × C) ∪ E − ({1/2} × C). As such[r] is non-differentiable on a dense set.
We say that B f is compactly generated if there is a compact set K such that its (future inextendible) generators enter K. For the notions of the next theorem not previous introduced we refer the reader to [24]. Observe that for the study of the development of time machines one is interested in the time dual version involving B p . Theorem 2.26. Assume that the null convergence condition holds. If B f is compactly generated and its generators are future complete then it is compact, C 3 , and generated by inextendible lightlike geodesics. Actually smooth if the metric is smooth, and analytic if the metric is analytic. Moreover, B f has zero Euler characteristic, it is generated by future complete lightlike lines and on B f In other words, denoting with n a lightlike tangent field to B f , for every X ∈ T B f , ∇ X n ∝ n and R(X, n)n ∝ n, that is, the second fundamental form vanishes on B f and the null genericity condition is violated everywhere on B f . In 2+1 spacetime dimensions either B f is a torus or a Klein bottle where the latter case is excluded if the spacetime is time orientable.
If it is know that if B f = H − (S) for some partial Cauchy surface S (e.g. see next section) then the condition on the geodesic completeness of B f can be dropped, for in this case one can use directly [24,Theor. 18]. We stress once again [24] that physically speaking it is incorrect to demand the validity of the null genericity condition on a compact set as done by some authors [31], thus its violation does not imply that the spacetime is unphysical.
Observe that B f belongs to the boundary of the chronology violating region, so if it is not compactly generated then the chronology violating region propagates to the boundary of spacetime. Thus this theorem establishes that either the formation of closed timelike curves happens as in the theorem, with a compact smooth B p with all its mentioned nice properties, or such CTC formation either violates energy conditions, extends to the boundary, or generates (geodesic) singularities.
Proof. The proof coincides with that of [24,Theor. 18] where this time the completeness of the future hypersurface must be assumed while there it was a consequence of it being a Cauchy horizon. If the spacetime dimension is three B f is two dimensional and the only compact closed surfaces with Euler characteristic zero are the Klein bottle and the torus. If the spacetime is orientable then B f is orientable thus the Klein bottle can be excluded.
In three spacetime dimensions one could obtain other interesting results by applying the Schwartz-Poincaré-Bendixson theorem to B f . In fact, observe that B f is at least C 3 thus its tangent vector field is C 2 as required by SPB's theorem. We conclude that under the assumption of Theorem 2.26 B f contains a closed causal curve (fountain) which is a minimal invariant set for the future (and past) lightlike flow on the horizon or the whole torus is itself a minimal invariant set. We recall that a minimal invariant sets is a closed minimal set which is left invariant by the future flow on B f . The concept makes sense on any imprisoned causal curve. Any minimal invariant set is generated by lightlike lines [11,19].
Any lightlike geodesic is no more lightlike if we open the light cones. However, in some cases the lightlike geodesic is stable in the sense that it gets simply moved aside, while in other cases it is unstable as it disappears completely. The next result does not assume null completeness but, rather, it relates it to the concept of stability.
) is compactly generated, then any geodesic on B f \B p belonging to one of its minimal invariant sets is future complete or it is unstable.
A better understanding of this theorem can be obtained from the proof.
Proof. Suppose that this geodesic γ is future incomplete, then there is a small timelike variation towards the future of γ which brings this curve to a timelike curve η such that η accumulates in the future to the same points to which accumulates γ, and hence accumulates on γ itself [23, Theor. 2.1] [13]. This fact implies that it is possible to construct a closed timelike curve σ in I + (γ) ∩ U where U is any neighborhood of γ. That is, γ is in the past boundary of a chronology violating class [q] and on the future boundary of another chronology violating class [r] (the two classes are different otherwise γ would belong to a chronology violating class, just take a timelike curve moving from [r] to [q] = [r] passing through a point of γ). Thus by opening slightly the cones, γ disappears but it cannot be recreated anywhere else since the two distinct classes join in a single class, thus showing that the previous configuration was unstable.
On might ask whether the violation of chronology near a point of B f ([r]) is a local or global phenomenon. The next result shows that if a minimal invariant set generator is incomplete in the past direction, then closed timelike curves can be found in any neighborhood of the generator.
Theorem 2.28. Let B f ([r]) be compactly generated and let γ be a lightlike geodesic on B f \B p belonging to one of its minimal invariant sets. Then either γ is past complete or for every neighborhood U ⊃ γ there is a closed timelike curve in U ∩ I − (γ) ∩ [r].
The proofs goes similarly to that of the previous theorem but reworked in the past direction.

The coincidence with previous definitions of boundary
The next result shows that, provided the chronal region is globally hyperbolic, the past Cauchy horizon of a suitable hypersurface is the future boundary of the chronology violating set. This result relates our definition of boundary with the more restrictive one given in some other papers [30].

Relationship between compact generation and compact construction
We have recalled that theorems on the non-existence of time machines are based on the observation that any creation of a region of chronology violation would lead to a Cauchy horizon which is compactly generated, namely, such that the generators followed in the past direction enter and get imprisoned in a compact set K. The idea is that the information on the production of closed timelike curves would propagate on spacetime along the generators of the horizon, so followed in the backward direction those generators have to enter the compact space region were the advanced civilization produced the timelike curves in the first place. This is the argument which is used to justify the assumption of 'compact generation of the horizon' in connection to the creation of time machines. It must be remarked that the generators being confined to the horizon cannot reach the Cauchy hypersurface, however, they do intersect the world tube of the compact region in which they are past imprisoned. In this sense the term 'space region' used in the previous paragraph is appropriate. Nevertheless, Amos Ori in a series of papers [27,28] has criticized the previous argument maintaining that the assumption of local time machine creation would have to be expressed by the following concept, which he terms compact construction. Here S 0 represents the region were the actions of the advanced civilization leading to the formation of closed timelike curves took place.
Actually, Ori uses "closed causal curves" in place of 'almost closed causal curves' in the above definition. The difference does not seem to be important: the idea is that almost closed (and possibly closed) causal curves would signal the development of closed timelike curves just behind the horizon. Ori shows that a compactly constructed time machine can be initiated with no violation of energy conditions [28].
The relative strength of compact generation and compact construction has remained open so far. One could suspect 'compact construction' to be a weaker property than 'compact generation', for the latter with its strength prevents the formation of time machines. In fact we are able to prove Theorem 3.2. Let S be a closed acausal hypersurface without edge (partial Cauchy hypersurface). If H + (S) is compactly generated then it is compactly constructed.
Proof. Let K be the imprisoning compact, we can assume that K ⊂ H + (S), otherwise replace K with K ∩ H + (S). Let C ⊂ I + (S) be another compact set, chosen so that K ⊂ Int C. Let us prove that S 0 := J − (C ∩ D + (S)) ∩ S is compact. Suppose not, then there is a sequence of past inextendible casual curves γ n with future endpoint p n ∈ C ∩ D + (S) which intersect S at q n with q n → ∞, meaning by this that the sequence q n escapes every compact subset of S. Following γ n in the future direction let r n ∈ ∂C ∩ D + (S) be the first point in C and let η n := γ n | qn→rn be the portion of γ n not in C saved for r n . Let r ∈ ∂C ∩ D + (S) be an accumulation point of r n . By the limit curve theorem [18] there is a past inextendible causal curve η with future endpoint r which does not intersect S (if it were to intersect it at some y ∈ S then a subsequence q ns would converge to y which is impossible since every subsequence escapes all compact sets). Being η the limit of curves contained in the closed set M\Int C it is also contained in this closed set and so does not intersect K. Observe that it is a causal curve which cannot enter D + (S) for otherwise it would be forced to reach S, thus it is entirely contained in H + (S). This fact proves that r ∈ H + (S). Since the horizon is achronal η is a lightlike geodesic, that is a generator (lightlike geodesics on the horizon cannot cross for it is easy to see that it would contradict achronality). This is a contradiction with compact generation since we have shown that η does not intersect K ⊂ IntC where every generator should enter. The contradiction proves that S 0 is compact. Let x ∈ K, and consider a sequence x k → x, x k ∈ I − (x). As a consequence, x k ∈ D + (S). For sufficiently large k, x k ∈ C which implies that x k ∈ D + (S 0 ) and consequently, x ∈ D + (S 0 ) which implies x ∈ H + (S 0 ). We have shown that K ⊂ H + (S 0 ) where K contains almost closed causal curves since it contains a minimal invariant set [19].

The case I + ([r]) = M and a singularity theorem
S. Hawking has suggested that the laws of physics prevent the formation of closed timelike curves in spacetime [9] (the chronology protection conjecture). According to this conjecture the effects preventing the formation of closed timelike curves could be quantistic in nature, in fact Hawking claims that the divergence of the stress energy tensor at the boundary of the chronology violating set would be a feature of this prevention mechanism.
Despite some work aimed at proving the chronology protection conjecture its present status remains quite unclear with some papers supporting it and other papers suggesting its failure [12,15,16,31,32]. Some people think that in order to solve the problem of the chronology protection conjecture a full theory of quantum gravity would be required [7,9].
A weak form of chronology protection would forbid the formation of closed timelike curves without denying the possibility that closed timelike curves could have been present since the very beginning of the universe. For this reason it is important to study spacetimes that originate causally from a chronology violating region [r], namely I + ([r]) = M.  . Since p ∈ M = I + (r) there is a timelike curve joining r to p and a causal curve joining p to [r]. By making a small variation starting near p we get a timelike curve from r to [r], and hence equivalently, from r to r passing arbitrarily close to p, thus p ∈ [r], a contradiction.
For the last equality it suffices to take the interior of the second one. The interesting fact is that M\[r] must admit a time function, provided null geodesic completeness and other reasonable physical conditions are satisfied (see Theorem 4.4). For more details on these conditions see [10]. It can be read as a singularity theorem: under fairly reasonable physical conditions if the spacetime outside the chronology violating region does not admit a time function then the spacetime is geodesically singular. Theorem 4.4 is a non-trivial generalization over the main theorem contained in [20]. Note that null geodesic completeness is required only on those geodesics intersecting M\[r]. These geodesics cannot be tangent to some geodesic generating the boundarẏ  Proof. Consider the spacetime N = M\[r] with the induced metric g N , and denote with J + N its causal relation. This spacetime is clearly chronological and in fact strongly causal. Indeed, if strong causality would fail at p ∈ N then there would be sequences p n , q n → p, and causal curves σ n of endpoints p n , q n , entirely contained in N, but all escaping and reentering some neighborhood of p. By an application of the limit curve theorem [1,18] on the spacetime M there would be an inextendible continuous causal curve σ passing through p and contained inN to which a reparametrized subsequence σ n converges uniformly on compact subsets (σ can possibly be closed). The curve σ must be achronal otherwise one would easily construct a closed timelike curve intersecting N (a piece of this curve would be a segment of some σ n thus intersecting N). Thus σ is a lightlike line and hence, by Lemma 4.3, it is entirely contained in N. By assumption σ is complete thus by null genericity and null convergence it has conjugate points, which is in contradiction with it being achronal. The contradiction proves that (N, g N ) is strongly causal. The next step is to prove that J + N is transitive. In this case N would be causally easy [21] and hence stably causal (thus admitting time functions). Suppose (x, y) ∈ J + N and (y, z) ∈ J + N . The transitivity of J + N is proved as done in [20,Theorem 5], observing that the limit curve passing through y constructed in that proof, necessarily contained inN, is either achronal and hence, by Lemma 4.3, entirely contained in N, which allows to apply that original argument, or non-achronal. In the latter case that argument of proof shows that (x, z) ∈ J + . Let us recall that J + = I + , thus there are neighborhoods U and V such that any timelike curve connecting U ∋ x, U ⊂ N to V ∋ z, V ⊂ N must stay in N, because otherwise there would be some w ∈ [r] such that x ′ ≤ w, with x ′ ∈ U. This is impossible because by Prop In a different work [22] I have argued, using entropic and homogeneity arguments, that our spacetime could indeed have been causally preceded by a region of chronology violation. In this picture the null hypersurface[r] would be generated by achronal inextendible lightlike geodesics, and would replace the usual Big Bang (which is usually taken as a spacelike hypersurface in the spacetime completion). Since[ r] would be generated by lightlike lines a rigidity mechanism would take place and several components of the Weyl tensor would vanish at the boundary (because the Weyl tensor causes focusing [10]). This fact is in accordance with Penrose's expectations on the beginning of the universe [29] (the Weyl tensor hypothesis) according to which, in order to solve the entropic problem of cosmology, the Weyl tensor must be small at the beginning of the Universe.

Conclusions
We have studied the boundary of the chronology violating set, defining its future and past parts and proving the reasonability of the definition. For instance, we have shown that the edges of these parts coincide and that the full boundary is obtained by gluing the future and past parts along their edges. We have shown that our definitions are compatible with a previous definition in the domain of applicability of the latter. We have studied other properties of these boundaries, including causal convexity, differentiability and smoothness under energy conditions. Theorem 2.26 clarified the connection with singularities. We have also proved that compactly generated horizons are compactly constructed. This results did not use the definition of chronological boundary but it is relevant in order to clarify no-go theorems on the creation of time machines. Finally, we have considered the circumstance in which there is just one chronology violating region at the beginning of the Universe, proving that under reasonable energy and genericity conditions either there is a time function outside it or the spacetime is singular.