On a duality between time and space cones

We give an exact mathematical construction of a spacelike order $<$, which is dual to the standard chronological order $\ll$ in the $n$-dimensional Minkowski space $M^n$, and we discuss its order-theoretic, geometrical as well as its topological implications, conjecturing a possible extension to curved spacetimes.


Introduction
Many important theorems, within the frame of general relativity, refer to spacelike properties, such as singularity theorems which require the existence of a smooth spacelike Cauchy surface Σ (Hawking et al., 1973) , schematic conformal diagrams depicting causal independence (for example, Penrose, 2007), etc. In all cases, spacelike is synonymous to locally acausal 1 , where there is no timelike relation or information traveling to the speed of light. In this article, we show that the structure of the null-cone is induced, in a topological sense, by a spacelike order which creates a spacelike orientation in an analogous way to the timelike orientation.
For an event x ∈ M n , we consider the following sets: For any plane P m (x), where m ∈ M n , m = 0 is the normal to P m (x), P m (x) = {y : g(m, y − x) = 0} and where g denotes the spacetime metric, we consider the half-planes: 1 It was highlighted to us by a reviewer of this article, and we consider it useful to mention this here as well, that spacelike is a purely local property, but acausal is a global property. For instance, the Lorentzian cylinder M = R 1 × S 1 , with metric −dt 2 + dθ 2 , and also the submanifold that is the image of the map f : R → M given by f (s) = ( 1 2 s, s): this is manifestly spacelike at all points, but f (s) ≪ f (s + 2π) for all s.
2 Here the word "cone" is used in a generalised sense, i.e. it is a cone on I × S n−2 .
1. P + m (x) = {y : g(m, y − x) ≥ 0 and y = x} and 2. P − m (x) = {y : g(m, y − x) ≤ 0 and y = x} For abbreviation, we will write P + (x) := P + m (x), P − (x) := P − m (x) and P (x) := P m (x). We observe that P + (x)∪P − (x) = {x} c , where the superscript c -here and throughout the text-denotes the complement of a set. So, for an event y ∈ M , Moreover, we define the following subspaces: It is standard (see (Penrose, 1972)) to consider two partial orders, the chronological order ≪ (which is irreflexive in M n ) and the causal order ≺, which is reflexive, two orders that are defined not only in M n but in general in any spacetime, as follows: 1. x ≪ y iff y ∈ C T + (x) and In addition, the reflexive relation horismos → is defined as x → y iff x ≺ y but not x ≪ y.

The weak interval topology
Consider the weak interval topology, which is constructed in an analogous way to the interval topology (Gierz et al., 1980), which however does not apply only to lattices. In fact, when restricted to the 2dimensional Minkowski space M 2 , under the causal order ≺, the weak interval topology coincides with the interval topology, but in general it will not be restricted to lattices. For its construction, we need a relation R defined on a set X. We then consider the sets as well as the collections The topology T in with respect to the relation →

The order on the space-cone and its induced topology
We define a partial spacelike order < dual to the chronological order ≪. This order is obviously not causal, but it brings an interesting duality between the time cone C T (x) and the space cone C S (x) of an event x. Through <, the "cone" C S (x) exhibits similar properties to C T under ≪. Since "chronological" comes from the word "chronos", which means time, we name < "chorological", as it refers to "choros", space.
Definition 0.1. For non causally-related events It follows that x < y iff x ∈ C S − (y). In addition, ≤ denotes < including the boundary, in a dual way as ≺ is to ≪, that is, x ≤ y iff y ∈ C LS + (x). We remark that < is a partial order; the transitivity is obvious, as soon as it is highlighted that < refers to events which are not causally related; thus, if x, y, z are mutually not causally related (x is not causally related to y, y is not causally related to z and x is not causally related to z), then x < y and y < z implies that x < z.  For the last result of our discussion, we will need to use Reed's definition of intersection topology (Reed, 1986): Definition 0.2. If T 1 and T 2 are two topologies on a set X, then the intersection topology T int with respect to T 1 and T 2 , is the topology on X such that the set {U 1 ∩ U 2 : U 1 ∈ T 1 , U 2 ∈ T 2 } forms a base for (X, T ).
The topology Z T is defined to be the intersection topology, according to Reed's definition, of the topologies T in ≤ and the natural topology of R n , in M n . This topology, in M , coincides with one of the three topologies that were suggested by Zeeman (Zeeman, 1967), as alternatives to his Fine topology. Zeeman introduced Zeno sequences (Zeeman, 1967) with respect to his "Fine" topology F ; a sequence {z n } n∈N which converges to some z in M n under the natural topology of R n and not under the topology F (or any other topology in the class of Zeeman topologies, (Göbel, 1976)) is called a Zeno sequence.
Agrawal-Shrivastava also showed that, within the ndimensional Minkowski space M n , for a Zeno sequence under topology Z T converging to z ∈ M n there exists a subsequence of this sequence whose image is closed under Z T but not under the natural topology of R n . In addition, within M n and for a nonempty open-set G in the natural topology of R n , if z ∈ G, then G contains a completed image of a Zeno sequence under Z T converging to z. With respect to the convergence of causal-curves, Low (2016) has shown that under the Path topology, i.e. under the general-relativistic analogue of Z T , the Limit Curve Theorem fails to hold (Low, 2016). Thus, the basic arguments for building contradiction in singularity theorems fail under the Path topology, as well. Low (2016) also establishes, with respect to the Path topology, that if one considers timelike paths, the notion of convergence is not affected by the choice of whether one uses the manifold topology or the path topology on a spacetime (Propositions 1 and 2). In the case of M n , one can restate this argument by substituting the Path topology with Z T and the manifold topology with the natural topology of R n . In particular, let T be the set of (endless) timelike curves; these are curves in M n where at each point the tangent vector is future pointing and timelike, (Penrose, 1972). Also, C denotes the set of (endless) causal curves; these are curves in M n where at each point the tangent vector is future pointing and timelike or null. Then the restriction of Z T in T gives a topological space homeomorphic to the restriction of T to the natural topology of R n , and also if γ n → γ, some γ ∈ T , is the restriction of T to the natural topology of R n , then for each x ∈ γ there exists a sequence The topology Z LT is defined to be the intersection topology, according to Reed's definition, of the topologies T in < and the natural topology of R n , in M n . This topology fully incorporates the causal structure of M n . This is so, because it admits a base of open sets of the form C LT (x) ∩ B ǫ (x), where B ǫ (x) is a ball in R n centered at x and of radius ǫ > 0. Low, again with respect to curved spacetimes, shows that the Path topology induces a strictly finer topology on C than the manifold topology does and that the restriction of the manifold topology in T is dense in the restriction of the manifold topology in C. Since there is no reference on a general relativistic analog of Z LT , we can make the following considerations with respect to the statements of part IV of (Low, 2016), before Proposition 3. Consider x ∈ γ and let a neighborhood of x in γ with respect to the natural topology of R n be a null-geodesic segment. Then there exists no sequence of timelike curves such that γ n ∈ γ with respect to the restriction of C in the natural topology but not in Z LT . So, the general relativistic analogue of Z LT will not induce a strictly finer topology on C than the manifold topology does.

Questions
It would be desirable if the results of Section 4 generalized to any curved spacetime, in the frame of general relativity. This hope comes for the following intuition. In a relativistic spacetime manifold, wherever there is spacetime, there are events and for every event there is a light-cone. Since our construction of T in ≤ is topological, depending exclusively on the interior (time-cone), boundary (light-cone) and exterior (space-cone) of an event x, independently of the geometry of the space-time, one could con- The question in this case is how could one define the general-relativistic analogue of the order ≤. How could one describe the general-relativistic analogues of the half-planes P + (x) and P − (x), that we examined in Section 2, since they will not be "flat" planes in the Euclidean sense anymore, but will follow the geometry of the particular spacetime manifold, so that their union will give {x} c . So, it would be vital to also express the general relativistic analogue of the normal m to the plane P (x) (Section 2), in a rigorous algebraic way; such an algebraic development should open further directions to our discussion about the duality between causal and locally acausal orders in a spacetime, a duality which might play a role for the passage from locality to non-locality (see, for example, (Vagenas, 2018)). An answer to such a question will also give a solution to the orderability problem (see (Papadopoulos, 2014)) in the particular case of the Path topology of Hawking-King-McCarthy. Similar questions may be asked for the general relativistic analogue of Z LT and a possible generalisation of the order < to curved spacetimes.