Journal of Gravity

Volume 2016 (2016), Article ID 6151726, 6 pages

http://dx.doi.org/10.1155/2016/6151726

## Spacetime Causal Structure and Dimension from Horismotic Relation

Horia Hulubei National Institute for Physics and Nuclear Engineering, No. 30, Strada Reactorului, P.O. BOX MG-6, 077125 Bucharest, Romania

Received 11 March 2016; Accepted 8 May 2016

Academic Editor: Anzhong Wang

Copyright © 2016 O. C. Stoica. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

A reflexive relation on a set can be a starting point in defining the causal structure of a spacetime in General Relativity and other relativistic theories of gravity. If we identify this relation as the relation between lightlike separated events (the horismos relation), we can construct in a natural way the entire causal structure: causal and chronological relations, causal curves, and a topology. By imposing a simple additional condition, the structure gains a definite number of dimensions. This construction works with both continuous and discrete spacetimes. The dimensionality is obtained also in the discrete case, so this approach can be suited to prove the fundamental conjecture of causal sets. Other simple conditions lead to a differentiable manifold with a conformal structure (the metric up to a scaling factor) as in Lorentzian manifolds. This structure provides a simple and general reconstruction of the spacetime in relativistic theories of gravity, which normally requires topological structure, differential structure, and geometric structure (which decomposes in the conformal structure, giving the causal relations and the volume element). Motivations for such a reconstruction come from relativistic theories of gravity, where the conformal structure is important, from the problem of singularities, and from Quantum Gravity, where various discretization methods are pursued, particularly in the causal sets approach.

#### 1. Introduction

In Lorentzian manifolds, the* causal relations* are defined as holding between events that can be joined by future oriented causal curves. Causal relations give the* causal structure* of a spacetime. In [1], the causal structure was used to recover the* horismos* and* chronology relations* of a spacetime (the relations between events that can be joined by future lightlike, resp., timelike curves). The causal structure is known to be sufficient to recover the metric of the spacetime up to a conformal factor. The conformal factor can be obtained if in addition we know a measure which gives the volume element [2–6]. This works for* distinguishing spacetimes*—spacetimes whose events can be distinguished by the chronological relations they have with the other events (e.g., spacetimes containing closed timelike curves are not distinguishing). Moreover, for distinguishing spacetimes, the causal structure can be obtained from the horismos relation [7].

The fact that the causal structure and a measure are enough to recover the geometry of spacetime in General Relativity and other relativistic theories of gravity and the hope that discretization may be the way to Quantum Gravity by providing an UV cutoff motivated the study of sets ordered by the causal order [4, 8, 9]. Another motivation was that a discrete structure could account for the black hole entropy [10, 11]. These reasons led in particular to the idea of* causal set*, defined as a set endowed with a partial order , which therefore is reflexive, antisymmetric, and transitive (in standard causal set articles and some General Relativity articles like [12] the notation “” is used, but in standard General Relativity articles and textbooks like [13] the notation “” is preferred). In addition, it is required that, for any , the cardinality of the set is finite [14–16]. In the causal set approach, the continuous spacetime is considered to be an effective limit of the causal set. The measure used to recover the volume element is given by the number of events in each region. As Sorkin put it, “order plus number equals geometry.”

However, causal sets do not have a definite dimension. One sort of dimension is the smallest dimension of a Minkowski spacetime in which the causal set can be embedded (flat conformal dimension), but there are more possible definitions of dimension, such as statistical and spectral dimensions [15, 17–19], none of them satisfactory enough for this problem. This is probably the main reason why it is so difficult to prove and even to formulate mathematically the fundamental conjecture of causal sets (*Hauptvermutung*) that the causal set can recover within reasonable approximation (yet to be defined) the manifold structure. In the limiting case of infinite event density uniformly distributed, the conjecture has been proven [20], but in general the problem remains open.

In the following, we consider sets of events endowed with a reflexive relation which represents the horismos relation. We do this in the most general settings, including both the continuous and the discrete cases. We show that from the horismos relation one can recover the topology, the causal structure, and, with simple additional requirements, the dimension of spacetime and the differential structure.

The paper is organized as follows. We start with the definitions and main properties of the horismotic sets in Section 2.1. From them, we derive the causal structure in Section 2.2 and show how to obtain a topology in Section 2.3. Then, based on the horismotic sets, we introduce the causal curves in Section 2.4. To recover the dimension, we start with a simple example in two dimensions, which contains the main ingredients in Section 3.1. Then, we introduce a notion of dimension on horismotic sets in Section 3.2, which allows us to construct light cone coordinates in Section 3.3. These allow straightforwardly recovering the structure of a topological manifold and, under reasonable conditions, the differential structure and the conformal structure in Section 3.4.

#### 2. Horismotic Sets

##### 2.1. Elementary Properties

*Definition 1 (horismotic set). *A* horismotic set * is a set whose elements one calls* events*, endowed with a binary relation , which is* reflexive* ( for any ). If , one says that and are in the* horismos* relation or simply that * horismos *. For an event , one defines its* future horismos* or* future light cone* as and its* past horismos* or* past light cone* as .

The horismos relation has the physical meaning that a light ray can be emitted from the event to ; thus, it represents the lightlike separation between events. This relation is not transitive.

*Definition 2. *The horismotic relation is* antisymmetric* (i.e., for any two events and from , from and follows ) if and only if for any event from , .

Proposition 3. *If the horismotic relation is antisymmetric, then for any two events and from , from it follows that .*

*Proof. *If , then, since and , it follows that and . Hence, and , and from antisymmetry, .

##### 2.2. Causal Structure

*Definition 4. *A horismotic chain between two events is a set of events , where is a nonnegative integer, so that , for all , and . The length of the chain is then defined to be . Let one define the* causal relation* between two events , by iff there is a horismotic chain joining and . One defines the* chronology relation * on by iff and not . The relation represents timelike separation between events. One also defines the relation by . Two events are* spacelike separated*, , iff neither nor .

*Definition 5. *For an event , one defines its* chronological future* by and its* chronological past* by . One defines its* causal future* by and its* causal past* by . One defines the* causal cone* of by , the* chronological cone* of by , and the* light cone* of by . One defines , , , and . Two events define a* chronological interval * and a* causal interval *.

Proposition 6. *Let . Then, , , and .*

*Proof. *The proof follows immediately from Definitions 1, 4, and 5.

Proposition 7. *The causal relations and are transitive.*

*Proof. *The proof follows immediately from the definitions of the relations and .

The causal relation is the smallest transitive extension of the horismos relation .

*Definition 8. *A horismotic set is said to be* future (past) distinguishing at an event * if for any , implies (resp., ). It is said to be* future (past) distinguishing* if it is future (past) distinguishing at all of its events. It is said to be* distinguishing* if it is both future and past distinguishing.

Many of the properties of the causal and chronological relations known from General Relativity and Lorentzian manifolds in general [12] can be derived in the settings of horismotic sets.

##### 2.3. The Topology

We can endow with a structure of topological space generated by finite intersections and unions of open sets the sets of the form . As an example, consider the spacetime of General Relativity and other relativistic theories of gravity. The sets of the form are the interiors of future and past light cones and are indeed open sets and generate the* Alexandrov interval topology*. This topology coincides with the manifold topology iff it is Hausdorff and iff the spacetime is* strongly causal* (at each event there is an open set so that timelike curves that leave do not return) [12].

But not any horismotic set has a definite dimension, nor it is locally homeomorphic to . Additional conditions are needed and will be provided in the following.

##### 2.4. Causal Curves

To define causal curves in Lorentzian manifolds, one usually imposes conditions on the vectors tangent to the curve [12]. However, by default a horismotic set does not have a differential structure, so here we will give a definition that does not require a differential and not even a topological structure.

*Definition 9. *Let denote any of the relations , , and on a horismotic set .

An* open curve with respect to the relation * defined on a horismotic set is a set of events so that the following two conditions hold: (1)The relation is* total* on ; that is, for any , , either or .(2)For any pair , , if there is an event so that and , then the restriction of the relation to the set is not total.A* loop or closed curve with respect to the relation * defined on a horismotic set is a set of events so that, for any event , the set is an open curve with respect to the same relation.

*Remark 10. *Note that usually a curve is defined as the image of a continuous injective function , where and is a topological space. Therefore, it is a topological subspace of and at the same time a totally ordered set, with the order induced by the order on the interval . Definition 9 is more general, since it applies to horismotic sets, in particular to both discrete and continuous spacetimes. In Section 2.3 the horismotic set was endowed with a topology, the Alexandrov interval topology, and a curve as in Definition 9 is still a topological subspace of , which has the property that it is totally ordered with respect to the relation . In the particular case when is a manifold with distinguishing causal structure, the notion of curve defined here coincides with the usual notion of curve.

For simplicity, in the following, by “curve” we will understand “open curve,” and by “loop,” “closed curve.” We denote by and the set of curves, respectively, loops, with respect to the relation . Let be two curves. If , then is said to be an* extension* of , and is named a* subcurve* of . If for any extension of the curve it follows that , we say that is an* inextensible curve*. If an extension of a curve is inextensible, we say that is a* maximal extension* of .

If , then defines two curves for which is an extension: , and . If , , then they define a curve , and we call it the* segment* of the curve determined by and .

A curve from is called* causal curve*. A curve from is called* chronological curve*. A curve from is called* lightlike curve*. Similar definitions are given for loops.

*Remark 11. *It is easy to see that , , , and . Also, if the horismos relation is antisymmetric, then there are no closed lightlike curves, . Even when the relation is antisymmetric, and are not necessarily antisymmetric, so if we want to avoid closed causal and chronological curves, we have to add this as a condition.

*Definition 12. *A spacetime on which there are no causal loops, that is, , is called* causal spacetime*. Similarly one defines a* chronological spacetime* by .

Proposition 13. *Let be a curve in , and . If is a causal curve, then . If is a chronological curve, then . If is a lightlike curve, then .*

*Proof. *It follows from the condition that the relation is total on any causal curve, is total on any chronological curve, and is total on any lightlike curve.

Corollary 14. *Let be a curve in , and . If is a causal curve, then . If is a chronological curve, then .*

*Proof. *It follows from Proposition 13.

Corollary 15. *The causal, chronological, and lightlike curves and loops are continuous with respect to the interval topology defined in Section 2.3.*

*Proof. *From Corollary 14, for any causal curve and , the curve . Since the interiors of the intervals of the form form a base for the interval topology, it follows that is continuous. Similarly, if is a loop, the curve obtained by removing a point of is a continuous curve.

#### 3. Recovering the Dimension

##### 3.1. Example: The Two-Dimensional Case

We look first at a simple example of recovering the conformal structure of a Lorentzian manifold in two dimensions, which later will be distilled and generalized.

Assume that through any event pass exactly two maximal lightlike lines, say and . For a Lorentzian manifold, the* global hyperbolicity condition* states that, for any two events , the set is compact. The notion of global hyperbolicity extends naturally to a general horismotic set , because it is also a topological space, as shown in Section 2.3. We assume that satisfies global hyperbolicity. In the two-dimensional case, this is equivalent to the condition that, for any event , intersects and intersects (see Figure 1). Let us see why the two conditions are equivalent. If, for example, would not intersect , then the set would not be compact. Because of the assumptions at the beginning of this paragraph, the intersection contains a unique event . Similarly, contains a unique event .