A Generalization of the Alexandrov and Path Topologies of Spacetime Via Linear Orders

• May 2016 • A Generalization of the Alexandrov & Path Topologies of Spacetime via Linear Orders Luis Martinez Dartmouth College [email protected] Abstract Let X be a topological space equipped with a collection P of continuous paths. Associated to the pair (X, P) we define a path topology that generalizes the P-topology of Hawking et al. [1] Moreover, we axiomatize a collection P that generates an order that we call an NRT full order. This in turn generalizes the chronology order of spacetime defined via timelike paths. We also investigate the associated Alexandrov topology using ideas of Penrose and Kronheimer[2]. We conclude by defining the notion of a strongly path ordered space (X, t, P) that generalizes the concept of strongly causal spacetimes and prove an equivalence involving these topologies. I. Introduction Fine topology, tF. His work was later taken by Hawking et al. and reconfigured so that Spacetimes have been studied as manifolds the theory would make more physical sense. equipped with a topology of a 4 dimensional One of Hawking and company’s main critiques space. However, there are some problems with was that while, tF was more physically pleas- this approach. E.C. Zeeman pointed out in his ing, the definitions used to define the latter 1966 paper that considering a manifold with (straight lines) has the information about iner- such properties has its drawbacks, listed below: tial observers a priori so the fact that the linear 1. The manifold topology is highly unphys- structure comes out of this theory is not very ical. Since the latter is considered to have surprising. the topology of a a 4 dimensional Eu- Hawking and company instead opted to clidean space, one cannot talk about open construct a similar topology, what they called neighborhoods, without referring to four the path topology tP, that they propose is more spheres, the latter of which have no phys- physical and produces more surprising results ical meaning in space time. with out so much a priori structure.[1] Among one of the flagship differences between t and 2. Four dimensional Euclidean spaces have P t is that while the two induce a Euclidean a topology incompatible with space time. F topology on timelike curves, the former for- The former is homogeneous at every goes the requirement that such timelike curves point (’event’) however, this is physically be smooth. In fact, t is the finest topology on not true, as at every event in space time, P a manifold M with such quality. More interest- light cones separate regions of space like ingly is the result that shows which particular separation and time like separation. paths are continuous in this tP topology. Such He suggested several improvements to this paths are referred to by Hawking as Feynman notion of spacetime. Specifically, he suggested paths which are informally defined as paths on a modification to the conventional notion of the space (or manifold to be precise) that are spacetimes, by instead considering the latter continuous (under the inherit topology) but are as manifolds with a new topology called the constrained to move within a light cone. The 1 • May 2016 • open sets under this topology are also worth uses Maudlin’s theory to study discrete spaces, noting, most notably light cones(including the she shows that Maudlin’s linear structure the- point of reference) are open in this topology ory can be easily integrated with the equivalent (a result that will be of interest later). Most, notions of Alexendrov-interval topologies phys- importantly is the relationship between tP and ically known as just the light cone structure. tF. Zeeman showed that tF was strictly finer The results of Hudetz work will not be dis- that the inherit topology on the space (or mani- cusses in further detail but it is important to fold), so one would assume that tP is also finer present, at least intuitavely, the connection that and maybe even comparable to tF. Hawking the Hudetz work has with the work of the au- showed that while the first assertion was true, thor the second is not, as there exists open sets in tF Hudetz’s study of causal spaces comes that when intersected with a continuous path • from the axiomatization of Minkowski give singleton points, which are not open. [1] geometry and order, where the notion David B. Malament’s work with the path of point events is given a structure that topology tF of Hawking et al. also deserves differentiates between the notions of be- much credit. While Hawking and others fore and after. While this notion may ap- proved that tF had physically useful proper- pear simple and intuative, it is, according ties and gave results useful in the study of to Hudetz, all that is needed to show space time, Malament showed that these sets how causal set theory is an appropri- of of continuous timelike curves gives the topo- ate methodology for studying discrete logical and differential structure of any space spaces.[4] time inherited with tF. However, the subtle The past literature most closely related to difference between Malament’s and Hawking’s the author’s present work (as well as the most study of spacetimes is that Malament relaxes vital) is the work of R. Penrose and E. Kron- some of the structure that Hawking deemed heimer on differential topology. The first of necessary. Malament hypothesized that requir- such works Techniques of Differential Topology ing a set to be strongly causal was unnecessary. in Relativity is the source of one of the main [5] This result is particularly important in the theorems of this paper. While this particular context of this paper, as seeing results while work of Penrose resides closer to the topic of removing structure may have interesting re- physics, it did provide some insight into what sults. However, Malament’s as well as this pa- a possible anzats for generalizing spacetimes per shows that some structure is vital, namely may look like. The most important result (at the notion of future and past distinguishing is least for the purpose of this paper) comes from the minimal level of structure that spacetimes Penrose’s attempt at studying "non-physical" (or general spaces) need to make the topology models of space time in order to understand tF physically interesting. the global structure of the latter. In this work, The reader may question if the theories de- Penrose presents a specialized form of one of veloped by the authors mentioned above can the main theorems presented in this work. The be applied to different kinds of spaces (man- two most notable results include: ifolds). Laurents Hudetz’s Linear Structures, Global Structure: Given a spacetime M Causal Sets and Topology proposes a method of • (note not a general space), three global taking Tim Maudlin’s study of linear structures properties of the space time are equiva- and applying it to the theory of causal sets for lent: [6] the purpose of studying discrete spaces. While – M is strongly causal this work will focus only on the study of con- – The Alexandrov topology agrees tinuous spaces, it is worth noting that Hudetz with the [inherit] topology approach to discrete spaces does produce phys- – The Alexandrov topology is Haus- ically interesting results involving structures dorff used by Hawking and others in the continuous The set of all point in M obeying strong analog. It is worth noting that while Hudetz • causality is an open set in the inherit 2 • May 2016 • topology. Definition II.2. Let X be a non-empty set. We This work by Penrose is extremely con- define the trivial topology on this set as the straining for our purposes as the setting of collection consisting of only X and Æ. most if not all the theorems shown is a space Definition II.3. Let X be a non empty set. We time. This extra structure, allows one to use the define the discrete topology on this set as the concepts of geodesics, length of paths, etc. It collection of all subsets of X, i.e. every single is his 1966 work alongside E. Kronheimer that subset of X is open under this topology. serves as the main foundation of this work. In On The Structure of Causal Spaces, Penrose and A set X can have different topologies de- Kronheimer describe a lot of the concepts that fined on it. However, not all topologies on a set are taken for granted in most studies of space X need be comparable to one another. By com- time. The work begins with a simple defini- parable we mean that given two topologies tions for causal, chronological, and horisomes on a set X say tA and tB then either tA ⊆ tB ordering and imposes this structure on a gen- or tB ⊆ tA. If neither is the case then we eral topological space. Most notably, the idea say the two topologies on the set X are non- of a chronological order being the only order comparable. If, on the other hand, two topolo- needed to define the other two (as a matter of gies on a set X are comparable, then we have fact, any one order suffices to define the other technical terms to describe their relationship: two) . This becomes highly important for later Definition II.4. We say that a topology on a works, as Penrose’s global strucutre results space X, t is finer than t if t ⊆ t . Simi- shown above requires more than the chrono- A B B A larly we say that a topology t is coarser than logical order for validity.

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