M\"obius structures and timed causal spaces on the circle

TL;DR

This paper explores a duality between hyperbolic spaces and spacetimes via Möbius structures on the circle, establishing a connection in 2D cases that may extend to more general settings.

Contribution

It demonstrates how Möbius structures on the circle can generate 2D spacetimes, providing a step toward understanding the duality between hyperbolic spaces and spacetimes.

Findings

Möbius structures on the circle lead to 2D spacetime models.

A class of Möbius structures includes those from hyperbolic spaces.

The work suggests a duality framework in 2D between hyperbolic spaces and spacetimes.

Abstract

We discuss a conjectural duality between hyperbolic spaces on one hand and spacetimes on the other hand, living on the opposite sides of the common absolute. This duality goes via M\"obius structures on the absolute, and it is easily recognized in the classical case of symmetric rank one spaces. In a general case, no trace of such duality is known. As a first step in this direction, we show how M\"obius structures on the circle from a large class including those which stem from hyperbolic spaces give rise to 2-dimensional spacetimes, which are axiomatic versions of de Sitter 2-space, and vice versa. The paper has two Appendices, one of which is written by V.Schroeder.

Equations

$$g(v)=x^2+y^2-z^2$$

$$|xy|\cdot |x^{\prime }z|=|xz|\cdot |x^{\prime }y|$$

$$|s{s}^{\prime }|=|\ln ⟨x,y,y^{\prime },x^{\prime }⟩|,$$

$$t=|\ln ⟨x,y,y^{\prime },x^{\prime }⟩|$$

$$⟨\alpha ,\beta ⟩=\frac{1}{2}\mathrm{Tr}(\alpha \beta )$$

$${\sigma }_1=\left[\begin{array}{cc}1& 0\\ 0& -1\end{array}\right],{\sigma }_2=\left[\begin{array}{cc}0& 1\\ 1& 0\end{array}\right],{\sigma }_3=\left[\begin{array}{cc}0& 1\\ -1& 0\end{array}\right],$$

$$x\mapsto \frac{ax+b}{cx+d}\text{for}\left[\begin{array}{cc}a& b\\ c& d\end{array}\right]\in G$$

$$K_3=\{g_3(t)=\exp (t{\sigma }_3)=\left[\begin{array}{cc}\cos t& \sin t\\ -\sin t& \cos t\end{array}\right]:\text{t∈R}\}$$

$$g_1(t)=\exp (t{\sigma }_1)=\left[\begin{array}{cc}{e}^t& 0\\ 0& e^{-t}.\end{array}\right]$$

$$\exp (t{\sigma }_2)=\left[\begin{array}{cc}\cosh t& \sinh t\\ \sinh t& \cosh t\end{array}\right],$$

$$\exp (t{\sigma }_2)a=\left\{\frac{\cosh t(-1)+\sinh t}{\sinh t(-1)+\cosh t},\frac{\cosh t+\sinh t}{\sinh t+\cosh t}\right\}=\{-1,1\}=a.$$

$$C_e^{\pm }=\{a\in \mathrm{aY}:\text{a⊂e±}\}.$$

Full text

Keywords: Möbius structures, cross-ratio, harmonic 4-tuples, hyperbolic spaces, spacetimes, de Sitter space

Mathematics Subject Classification: 51B10, 53C50

1 Introduction

It is classical that the quadratic form

[TABLE]

on $\mathbb{R}^{3}$, $v=(x,y,z)\in\mathbb{R}^{3}$, induces on any connected component of the set $g(v)=-1$ a Riemannian metric of the hyperbolic plane $\operatorname{H}^{2}$, while on the set $g(v)=1$ a Lorentz metric of de Sitter 2-space $\operatorname{dS}^{2}$. The (set of lines in the) cone $g(v)=0$ serves as the common absolute $S^{1}$ of the both $\operatorname{H}^{2}$ and $\operatorname{dS}^{2}$. A similar picture takes place in any dimension and even for all rank one symmetric spaces of noncompact type.

In other words, we observe a life on the other side of the absolute $S^{1}$ of $\operatorname{H}^{2}$ that is de Sitter space $\operatorname{dS}^{2}$. For mathematical aspects of the duality between hyperbolic spaces $\operatorname{H}^{n+1}$ and de Sitter spacetimes $\operatorname{dS}^{n+1}$ see e.g. [Ge], [Yu]. Interplay between geometry of hyperbolic surfaces and Lorentz (2+1)-spaces is exploited in the famous paper [Mes], see also [A-S]. Duality for quadratic forms of arbitrary signature is discussed in [Ro]. For physical aspects of de Sitter spaces see e.g. [SSV] and references therein.

In sect. 2, we describe this duality in intrinsic terms. The basic feature is the canonical Möbius structure $M_{0}$ on the absolute $S^{1}$, which governs its both sides $\operatorname{H}^{2}$ and $\operatorname{dS}^{2}$. In particular, the isometry groups of $\operatorname{H}^{2}$ and $\operatorname{dS}^{2}$ coincide with the group of Möbius automorphisms of $M_{0}$. We show how to recover the hyperbolic plane $\operatorname{H}^{2}$ and de Sitter 2-space $\operatorname{dS}^{2}$ purely out of $M_{0}$.

Moreover, we explain a mechanism of the passage from $\operatorname{H}^{2}$ to $\operatorname{dS}^{2}$ and back. Shortly, $\operatorname{H}^{2}$ is the homogeneous space of the $M_{0}$-automorphism group $\operatorname{PSL}_{2}(\mathbb{R})$ over a compact elliptic subgroup isomorphic to $S^{1}$, while $\operatorname{dS}^{2}$ is the homogeneous space of $\operatorname{PSL}_{2}(\mathbb{R})$ over a (closed) hyperbolic subgroup isomorphic to $\mathbb{R}$.

This rises a bold question: Is there any life (a spacetime) on the other side of the absolute, i.e., the boundary at infinity, of any Gromov hyperbolic space with the same symmetry group? The main result of the paper is the answer “yes” for a large class of hyperbolic spaces with the absolute $S^{1}$, see Theorem 1.1.

A Möbius structure on a set $X$ is a class of semi-metrics having one and the same cross-ratio on any given ordered 4-tuple of distinct points in $X$, see sect. LABEL:sect:moeb. Every hyperbolic space $Y$ induces on its boundary at infinity $X=\partial_{\infty}Y$ a Möbius structure which encodes most essential properties of $Y$ and in a number of cases allows to recover $Y$ completely, e.g., in the case $Y$ is a rank one symmetric space of noncompact type, see [BS2], [BS3]. In sect. LABEL:subsect:boundary_continuous, we explain this for the class of boundary continuous hyperbolic spaces. In Appendix 1, sect. LABEL:sect:appendix_1, we show that every proper Gromov hyperbolic $\operatorname{CAT}(0)$ space is boundary continuous.

We axiomatically describe a class $\mathcal{M}$ of monotone Möbius structures on the circle $S^{1}$, see sect. LABEL:sect:monotone. The class $\mathcal{M}$ includes every Möbius structure $M$ which is induced on $S^{1}$ by a hyperbolic $\operatorname{CAT}(0)$ surface $Y$ without singular points, see Theorem LABEL:thm:without_singular. In particular, the isometry group of $Y$ is included in the group of Möbius automorphisms of $M$. Furthermore, the canonical Möbius structure $M_{0}$ is the most symmetric representative from $\mathcal{M}$.

On the other hand, the set $\operatorname{aY}$ of unordered pairs of distinct points on the circle $X=S^{1}$ has a natural causal structure, which is independent of anything else, see sect. 2. Points of $\operatorname{aY}$ are called events. There is a large class $\mathcal{T}$ of 2-dimensional spacetimes compatible with that causal structure, and we characterize it axiomatically in sect. 3. Any spacetime $T\in\mathcal{T}$ is a triple $T=(\operatorname{aY},\mathcal{H},t)$, where $\mathcal{H}$ is a class of timelike curves in $\operatorname{aY}$ called timelike lines, which are actually timelike geodesics, and $t$ is the time between events in the causal relation. The spacetime $T\in\mathcal{T}$ is called timed causal space. We prove

Theorem 1.1**.**

There are natural mutually inverse maps $\widehat{T}:\mathcal{M}\to\mathcal{T}$ and $\widehat{M}:\mathcal{T}\to\mathcal{M}$ such that the groups of automorphisms of any $M\in\mathcal{M}$ and of the respective $T=\widehat{T}(M)\in\mathcal{T}$ are canonically isomorphic.

It follows from constructions of sect. 2 that the canonical Möbius structure $M_{0}$ on $S^{1}$ determines de Sitter space $\operatorname{dS}^{2}$, that is, $\widehat{T}(M_{0})=\operatorname{dS}^{2}$ and $\widehat{M}(\operatorname{dS}^{2})=M_{0}$. In other words, Theorem 1.1 says that a monotone Möbius structure on $S^{1}$ on one hand, and the respective timed causal space with the absolute $S^{1}$ on the other hand, are different sides of one and the same phenomenon also in a general case.

The fundamental feature of spacetimes is the time inequality. In section LABEL:sect:time_inequality, we discuss a hierarchy of time conditions, in particular, we introduce the weak time inequality, and show that every timed causal space $T\in\mathcal{T}$ satisfies the weak time inequality, see Theorem LABEL:thm:wti.

In sect. LABEL:subsect:monotone_vp, we introduce Increment Axiom (I) which implies the time inequality, and show that the subset $\mathcal{I}\subset\mathcal{M}$ of Möbius structures satisfying (I) contains the canonical structure $M_{0}$, $M_{0}\in\mathcal{I}$, (Proposition LABEL:pro:canonical_vp) with a neighborhood of $M_{0}$ in the fine topology (Proposition LABEL:pro:pertubed_canonical_vp).

In sect. LABEL:subsect:convex_moeb, we introduce Convexity Axiom (C) for monotone Möbius structures $M\in\mathcal{M}$, which implies convexity of a functional $F_{ab}$ playing an important role in the hierarchy of time conditions and show that the subset $\mathcal{C}\subset\mathcal{M}$ of convex Möbius structures contains $M_{0}$ (Proposition LABEL:pro:canonical_convex).

The spacetimes of the class $\mathcal{T}$ are related to de Sitter 2-space $\operatorname{dS}^{2}$ in a sense at least as hyperbolic $\operatorname{CAT}(0)$ surfaces without singular points with the absolute $S^{1}$ are related to the hyperbolic plane $\operatorname{H}^{2}$. If one would extend results of this paper to more general hyperbolic spaces even with 1-dimension boundary at infinity, then this potentially could produce new interesting classes of spacetimes e.g. having a branching time (timelike lines).

Acknowledgments. The author is very much thankful to Prof.Dr. Viktor Schroeder for attention to this paper and valuable remarks. Especially, for pointing me out that Axiom (t6) from Sect. 3.3 follows from the other axioms of timed causal spaces. This is explained in details in Appendix 2 written by Viktor Schroeder.

Contents

1. 1 Introduction

2. 2 On the other side of the absolute

3. 2.1 Recovering the hyperbolic plane $\operatorname{H}^{2}$

4. 2.2 Recovering de Sitter space $\operatorname{dS}^{2}$

5. 2.3 Automorphisms of $M_{0}$

6. 3 Timed causal spaces on the circle

7. 3.1 The canonical causality structure

8. 3.2 Timelike lines and a causal space

9. 3.3 Timed causal space

2 On the other side of the absolute

This section serves as a motivation, contains no new result, and its constructions are widely known. Here, we show how the both sides $\operatorname{H}^{2}$ and $\operatorname{dS}^{2}$ of the common absolute $S^{1}$ can be recovered out of the canonical Möbius structure $M_{0}$ on $S^{1}$. It is common to define $\operatorname{H}^{2}$ and $\operatorname{dS}^{2}$ in the quotient of $\mathbb{R}^{3}$ by the antipodal map $x\mapsto-x$. This does not affect $\operatorname{H}^{2}$, while $\operatorname{dS}^{2}$ becomes a nontrivial line bundle over $S^{1}$, that is, the open Möbius band.

2.1 Recovering the hyperbolic plane $\operatorname{H}^{2}$

The canonical Möbius structure $M_{0}$ on the circle $S^{1}$ is determined by the condition that any its representative with infinitely remote point is a standard metric (up to a positive factor) on $\widehat{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ extended in the sense that the distance between any $x\in\mathbb{R}$ and $\infty$ is infinite.

To recover $\operatorname{H}^{2}$ from $M_{0}$, we consider the space $Y$ of all Möbius involutions $s:S^{1}\to S^{1}$ with respect to $M_{0}$ without fixed points. The space $Y$ serves as the underlying space for $\operatorname{H}^{2}$, and what remains to do is to introduce a respective metric on $Y$.

A line in $Y$ is determined by a pair $x$, $x^{\prime}\in S^{1}$ of distinct points and consists of all involutions $s\in Y$ which permute $x$, $x^{\prime}$, $sx=x^{\prime}$. Given two distinct points $s$, $s^{\prime}\in Y$, the compositions $s^{\prime}s$, $ss^{\prime}:S^{1}\to S^{1}$ have one and the same fixed point set consisting of two distinct points $x$, $x^{\prime}\in S^{1}$. Thus there is a uniquely determined line in $Y$ through $s$, $s^{\prime}$.

We say that an ordered 4-tuple $q=(x,x^{\prime},y,z)\in(S^{1})^{4}$ of pairwise distinct points is harmonic if

[TABLE]

for some and hence any metric on $S^{1}$ from $M_{0}$. For the canonical Möbius structure $M_{0}$, harmonicity of $(x,x^{\prime},y,z)$ is equivalent to that the geodesic lines $xx^{\prime}$, $yz\subset\operatorname{H}^{2}$ are mutually orthogonal.

A sphere $S$ between $x$, $x^{\prime}\in S^{1}$ is of a pair $(y,z)\subset S^{1}$ such that the 4-tuple $(x,x^{\prime},y,z)$ is harmonic. We take spheres $S$, $S^{\prime}\subset S^{1}$ between $x$, $x^{\prime}$ such that $S$ is invariant under $s$, $s(S)=S$, and $S^{\prime}$ is invariant under $s^{\prime}$, $s^{\prime}(S^{\prime})=S^{\prime}$. The spheres $S$, $S^{\prime}$ with these properties exist and are uniquely determined. Now, we take $y\in S$, $y^{\prime}\in S^{\prime}$ and put

[TABLE]

where $\langle x,y,y^{\prime},x^{\prime}\rangle=\frac{|xy^{\prime}|\cdot|yx^{\prime}|}{|xy|\cdot|y^{\prime}x^{\prime}|}$ is the cross-ratio of the 4-tuple $(x,y,y^{\prime},x^{\prime})$. This is well defined and independent of the choice $y\in S$, $y^{\prime}\in S^{\prime}$. It is easy to show that $|ss^{\prime}|$ is the distance in the geometry of $\operatorname{H}^{2}$, see sect. 2.3.

Remark 2.1*.*

This construction is easily extended to any rank one symmetric space of noncompact type, see [BS2].

2.2 Recovering de Sitter space $\operatorname{dS}^{2}$

Let $\operatorname{aY}$ be the space of unordered pairs $(x,y)\sim(y,x)$ of distinct points on $S^{1}$ with the induced from $S^{1}$ topology, that is, $\operatorname{aY}=S^{1}\times S^{1}\setminus\Delta/\sim$, where \Delta=\{(x,x):\,\text{x\in S^{1}}\} is the diagonal. Then $\operatorname{aY}$ is a nontrivial $\mathbb{R}$-bundle over $\mathbb{R}\!\operatorname{P}^{1}\approx S^{1}$, i.e., $\operatorname{aY}$ is the open Möbius band. In this case, $S^{1}$ is the boundary of $\operatorname{aY}$ at infinity, $\partial_{\infty}\operatorname{aY}=S^{1}$. Points of $\operatorname{aY}$ are called events.

We say that two events $e$, $e^{\prime}\in\operatorname{aY}$ are in the causal relation if and only if $e$, $e^{\prime}$ do not separate each other as pairs of points in $S^{1}$. This defines the canonical causality structure on $\operatorname{aY}$.

A light line in $\operatorname{aY}$ is determined by any $x\in S^{1}$ and consists of all events $a=(x,x^{\prime})\in\operatorname{aY}$, $x^{\prime}\in S^{1}\setminus x$. For this light line $p_{x}$, $x$ is the unique point at infinity. Two distinct light lines $p_{x}$, $p_{y}$ have a unique common event $(x,y)\in\operatorname{aY}$, and any two events on a light line are in the causal relation.

The canonical causality structure as well as light lines are inherent in $\operatorname{aY}$, and they do not depend of anything else.

Remark 2.2*.*

In higher dimensional case, a causality structure can be defined similarly, but then it depends on the Möbius structure because events are codimension one spheres in $S^{n}$.

A timelike line in $\operatorname{aY}$ is determined by any event $e\in\operatorname{aY}$ and consists of all $a\in\operatorname{aY}$ such that the 4-tuple $(e,a)$ is harmonic. For the timelike line $h_{e}\subset\operatorname{aY}$ determined by $e=(x,x^{\prime})$, the points $x$, $x^{\prime}\in S^{1}$ are the ends of $h_{e}$ at infinity. It follows from definitions that $a\in h_{e}$ if and only if $e\in h_{a}$.

Any two events on a timelike line are in the causal relation. Conversely, for any two events $a$, $a^{\prime}\in\operatorname{aY}$ which are in the causal relation and not on a light line there is a unique timelike line (the common perpendicular) $h_{e}$ with $a$, $a^{\prime}\in h_{e}$ (this amounts to existence and uniqueness of a common perpendicular to divergent geodesics in $\operatorname{H}^{2}$. For a (de Sitter) proof see Corollary LABEL:cor:common_perpendicular_prescribed and Lemma LABEL:lem:unique_common_perpendicular). Let $a=(y,z)$, $a^{\prime}=(y^{\prime},z^{\prime})$, $e=(x,x^{\prime})$ in this case. Then the time $t=t(a,a^{\prime})$ between the events $a$, $a^{\prime}$ is defined by formula (2)

[TABLE]

(note that $a$, $a^{\prime}$ are sphere between $x$, $x^{\prime}$).

It follows that two timelike lines $h_{e}$, $h_{e^{\prime}}$ intersect each other if and only if the events $e$, $e^{\prime}\in\operatorname{aY}$ are in the causal relation and not on a light line. In this case, the intersection $h_{e}\cap h_{e^{\prime}}$ is a unique event.

An elliptic line in $\operatorname{aY}$ is determined by any Möbius involution without fixed points $s\in Y$ and consists of all events $a\in\operatorname{aY}$ such that $sa=a$. No two distinct events on an elliptic line are in the causal relation.

Remark 2.3*.*

The last definition make sense only for the canonical Möbius structure $M_{0}$ because in a general case a Möbius structure may not admit any Möbius involution without fixed points.

2.3 Automorphisms of $M_{0}$

To introduce a metric structure on $\operatorname{aY}$ we consider the Lie algebra $\mathfrak{g}$ of the Lie group $G=\operatorname{SL}_{2}(\mathbb{R})$. Given $\alpha$, $\beta\in\mathfrak{g}$ we have the Killing form

[TABLE]

as a scalar product. Note that the matrices $\sigma_{1},\sigma_{2},\sigma_{3}\in\mathfrak{g}$,

[TABLE]

are mutually orthogonal and $\|\sigma_{1}\|^{2}=\langle\sigma_{1},\sigma_{1}\rangle=1=\|\sigma_{2}\|^{2}$, $\|\sigma_{3}\|^{2}=-1$.

The group $G$ acts on $\widehat{\mathbb{R}}=\mathbb{R}\cup\{\infty\}$ by linear-fractional transformations

[TABLE]

which are Möbius with respect to the canonical Möbius structure $M_{0}$. The action is not effective with the kernel $\mathbb{Z}_{2}=\{\pm\operatorname{id}\}$, $G/\mathbb{Z}_{2}=\operatorname{PSL}_{2}(\mathbb{R})$. The group $\operatorname{PSL}_{2}(\mathbb{R})$ with the left invariant Lorentz metric (3) is anti de Sitter 3-space $\operatorname{AdS}^{3}$.

We denote by K_{i}=\{\exp(t\sigma_{i}):\,\text{t\in\mathbb{R}}\}, $i=1,2,3$, a 1-parametric subgroup in $G$, and by $\widehat{K}_{i}$ its image in $\operatorname{PSL}_{2}(\mathbb{R})$. Note that $\widehat{K}_{i}=K_{i}$ for $i=1,2$ and that $\widehat{K}_{3}=K_{3}/\mathbb{Z}_{2}$.

The space $Y$ of Möbius involutions $s:S^{1}\to S^{1}$ without fixed points can be identified with the homogeneous space $G/K_{3}=\operatorname{PSL}_{2}(\mathbb{R})/\widehat{K}_{3}$ because the group

[TABLE]

stabilizes $s=g_{3}(\frac{\pi}{2})=\begin{bmatrix}0&1\\ -1&0\end{bmatrix}$ which acts on $\widehat{\mathbb{R}}$ as the Möbius involution $s(x)=-\frac{1}{x}$ without fixed points. The space $G/K_{3}$ carries a left-invariant Riemannian metric $h_{3}$ originated from the subspace $L_{3}\subset\mathfrak{g}$ spanned by $\sigma_{1}$, $\sigma_{2}$, and $(G/K_{3},h_{3})$ is isometric to $\operatorname{H}^{2}$. To see that we compute the respective Riemannian distance between two involutions $s_{1}$, $s_{2}\in Y$. By conjugation we can assume that $s_{1}=s$, $s_{2}=s^{\prime}$, where $s^{\prime}=g_{1}(t)\cdot s\cdot g_{1}^{-1}(t)$ for some $t\in\mathbb{R}$,

[TABLE]

Then $s^{\prime}=\begin{bmatrix}0&e^{2t}\\ -e^{-2t}&0\end{bmatrix}$ and $s^{\prime}(x)=-\frac{e^{4t}}{x}$. The curve $t\mapsto g_{1}(t)$ is a unit speed geodesic in $G$. While projected to $\operatorname{PSL}_{2}(\mathbb{R})$ the speed is doubled because of linear-fractional action of $\operatorname{PSL}_{2}(\mathbb{R})$, so we have $|ss^{\prime}|=2t$. In the upper half-plane model of $\operatorname{H}^{2}$ the involution $s$ fixes $i=(0,1)$ with Euclidean distance $|0i|_{e}=1$ and $s^{\prime}$ fixes $ie^{2t}=(0,e^{2t})$ with Euclidean distance $|0ie^{2t}|_{e}=e^{2t}$, hence $|ss^{\prime}|=2t$ equals the $\operatorname{H}^{2}$-distance $|(0,1)(0,e^{2t})|=\ln\frac{|0ie^{2t}|_{e}}{|0i|_{e}}=2t$.

The action of $\operatorname{PSL}_{2}(\mathbb{R})$ on $\widehat{\mathbb{R}}\approx S^{1}$ induces the standard action of $\operatorname{PSL}_{2}(\mathbb{R})$ on $\operatorname{aY}$. Note that $a=\{-1,1\}\in\operatorname{aY}$ is a fixed point for $K_{2}$ because

[TABLE]

and

[TABLE]

It follows that $\operatorname{aY}$ can be identified with the homogeneous space $\operatorname{PSL}_{2}(\mathbb{R})/K_{2}$, or similarly with $\operatorname{PSL}_{2}(\mathbb{R})/K_{1}$. The space $\operatorname{aY}=\operatorname{PSL}_{2}(\mathbb{R})/K_{2}$ carries a left-invariant Lorentz metric $h_{2}$ originated from the subspace $L_{2}\subset\mathfrak{g}$ spanned by $\sigma_{1}$, $\sigma_{3}$, and $\operatorname{dS}^{2}=(\operatorname{aY},h_{2})$.

In the rest of the paper, we explain how a Möbius structure $M$ from a large class of structures on the circle gives rise to a spacetime, and vice versa, without any assumption on symmetries of $M$.

3 Timed causal spaces on the circle

In this section we list axioms for timed causal spaces on the circle.

3.1 The canonical causality structure

Recall that on the space $\operatorname{aY}$ of unordered pairs of distinct point in $X=S^{1}$, which is homeomorphic to the open Möbius band, we have the canonical causal structure. That is, events $e$, $e^{\prime}\in\operatorname{aY}$ are in the causal relation if and only if they do not separate each other as pairs of points in $X$. In the opposite case, we also say that events $e$, $e^{\prime}\in\operatorname{aY}$ separate each other. In the case $e$, $e^{\prime}\in\operatorname{aY}$ are in the causal relation and not on a light line, we say that the events $e$, $e^{\prime}$ are in the strong causal relation.

The canonical causal structure and light lines inherent to $\operatorname{aY}$, see sect. 2.2.

For a fixed event $e\in\operatorname{aY}$ the set $C_{e}$ of all $e^{\prime}\in\operatorname{aY}$ in the causal relation with $e$ is called the causal cone. The pair $e\subset X$ decomposes $X$ into two closed arcs, which we denote by $e^{+}$, $e^{-}$, with $e^{+}\cap e^{-}=e$. Every $a\in\operatorname{aY}$ with $a\subset e^{\pm}$ is in the causal relation with $e$. We let

[TABLE]

Therefore, a choice of $e^{+}$, $e^{-}$ induces the decomposition $C_{e}=C_{e}^{+}\cup C_{e}^{-}$ of the causal cone $C_{e}$ into the future cone $C_{e}^{+}$ and the past cone $C_{e}^{-}$ with $C_{e}^{+}\cap C_{e}^{-}=e$, and moreover introduces a partial order on $\operatorname{aY}$ in the following way. Every $a\in C_{e}^{\pm}$ decomposes $X$ into two closed arcs, and if $a\neq e$, we canonically define $a_{e}^{\pm}$ as one of them that does not contain $e$, otherwise $a_{e}^{\pm}=e^{\pm}$. Now we define: $a\leq_{e}a^{\prime}$ if and only if one of the following holds

• •

$a\in C_{e}^{-}$, $a^{\prime}\in C_{e}^{+}$

• •

$a^{\prime}\subset a_{e}^{+}$ if $a,a^{\prime}\in C_{e}^{+}$ and $a\subset(a^{\prime})_{e}^{-}$ if $a,a^{\prime}\in C_{e}^{-}$.

As usual, we say that $a<_{e}a^{\prime}$, if $a\leq_{e}a^{\prime}$ and $a\neq a^{\prime}$.

Note that there is no global partial order on $\operatorname{aY}$ compatible with the canonical causal structure, and the order above only appears if an event $e\in\operatorname{aY}$, future $e^{+}$ and past $e^{-}$ arcs are chosen.

3.2 Timelike lines and a causal space

The notion of a timelike line is not inherent in $\operatorname{aY}$, and we define this notion axiomatically.

Axioms for timelike lines

• (h1)

every event $e\in\operatorname{aY}$ uniquely determines a timelike line $h_{e}\subset\operatorname{aY}$, and every timelike line in $\operatorname{aY}$ is of form $h_{e}$ for some $e\in\operatorname{aY}$;

• (h2)

any event $a\in h_{e}$ separates $e$;

• (h3)

any two events on a timelike line are in the causal relation;

• (h4)

for any point $x\in X\setminus e$ there is a unique event $x_{e}=(x,y)\in h_{e}$;

• (h5)

if an event $a\in\operatorname{aY}$ is on a timelike line $h_{e}$, then $e\in h_{a}$;

• (h6)

for any two distinct events $a$, $a^{\prime}\in\operatorname{aY}$ there is at most one timelike line $h_{e}$ with $a$, $a^{\prime}\in h_{e}$.

The space $\operatorname{aY}$ with a fixed collection $\mathcal{H}$ of subsets satisfying the axioms of timelike lines is called a causal space. We use notation $(\operatorname{aY},\mathcal{H})$ for a causal space. In view of Axiom (h1), we say that an event $e\in\operatorname{aY}$ and the timelike line $h_{e}\subset\operatorname{aY}$ are dual to each other.

It follows from (h4) that for every event $e=(z,u)\in\operatorname{aY}$ we have a well defined map $\rho_{e}:X\to X$ given by $\rho_{e}(z)=z$, $\rho_{e}(u)=u$, and $(x,\rho_{e}(x))=x_{e}$ for every $x\in X\setminus e$. The map $\rho_{e}$ is called the reflection with respect to $e$.

Lemma 3.1**.**

For every $e=(z,u)\in\operatorname{aY}$, the map $\rho_{e}:X\to X$ is an involutive homeomorphism.

Proof.

By definition, $\rho_{e}^{2}(z)=z$, $\rho_{e}^{2}(u)=u$. Let $y=\rho_{e}(x)$ for $x\in X\setminus e$. Then events $(x,y),(\rho_{e}(y),y)\in h_{e}$, hence $\rho_{e}(y)=x$ by (h4). Thus $\rho_{e}^{-1}=\rho_{e}$, and $\rho_{e}$ is a bijection.

The event $e$ decomposes $X$ into two closed arcs $e^{+}$, $e^{-}$ with $X=e^{+}\cup e^{-}$, $e^{+}\cap e^{-}=e$. An orientation of $X$ determines linear orders on $e^{+}$, $e^{-}$. It follows from (h2) and (h3) that $\rho_{e}:e^{+}\to e^{-}$ reverses the orders. Hence, $\rho_{e}$ is continuous and, therefore, a homeomorphism. ∎

Proposition 3.2**.**

Let $(\operatorname{aY},\mathcal{H})$ be a causal space. Then

• (a)

for any $e\in\operatorname{aY}$ the line $h_{e}$ is homeomorphic (in the induced from $\operatorname{aY}$ topology) to $\mathbb{R}$, and the boundary of the closure $\overline{h}_{e}\subset\operatorname{aY}\cap\partial_{\infty}\operatorname{aY}$ is $e$, $\partial\overline{h}_{e}=e$;

• (b)

for any two distinct events $a$, $a^{\prime}\in\operatorname{aY}$ there is a timelike line $h$ including $a$, $a^{\prime}$ if and only if $a$ and $a^{\prime}$ are in the strong causal relation. In this case, $h$ is unique with this property;

• (c)

any two distinct timelike lines $h_{e}$, $h_{e^{\prime}}\in\mathcal{H}$ have a common event $a$ if and only if $e$, $e^{\prime}$ are in the strong causal relation. In this case, $a$ is unique;

• (d)

a light line $p_{x}\subset\operatorname{aY}$ intersects a timelike line $h_{e}\subset\operatorname{aY}$ if and only if $x\not\in e$. In this case, the common event $a\in p_{x}\cap h_{e}$ is unique.

Proof.

(a) Let $\rho_{e}:X\to X$ be the reflection with respect to $e$, $e^{+}\subset X$ one of the two closed arcs, in which $e$ decomposes $X$. Then the map $\operatorname{int}e^{+}\to h_{e}$, $x\mapsto(x,\rho_{e}(x))$ is an order preserving bijection. Extended to $e^{+}$, it gives an order preserving bijection to $\overline{h}_{e}=h_{e}\cup e$. Thus $h_{e}$ is homeomorphic to $\operatorname{int}e^{+}\approx\mathbb{R}$, and $\partial\overline{h}_{e}=e$.

(b) Any distinct $a$, $a^{\prime}\in\operatorname{aY}$ on a timelike line $h$ are in the causal relation by (h3), and by (h4) they are not on a light line. Hence, $a$, $a^{\prime}$ are in the strong causal relation. Conversely, assume that events $a$, $a^{\prime}\in\operatorname{aY}$ are in the strong causal relation. Let $\rho=\rho_{a}\circ\rho_{a^{\prime}}$ be the composition of respective reflections, $a^{+}\subset X$ the closed arc determined by $a$ that does not contain $a^{\prime}$. Then $\rho(a^{+})\subset\operatorname{int}a^{+}$, and thus there is a fixed point $x\in\operatorname{int}a^{+}$ of $\rho$, $\rho(x)=x$. It follows that the both reflections $\rho_{a}$, $\rho_{a^{\prime}}$ preserve the event $e=(x,y)$, where $y=\rho_{a^{\prime}}(x)$. Hence, $e\in h_{a}\cap h_{a^{\prime}}$, and by (h5), $a,a^{\prime}\in h_{e}$. By (h6), $h_{e}$ is unique with this property.

(c) By duality (h5), this is a reformulation of (b).

(d) This immediately follows from (h2) and (h4). ∎

Remark 3.3*.*

Axiom (h6) is not used in Lemma 3.1, and it is only used in Proposition 3.2 to prove the uniqueness in (b) and (c). Thus all the conclusions of Proposition 3.2 except for the uniqueness in (b) and (c) hold true without Axiom (h6). We use this remark in sect. LABEL:subsect:timelike_lines.

3.3 Timed causal space

The notion of a time is also defined axiomatically.

The time axioms

• (t1)

A time $t(e,e^{\prime})\geq 0$ between two events $e$, $e^{\prime}\in\operatorname{aY}$ is determined if and only if $e$, $e^{\prime}$ are in the causal relation;

• (t2)

$t(e,e^{\prime})=0$ if and only if $e$, $e^{\prime}$ are events on a light line;

• (t3)

$t(e,e^{\prime})=t(e^{\prime},e)$ whenever it is defined;

• (t4)

timelike lines are $t$-geodesics:

(a) if $e,e^{\prime},e^{\prime\prime}\in h_{a}$ are events on a timelike line such that $e\leq e^{\prime}\leq e^{\prime\prime}$, then $t(e,e^{\prime})+t(e^{\prime},e^{\prime\prime})=t(e,e^{\prime\prime})$;

(b) for every $e\in h_{a}$ and every $s>0$ there are $e_{\pm}\in h_{a}\cap C_{e}^{\pm}$ with $t(e,e_{\pm})=s$;

• (t5)

for any events $e=(x,y)$, $d=(z,u)$ in the strong causal relation, we have $t(z_{e},u_{e})=t(x_{d},y_{d})$;

• (t6)

for any $e=(x,y),d=(z,u)\in h_{e}$ the 4-tuple $(d,e)$ is harmonic in the sense that $t(y_{a},u_{a})=t(y_{b},z_{b})$, where $a=(x,z)$, $b=(x,u)$.

A timed causal space is defined as $T=(\operatorname{aY},\mathcal{H},t)$, where $t$ is a time on the causal space $(\operatorname{aY},\mathcal{H})$. This is a version of Busemann (locally) timelike spaces, see [Bus], and also an axiomatic version of the de Sitter space $\operatorname{dS}^{2}$. Since $\operatorname{dS}^{2}$ is recovered from the canonical Möbius structure $M_{0}$ on the circle, see sect. 2.2, it follows from results of sect. LABEL:sect:monotone (see Proposition LABEL:pro:safisfies_hypline_axiom, Lemma LABEL:lem:unique_common_perpendicular and Proposition LABEL:pro:satisfies_time_axioms), that $\operatorname{dS}^{2}$ is a timed causal space.

We denote by $\mathcal{T}$ the set of all timed causal spaces $(\operatorname{aY},\mathcal{H},t)$, where the collection $\mathcal{H}$ of timelike line satisfies Axioms (h1)–(h6), and the time $t$ satisfies Axioms (t1)–(t6). A $T$-automorphism, $T=(\operatorname{aY},\mathcal{H},t)\in\mathcal{T}$, is a bijection $g:\operatorname{aY}\to\operatorname{aY}$ that preserves the timelike lines $\mathcal{H}$ and the time $t$, $t(g(e),g(e^{\prime}))=t(e,e^{\prime})$ whenever $t(e,e^{\prime})$ is defined (we do not require to preserve the causality structure because this is automatic).

Remark 3.4*.*

I am not satisfied with a terminology from e.g. [Bus], [PY], where the term “timelike (metric) space” is used, because a respective object is never a metric space and its basic feature is a causality relation. On the other hand to say “timelike causal space” sounds a little bit tautologically. Thus I use the term “timed causal space” instead.