A primer on the (2 + 1) Einstein universe

Thierry Barbot, Virginie Charette, Todd Drumm, William M. Goldman, and Karin Melnick ^{∗}

Abstract. The Einstein universe is the conformal compactification of Minkowski space. It also arises as the ideal boundary of anti-de Sitter space. The purpose of this article is to develop the synthetic geometry of the Einstein universe in terms of its homogeneous submanifolds and causal structure, with particular emphasis on dimension 2+1, in which there is a rich interplay with symplectic geometry.

Mathematics Subject Classification (2000). Primary 53-XX; Secondary 83-XX.

Keywords. Minkowski space, spacetime, Lorentzian manifold, conformal structure, Lie algebra, symplectic vector space

Contents

1 Introduction 3

2 Synthetic geometry of Ein^{n,1} 4 2.1 Lorentzian vector spaces . . . . . . . . . . . . . . . . . . . . . . . . 4 2.2 Minkowski space . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 2.3 Einstein space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7 2.4 2-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.5 3-dimensional case . . . . . . . . . . . . . . . . . . . . . . . . . . . 10

3 Ein^{n,1} as the conformal compactification of 𝔼^{n,1} 12 3.1 The conformal Riemannian sphere . . . . . . . . . . . . . . . . . . 12 3.2 The conformal Lorentzian quadric . . . . . . . . . . . . . . . . . . 14 3.3 Involutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

3.3.1 Inverting photons . . . . . . . . . . . . . . . . . . . . . . . . 16 3.3.2 Extending planes in 𝔼^{2,1} to Ein^{2,1} . . . . . . . . . . . . . . 16

4 Causal geometry 17 4.1 Time orientation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 17 4.2 Future and past . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19 4.3 Geometry of the universal covering . . . . . . . . . . . . . . . . . . 20 4.4 Improper points of Minkowski patches . . . . . . . . . . . . . . . . 21

∗Goldman is grateful to the Erwin Schrodinger Institute (Vienna) for hospitality during the writing of this paper, and to National Science Foundation grant DMS-0405605. Charette is grateful for support from NSERC.

5 Four-dimensional real symplectic vector spaces 22 5.1 The inner product on the second exterior power . . . . . . . . . . . 23 5.2 Lagrangian subspaces and the Einstein universe . . . . . . . . . . . 24

5.2.1 Complete flags . . . . . . . . . . . . . . . . . . . . . . . . . 24 5.2.2 Pairs of Lagrangian planes . . . . . . . . . . . . . . . . . . . 24

5.3 Symplectic planes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 5.4 Positive complex structures and the Siegel space . . . . . . . . . . 25 5.5 The contact projective structure on photons . . . . . . . . . . . . . 27

5.5.1 Construction of the contact structure . . . . . . . . . . . . 27 5.5.2 The contact structure and polarity . . . . . . . . . . . . . . 28 5.5.3 Relation with positive complex structures on ℝ^{4} . . . . . . 28

5.6 The Maslov cycle . . . . . . . . . . . . . . . . . . . . . . . . . . . . 28 5.7 Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

6 Lie theory of Pho^{2,1} and Ein^{2,1} 29

6.1 Structure theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 6.1.1 Cartan subalgebras . . . . . . . . . . . . . . . . . . . . . . . 30 6.1.2 Positive and negative roots . . . . . . . . . . . . . . . . . . 31 6.1.3 Root space decomposition . . . . . . . . . . . . . . . . . . . 31

6.2 Symplectic splittings . . . . . . . . . . . . . . . . . . . . . . . . . . 31 6.3 The Orthogonal Representation of 𝔰𝔭(4, ℝ) . . . . . . . . . . . . . . 32 6.4 Parabolic subalgebras . . . . . . . . . . . . . . . . . . . . . . . . . 33

6.4.1 The Borel subalgebra and Flag^{2,1} . . . . . . . . . . . . . . . 33 6.4.2 The parabolic subgroup corresponding to Pho^{2,1} . . . . . . 33 6.4.3 The parabolic subgroup corresponding to Ein^{2,1} . . . . . . 33

6.5 Weyl groups . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

7 Three kinds of dynamics 34 7.1 Projective singular limits . . . . . . . . . . . . . . . . . . . . . . . 35 7.2 Cartan’s decomposition G = KAK . . . . . . . . . . . . . . . . . . 35

7.2.1 Sequences in Sp(4, ℝ) . . . . . . . . . . . . . . . . . . . . . 36 7.2.2 Sequences in SO^{+}(3, 2) . . . . . . . . . . . . . . . . . . . . . 37

7.3 Maximal domains of properness . . . . . . . . . . . . . . . . . . . . 37 7.3.1 Action on Ein^{2,1} . . . . . . . . . . . . . . . . . . . . . . . . 38 7.3.2 Action on ℙ(V) . . . . . . . . . . . . . . . . . . . . . . . . . 39 7.3.3 Action on the flag manifold . . . . . . . . . . . . . . . . . . 39

8 Crooked surfaces 40 8.1 Crooked planes in Minkowski space . . . . . . . . . . . . . . . . . . 40 8.2 An example . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41 8.3 Topology of a crooked surface . . . . . . . . . . . . . . . . . . . . . 42 8.4 Automorphisms of a crooked surface . . . . . . . . . . . . . . . . . 47 9 Construction of discrete groups 48

9.1 Spine reflections . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48 9.2 Actions on photon space . . . . . . . . . . . . . . . . . . . . . . . . 50 9.3 Some questions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51

1. Introduction

We will explore the geometry of the conformal compactification of Minkowski (n + 1)–space inside of ℝ^{n,2}. We shall call this conformal compactification Ein^{n,1}, or the Einstein universe, and its universal cover will be denoted Ẽin^{n,1}. The Einstein universe is a homogeneous space G/P, where G = PO(n, 2), and P is a parabolic subgroup. When n = 3, then G is locally isomorphic to Sp(4, ℝ).

The origin of the terminology “Einstein universe” is that A. Einstein himself considered as a paradigmatic universe the product S^{3}×ℝ endowed with the Lorentz metric ds_{0}^{2} − dt^{2}, where ds_{0}^{2} is the usual constant curvature Riemannian metric on S^{3}. The conformal transformations preserve the class of lightlike geodesics and provide a more flexible geometry than that given by the metric tensor.

Our motivation is to understand conformally flat Lorentz manifolds and the Lorentzian analog of Kleinian groups. Such manifolds are locally homogeneous geometric structures modeled on Ein^{2,1}.

The Einstein universe Ein^{n,1} is the conformal compactification of Minkowski space 𝔼^{n,1} in the same sense that the n-sphere

S^{n} = 𝔼^{n} ∪ {∞}

conformally compactifies Euclidean space 𝔼^{n}; in particular, a Lorentzian analog of the following theorem holds (see [11]):

Theorem 1.0.1 (Liouville’s theorem). Suppose n ≥ 3. Then every conformal map U ⟶^{f} 𝔼^{n} defined on a nonempty connected subdomain U ⊂ 𝔼^{n} extends to a conformal automorphism f̄ of S^{n}. Furthermore f̄ lies in the group PO(n + 1, 1) generated by inversions in hyperspheres and Euclidean isometries.

Our viewpoint involves various geometric objects in Einstein space: points are organized into 1-dimensional submanifolds which we call photons, as they are lightlike geodesics. Photons in turn form various subvarieties, such as lightcones and hyperspheres. For example, a lightcone is the union of all photons through a given point. Hyperspheres fall into two types, depending on the signature of the induced conformal metric. Einstein hyperspheres are Lorentzian, and are models of Ein^{n−1,1}, while spacelike hyperspheres are models of S^{n} with conformal Euclidean geometry.

The Einstein universe Ein^{n,1} can be constructed by projectivizing the nullcone in the inner product space ℝ^{n+1,2} defined by a symmetric bilinear form of type (n+ 1, 2). Thus the points of Ein^{n,1} are null lines in ℝ^{n+1,2}, and photons correspond to isotropic 2-planes. Linear hyperplanes H in ℝ^{n+1,2} determine lightcones, Einstein hyperspheres, and spacelike hyperspheres, respectively, depending on whether the restriction of the bilinear form to H is degenerate, type (n, 2), or Lorentzian, respectively.

Section 4 discusses causality in Einstein space. Section 5 is specific to dimension 3, where the conformal Lorentz group is locally isomorphic to the group of linear symplectomorphisms of ℝ^{4}. This establishes a close relationship between the symplectic geometry of ℝ^{4} (and hence the contact geometry of ℝℙ^{3}) and the conformal Lorentzian geometry of Ein^{2,1}. Section 6 reinterprets these synthetic geometries in terms of the structure theory of Lie algebras. Section 7 discusses the dynamical theory of discrete subgroups of Ein^{2,1} due to Frances [13], and begun by Kulkarni [19]. Section 8 discusses the crooked planes, discovered by Drumm [8], in the context of Ein^{2,1}; their closures, called crooked surfaces are studied and shown to be Klein bottles invariant under the Cartan subgroup of SO(3, 2). The paper concludes with a brief description of discrete groups of conformal transformations and some open questions.

Much of this work was motivated by the thesis of Charles Frances [11], which contains many constructions and examples, his paper [13] on Lorentzian Kleinian groups, and his note [11] on compactifying crooked planes. We are grateful to Charles Frances and Anna Wienhard for many useful discussions.

We are also grateful to the many institutions where we have been able to meet to discuss the mathematics in this paper. In particular, we are grateful for the hospitality provided by the Banff International Research Station [5] where all of us were able to meet for a workshop in November 2004, the workshop in Oostende, Belgium in May 2005 on “Discrete groups and geometric structures,” the miniconference in Lorentzian geometry at the E.N.S. Lyon in July 2005, the special semester at the Newton Institute in Cambridge in Fall 2005, the special semester at the Erwin Schrödinger Institute in Fall 2005, and a seminar at the University of Maryland in summer 2006, when the writing began.

2. Synthetic geometry of Ein^{n,1}

In this section we develop the basic synthetic geometry of Einstein space, or the Einstein universe, starting with the geometry of Minkowski space 𝔼^{n,1}.

2.1. Lorentzian vector spaces. We consider real inner product spaces, that is, vector spaces V over ℝ with a nondegenerate symmetric bilinear form ⟨, ⟩. A nonsingular symmetric n × n-matrix B defines a symmetric bilinear form on ℝ^{n} by the rule:

⟨u, v⟩_{B} := u^{†}Bv.

where u^{†} denotes the transpose of the vector u. We shall denote by ℝ^{p,q} a real inner product space whose inner product is of type (p, q). For example, if

u = [ u_{1} ; ⋮ ; u_{p} ; u_{p+1} ; ⋮ ; u_{p+q}] , v = [ v_{1} ; ⋮ ; v_{p} ; v_{p+1} ; ⋮ ; v_{p+q}] ,

then

⟨u, v⟩ := u_{1}v_{1} + ⋯ + u_{p}v_{p} − u_{p+1}v_{p+1} − ⋯ − u_{p+q}v_{p+q}

defines a type (p, q) inner product, induced by the matrix I_{p} ⊕ − I_{q} on ℝ^{p+q}. The group of linear automorphisms of ℝ^{p,q} is O(p, q).

If B is positive definite—that is, q = 0—then we say that the inner product space (V, ⟨, ⟩) is Euclidean. If q = 1, then (V, ⟨, ⟩) is Lorentzian. We may omit reference to the bilinear form if it is clear from context.

If V is Lorentzian, and v ∈ V, then v is:

• timelike if ⟨v, v⟩ < 0;

• lightlike (or null or isotropic) if ⟨v, v⟩ = 0;

• causal if ⟨v, v⟩ ≤ 0;

• spacelike if ⟨v, v⟩ > 0.

The nullcone 𝔑(V) in V consists of all null vectors.

If W ⊂ V, then define its orthogonal complement:

W^{⊥} := {v ∈ V | ⟨v, w⟩ = 0 ∀ w ∈ W }.

The hyperplane v^{⊥} is null (respectively, timelike, spacelike) if v is null (respectively spacelike, timelike).

In the sequel, according to the object of study, we will consider several symmetric n × n-matrices and the associated type (p, q) symmetric bilinear forms. For different bilinear forms, different subgroups of O(p, q) are more apparent. For example:

• Using the diagonal matrix

I_{p} ⊕ − I_{q}

invariance under the maximal compact subgroup

O(p) × O(q) ⊂ O(p, q)

is more apparent.

• Under the bilinear form defined by the matrix

I_{p−q} ⊕ ⨁^{q} −1/2 · [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}]

(if p ≥ q), invariance under the Cartan subgroup

{I_{p−q}} × ∏^{q} O(1, 1)

is more apparent.

• Another bilinear form which we use in the last two sections is:

I_{p−1} ⊕ − I_{q−1} ⊕ − 1/2 · [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}]

which is useful in extending subgroups of O(p − 1, q − 1) to O(p, q).

2.2. Minkowski space. Euclidean space 𝔼^{n} is the model space for Euclidean geometry, and can be characterized up to isometry as a simply connected, geodesically complete, flat Riemannian manifold. For us, it will be simpler to describe it as an affine space whose underlying vector space of translations is a Euclidean inner product space ℝ^{n}. That means 𝔼^{n} comes equipped with a simply transitive vector space of translations

p ↦ p + v

where p ∈ 𝔼^{n} is a point and v ∈ ℝ^{n} is a vector representing a parallel displacement. Under this simply transitive ℝ^{n}-action, each tangent space T_{p}(𝔼^{n}) naturally identifies with the vector space ℝ^{n}. The Euclidean inner product on ℝ^{n} defines a positive definite symmetric bilinear form on each tangent space—that is, a Riemannian metric.

Minkowski space 𝔼^{n,1} is the Lorentzian analog. It is characterized up to isometry as a simply connected, geodesically complete, flat Lorentzian manifold. Equivalently, it is an affine space whose underlying vector space of translations is ℝ^{n,1}.

The geodesics in 𝔼^{n,1} are paths of the form

ℝ ⟶^{γ} 𝔼^{n,1}
t ⟼ p_{0} + tv

where p_{0} ∈ 𝔼^{n,1} is a point and v ∈ ℝ^{n,1} is a vector. A path γ as above is timelike, lightlike, or spacelike, if the velocity v is timelike, lightlike, or spacelike, respectively.

Let p ∈ 𝔼^{n,1}. The affine lightcone L^{aff}(p) at p is defined as the union of all lightlike geodesics through p:

L^{aff}(p) := {p + v ∈ 𝔼^{n,1} | ⟨v, v⟩ = 0}.

Equivalently L^{aff}(p) = p + 𝔑 where 𝔑 ⊂ ℝ^{n,1} denotes the nullcone in ℝ^{n,1}. The hypersurface L^{aff}(p) is ruled by lightlike geodesics; it is singular only at {p}. The Lorentz form on 𝔼^{n,1} restricts to a degenerate metric on L^{aff}(p) \ {p}.

A lightlike geodesic ℓ ⊂ 𝔼^{n,1} lies in a unique null affine hyperplane. (We denote this ℓ^{⊥}, slightly abusing notation.) That is, writing ℓ = p + ℝv, where v ∈ ℝ^{n,1} is a lightlike vector, the null hyperplane p + v^{⊥} is independent of the choices of p and v used to define ℓ.

The de Sitter hypersphere of radius r centered at p is defined as

S_{r}(p) := {p + v ∈ 𝔼^{n,1} | ⟨v, v⟩ = r^{2}}.

The Lorentz metric on 𝔼^{n,1} restricts to a Lorentz metric on S_{r}(p) having constant sectional curvature 1/r^{2}. It is geodesically complete and homeomorphic to S^{n−1} × ℝ. It is a model for de Sitter space dS^{n−1,1}.

As in Euclidean space, a homothety (centered at x_{0}) is any map conjugate by a translation to scalar multiplication:

𝔼^{n,1} ⟶ 𝔼^{n,1}
x ⟼ x_{0} + r(x − x_{0}).

A Minkowski similarity transformation is a composition of an isometry of 𝔼^{n,1} with a homothety:

f : x ⟼ rA(x) + b.

where A ∈ O(n, 1), r > 0 and b ∈ ℝ^{n,1} defines a translation. Denote the group of similarity transformations of 𝔼^{n,1} by Sim(𝔼^{n,1}).

2.3. Einstein space. Einstein space Ein^{n,1} is the projectivized nullcone of ℝ^{n+1,2}. The nullcone is

𝔑^{n+1,2} := {v ∈ ℝ^{n+1,2} | ⟨v, v⟩ = 0}

and the (n + 1)-dimensional Einstein universe Ein^{n,1} is the image of 𝔑^{n+1,2} − {0} under projectivization:

ℝ^{n+1,2} − {0} ⟶^{ℙ} ℝℙ^{n+2}.

In the sequel, for notational convenience, we will denote ℙ as a map from ℝ^{n+1,2}, implicitly assuming that the origin 0 is removed from any subset of ℝ^{n+1,2} on which we apply ℙ.

The double covering Êin^{n,1} is defined as the quotient of the nullcone 𝔑^{n+1,2} by the action by positive scalar multiplications. For many purposes the double covering may be more useful than Ein^{n,1}, itself. We will also consider the universal covering Ẽin^{n,1} in §4.

Writing the bilinear form on ℝ^{n+1,2} as I_{n+1} ⊕ − I_{2}, that is,

⟨v, v⟩ = v_{1}^{2} + ⋯ + v_{n+1}^{2} − v_{n+2}^{2} − v_{n+3}^{2},

the nullcone is defined by

v_{1}^{2} + ⋯ + v_{n+1}^{2} = v_{n+2}^{2} + v_{n+3}^{2}.

This common value is always nonnegative, and if it is zero, then v = 0 and v does not correspond to a point in Ein^{n,1}. Dividing by the positive number \sqrt{v_{n+2}^{2} + v_{n+3}^{2}} we may assume that

v_{1}^{2} + ⋯ + v_{n+1}^{2} = v_{n+2}^{2} + v_{n+3}^{2} = 1

which describes the product S^{n} × S^{1}. Thus

Êin^{n,1} ≈ S^{n} × S^{1}.

Scalar multiplication by −1 acts by the antipodal map on both the S^{n} and the S^{1}-factor. On the S^{1}-factor the antipodal map is a translation of order two, so the quotient

Ein^{n,1} = Êin^{n,1}/{±1}

is homeomorphic to the mapping torus of the antipodal map on S^{n}. When n is even, Ein^{n,1} is nonorientable and Êin^{n,1} is an orientable double covering. If n is odd, then Ein^{n,1} is orientable.

The objects in the synthetic geometry of Ein^{n,1} are the following collections of points in Ein^{n,1}:

• Photons are projectivizations of totally isotropic 2-planes. We denote the space of photons by Pho^{n,1}. A photon enjoys the natural structure of a real projective line: each photon ϕ ∈ Pho^{n,1} admits projective parametrizations, which are diffeomorphisms of ϕ with ℝℙ^{1} such that if g is an automorphism of Ein^{n,1} preserving ϕ, then g|_{ϕ} corresponds to a projective transformation of ℝℙ^{1}. The projective parametrizations are unique up to post-composition with transformations in PGL(2, ℝ).

• Lightcones are singular hypersurfaces. Given any point p ∈ Ein^{n,1}, the lightcone L(p) with vertex p is the union of all photons containing p:

L(p) := ⋃{ϕ ∈ Pho^{n,1} | p ∈ ϕ}.

The lightcone L(p) can be equivalently defined as the projectivization of the orthogonal complement p^{⊥} ∩ 𝔑^{n+1,2}. The only singular point on L(p) is p, and L(p) \ {p} is homeomorphic to S^{n−1} × ℝ.

• The Minkowski patch Min(p) determined by an element p of Ein^{n,1} is the complement of L(p) and has the natural structure of Minkowski space 𝔼^{n,1}, as will be explained in §3 below. In the double cover, a point p̂ determines two Minkowski patches:

Min^{+}(p̂) := {q̂ ∈ Êin^{n,1} | ⟨p, q⟩ > 0 ∀p, q ∈ ℝ^{n+1,2} representing p̂, q̂}
Min^{−}(p̂) := {q̂ ∈ Êin^{n,1} | ⟨p, q⟩ < 0 ∀p, q ∈ ℝ^{n+1,2} representing p̂, q̂}.

• There are two different types of hyperspheres.

– Einstein hyperspheres are closures in Ein^{n,1} of de Sitter hyperspheres S_{r}(p) in Minkowski patches as defined in §2.2. Equivalently, they are projectivizations of v^{⊥} ∩ 𝔑^{n+1,2} for spacelike vectors v.

– Spacelike hyperspheres are one-point compactifications of spacelike hyperplanes like ℝ^{n} in a Minkowski patch ℝ^{n,1} ⊂ Ein^{n,1}. Equivalently, they are projectivizations of v^{⊥} ∩ 𝔑^{n+1,2} for timelike vectors v.

• An anti-de Sitter space AdS^{n,1} is one component of the complement of an Einstein hypersphere Ein^{n−1,1} ⊂ Ein^{n,1}. It is homeomorphic to S^{1} × ℝ^{n}. Its ideal boundary is Ein^{n−1,1}.

2.4. 2-dimensional case. Because of its special significance, we discuss in detail the geometry of the 2-dimensional Einstein universe Ein^{1,1}.

• Ein^{1,1} is diffeomorphic to a 2-torus.

• Each lightcone L(p) consists of two photons which intersect at p.

• Ein^{1,1} has two foliations F_{−} and F_{+} by photons, and the lightcone L(p) is the union of the leaves through p of the respective foliations.

• The leaf space of each foliation naturally identifies with ℝℙ^{1}, and the map-

ping

Ein^{1,1} ⟶ ℝℙ^{1} × ℝℙ^{1}

is equivariant with respect to the isomorphism

O(2, 2) ⟶^{≅} PGL(2, ℝ) × PGL(2, ℝ).

Here is a useful model (compare Pratoussevitch [26]): The space Mat_{2}(ℝ) of 2 × 2 real matrices with the bilinear form associated to the determinant gives an isomorphism of inner product spaces:

Mat_{2}(ℝ) ⟶ ℝ^{2,2}

[\begin{matrix}m_{11} & m_{12} \\ m_{21} & m_{22} \end{matrix}] ⟼ [\begin{matrix}m_{11} \\ m_{12} \\ m_{21} \\ m_{22} \end{matrix}]

where ℝ^{2,2} is given the bilinear form defined by

\frac{1}{2} [\begin{matrix}0 & 0 & 0 & 1 \\ 0 & 0 & −1 & 0 \\ 0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \end{matrix}] .

The group GL(2, ℝ) × GL(2, ℝ) acts on Mat_{2}(ℝ) by:

X ⟶^{(A,B)} AXB^{−1}

and induces a local isomorphism

SL_{±}(2, ℝ) × SL_{±}(2, ℝ) ⟶ O(2, 2)

where

SL_{±}(2, ℝ) := {A ∈ GL(2, ℝ) | det(A) = ±1}.

Here we will briefly introduce stems, which are pieces of crooked planes, as will be discussed in §8 below. Let p_{0}, p_{∞} ∈ Ein^{1,1} be two points not contained in a common photon. Their lightcones intersect in two points p_{1} and p_{2}, and the union

L(p_{0}) ∪ L(p_{∞}) ⊂ Ein^{1,1}

comprises four photons intersecting in the four points p_{0}, p_{∞}, p_{1}, p_{2}, such that each point lies on two photons and each photon contains two of these points. This stem configuration of four points and four photons can be represented schematically as in Figure 1 below.

Figure 1. Stem Configuration

The complement

Ein^{1,1} \(L(p_{0}) ∪ L(p_{∞}))

consists of four quadrilateral regions (see Figure 2). In §8 the union S of two non-adjacent quadrilateral regions will be studied; this is the stem of a crooked surface. Such a set is bounded by the four photons of L(p_{0}) ∪ L(p_{∞}).

2.5. 3-dimensional case. Here we present several observations particular to the case of Ein^{2,1}.

• We will see that Pho^{2,1} identifies naturally with a 3-dimensional real projective space (§5.5).

• A lightcone in Ein^{2,1} is homeomorphic to a pinched torus.

Figure 2. Two lightcones in Ein^{1,1}

• Suppose p ≠ q. Define

C(p, q) := L(p) ∩ L(q).

If p and q are incident,—that is, they lie on a common photon—then C(p, q) is the unique photon containing them. Otherwise C(p, q) is a submanifold that we will call a spacelike circle. Spacelike circles are projectivized nullcones of linear subspaces of ℝ^{3,2} of type (2, 1). The closure of a spacelike geodesic in 𝔼^{2,1} is a spacelike circle.

• A timelike circle is the projectivized nullcone of a linear subspace of ℝ^{3,2} of metric type (1, 2).

• Einstein hyperspheres in Ein^{2,1} are copies of Ein^{1,1}. In addition to their two rulings by photons, they have a foliation by spacelike circles.

• Lightcones may intersect Einstein hyperspheres in two different ways. These correspond to intersections of degenerate linear hyperplanes in ℝ^{3,2} with linear hyperplanes of type (2, 2). Let u, v ∈ ℝ^{3,2} be vectors such that u^{⊥} is degenerate, so u determines a lightcone L, and v^{⊥} has type (2, 2), so v defines the Einstein hypersphere H. In terms of inner products,

⟨u, u⟩ = 0, ⟨v, v⟩ > 0.

If ⟨u, v⟩ ≠ 0, then u, v span a nondegenerate subspace of signature (1, 1). In that case L∩H is a spacelike circle. If ⟨u, v⟩ = 0, then u, v span a degenerate subspace and the intersection is a lightcone in H, which is a union of two distinct but incident photons.

• Similarly, lightcones intersect spacelike hyperspheres in two different ways. The generic intersection is a spacelike circle, and the non-generic intersection is a single point, such as the intersection of L(0) with the spacelike plane z = 0 in ℝ^{2,1}.

• A pointed photon is a pair (p, ϕ) ∈ Ein^{2,1} × Pho^{2,1} such that p ∈ ϕ. Such a pair naturally extends to a triple

p ∈ ϕ ⊂ L(p)

which corresponds to an isotropic flag, that is, a linear filtration of ℝ^{3,2}

0 ⊂ ℓ_{p} ⊂ P_{ϕ} ⊂ (ℓ_{p})^{⊥} ⊂ ℝ^{3,2},

where ℓ_{p} is the 1-dimensional linear subspace corresponding to p; P_{ϕ} is the 2-dimensional isotropic subspace corresponding to ϕ; and (ℓ_{p})^{⊥} is the orthogonal subspace of ℓ_{p}. These objects form a homogeneous space, an incidence variety, denoted Flag^{2,1}, of O(3, 2), which fibers both over Ein^{2,1} and Pho^{2,1}. The fiber of the fibration Flag^{2,1} ⟶ Ein^{2,1} over a point p is the collection of all photons through p. The fiber of the fibration Flag^{2,1} ⟶ Pho^{2,1} over a photon ϕ identifies with all the points of ϕ. Both fibrations are circle bundles.

3. Ein^{n,1} as the conformal compactification of 𝔼^{n,1}

Now we shall describe the geometry of Ein^{n,1} as the compactification of Minkowski space 𝔼^{n,1}. We begin with the Euclidean analog.

3.1. The conformal Riemannian sphere. The standard conformal

compactification of Euclidean space 𝔼^{n} is topologically the one-point compactification, the n-dimensional sphere. The conformal Riemannian sphere S^{n} is the projectivization ℙ(𝔑^{n+1,1}) of the nullcone of ℝ^{n+1,1}.

For U ⊂ S^{n} an arbitrary open set, any local section

U ⟶^{σ} ℝ^{n+1,1} \ {0}

of the restriction of the projectivization map to U determines a pullback of the Lorentz metric on 𝔼^{n+1,1} to a Riemannian metric g_{σ} on U. This metric depends on σ, but its conformal class is independent of σ. Every section is σ′ = fσ for some non-vanishing function f : U → ℝ. Then

g_{σ′} = f^{2}g_{σ}

so the pullbacks are conformally equivalent. Hence the metrics g_{σ} altogether define a canonical conformal structure on S^{n}.

The orthogonal group O(n+1, 1) leaves invariant the nullcone 𝔑^{n+1,1} ⊂ ℝ^{n+1,1}. The projectivization

S^{n} = ℙ(𝔑^{n+1,1})

is invariant under the projective orthogonal group PO(n + 1, 1), which is its conformal automorphism group.

Let

S^{n} ⟶^{σ} 𝔑^{n+1,1} ⊆ ℝ^{n+1,1} \ {0}

be the section taking values in the unit Euclidean sphere. Then the metric g_{σ} is the usual O(n + 1)-invariant spherical metric.

Euclidean space 𝔼^{n} embeds in S^{n} via a spherical paraboloid in the nullcone 𝔑^{n+1,1}. Namely consider the quadratic form on ℝ^{n+1,1} defined by

I_{n} ⊕ − 1/2 · [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] = [\begin{matrix}I & n \\ & 0 & −1/2 \\ & −1/2 & 0 \end{matrix}] .

The map

𝔼^{n} → 𝔑^{n+1,1} ⊂ ℝ^{n+1,1}
x ⟼ [\begin{matrix}x \\ ⟨x, x⟩ \\ 1 \end{matrix}]  (1)

composed with projection 𝔑^{n+1,1} ⟶^{ℙ} S^{n} is an embedding E of 𝔼^{n} into S^{n}, which is conformal.

The Euclidean similarity transformation

f_{r,A,b} : x ⟼ rAx + b

where r ∈ ℝ_{+}, A ∈ O(n), and b ∈ ℝ^{n}, is represented by

F_{r,A,b} := [\begin{matrix}I_{n} & 0 & b \\ 2b^{†} & 1 & ⟨b, b⟩ \\ 0 & 0 & 1 \end{matrix}] · [\begin{matrix}A & 0 & 0 \\ 0 & r & 0 \\ 0 & 0 & r^{−1} \end{matrix}] ∈ O(n + 1, 1).  (2)

That is, for every x ∈ 𝔼^{n},

F_{r,A,b}E(x) = E(f_{r,A,b}(x)).

Inversion in the unit sphere ⟨v, v⟩ = 1 of 𝔼^{n} is represented by the element

I_{n} ⊕ [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}]

which acts on 𝔼^{n} \ {0} by:

ι : x ↦ \frac{1}{⟨x, x⟩}x.

The origin is mapped to the point (called ∞) having homogeneous coordinates

[\begin{matrix}0_{n} \\ 1 \\ 0 \end{matrix}]

where 0_{n} ∈ ℝ^{n} is the zero vector.

The map E^{−1} is a coordinate chart on the open set

𝔼^{n} = S^{n} \ {∞}

and E^{−1} ∘ ι is a coordinate chart on the open set (𝔼^{n} ∪ {∞}) \ {0} = S^{n} \ {0}.

3.2. The conformal Lorentzian quadric. Consider now the inner

product space ℝ^{n+1,2}. Here it will be convenient to use the inner product

⟨u, v⟩ := u_{1}v_{1} + … + u_{n}v_{n} − u_{n+1}v_{n+1} − \frac{1}{2}u_{n+2}v_{n+3} − \frac{1}{2}u_{n+3}v_{n+2}

= u^{†}( I_{n} ⊕ − I_{1} ⊕ − 1/2 · [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] )v.

In analogy with the Riemannian case, consider the embedding E : 𝔼^{n,1} → Ein^{n,1} via a hyperbolic paraboloid defined by (1) as above, where the Lorentzian inner product on 𝔼^{n,1} is defined by Q = I_{n} ⊕ − I_{1}. The procedure used previously in the Riemannian case naturally defines an O(n + 1, 2)-invariant conformal Lorentzian structure on Ein^{n,1}, and the embedding we have just defined is conformal.

Minkowski similarities f_{r,A,b} map into O(n + 1, 2) as in the formula (2), where r ∈ ℝ_{+}; A ∈ O(n + 1, 1); b ∈ ℝ^{n,1}; ⟨, ⟩ is the Lorentzian inner product on ℝ^{n,1}; and 2b^{†} is replaced by 2b^{†}Q.

The conformal compactification of Euclidean space is the one-point compactification; the compactification of Minkowski space, however, is more complicated, requiring the addition of more than a single point. Let p_{0} ∈ Ein^{n,1} denote the origin, corresponding to

[\begin{matrix}0_{n+1} \\ 0 \\ 1 \end{matrix}] .

To see what lies at infinity, consider the Lorentzian inversion in the unit sphere

defined by the matrix I_{n+1} ⊕ [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}], which is given on 𝔼^{n,1} by the formula

ι : x ⟼ \frac{1}{⟨x, x⟩}x.  (3)

Here the whole affine lightcone L^{aff}(p_{0}) is thrown to infinity. We distinguish the points on ι(L^{aff}(p_{0})):

• The improper point p_{∞} is the image ι(p_{0}). It is represented in homogeneous coordinates by

[\begin{matrix}0_{n+1} \\ 1 \\ 0 \end{matrix}] .

• The generic point on ι(L^{aff}(p_{0})) has homogeneous coordinates

[\begin{matrix}v \\ 1 \\ 0 \end{matrix}]

where 0 ≠ v ∈ ℝ^{n,1}; it equals ι(E(v)).

We have described all the points in

𝔼^{n,1} ∪ ι(𝔼^{n,1})

which are the points defined by vectors v ∈ ℝ^{n+1,2} with coordinates v_{n+2} ≠ 0 or v_{n+3} ≠ 0. It remains to consider points having homogeneous coordinates

[\begin{matrix}v \\ 0 \\ 0 \end{matrix}]

where necessarily ⟨v, v⟩ = 0. This equation describes the nullcone in ℝ^{n,1}; its projectivization is a spacelike sphere S_{∞}, which we call the ideal sphere. When n = 2, we call this the ideal circle and its elements ideal points. Each ideal point is the endpoint of a unique null geodesic from the origin; the union of that null geodesic with the ideal point is a photon through the origin. Every photon through the origin arises in this way. The ideal sphere is fixed by the inversion ι.

The union of the ideal sphere S_{∞} with ι(L^{aff}(p_{0})) is the lightcone L(p_{∞}) of the improper point. Photons in L(p_{∞}) are called ideal photons. Minkowski space 𝔼^{n,1} is thus the complement of a lightcone L(p_{∞}) in Ein^{n,1}. This fact motivated the earlier definition of a Minkowski patch Min(p) as the complement in Ein^{n,1} of a lightcone L(p).

Changing a Lorentzian metric by a non-constant scalar factor modifies timelike and spacelike geodesics, but not images of null geodesics (see for example [3], p. 307). Hence the notion of (non-parametrized) null geodesic is well-defined in a conformal Lorentzian manifold. For Ein^{n,1}, the null geodesics are photons.

3.3. Involutions. When n is even, involutions in SO(n + 1, 2) ≅ PO(n + 1, 2) correspond to nondegenerate splittings of ℝ^{n+1,2}. For any involution in PO(3, 2), the fixed point set in Ein^{2,1} must be one of the following:

• the empty set ∅;

• a spacelike hypersphere;

• a timelike circle;

• the union of a spacelike circle with two points;

• an Einstein hypersphere.

In the case that Fix(f) is disconnected and equals

{p_{1}, p_{2}} ∪ S

where p_{1}, p_{2} ∈ Ein^{2,1}, and S ⊂ Ein^{2,1} is a spacelike circle, then

S = L(p_{1}) ∩ L(p_{2}).

Conversely, given any two non-incident points p_{1}, p_{2}, there is a unique involution fixing p_{1}, p_{2} and the spacelike circle L(p_{1}) ∩ L(p_{2}).

3.3.1. Inverting photons. Let p_{∞} be the improper point, as above. A photon in Ein^{2,1} either lies on the ideal lightcone L(p_{∞}), or it intersects the spacelike plane S_{0} consisting of all

p = [\begin{matrix}x \\ y \\ z \end{matrix}]

for which z = 0. Suppose ϕ is a photon intersecting S_{0} in the point p_{0} with polar coordinates

p_{0} = [\begin{matrix}r_{0} cos(ψ) \\ r_{0} sin(ψ) \\ 0 \end{matrix}] ∈ S_{0} ⊂ 𝔼^{2,1}.

Let v_{0} be the null vector

v_{0} = [\begin{matrix}cos(θ) \\ sin(θ) \\ 1 \end{matrix}]

and consider the parametrized lightlike geodesic

ϕ(t) := p_{0} + tv_{0}

for t ∈ ℝ. Then inversion ι maps ϕ(t) to

(ι ∘ ϕ)(t) = ι(p_{0}) + t̃[\begin{matrix}− cos(θ − 2ψ) \\ sin(θ − 2ψ) \\ 1 \end{matrix}]

where

t̃:= \frac{t}{r_{0}^{2} + 2r_{0} cos(θ − ψ)t} .

Observe that ι leaves invariant the spacelike plane S_{0} and acts by Euclidean inversion on that plane.

3.3.2. Extending planes in 𝔼^{2,1} to Ein^{2,1}.

• The closure of a null plane P in 𝔼^{2,1} is a lightcone and its frontier P̄ \ P is an ideal photon. Conversely a lightcone with vertex on the ideal circle S_{∞} is the closure of a null plane containing p_{0}, while a lightcone with vertex on

L(p_{∞}) \ (S_{∞} ∪ {p_{∞}})

is the closure of a null plane not containing p_{0}.

• The closure of a spacelike plane in 𝔼^{2,1} is a spacelike sphere and its frontier is the improper point p_{∞}.

• The closure of a timelike plane in 𝔼^{2,1} is an Einstein hypersphere and its frontier is a union of two ideal photons (which intersect in p_{∞}).

• The closure of a timelike (respectively spacelike) geodesic in 𝔼^{2,1} is a timelike (respectively spacelike) circle containing p_{∞}, and p_{∞} is its frontier.

Consider the inversion on the lightcone of p_{0}:

ι ([\begin{matrix}t sin θ \\ t cos θ \\ t \\ 0 \\ 1 \end{matrix}]) = [\begin{matrix}t sin θ \\ t cos θ \\ t \\ 1 \\ 0 \end{matrix}] .

The entire image of the light cone L(p_{0}) lies outside the Minkowski patch 𝔼^{2,1}.

Let us now look at the image of a timelike line in 𝔼^{2,1} under the inversion. For example,

ι ([\begin{matrix}0 \\ 0 \\ t \\ −t^{2} \\ 1 \end{matrix}]) = [\begin{matrix}0 \\ 0 \\ t \\ 1 \\ −t^{2} \end{matrix}] ∼ [\begin{matrix}0 \\ 0 \\ −1/t \\ −1/t^{2} \\ 1 \end{matrix}] = [\begin{matrix}0 \\ 0 \\ s \\ −s^{2} \\ 1 \end{matrix}]

where s = −1/t. That is, the inversion maps the timelike line minus the origin to itself, albeit with a change in the parametrization.

4. Causal geometry

In §3.2 we observed that Ein^{n,1} is naturally equipped with a conformal structure. This structure lifts to the double cover Êin^{n,1}. As in the Riemannian case in §3.1, a global representative of the conformal structure on Êin^{n,1} is the pullback by a global section σ : Êin^{n,1} → ℝ^{n+1,2} of the ambient quadratic form of ℝ^{n+1,2}. The section σ : Êin^{n,1} → ℝ^{n+1,2} taking values in the set where

v_{1}^{2} + ⋯ + v_{n+1}^{2} = v_{n+2}^{2} + v_{n+3}^{2} = 1

exhibits a homeomorphism Êin^{n,1} ≅ S^{n} × S^{1} as in §2.3; it is now apparent that Êin^{n,1} is conformally equivalent to S^{n} × S^{1} endowed with the Lorentz metric ds_{0}^{2} − dθ^{2}, where ds_{0}^{2} and dθ^{2} are the usual round metrics on the spheres S^{n} and S^{1} of radius one.

In the following, elements of S^{n}×S^{1} are denoted by (φ, θ). In these coordinates, we distinguish the timelike vector field η = ∂_{θ} tangent to the fibers {∗} × S^{1}.

4.1. Time orientation. First consider Minkowski space 𝔼^{n,1} with underlying vector space ℝ^{n,1} equipped with the inner product:

⟨u, v⟩ := u_{1}v_{1} + ⋯ + u_{n}v_{n} − u_{n+1}v_{n+1}.

A vector u in ℝ^{n,1} is causal if u_{n+1}^{2} ≥ u_{1}^{2} + … + u_{n}^{2}. It is future-oriented (respectively past-oriented) if the coordinate u_{n+1} is positive (respectively negative); equivalently, u is future-oriented if its inner product with

η_{0} = [0 ⋮ 0 1]

is negative.

The key point is that the choice of the coordinate u_{n+1}—equivalently, of an everywhere timelike vector field like η_{0}—defines a decomposition of every affine lightcone L^{aff}(p) in three parts:

• {p};

• The future lightcone L_{+}^{aff}(p) of elements p + v where v is a future-oriented null vector;

• The past lightcone L_{−}^{aff}(p) of elements p + v where v is a past-oriented null vector.

The above choice is equivalent to a continuous choice of one of the connected components of the set of timelike vectors based at each x ∈ 𝔼^{n,1}; timelike vectors in these components are designated future-oriented. In other words, η_{0} defines a time orientation on 𝔼^{n,1}.

To import this notion to Êin^{n,1}, replace η_{0} by the vector field η on Êin^{n,1}. Then a causal tangent vector v to Êin^{n,1} is future-oriented (respectively past-oriented) if the inner product ⟨v, η⟩ is negative (respectively positive).

We already observed in §2.3 that the antipodal map is (φ, θ) ↦ (−φ, −θ) on S^{n} × S^{1}; in particular, it preserves the timelike vector field η, which then descends to a well-defined vector field on Ein^{n,1}, so that Ein^{n,1} is time oriented, for all integers n.

Remark 4.1.1. The Einstein universe does not have a preferred Lorentz metric in its conformal class. The definition above is nonetheless valid since it involves only signs of inner products and hence is independent of the choice of metric in the conformal class.

The group O(n + 1, 2) has four connected components. More precisely, let SO(n + 1, 2) be the subgroup of O(n + 1, 2) formed by elements with determinant 1; these are the orientation-preserving conformal transformations of Êin^{n,1}. Let O^{+}(n + 1, 2) be the subgroup comprising the elements preserving the time orientation of Êin^{n,1}. The identity component of O(n + 1, 2) is the intersection

SO^{+}(n + 1, 2) = SO(n + 1, 2) ∩ O^{+}(n + 1, 2).

Moreover, SO(n + 1, 2) and O^{+}(n + 1, 2) each have two connected components.

The center of O(n + 1, 2) has order two and is generated by the antipodal map, which belongs to SO(n + 1, 2) if and only if n is odd. Hence the center of SO(n + 1, 2) is trivial if n is even—in particular, when n = 2. On the other hand, the antipodal map always preserves the time orientation.

The antipodal map is the only element of O(n + 1, 2) acting trivially on Ein^{n,1}. Hence the group of conformal transformations of Ein^{n,1} is PO(n+1, 2), the quotient of O(n + 1, 2) by its center. When n is even, PO(n + 1, 2) is isomorphic to SO(n + 1, 2).

4.2. Future and past. A C^{1}-immersion

[0, 1] ⟶^{c} 𝔼^{1,n}

is a causal curve (respectively a timelike curve) if the tangent vectors c′(t) are all causal (respectively timelike). This notion extends to any conformally Lorentzian space—in particular, to Ein^{n,1}, Êin^{n,1}, or Ẽin^{n,1}. Furthermore, a causal curve c is future-oriented (respectively past-oriented) if all the tangent vectors c′(t) are future-oriented (respectively past-oriented).

Let A be a subset of 𝔼^{n,1}, Ein^{n,1}, Êin^{n,1}, or Ẽin^{n,1}. The future I^{+}(A) (respectively the past I^{−}(A)) of A is the set comprising endpoints c(1) of future-oriented (respectively past-oriented) timelike curves with starting point c(0) in A. The causal future J^{+}(A) (respectively the causal past J^{−}(A)) of A is the set comprising endpoints c(1) of future-oriented (respectively past-oriented) causal curves with starting point c(0) in A. Two points p, p′ are causally related if one belongs to the causal future of the other: p′ ∈ J^{±}(p). The notion of future and past in 𝔼^{n,1} is quite easy to understand: p′ belongs to the future I^{+}(p) of p if and only if p′ − p is a future-oriented timelike element of ℝ^{n,1}.

Thanks to the conformal model, these notions are also quite easy to understand in Ein^{n,1}, Êin^{n,1}, or Ẽin^{n,1}: let d_{n} be the spherical distance on the homogeneous Riemannian sphere S^{n} of radius 1. The universal covering Ẽin^{n,1} is conformally isometric to the Riemannian product S^{n} ×ℝ where the real line ℝ is endowed with the negative quadratic form −dθ^{2}. Hence, the image of any causal, C^{1}, immersed curve in Ẽin^{n,1} ≈ S^{n} × ℝ is the graph of a map f : I → S^{n} where I is an interval in ℝ and where f is 1-Lipschitz —that is, for all θ, θ′ in ℝ:

d_{n}(f(θ), f(θ′)) ≤ |θ − θ′|.

Moreover, the causal curve is timelike if and only if the map f is contracting—that is, satisfies

d_{n}(f(θ), f(θ′)) < |θ − θ′|.

It follows that the future of an element (φ_{0}, θ_{0}) of Ẽin^{n,1} ≈ S^{n} × ℝ is:

I^{+}(φ_{0}, θ_{0}) = {(φ, θ) | θ − θ_{0} > d_{n}(φ, φ_{0})}

and the causal future J^{+}(p) of an element p of Ẽin^{n,1} is the closure of the future I^{+}(p):

J^{+}(φ_{0}, θ_{0}) = {(φ, θ) | θ − θ_{0} ≥ d_{n}(φ, φ_{0})}.

As a corollary, the future I^{+}(A) of a nonempty subset A of Ein^{n,1} or Êin^{n,1} is the entire spacetime. In other words, the notion of past or future is relevant in Ẽin^{n,1}, but not in Ein^{n,1} or Êin^{n,1}.

There is, however, a relative notion of past and future still relevant in Êin^{n,1} that will be useful later when considering crooked planes and surfaces: let p̂, p̂′ be two elements of Êin^{n,1} such that p̂′ ≠ ±p̂. First observe that the intersection Min^{+}(p̂) ∩ Min^{+}(p̂′) is never empty. Let p_{∞} be any element of this intersection, so Min^{+}(p̂_{∞}) contains p̂ and p̂′. The time orientation on Êin^{n,1} induces a time orientation on such a Minkowski patch Min^{+}(p̂_{∞}).

Fact 4.2.1. The points p̂′ and p̂ are causally related in Min^{+}(p̂_{∞}) if and only if, for any lifts p, p′ of p̂, p̂′, respectively, to ℝ^{n+1,2}, the inner product ⟨p, p′⟩ is positive.

Hence, if p̂ and p̂′ are causally related in some Minkowski patch, then they are causally related in any Minkowski patch containing both of them. Therefore, (slightly abusing language) we use the following convention: two elements p̂, p̂′ of Êin^{n,1} are causally related if the inner product ⟨p, p′⟩ in ℝ^{n+1,2} is positive for any lifts p, p′.

4.3. Geometry of the universal covering. The geometrical under-

standing of the embedding of Minkowski space in the Einstein universe can be a challenge. In particular, the closure in Ein^{n,1} of a subset of a Minkowski patch may be not obvious, as we will see for crooked planes. This difficulty arises from the nontrivial topology of Ein^{n,1}.

On the other hand, the topology of the universal covering Ẽin^{n,1} is easy to visualize; indeed, the map

Ẽin^{n,1} ≈ S^{n} × ℝ ⟶^{S} ℝ^{n+1} \ {0}
S : (φ, θ) ⟼ exp(θ)φ

is an embedding. Therefore, Ẽin^{n,1} can be considered as a subset of ℝ^{n+1}—one that is particularly easy to visualize when n = 2. Observe that the map S is O(n + 1)-equivariant for the natural actions on Ẽin^{n,1} and ℝ^{n+1}.

The antipodal map

(φ, θ) ⟼ (−φ, −θ)

lifts to the automorphism α of

Ẽin^{n,1} ≈ S^{n} × ℝ,

defined by

(φ, θ) ⟼^{α} (−φ, θ + π).

In the coordinates Ẽin^{n,1} ≈ ℝ^{n+1} \ {0} this lifting α is expressed by x → −λx, where λ = exp(π).

Since null geodesics in Ein^{n,1} are photons, the images by S of null geodesics of Ẽin^{n,1} are curves in ℝ^{n+1} \ {0} characterized by the following properties:

• They are contained in 2-dimensional linear subspaces;

• Each is a logarithmic spiral in the 2-plane containing it.

Hence, for n = 2, the lightcone of an element p of Ẽin^{2,1} (that is, the union of the null geodesics containing p) is a singular surface of revolution in ℝ^{3} obtained by rotating a spiral contained in a vertical 2-plane around an axis of the plane. In particular, for every x in Ẽin^{n,1} ≈ ℝ^{n+1} \ {0}, every null geodesic containing x contains α(x) = −λx. The image α(x) = −λx is uniquely characterized by the following properties, so that it can be called the first future-conjugate point to x:

• It belongs to the causal future J^{+}(x);

• For any y ∈ J^{+}(x) such that y belongs to all null geodesics containing x, we have α(x) ∈ J^{−}(y).

All these considerations allow us to visualize how Minkowski patches embed in ℝ^{n+1} \ {0} (see Figure 3): let p̃ ∈ Ẽin^{n,1} and p̂ be its projection to Êin^{n,1}. The Minkowski patch Min^{+}(p̂) is the projection in Êin^{n,1} of I^{+}(p̃) \ J^{+}(α(p̃)), which can also be defined as I^{+}(p̃) ∩ I^{−}(α^{2}(p̃)). The projection in Êin^{n,1} of

Ẽin^{n,1} \ (J^{+}(p̃) ∪ J^{−}(p̃))

is the Minkowski patch Min^{−}(p̂), which is the set of points non-causally related to p̂.

4.4. Improper points of Minkowski patches. We previously defined

the improper point p_{∞} associated to a Minkowski patch in Ein^{n,1}: it is the unique point such that the Minkowski patch is Min(p_{∞}).

In the double-covering Êin^{n,1}, to every Minkowski patch are attached two improper points:

• the spatial improper point, the unique element p_{∞}^{sp} such that the given Minkowski patch is Min^{−}(p_{∞}^{sp});

• the timelike improper point, the unique element p_{∞}^{ti} such that the given Minkowski patch is Min^{+}(p_{∞}^{ti}).

Min^{+}(p̂) p̂ α(p̂) α^{2}(p̂)

Figure 3. A Minkowski patch in Ẽin^{1,1}

Let Min^{+}(p_{∞}^{ti}) = Min^{−}(p_{∞}^{sp}) be a Minkowski patch in Êin^{n,1}. Let

ℝ ⟶^{γ} Min^{+}(p_{∞}^{ti}) ≈ 𝔼^{n,1}

be a geodesic. Denote by Γ the image of γ, and by Γ̄ the closure in Êin^{n,1} of Γ.

• If γ is spacelike, then

Γ̄ = Γ ∪ {p_{∞}^{sp}}.

• If γ is timelike, then

Γ̄ = Γ ∪ {p_{∞}^{ti}}.

• If γ is lightlike, then Γ̄ is a photon avoiding p_{∞}^{sp} and p_{∞}^{ti}.

5. Four-dimensional real symplectic vector spaces

In spatial dimension n = 2, Einstein space Ein^{2,1} admits an alternate description as the Lagrangian Grassmannian, the manifold Lag(V) of Lagrangian 2-planes in a real symplectic vector space V of dimension 4. There results a kind of duality between the conformal Lorentzian geometry of Ein^{2,1} and the symplectic geometry of ℝ^{4}. Photons correspond to linear pencils of Lagrangian 2-planes (that is, families of Lagrangian subspaces passing through a given line). The corresponding local isomorphism

Sp(4, ℝ) ⟶ O(3, 2)

manifests the isomorphism of root systems of type B_{2} (the odd-dimensional orthogonal Lie algebras) and C_{2} (the symplectic Lie algebras) of rank 2. We present this correspondence below.

5.1. The inner product on the second exterior power. Begin

with a four-dimensional vector space V over ℝ and choose a fixed generator

vol ∈ Λ^{4}(V).

The group of automorphisms of (V, vol) is the special linear group SL(V).

The second exterior power Λ^{2}(V) has dimension 6. The action of SL(V) on V induces an action on Λ^{2}(V) which preserves the bilinear form

Λ^{2}(V) × Λ^{2}(V) ⟶^{𝔹} ℝ

defined by:

α_{1} ∧ α_{2} = −𝔹(α_{1}, α_{2}) vol .

This bilinear form satisfies the following properties:

• 𝔹 is symmetric;

• 𝔹 is nondegenerate;

• 𝔹 is split—that is, of type (3, 3).

(That 𝔹 is split follows from the fact that any orientation-reversing linear automorphism of V maps 𝔹 to its negative.)

The resulting homomorphism

SL(4, ℝ) ⟶ SO(3, 3)  (4)

is a local isomorphism of Lie groups, with kernel {± I} and image the identity component of SO(3, 3).

Consider a symplectic form ω on V —that is, a skew-symmetric nondegenerate bilinear form on V. Since 𝔹 is nondegenerate, ω defines a dual exterior bivector ω^{∗} ∈ Λ^{2}(V) by

ω(v_{1}, v_{2}) = 𝔹(v_{1} ∧ v_{2}, ω^{∗}).

We will assume that

ω^{∗} ∧ ω^{∗} = 2 vol .  (5)

Thus 𝔹(ω^{∗}, ω^{∗}) = −2 < 0, so that its symplectic complement

W_{0} := (ω^{∗})^{⊥} ⊂ Λ^{2}(V)

is an inner product space of type (3, 2). Now the local isomorphism (4) restricts to a local isomorphism

Sp(4, ℝ) ⟶ SO(3, 2)  (6)

with kernel {± I_{4}} and image the identity component of SO(3, 2).

5.2. Lagrangian subspaces and the Einstein universe. Let V,

ω, 𝔹, ω^{∗}, and W_{0} be as above. The projectivization of the null cone in W_{0} is equivalent to Ein^{2,1}. Points in Ein^{2,1} correspond to Lagrangian planes in V — that is, 2-dimensional linear subspaces P ⊂ V such that the restriction ω|_{P} ≡ 0. Explicitly, if v_{1}, v_{2} constitute a basis for P, then the line generated by the bivector

w = v_{1} ∧ v_{2} ∈ Λ^{2}(V)

is independent of the choice of basis for P. Furthermore, w is null with respect to 𝔹 and orthogonal to ω^{∗}, so w generates a null line in W_{0} ≅ ℝ^{3,2}, and hence defines a point in Ein^{2,1}.

For the reverse correspondence, first note that a point of Ein^{2,1} ≅ ℙ(𝔑(W_{0})) is represented by a vector a ∈ W_{0} such that a ∧ a = 0. Elements a ∈ Λ^{2}V with a ∧ a = 0 are exactly the decomposable ones—that is, those that can be written a = v_{1} ∧v_{2} for v_{1}, v_{2} ∈ V. Then the condition a ⊥ ω^{∗} is equivalent by construction to ω(v_{1}, v_{2}) = 0, so a represents a Lagrangian plane, span{v_{1}, v_{2}}, in V. Thus Lagrangian 2-planes in V correspond to isotropic lines in W_{0} ≅ ℝ^{3,2}.

For a point q ∈ Ein^{2,1}, denote by L_{q} the corresponding Lagrangian plane in V.

5.2.1. Complete flags. A photon ϕ in Ein^{2,1} corresponds to a line ℓ_{ϕ} in V, where

ℓ_{ϕ} = ⋂_{p∈ϕ} L_{p}.

A pointed photon (p, ϕ), as defined in §2.5, corresponds to a pair of linear subspaces

ℓ_{ϕ} ⊂ L_{p}  (7)

where ℓ_{ϕ} ⊂ V is the line corresponding to ϕ and where L_{p} ⊂ V is the Lagrangian plane of corresponding to p. Recall that the incidence relation p ∈ ϕ extends to

p ∈ ϕ ⊂ L(p),

corresponding to the complete linear flag

0 ⊂ ℓ_{p} ⊂ P_{ϕ} ⊂ (ℓ_{p})^{⊥} ⊂ W_{0}

where P_{ϕ} is the null plane projectivizing to ϕ. The linear inclusion (7) extends to a linear flag

0 ⊂ ℓ_{ϕ} ⊂ L_{p} ⊂ (ℓ_{ϕ})^{⊥} ⊂ V

where now (ℓ_{ϕ})^{⊥} denotes the symplectic orthogonal of ℓ_{ϕ}. Clearly the lightcone L(p) corresponds to the linear hyperplane (ℓ_{ϕ})^{⊥} ⊂ V.

5.2.2. Pairs of Lagrangian planes. Distinct Lagrangian subspaces L_{1}, L_{2} may intersect in either a line or in 0. If L_{1} ∩ L_{2} ≠ 0, the corresponding points p_{1}, p_{2} ∈ Ein^{2,1} are incident. Otherwise

V = L_{1} ⊕ L_{2}

and the linear involution of V

θ = I_{L_{1}} ⊕ − I_{L_{2}}

is anti-symplectic :

ω(θ(v_{1}), θ(v_{2})) = −ω(v_{1}, v_{2}).

The corresponding involution of Ein^{2,1} fixes the two points p_{1}, p_{2} and the spacelike circle L(p_{1}) ∩ L(p_{2}). It induces a time-reversing involution of Ein^{2,1}.

5.3. Symplectic planes. Let P ⊂ V be a symplectic plane, that is, one for which the restriction ω|_{P} is nonzero (and hence nondegenerate). Its symplectic complement P^{⊥} is also a symplectic plane, and

V = P ⊕ P^{⊥}

is a symplectic direct sum decomposition.

Choose a basis {u_{1}, u_{2}} for P. We may assume that ω(u_{1}, u_{2}) = 1. Then

𝔹(u_{1} ∧ u_{2}, ω^{∗}) = 1

and

υ_{P} := 2u_{1} ∧ u_{2} + ω^{∗}

lies in (ω^{∗})^{⊥} since 𝔹(ω^{∗}, ω^{∗}) = −2. Furthermore

𝔹(υ_{P} , υ_{P}) = 𝔹(2u_{1} ∧ u_{2}, 2u_{1} ∧ u_{2}) + 2 𝔹(2u_{1} ∧ u_{2}, ω^{∗}) + 𝔹(ω^{∗}, ω^{∗})
= 0 + 4 − 2
= 2.

whence υ_{P} is a positive vector in W_{0} ≅ ℝ^{3,2}. In particular ℙ(υ_{P}^{⊥} ∩ 𝔑(W_{0})) is an Einstein hypersphere.

The two symplectic involutions leaving P (and necessarily also P^{⊥}) invariant

±( I |_{P} ⊕ − I |_{P^{⊥}})

induce maps fixing υ_{P}, and acting by −1 on (υ_{P})^{⊥}. The corresponding eigenspace decomposition is ℝ^{1,0} ⊕ ℝ^{2,2} and the corresponding conformal involution in Ein^{2,1} fixes an Einstein hypersphere.

5.4. Positive complex structures and the Siegel space. Not ev-

ery involution of Ein^{2,1} arises from a linear involution of V. Particularly important are those which arise from compatible complex structures, defined as follows. A complex structure on V is an automorphism V ⟶^{J} V such that J ∘ J = − I. The pair (V, J) then inherits the structure of a complex vector space for which V is the underlying real vector space. The complex structure J is compatible with the symplectic vector space (V, ω) when

ω(J x, J y) = ω(x, y).

(In the language of complex differential geometry, the exterior 2-form ω has Hodge type (1, 1) on the complex vector space (V, J).) Moreover

V × V ⟶ ℂ
(v, w) ⟼ ω(v, J w) + iω(v, w)

defines a Hermitian form on (V, J).

A compatible complex structure J on (V, ω) is positive if ω(v, J v) > 0 whenever v ≠ 0. Equivalently, the symmetric bilinear form defined by

v · w := ω(v, J w)

is positive definite. This is in turn equivalent to the above Hermitian form being positive definite.

The positive compatible complex structures on V are parametrized by the symmetric space of Sp(4, ℝ). A convenient model is the Siegel upper-half space 𝔖_{2}, which can be realized as the domain of 2 × 2 complex symmetric matrices with positive definite imaginary part (Siegel [29]).

A matrix M ∈ Sp(4, ℝ) acts on a complex structure J by

J ↦ M J M^{−1}

and the stabilizer of any J is conjugate to U(2), the group of unitary transformations of ℂ^{2}. Let the symplectic structure ω be defined by the 2 × 2-block matrix

J := [\begin{matrix}0_{2} & − I_{2} \\ I_{2} & 0_{2} \end{matrix}] .

This matrix also defines a complex structure. Write M as a block matrix with

M = [\begin{matrix}A & B \\ C & D \end{matrix}]

where the blocks A, B, C, D are 2 × 2 real matrices. Because M ∈ Sp(4, ℝ),

M^{†} J M = J .  (8)

The condition that M preserves the complex structure J means that M commutes with J, which together with (8), means that

M^{†}M = I_{4},

that is, M ∈ O(4). Thus the stabilizer of the pair (ω, J) is Sp(4, ℝ) ∩ O(4), which identifies with the unitary group U(2) as follows.

If M commutes with J, then its block entries satisfy

B = −C, D = A.

Relabelling X = A and Y = C, then

M = [\begin{matrix}X & −Y \\ Y & X \end{matrix} ]

corresponds to a complex matrix Z = X + iY. This matrix is symplectic if and only if Z is unitary,

Z̄^{†}Z = I_{2} .

5.5. The contact projective structure on photons. The points of

a photon correspond to Lagrangian planes in V intersecting in a common line. Therefore, photons correspond to linear 1-dimensional subspaces in V, and the photon space Pho^{2,1} identifies with the projective space ℙ(V). This space has a natural contact geometry defined below.

Recall that a contact structure on a manifold M^{2n+1} is a vector subbundle E ⊂ T M of codimension one that is maximally non-integrable: E is locally the kernel of a nonsingular 1-form α such that α ∧ (dα)^{n} is nondegenerate at every point. This condition is independent of the 1-form α defining E, and is equivalent to the condition that any two points in the same path-component can be joined by a smooth curve with velocity field in E. The 1-form α is called a contact 1-form defining E. For more details on contact geometry, see [23, 16, 30].

The restriction of dα to E is a nondegenerate exterior 2-form, making E into a symplectic vector bundle. Such a vector bundle always admits a compatible complex structure J_{E} : E ⟶ E (an automorphism such that J_{E} ∘ J_{E} = − I), which gives E the structure of a Hermitian vector bundle. The contact structure we define on photon space ℙ(ℝ^{4}) ≅ Pho^{2,1} will have such Hermitian structures and contact 1-forms arising from compatible complex structures on the symplectic vector space ℝ^{4}.

5.5.1. Construction of the contact structure. Let v ∈ V be nonzero, and denote the corresponding line by [v] ∈ ℙ(V). The tangent space T_{[v]}ℙ(V) naturally identifies with Hom([v], V/[v]) ([v] ⊂ V denotes the 1-dimensional subspace of V, as well). If V_{1} ⊂ V is a hyperplane complementary to [v], then an affine patch for ℙ(V) containing [v] is given by

Hom([v], V_{1}) ⟶^{A_{V_{1}}} ℙ(V)
ϕ ⟼ [v + ϕ(v)].

That is, A_{V_{1}}(ϕ) is the graph of the linear map ϕ in V = [v] ⊕ V_{1}. This affine patch defines an isomorphism

T_{[v]}ℙ(V) ⟶ Hom([v], V_{1}) ≅ Hom([v], V/[v])

that is independent of the choice of V_{1}. Now, since ω is skew-symmetric, symplectic product with v defines a linear functional

V/[v] ⟶^{α_{v}} ℝ
u ⟼ ω(u, v).

The hyperplane field

[v] ⟼ {φ : α_{v} ∘ φ = 0}

is a well-defined contact plane field on ℙ(V). It posseses a unique transverse orientation; we denote a contact 1-form for this hyperplane field by α.

5.5.2. The contact structure and polarity. The contact structure and the projective geometry of ℙ(V) interact with each other in an interesting way. If p ∈ ℙ(V), then the contact structure at p is a hyperplane E_{p} ⊂ T_{p}ℙ(V). There is a unique projective hyperplane H = H(p) tangent to E_{p} at p. Conversely, suppose H ⊂ ℙ(V) is a projective hyperplane. The contact plane field is transverse to H everywhere but one point, and that point p is the unique point for which H = H(p). This correspondence results from the correspondence between a line ℓ ⊂ V and its symplectic orthogonal ℓ^{⊥} ⊂ V.

The above correspondence is an instance of a polarity in projective geometry. A polarity of a projective space ℙ(V) is a projective isomorphism between ℙ(V) and its dual ℙ(V)^{∗} := ℙ(V^{∗}), arising from a nondegenerate bilinear form on V, which can be either symmetric or skew-symmetric.

Another correspondence is between the set of photons through a given point p ∈ Ein^{2,1} and the set of 1-dimensional linear suspaces of the Lagrangian plane L_{p} ⊂ V. The latter set projects to a projective line in ℙ(V) tangent to the contact plane field, a contact projective line. All contact projective lines arise from points in Ein^{2,1} in this way.

5.5.3. Relation with positive complex structures on ℝ^{4}. A compatible positive complex structure J defines a contact vector field for the contact structure as follows. Let Ω ⊂ ℙ(V) be a subdomain. For any nonzero v ∈ V, the map v ⟼ J(v) defines an element of Hom([v], V/[v]), that is, a tangent vector in T_{[v]}ℙ(V). The resulting vector field ξ_{J} satisfies α(ξ_{J}) > 0 for any 1-form α defining the contact structure, since ω(v, J v) > 0 for nonzero v ∈ V. More generally, for any smooth map J : Ω ⟶ 𝔖_{2}, this construction defines a contact vector field.

5.6. The Maslov cycle. Given a 2n-dimensional symplectic vector space V over ℝ, the set Lag(V) of Lagrangian subspaces of V is a compact homogeneous space. It identifies with U(n)/ O(n), given a choice of a positive compatible complex structure on V ≅ ℝ^{2n}. The fundamental group

π_{1}( Lag(V)) ≅ ℤ.

An explicit isomorphism is given by the Maslov index, which associates to a loop γ in Lag(V) an integer. (See McDuff-Salamon [23], §2.4 for a general discussion.)

Let W ∈ Lag(V) be a Lagrangian subspace. The Maslov cycle Maslov_{W} (V) associated to W is the subset of Lag(V) consisting of W′ such that

W ∩ W′ ≠ 0.

Although it is not a submanifold, Maslov_{W} (V) carries a natural co-orientation (orientation of its conormal bundle) and defines a cycle whose homology class generates H_{N−1}(Lag(V), ℤ) where

N = \frac{n(n + 1)}{2} = dim ( Lag(V)).

The Maslov index of a loop γ is the oriented intersection number of γ with the Maslov cycle (after γ is homotoped to be transverse to Maslov_{W} (V)). If p ∈ Ein^{2,1} corresponds to a Lagrangian subspace W ⊂ V, then the Maslov cycle Maslov_{W} (V) corresponds to the lightcone L(p). (We thank A. Wienhard for this observation.)

5.7. Summary. We now have a dictionary between the symplectic geometry of ℝ_{ω}^{4} and the orthogonal geometry of ℝ^{3,2}:

Symplectic ℝ_{ω}^{4} and contact ℙ(V)  Pseudo-Riemannian ℝ^{3,2} and Ein^{2,1}
Lagrangian planes L ⊂ ℝ_{ω}^{4}  Points p ∈ Ein^{2,1}
Contact projective lines in ℙ(V)  Points p ∈ Ein^{2,1}
Lines ℓ ⊂ ℝ_{ω}^{4}  Photons ϕ
Hyperplanes ℓ^{⊥} ⊂ ℝ_{ω}^{4}  Lightcones
Symplectic planes (splittings) in ℝ_{ω}^{4}  Einstein hyperspheres
Linear symplectic automorphisms  time-preserving conformal automorphisms
Linear anti-symplectic automorphisms time-reversing conformal automorphisms
Flags ℓ ⊂ L ⊂ ℓ^{⊥} in ℝ_{ω}^{4}  Incident pairs p ∈ ϕ ⊂ L(p)
Positive compatible complex structures Free involutions of Ein^{2,1}
Lagrangian splittings V = L_{1} ⊕ L_{2}  Nonincident pairs of points
Lagrangian splittings V = L_{1} ⊕ L_{2}  Spacelike circles

6. Lie theory of Pho^{2,1} and Ein^{2,1}

This section treats the structure of the Lie algebra 𝔰𝔭(4, ℝ) and the isomorphism with 𝔬(3, 2). We relate differential-geometric properties of the homogeneous spaces Ein^{2,1} and Pho^{2,1} with the Lie algebra representations corresponding to the isotropy. This section develops the structure theory (Cartan subalgebras, roots, parabolic subalgebras) and relates these algebraic notions to the synthetic geometry of the three parabolic homogenous spaces Ein^{2,1}, Pho^{2,1} and Flag^{2,1}. Finally, we discuss the geometric significance of the Weyl group of Sp(4, ℝ) and SO(2, 3).

6.1. Structure theory. Let V ≅ ℝ^{4}, equipped with the symplectic form ω, as above. We consider a symplectic basis e_{1}, e_{2}, e_{3}, e_{4} in which ω is

J = [\begin{matrix}0 & −1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 0 & −1 \\ 0 & 0 & 1 & 0 \end{matrix}]

The Lie algebra 𝔤 = 𝔰𝔭(4, ℝ) consists of all 4 × 4 real matrices M satisfying

M^{†} J + J M = 0,

that is,

M = [\begin{matrix}a & a_{12} & r_{11} & r_{12} \\ a_{21} & −a & r_{21} & r_{22} \\ −r_{22} & r_{12} & b & b_{12} \\ r_{21} & −r_{11} & b_{21} & −b \end{matrix}]  (9)

where a, b, a_{ij}, b_{ij}, r_{ij} ∈ ℝ.

6.1.1. Cartan subalgebras. A Cartan subalgebra 𝔞 of 𝔰𝔭(4, ℝ) is the subalgebra stabilizing the four coordinate lines ℝe_{i} for i = 1, 2, 3, 4, and comprises the diagonal matrices

H(a, b) := [\begin{matrix}a & 0 & 0 & 0 \\ 0 & −a & 0 & 0 \\ 0 & 0 & b & 0 \\ 0 & 0 & 0 & −b \end{matrix}]

for a, b ∈ ℝ. The calculation

[H, M] = [\begin{matrix}0 & (2a)a_{12} & (a − b)r_{11} & (a + b)r_{12} \\ (−2a)a_{21} & 0 & (−a − b)r_{21} & (−a + b)r_{22} \\ (a − b)r_{22} & (−a − b)r_{12} & 0 & (2b)b_{12} \\ (a + b)r_{21} & (−a + b)r_{11} & (−2b)b_{21} & 0 \end{matrix}]

implies that the eight linear functionals assigning to H(a, b) the values

2a, −2a, 2b, −2b, a − b, a + b, −a − b, −a + b

define the root system

Δ := {(2, 0), (−2, 0), (0, 2), (0, −2),
(1, −1), (1, 1), (−1, −1), (−1, 1)} ⊂ 𝔞^{∗}

pictured below.

Figure 4. Root diagram of 𝔰𝔭(4, ℝ)

6.1.2. Positive and negative roots. A vector v_{0} ∈ 𝔞 such that λ(v_{0}) ≠ 0 for all roots λ ∈ Δ partitions Δ into positive roots Δ_{+} and negative roots Δ_{−} depending on whether λ(v_{0}) > 0 or λ(v_{0}) < 0 respectively. For example,

v_{0} = [\begin{matrix}1 \\ 2 \end{matrix}]

partitions Δ into

Δ_{+} = { (2, 0), (1, 1), (0, 2), (−1, 1) }
Δ_{−} = { (−2, 0), (−1, −1), (0, −2), (1, −1) }.

The positive roots

α := (2, 0), β := (−1, 1)

form a pair of simple positive roots in the sense that every λ ∈ Δ_{+} is a positive integral linear combination of α and β. Explicitly:

Δ_{+} = {α, α + β, α + 2β, β}.

6.1.3. Root space decomposition. For any root λ ∈ Δ, define the root space

𝔤_{λ} := {X ∈ 𝔤 | [H, X] = λ(H)X}.

In 𝔤 = 𝔰𝔭(4, ℝ), each root space is one-dimensional, and the elements X_{λ} ∈ 𝔤_{λ} are called root elements. The Lie algebra decomposes as a direct sum of vector spaces:

𝔤 = 𝔞 ⊕ ⨁_{λ∈Δ} 𝔤_{λ}.

For more details, see Samelson [28].

6.2. Symplectic splittings. The basis vectors e_{1}, e_{2} span a symplectic plane P ⊂ V and e_{3}, e_{4} span its symplectic complement P^{⊥} ⊂ V. These planes define a symplectic direct sum decomposition

V = P ⊕ P^{⊥}.

The subalgebra 𝔥_{P} ⊂ 𝔰𝔭(4, ℝ) preserving P also preserves P^{⊥} and consists of matrices of the form (9) that are block-diagonal:

[\begin{matrix}a & a_{12} & 0 & 0 \\ a_{21} & −a & 0 & 0 \\ 0 & 0 & b & b_{12} \\ 0 & 0 & b_{21} & −b \end{matrix}] .

Thus

𝔥_{P} ≅ 𝔰𝔭(2, ℝ) ⊕ 𝔰𝔭(2, ℝ)
≅ 𝔰𝔩(2, ℝ) ⊕ 𝔰𝔩(2, ℝ).

The Cartan subalgebra 𝔞 of 𝔰𝔭(4, ℝ) is also a Cartan subalgebra of 𝔥_{P}, but only the four long roots

Δ′ = {(±2, 0), (0, ±2)} = {±α, ±(α + 2β)}

are roots of 𝔥_{P}. In particular 𝔥_{P} decomposes as

𝔥_{P} = 𝔞 ⊕ ⨁_{λ∈Δ′} 𝔤_{λ}.

6.3. The Orthogonal Representation of 𝔰𝔭(4, ℝ). Let e_{1}, …, e_{4}

be a symplectic basis for V as above and

vol := e_{1} ∧ e_{2} ∧ e_{3} ∧ e_{4}

a volume element for V. A convenient basis for Λ^{2}V is:

f_{1} := e_{1} ∧ e_{3}
f_{2} := e_{2} ∧ e_{3}
f_{3} := \frac{1}{\sqrt{2}}(e_{1} ∧ e_{2} − e_{3} ∧ e_{4})
f_{4} := e_{4} ∧ e_{1}
f_{5} := e_{2} ∧ e_{4}  (10)

for which the matrix

[\begin{matrix}0 & 0 & 0 & 0 & 1 \\ 0 & 0 & 0 & 1 & 0 \\ 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 \end{matrix}]

defines the bilinear form 𝔹 associated to this volume element.

The matrix M defined in (9) above maps to

M̃ = [\begin{matrix}a + b & a_{12} & r_{12} & −b_{12} & 0 \\ a_{21} & −a + b & r_{22} & 0 & b_{12} \\ r_{21} & r_{11} & 0 & −r_{22} & −r_{12} \\ −b_{21} & 0 & −r_{11} & a − b & −a_{12} \\ 0 & b_{21} & −r_{21} & −a_{21} & −a − b \end{matrix}] ∈ 𝔰𝔬(3, 2).  (11)

For a fixed symplectic plane P ⊂ V, such as the one spanned by e_{1} and e_{2}, denote by P ∧P^{⊥} the subspace of Λ^{2}V of elements that can be written in the form ∑_{i} v_{i} ∧w_{i}, where v_{i} ∈ P and w_{i} ∈ P^{⊥} for all i. The restriction of the bilinear form 𝔹 to this subspace, which has basis {f_{1}, f_{2}, f_{4}, f_{5}}, is type (2, 2). Its stabilizer is the image 𝔥̃_{P} of 𝔥_{P} in 𝔬(3, 2). Note that this image is isomorphic to

𝔬(2, 2) ≅ 𝔰𝔩(2, ℝ) ⊕ 𝔰𝔩(2, ℝ).

6.4. Parabolic subalgebras. The homogeneous spaces Ein^{2,1}, Pho^{2,1} and

Flag^{2,1} identify with quotients G/P of G = Sp(4, ℝ) where P ⊂ G is a proper parabolic subgroup. When G is algebraic, then any parabolic subgroup P of G is algebraic, and the quotient G/P is a compact projective variety. See Chapter 7 of [18] for more details.

As usual, working with Lie algebras is more convenient. We denote the corresponding parabolic subalgebras by 𝔭, and they are indexed by subsets S ⊂ Π^{−} of the set Π^{−} := {−α, −β} of simple negative roots, as follows.

The Borel subalgebra or minimal parabolic subalgebra corresponds to S = ∅ and is defined as

𝔭_{∅} := 𝔭_{S} = 𝔞 ⊕ ⨁_{λ∈Δ^{+}} 𝔤_{λ}.

In general, let S̃ be the set of finite sums of elements of S. The parabolic subalgebra determined by S is

𝔭_{S} := 𝔭_{∅} ⊕ ⨁_{λ∈S̃} 𝔤_{λ}.

6.4.1. The Borel subalgebra and Flag^{2,1}. Let 𝔭_{∅} be the Borel subalgebra defined above. The corresponding Lie subgroup P_{∅} is the stabilizer of a unique pointed photon, equivalently, an isotropic flag, in Flag^{2,1}; thus Flag^{2,1} identifies with the homogeneous space G/P_{∅}. The subalgebra

𝔲_{∅} := ∑_{λ∈Δ_{+}} 𝔤_{λ} ⊂ 𝔰𝔭(4, ℝ)

is the Lie algebra of the unipotent radical of P_{∅} and is 3-step nilpotent. A realization of the corresponding group is the group generated by the translations of 𝔼^{2,1} and a unipotent one-parameter subgroup of SO(2, 1).

6.4.2. The parabolic subgroup corresponding to Pho^{2,1}. Now let S = {−α}; the corresponding parabolic subalgebra 𝔭_{α} is the stabilizer subalgebra of a line in V, or, equivalently, of a point in ℙ(V). In 𝔬(3, 2) this parabolic is the stabilizer of a null plane in ℝ^{3,2}, or, equivalently, of a photon in Ein^{2,1}.

6.4.3. The parabolic subgroup corresponding to Ein^{2,1}. Now let S = {−β}; the corresponding parabolic subalgebra 𝔭_{β} is the stabilizer subalgebra of a Lagrangian plane in V, or, equivalently, a contact projective line in ℙ(V). In 𝔬(3, 2), this parabolic is the stabilizer of a null line in ℝ^{3,2}, or, equivalently, of a point in Ein^{2,1}.

6.5. Weyl groups. The Weyl group W of Sp(4, ℝ) is isomorphic to a dihedral group of order 8 (see Figure 4). It acts by permutations on elements of the quadruples in ℙ(V) corresponding to a basis of V.

Let A be the connected subgroup of Sp(4, ℝ), with Lie algebra 𝔞. In the symplectic basis e_{1}, …, e_{4}, it consists of matrices of the form

[ \begin{matrix}a_{1} \\ & a_{1}^{−1} \\ & & a_{2} \\ & & & a_{2}^{−1} \end{matrix} ] a_{1}, a_{2} > 0.

The semigroup A^{+} ⊂ A with a_{2} > a_{1} > 1 corresponds to an open Weyl chamber in 𝔞. For i = 1, 2, 3, 4, let H_{i} be the image in ℙ(V) of the hyperplane spanned by e_{j} for j ≠ i. The point [e_{3}] ∈ ℙ(V) is an attracting fixed point for all sequences in A^{+}, and [e_{4}] is a repelling fixed point: Any unbounded a_{n} ∈ A^{+} converges uniformly on compact subsets of ℙ(V)\H_{3} to the constant map [e_{3}], while a_{n}^{−1} converges to [e_{4}] uniformly on compact subsets of ℙ(V)\H_{2}. On H_{3}\(H_{3} ∩ H_{1}), an unbounded sequence {a_{n}} converges to [e_{1}], while on H_{4}\(H_{4} ∩ H_{2}), the inverses a_{n}^{−1} converge to [e_{2}].

We will call the point [e_{1}] a codimension-one attracting fixed point for sequences in A^{+} and [e_{2}] a codimension-one repelling fixed point. Every Weyl chamber has associated to it a dynamical quadruple like ([e_{3}], [e_{4}], [e_{1}], [e_{2}]), consisting of an attracting fixed point, a repelling fixed point, a codimension-one attracting fixed point, and a codimension-one repelling fixed point.

Conversely, given a symplectic basis v_{1}, …, v_{4}, the intersection of the stabilizers in Sp(4, ℝ) of the lines ℝv_{i} is a Cartan subgroup A. The elements a ∈ A such that ([v_{1}], …, [v_{4}]) is a dynamical quadruple for the sequence a^{n} form a semigroup A^{+} that is an open Weyl chamber in A.

The Weyl group acts as a group of permutations of such a quadruple. These permutations must preserve a stem configuration as in Figure 1, where now two points are connected by an edge if the corresponding lines in V are in a common Lagrangian plane, or, equivalently, the two points of ℙ(V) span a line tangent to the contact structure. The permissible permutations are those preserving the partition {v_{1}, v_{2}}|{v_{3}, v_{4}}.

In O(3, 2), the Weyl group consists of permutations of four points p_{1}, …, p_{4} of Ein^{2,1} in a stem configuration that preserve the configuration. A Weyl chamber again corresponds to a dynamical quadruple (p_{1}, …, p_{4}) of fixed points, where now sequences a_{n} ∈ A^{+} converge to the constant map p_{1} on the complement of L(p_{2}) and to p_{3} on L(p_{2})\(L(p_{4}) ∩ L(p_{2})); the inverse sequence converges to p_{2} on the complement of L(p_{1}) and to p_{4} on L(p_{1})\(L(p_{1}) ∩ L(p_{3})).

7. Three kinds of dynamics

In this section, we present the ways sequences in Sp(4, ℝ) can diverge to infinity in terms of projective singular limits. In [13], Frances defines a trichotomy for sequences diverging to infinity in O(3, 2): they have bounded, mixed, or balanced distortion. He introduces limit sets for such sequences and finds maximal domains of proper discontinuity for certain subgroups of O(3, 2). We translate Frances’ trichotomy to Sp(4, ℝ), along with the associated limit sets and maximal domains of properness.

7.1. Projective singular limits. Let E be a finite-dimensional vector space, and let (g_{n})_{n∈ℕ} be a sequence of elements of GL(E). This sequence induces a sequence (ḡ_{n})_{n∈ℕ} of projective transformations of ℙ(E). Let ∥ · ∥ be an auxiliary Euclidean norm on E and let ∥·∥_{∞} be the associated operator norm on the space of endomorphisms End(E). The division of g_{n} by its norm ∥g_{n}∥_{∞} does not modify the projective transformation ḡ_{n}. Hence we can assume that g_{n} belongs to the ∥ · ∥_{∞}- unit sphere of End(E). This sphere is compact, so (g_{n})_{n∈ℕ} admits accumulation points. Up to a subsequence, we can assume that (g_{n})_{n∈ℕ} converges to an element g_{∞} of the ∥ · ∥_{∞}-unit sphere. Let I be the image of g_{∞}, and let L be the kernel of g_{∞}. Let

ḡ_{∞} : ℙ(E) \ ℙ(L) → ℙ(I) ⊂ ℙ(E)

be the induced map.

Proposition 7.1.1. For any compact K ⊂ ℙ(E) \ ℙ(L), the restriction of the sequence (ḡ_{n})_{(n∈ℕ)} on K converges uniformly to the restriction on K of ḡ_{∞}.

Corollary 7.1.2. Let Γ be a discrete subgroup of PGL(E). Let Ω be the open subset of ℙ(E) formed by points admitting a neighborhood U such that, for any sequence (g_{n}) in Γ with accumulation point g_{∞} having image I and kernel L,

U ∩ ℙ(L) = U ∩ ℙ(I) = ∅.

Then Γ acts properly discontinuously on Ω.

In fact, the condition U ∩ ℙ(L) = ∅ is sufficient to define Ω (as is U ∩ ℙ(I) = ∅). To see this, note that if g_{n} → ∞ with

g_{n}/∥g_{n}∥_{∞} ⟶ g_{∞}
g_{n}^{−1}/∥g_{n}^{−1}∥_{∞} ⟶ g_{∞}^{−},

then

g_{∞} ∘ g_{∞}^{−} = g_{∞}^{−} ∘ g_{∞} = 0.

Hence

Im(g_{∞}) ⊆ Ker(g_{∞}^{−}) and Im(g_{∞}^{−}) ⊆ Ker(g_{∞}).

7.2. Cartan’s decomposition G = KAK. When (g_{n})_{n∈ℕ} is a se-

quence in a semisimple Lie group G, a very convenient way to identify the accumulation points ḡ_{∞} is to use the KAK-decomposition in G: first select the Euclidean norm ∥ · ∥ so that it is preserved by the maximal compact subgroup K of G. Decompose every g_{n} in the form k_{n}a_{n}k′_{n}, where k_{n} and k′_{n} belong to K, and a_{n} belongs to a fixed Cartan subgroup. We can furthermore require that a_{n} is the image by the exponential of an element of the closure of a Weyl chamber. Up to a subsequence, k_{n} and k′_{n} admit limits k_{∞} and k′_{∞}, respectively. Composition on the right or on the left by an element of K does not change the operator norm, so g_{n} has ∥ · ∥_{∞}-norm 1 if and only if a_{n} has ∥ · ∥_{∞}-norm 1. Let a_{∞} be an accumulation point of (a_{n})_{n∈ℕ}. Then

g_{∞} = k_{∞}a_{∞}k′_{∞}.

The kernel of g_{∞} is the image by (k′_{∞})^{−1} of the kernel of a_{∞}, and the image of g_{∞} is the image by k_{∞} of the image of a_{∞}. Hence, in order to find the singular projective limit ḡ_{∞}, the main task is to find the limit a_{∞}, and this problem is particularly easy when the rank of G is small.

7.2.1. Sequences in Sp(4, ℝ). The image by the exponential map of a Weyl chamber in 𝔰𝔭(4, ℝ) is the semigroup A^{+} ⊂ A of matrices (see §6.5):

A(α_{1}, α_{2}) = [ \begin{matrix}exp(α_{1}) \\ & exp(−α_{1}) \\ & & exp(α_{2}) \\ & & & exp(−α_{2}) \end{matrix} ] α_{2} > α_{1} > 0.

The operator norm of A(α_{1}, α_{2}) is exp(α_{2}). We therefore can distinguish three kinds of dynamical behaviour for a sequence (A(α_{1}^{(n)}, α_{2}^{(n)}))_{n∈ℕ}:

• no distortion: when α_{1}^{(n)} and α_{2}^{(n)} remain bounded,

• bounded distortion: when α_{1}^{(n)} and α_{2}^{(n)} are unbounded, but the difference α_{2}^{(n)} − α_{1}^{(n)} is bounded,

• unbounded distortion: when the sequences α_{1}^{(n)} and α_{2}^{(n)} − α_{1}^{(n)} are unbounded.

This distinction extends to any sequence (g_{n})_{n∈ℕ} in Sp(4, ℝ). Assume that the sequence (g_{n}/∥g_{n}∥_{∞})_{n∈ℕ} converges to a limit g_{∞}. Then:

• For no distortion, the limit g_{∞} is not singular—the sequence (g_{n})_{n∈ℕ} converges in Sp(4, ℝ).

• For bounded distortion, the kernel L and the image I are 2-dimensional. More precisely, they are Lagrangian subspaces of V. The singular projective transformation ḡ_{∞} is defined in the complement of a projective line and takes values in a projective line; these projective lines are both tangent everywhere to the contact structure.

• For unbounded distortion, the singular projective transformation ḡ_{∞} is defined in the complement of a projective hyperplane and admits only one value.

7.2.2. Sequences in SO^{+}(3, 2). The Weyl chamber of SO^{+}(3, 2) is simply the image of the Weyl chamber of 𝔰𝔭(4, ℝ) by the differential of the homomorphism

Sp(4, ℝ) → SO^{+}(3, 2)

defined in §6.2. More precisely, the image of an element A(α_{1}, α_{2}) of A^{+} is A′(a_{1}, a_{2}) where

a_{1} = α_{1} + α_{2}, a_{2} = α_{2} − α_{1}

and:

A′(a_{1}, a_{2}) = [ \begin{matrix}exp(a_{1}) \\ & exp(a_{2}) \\ & & 1 \\ & & & exp(−a_{2}) \\ & & & & exp(−a_{1}) \end{matrix} ] a_{1} > a_{2} > 0.

The KAK decomposition of Sp(4, ℝ) above corresponds under the homomorphism to a KAK decomposition of SO^{+}(3, 2). Reasoning as in the previous section, we distinguish three cases:

• no distortion: when a_{1}^{(n)} and a_{2}^{(n)} remain bounded,

• balanced distortion: when a_{1}^{(n)} and a_{2}^{(n)} are unbounded, but the difference a_{1}^{(n)} − a_{2}^{(n)} is bounded,

• unbalanced distortion: when the sequences a_{1}^{(n)} and a_{1}^{(n)} − a_{2}^{(n)} are unbounded.

The dynamical analysis is similar, but we restrict to the closed subset Ein^{2,1} of ℙ(ℝ^{3,2}):

• No distortion corresponds to sequences (g_{n})_{n∈ℕ} converging in SO^{+}(3, 2).

• For balanced distortion, the intersection between ℙ(L) and Ein^{2,1}, and the intersection between ℙ(I) and Ein^{2,1} are both photons. Hence the restriction of the singular projective transformation ḡ_{∞} to Ein^{2,1} is defined in the complement of a photon and takes value in a photon.

• For unbalanced distortion, the singular projective transformation ḡ_{∞} is defined in the complement of a lightcone and admits only one value.

7.3. Maximal domains of properness. Most of the time, applying

directly Proposition 7.1.1 and Corollary 7.1.2 to a discrete subgroup Γ of Sp(4, ℝ) or SO^{+}(3, 2) in order to find domains where the action of Γ is proper is far from optimal.

Through the morphism Sp(4, ℝ) → SO^{+}(3, 2), a sequence in Sp(4, ℝ) can also be considered as a sequence in SO(3, 2). Observe that our terminology is coherent: a sequence has no distortion in Sp(4, ℝ) if and only if it has no distortion in SO^{+}(3, 2). Observe also that since

a_{1} = α_{1} + α_{2},
a_{2} = α_{2} − α_{1},

a sequence with bounded distortion in Sp(4, ℝ) is unbalanced in SO^{+}(3, 2), and a sequence with balanced distortion in SO^{+}(3, 2) is unbounded in Sp(4, ℝ). In summary, we distinguish three different kinds of non-converging dynamics, covering all the possibilities:

Definition 7.3.1. A sequence (g_{n})_{n∈ℕ} of elements of Sp(4, ℝ) escaping from any compact subset in Sp(4, ℝ) has:

• bounded distortion if the coefficient a_{2}^{(n)} = α_{2}^{(n)} − α_{1}^{(n)} is bounded,

• balanced distortion if the coefficient α_{2}^{(n)} = (a_{1}^{(n)} + a_{2}^{(n)})/2 is bounded,

• mixed distortion if all the coefficients a_{1}^{(n)}, a_{2}^{(n)}, α_{1}^{(n)}, α_{2}^{(n)} are unbounded.

7.3.1. Action on Ein^{2,1}. The dynamical analysis can be refined in the mixed distortion case. In [13], C. Frances proved:

Proposition 7.3.2. Let (g_{n})_{n∈ℕ} be a sequence of elements of SO^{+}(3, 2) with mixed distortion, such that the sequence (g_{n}/∥g_{n}∥_{∞})_{n∈ℕ} converges to an endomorphism g_{∞}. Then there are photons Δ^{−} and Δ^{+} in Ein^{2,1} such that, for any sequence (p_{n})_{n∈ℕ} in Ein^{2,1} converging to an element of Ein^{2,1} \Δ^{−}, all the accumulation points of (g_{n}(p_{n}))_{n∈ℕ} belong to Δ^{+}.

As a corollary (§4.1 in [13]):

Corollary 7.3.3. Let Γ be a discrete subgroup of SO^{+}(3, 2). Let Ω_{0} be the union of all open domains U in Ein^{2,1} such that, for any accumulation point g_{∞}, with kernel L and image I, of a sequence (g_{n}/∥g_{n}∥_{∞})_{n∈ℕ} with g_{n} ∈ SO^{+}(3, 2):

• When (g_{n})_{n∈ℕ} has balanced distortion, U is disjoint from the photons ℙ(L) ∩ Ein^{2,1} and ℙ(I) ∩ Ein^{2,1};

• When (g_{n})_{n∈ℕ} has bounded distortion, U is disjoint from the lightcone ℙ(L)∩ Ein^{2,1};

• When (g_{n})_{n∈ℕ} has mixed distortion, U is disjoint from the photons Δ_{−} and Δ_{+}.

Then the action of Γ on Ω_{0} is properly discontinuous.

Observe that the domain Ω_{0} is in general bigger than the domain Ω appearing in Corollary 7.1.2. An interesting case is that in which Ω_{0} is obtained by removing only photons:

Proposition 7.3.4 (Frances [13]). A discrete subgroup Γ of SO^{+}(3, 2) does not contain sequences with bounded distortion if and only if its action on ℙ(ℝ^{3,2})\Ein^{2,1} is properly discontinuous.

Frances calls such a subgroup a of the first kind. The following suggests that the domain Ω_{0} is optimal.

Proposition 7.3.5 (Frances [13]). Let Γ be a discrete, Zariski dense subgroup of SO^{+}(3, 2) which does not contain sequences with bounded distortion. Then Ω_{0} is the unique maximal open subset of Ein^{2,1} on which Γ acts properly.

7.3.2. Action on ℙ(V ). A similar analysis should be done when Γ is considered a discrete subgroup of Sp(4, ℝ) instead of SO^{+}(3, 2). The following proposition is analogous to Proposition 7.3.2:

Proposition 7.3.6. Let (g_{n})_{n∈ℕ} be a sequence of elements of Sp(V) with mixed distortion, such that the sequence (g_{n}/∥g_{n}∥_{∞})_{n∈ℕ} converges to an endomorphism g_{∞} of V. Then there are contact projective lines Δ^{−} and Δ^{+} in ℙ(V) such that, for any sequence

(p_{n})_{n∈ℕ} ∈ ℙ(V)

converging to an element of ℙ(V) \ Δ^{−}, all the accumulation points of (g_{n}(p_{n}))_{n∈ℕ} belong to Δ^{+}.

We can then define a subset Ω_{1} of ℙ(V) as the interior of the subset obtained after removing limit contact projective lines associated to subsequences of Γ with bounded or mixed distortion, and removing projective hyperplanes associated to subsequences with balanced distortion. Then it is easy to prove that the action of Γ on Ω_{1} is properly discontinuous.

An interesting case is that in which we remove only projective lines, and no hypersurfaces—the case in which Γ has no subsequence with balanced distortion. Frances calls such Γ groups of the second kind. The following questions arise from comparison with Propositions 7.3.5 and 7.3.4:

Question: Can groups of the second kind be defined as groups acting properly on some associated space?

Question: Is Ω_{1} the unique maximal open subset of ℙ(V) on which the action of Γ is proper, at least if Γ is Zariski dense?

7.3.3. Action on the flag manifold. Now consider the action of Sp(4, ℝ) on the flag manifold Flag^{2,1}. Let v, w ∈ V be such that ω(v, w) = 0, so v and w span a Lagrangian plane. Let

Flag^{2,1} ⟶^{ρ_{1}} Pho^{2,1}
Flag^{2,1} ⟶^{ρ_{2}} Ein^{2,1}

be the natural projections. Let g_{n} be a sequence in Sp(4, ℝ) diverging to infinity with mixed distortion. We invite the reader to verify the following statements:

• There are a flag q^{+} ∈ Flag^{2,1} and points [v] ∈ ℙ(V) and z ∈ Ein^{2,1} such that,

on the complement of

ρ_{1}^{−1}([v^{⊥}]) ∪ ρ_{2}^{−1}(L(z))

the sequence g_{n} converges uniformly to the constant map q^{+}.

• There are contact projective lines α^{+},α^{−} in ℙ(V) and photons β^{+},β^{−} in Ein^{2,1} such that, on the complement of

ρ_{1}^{−1}(α^{−}) ∪ ρ_{2}^{−1}(β^{−})

all accumulation points of g_{n} lie in

ρ_{1}^{−1}(α^{+}) ∩ ρ_{2}^{−1}(β^{+}).

This intersection is homeomorphic to a wedge of two circles.

8. Crooked surfaces

Crooked planes were introduced by Drumm [8, 9, 10] to investigate discrete groups of Lorentzian transformations which act freely and properly on 𝔼^{2,1}. He used crooked planes to construct fundamental polyhedra for such actions; they play a role analogous to equidistant surfaces bounding Dirichlet fundamental domains in Hadamard manifolds. This section discusses the conformal compactification of a crooked plane and its automorphisms.

8.1. Crooked planes in Minkowski space. For a detailed description

of crooked planes, see Drumm-Goldman [10]. We quickly summarize the basic results here.

Consider 𝔼^{2,1} with the Lorentz metric from the inner product I_{2} ⊕−I_{1} on ℝ^{2,1}. A crooked plane C is a surface in 𝔼^{2,1} that divides 𝔼^{2,1} into two cells, called crooked half-spaces. It is a piecewise linear surface composed of four 2-dimensional faces, joined along four rays, which all meet at a point p, called the vertex. The four rays have endpoint p, and form two lightlike geodesics, which we denote ℓ_{1} and ℓ_{2}. Two of the faces are null half-planes W_{1} and W_{2}, bounded by ℓ_{1} and ℓ_{2} respectively, which we call wings. The two remaining faces consist of the intersection between J^{±}(p) and the timelike plane P containing ℓ_{1} and ℓ_{2}; their union is the stem of C. The timelike plane P is the orthogonal complement of a unique spacelike line P^{⊥}(p) containing p, called the spine of C.

To define a crooked plane, we first define the wings, stem, and spine. A lightlike geodesic ℓ = p + ℝv lies in a unique null plane ℓ^{⊥} (§2.2). The ambient orientation of ℝ^{2,1} distinguishes a component of ℓ^{⊥} \ ℓ as follows. Let u ∈ ℝ^{2,1} be a timelike vector such that ⟨u, v⟩ < 0. Then each component of ℓ^{⊥} \ ℓ defined by

W^{+}(ℓ) := {p + w ∈ ℓ^{⊥} | det(u, v, w) > 0}
W^{−}(ℓ) := {p + w ∈ ℓ^{⊥} | det(u, v, w) < 0}

is independent of the choices above. In particular, every orientation-preserving isometry f of 𝔼^{2,1} maps

W^{+}(ℓ) ⟶ W^{+}(f(ℓ))
W^{−}(ℓ) ⟶ W^{−}(f(ℓ))

and every orientation-reversing isometry f maps

W^{+}(ℓ) ⟶ W^{−}(f(ℓ))
W^{−}(ℓ) ⟶ W^{+}(f(ℓ)).

Given two lightlike geodesics ℓ_{1}, ℓ_{2} containing p, the stem is defined as

S(ℓ_{1}, ℓ_{2}) := J^{±}(p) ∩ (p + span{ℓ_{1} − p, ℓ_{2} − p}).

The spine is

σ = p + (S(ℓ_{1}, ℓ_{2}) − p)^{⊥}.

Compare Drumm-Goldman [10].

The positively-oriented crooked plane with vertex p and stem S(ℓ_{1}, ℓ_{2}) is the union

W^{+}(ℓ_{1}) ∪ S(ℓ_{1}, ℓ_{2}) ∪ W^{+}(ℓ_{2}).

Similarly, the negatively-oriented crooked plane with vertex p and stem S(ℓ_{1}, ℓ_{2}) is

W^{−}(ℓ_{1}) ∪ S(ℓ_{1}, ℓ_{2}) ∪ W^{−}(ℓ_{2}).

Given an orientation on 𝔼^{2,1}, a positively-oriented crooked plane is determined by its vertex and its spine. Conversely, every point p and spacelike line σ containing p determines a unique positively- or negatively-oriented crooked plane.

A crooked plane C is homeomorphic to ℝ^{2}, and the complement 𝔼^{2,1}\C consists of two components, each homeomorphic to ℝ^{3}. The components of the complement of a crooked plane are called open crooked half-spaces and their closures closed crooked half-spaces. The spine of C is the unique spacelike line contained in C.

8.2. An example. Here is an example of a crooked plane with vertex the origin and spine the x-axis:

p = [\begin{matrix}0 \\ 0 \\ 0 \end{matrix}] , σ = ℝ [\begin{matrix}1 \\ 0 \\ 0 \end{matrix}] .

The lightlike geodesics are

ℓ_{1} = ℝ [\begin{matrix}0 \\ −1 \\ 1 \end{matrix}] , ℓ_{2} = ℝ [\begin{matrix}0 \\ 1 \\ 1 \end{matrix}] ,

the stem is

\{[\begin{matrix}0 \\ y \\ z \end{matrix}] : y^{2} − z^{2} ≤ 0\}

and the wings are

W_{1} = \{[\begin{matrix}x \\ y \\ −y \end{matrix}] : x ≥ 0, y ∈ ℝ\}

W_{2} = \{[\begin{matrix}x \\ y \\ y \end{matrix}] : x ≤ 0, y ∈ ℝ\} .

The identity component of Isom(𝔼^{2,1}) acts transitively on the space of pairs of vertices and unit spacelike vectors, so it is transitive on positively-oriented and negatively-oriented crooked planes. An orientation-reversing isometry exchanges positively- and negatively-oriented crooked planes, so Isom(𝔼^{2,1}) acts transitively on the set of all crooked planes.

8.3. Topology of a crooked surface. The closures of crooked planes in

Minkowski patches are crooked surfaces. These were studied in Frances [12]. In this section we describe the topology of a crooked surface.

Let C ⊂ 𝔼^{2,1} be a crooked plane.

Theorem 8.3.1. The closure \overline{C} ∈ Ein^{2,1} is a topological submanifold homeomorphic to a Klein bottle. The lift of \overline{C} to the double covering Êin^{2,1} is the oriented double covering of \overline{C} and is homeomorphic to a torus.

Proof. Since the isometry group of Minkowski space acts transitively on crooked planes, it suffices to consider the single crooked plane C defined in §8.2.

Recall the stratification of Ein^{2,1} from §3.2. Write the nullcone 𝔑^{3,2} of ℝ^{3,2} as

[\begin{matrix}X \\ Y \\ Z \\ U \\ V \end{matrix}] where X^{2} + Y^{2} − Z^{2} − UV = 0.

The homogeneous coordinates of points in the stem S(C) satisfy

X = 0,  Y^{2} − Z^{2} ≤ 0,  V ≠ 0

and thus the closure of the stem \overline{S(C)} is defined by (homogeneous) inequalities

X = 0, Y^{2} − Z^{2} ≤ 0.

The two lightlike geodesics

ℓ_{1} = ℝ [\begin{matrix}0 \\ −1 \\ 1 \end{matrix}] , ℓ_{2} = ℝ [\begin{matrix}0 \\ 1 \\ 1 \end{matrix}]

defining S(C) extend to photons ϕ_{1}, ϕ_{2} with ideal points represented in homogeneous coordinates

p_{1} = [\begin{matrix}0 \\ −1 \\ 1 \\ 0 \\ 0 \end{matrix}] , p_{2} = [\begin{matrix}0 \\ 1 \\ 1 \\ 0 \\ 0 \end{matrix}] .

The closures of the corresponding wings W_{1}, W_{2} are described in homogeneous coordinates by:

\overline{W_{1}} = \{[\begin{matrix}X \\ −Y \\ Y \\ U \\ V \end{matrix} ] : X^{2} − UV = 0, XV ≥ 0\}

\overline{W_{2}} = \{[\begin{matrix}X \\ Y \\ Y \\ U \\ V \end{matrix}] : X^{2} − UV = 0, XV ≤ 0\} .

The closure of each wing intersects the ideal lightcone L(p_{∞}) (described by V = 0) in the photons:

ψ_{1} = \{[\begin{matrix}0 \\ −Y \\ Y \\ U \\ 0 \end{matrix} ] : Y, U ∈ ℝ\}

ψ_{2} = \{[\begin{matrix}0 \\ Y \\ Y \\ U \\ 0 \end{matrix}] : Y, U ∈ ℝ\} .

Thus the crooked surface \overline{C} decomposes into the following strata:

• four points in a stem configuration: the vertex p_{0}, the improper point p_{∞}, and the two ideal points p_{1} and p_{2}; • eight line segments, the components of

ϕ_{1} \ {p_{0}, p_{1}}
ϕ_{2} \ {p_{0}, p_{2}}
ψ_{1} \ {p_{∞}, p_{1}}
ψ_{2} \ {p_{∞}, p_{2}};

• two null-half planes, the interiors of the wings W_{1}, W_{2};

• the two components of the interior of the stem S.

Recall that the inversion in the unit sphere ι = I_{3} ⊕ [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}] fixes p_{1} and p_{2}, and

interchanges p_{0} and p_{∞}. Moreover ι interchanges ϕ_{i} with ψ_{i}, i = 1, 2. Finally ι leaves invariant the interior of each W_{i} and interchanges the two components of the interior of S.

The original crooked plane equals

{p_{0}} ∪ ϕ_{1} \ {p_{1}} ∪ ϕ_{2} \ {p_{2}} ∪ int(W_{1}) ∪ int(W_{2}) ∪ int(S)

and is homeomorphic to ℝ^{2}. The homeomorphism is depicted schematically in Figure 5. The interiors of W_{1}, W_{2}, and S correspond to the four quadrants in ℝ^{2}. The wing W_{i} is bounded by the two segments of ϕ_{i}, whereas each component of S is bounded by one segment of ϕ_{1} and one segment of ϕ_{2}. These four segments correspond to the four coordinate rays in ℝ^{2}.

Now we can see that C is a topological manifold: points in int(W_{1}), int(W_{2}), or int(S) have coordinate neighborhoods in these faces. Interior points of the segments have two half-disc neighborhoods, one from a wing and one from the stem. The vertex p_{0} has four quarter-disc neighborhoods, one from each wing, and one from each component of the stem. (See Figure 5.)

Figure 5. Flattening a crooked plane around its vertex

Coordinate charts for the improper point p_{∞} and points in ψ_{i} \ {p_{∞}, p_{i}} are obtained by composing the above charts with the inversion ι. It remains to find coordinate charts near the ideal points p_{1}, p_{2}. Consider first the case of p_{1}. The linear functionals on ℝ^{3,2} defined by

T = Y − Z
W = Y + Z

are null since the defining quadratic form factors:

X^{2} + Y^{2} − Z^{2} − UV = X^{2} + T W − UV.

Working in the affine patch defined by T ≠ 0 with inhomogeneous coordinates

ξ := \frac{X}{T}
η := \frac{Y}{T}
ω := \frac{W}{T}
υ := \frac{U}{T}
ν := \frac{V}{T}

the nullcone is defined by:

ξ^{2} + ω − υν = 0

whence

ω = −ξ^{2} + υν

and (ξ, υ, ν) ∈ ℝ^{3} is a coordinate chart for this patch on Ein^{2,1}.

In these coordinates, p_{1} is the origin (0, 0, 0), ϕ_{1} is the line ξ = υ = 0, and ψ_{1} is the line ξ = ν = 0. The wing W_{2} misses this patch, but both S and W_{1} intersect it. In these coordinates S is defined by

ξ = 0, ω ≤ 0

and W_{1} is defined by

ξ ≤ 0, ω = 0.

Since on W_{1}

υν = ξ^{2} ≥ 0

this portion of W_{1} in this patch has two components

υ, ν < 0
υ, ν > 0

and the projection (υ, ν) defines a coordinate chart for a neighborhood of p_{1}. (Compare Figure 6.)

Figure 6. Flattening a crooked surface around an ideal point p_{1}

The case of p_{2} is completely analogous. It follows that \overline{C} is a closed surface with cell decomposition with four 0-cells, eight 1-cells and four 2-cells. Therefore

χ(\overline{C}) = 4 − 8 + 4 = 0

and \overline{C} is homeomorphic to either a torus or a Klein bottle.

To see that \overline{C} is nonorientable, consider a photon, for example ϕ_{1}. Parallel translate the null geodesic ϕ_{1} \ {p_{1}} to a null geodesic ℓ lying on the wing W_{1} and disjoint from ϕ_{1} \ \{p_{1}\}. Its closure ℓ̄ = ℓ ∪ \{p_{1}\} is a photon on \overline{W}_{1} ⊂ \overline{C} which intersects ϕ_{1} transversely with intersection number 1. Thus the self-intersection number

ϕ_{1} · ϕ_{1} = 1

so ϕ_{1} ⊂ \overline{C} is an orientation-reversing loop. Thus \overline{C} is nonorientable, and homeomorphic to a Klein bottle.

Next we describe the stratification of a crooked surface in the double covering Êin^{2,1}. Recall from §4.4 that a Minkowski patch in Êin^{2,1} has both a spatial and a timelike improper point. Let C be a crooked plane of 𝔼^{2,1}, embedded in a Minkowski patch Min^{+}(p_{∞}), so p_{∞} = p_{∞}^{ti}, the timelike improper point of this patch. Denote by p_{∞}^{sp} the spatial improper point.

The closure \overline{C} of C in Êin^{2,1} decomposes into the following strata:

• seven points: p_{0}, p_{∞}^{ti}, p_{∞}^{sp}, p_{1}^{±}, p_{2}^{±};

• twelve photon segments:

ϕ_{i}^{±}, connecting p_{0} to p_{i}^{±}
α_{i}^{±}, connecting p_{∞}^{ti} to p_{i}^{±}
β_{i}^{±}, connecting p_{∞}^{𝔰𝔭} to p_{i}^{±};

• two null half-planes, the interiors of W_{1} and W_{2}. The wing W_{i} is bounded by the curves ϕ_{i}^{±} and β_{i}^{±};

• the two components of the interior of the stem S. The stem is bounded by the curves ϕ_{i}^{±} and α_{i}^{±}, for i = 1, 2.

The saturation of \overline{C} by the antipodal map on Êin^{2,1} is the lift of a crooked surface from Ein^{2,1}. The interested reader can verify that it is homeomorphic to a torus.

8.4. Automorphisms of a crooked surface. Let C be the positively-

oriented crooked plane of Section 8.2, and \overline{C} the associated crooked surface in Ein^{2,1}. First, C is invariant by all positive homotheties centered at the origin, because each of the wings and the stem are. Second, it is invariant by the 1dimensional group of linear hyperbolic isometries of Minkowski space preserving the lightlike lines bounding the stem. The subgroup A, which can be viewed as the subgroup of SO(3, 2) acting by positive homotheties and positive linear hyperbolic isometries of Minkowski space, then preserves C, and hence \overline{C}. The element

s_{0} = ( \begin{matrix}1 \\ & −1 \\ & & −1 \\ & & & 1 \\ & & & & 1 \end{matrix} )

is a reflection in the spine, and also preserves \overline{C}. Note that s_{0} is time-reversing. Then we have

ℤ_{2} ⋉ A ≅ ℤ_{2} ⋉ (ℝ_{>0}^{∗})^{2} ⊂ Aut(\overline{C}).

Next let ℓ_{1}, ℓ_{2} be the two lightlike geodesics bounding the stem (alternatively bounding the wings) of C. As above, the inversion ι leaves invariant C \ (ℓ_{1} ∪ ℓ_{2}). In fact, the element

s_{1} = ( \begin{matrix}−1 \\ & −1 \\ & & −1 \\ & & & & 1 \\ & & & 1 \end{matrix} )

is an automorphism of \overline{C}. The involution

s_{2} = ( \begin{matrix}−1 \\ & 1 \\ & & −1 \\ & & & 1 \\ & & & & 1 \end{matrix} )

also preserves \overline{C} and exchanges the ideal points p_{1} and p_{2}. The involutions s_{0}, s_{1}, and s_{2} pairwise commute, and each product is also an involution, so we have

G := ℤ_{2}^{3} ⋉ (ℝ_{>0}^{∗})^{2} ⊂ Aut(\overline{C})

To any crooked surface can be associated a quadruple of points in a stem configuration. The stabilizer of a stem configuration in SO(3, 2) ≅ PO(3, 2) is N(A), the normalizer of a Cartan subgroup A. Suppose that the points (p_{0}, p_{1}, p_{2}, p_{∞}) are associated to \overline{C}. As above, a neighborhood of p_{0} in \overline{C} is not diffeomorphic to a neighborhood of p_{1} in \overline{C}, so any automorphism must in fact belong to the subgroup N′(A) preserving each pair {p_{0}, p_{∞}} and {p_{1}, p_{2}}. Each g ∈ N′(A) either preserves \overline{C} or carries it to its opposite, the closure of the negatively-oriented crooked plane having the same vertex and spine as C. Now it is not hard to verify that the full automorphism group of \overline{C} in SO(3, 2) is G.

9. Construction of discrete groups

A complete flat Lorentzian manifold is a quotient 𝔼^{n,1}/Γ, where Γ acts freely and properly discontinuously on 𝔼^{n,1}. When n = 2, Fried and Goldman [15] showed that unless Γ is solvable, projection on O(2, 1) is necessarily injective and, furthermore, this linear part is a discrete subgroup Γ_{0} ⊂ O(2, 1)[1, 6, 24].

In this section we identify 𝔼^{2,1} with its usual embedding in Ein^{2,1}, so that we consider such Γ as discrete subgroups of SO(3, 2). We will look at the resulting actions on Einstein space, as well as on photon space. At the end of the section, we list some open questions.

9.1. Spine reflections. In §8.4, we described the automorphism group of a crooked surface. We recall some of the basic facts about the reflection in the spine of a crooked surface, which is discussed in §3.3 and §5.2.2, and which is denoted s_{0} in the example above. Take the inner product on ℝ^{3,2} to be given by the matrix

I_{2} ⊕ − I_{1} ⊕ (−\frac{1}{2}) [\begin{matrix}0 & 1 \\ 1 & 0 \end{matrix}]

and identify 𝔼^{2,1} with its usual embedding in the Minkowski patch determined by the improper point p_{∞}. Let C be the crooked plane determined by the stem configuration (p_{0}, p_{1}, p_{2}, p_{∞}) as in §8.2, with

p_{1} = [\begin{matrix}0 \\ −1 \\ 1 \\ 0 \\ 0 \end{matrix}] and p_{2} = [\begin{matrix}0 \\ 1 \\ 1 \\ 0 \\ 0 \end{matrix}] .

Then s_{0} is an orientation-preserving, time-reversing involution having fixed set

Fix(s_{0}) = {p_{1}, p_{2}} ∪ (L(p_{1}) ∩ L(p_{2})).

In the Minkowski patch, ⟨s_{0}⟩ interchanges the two components of the complement of C.

If a set of crooked planes in 𝔼^{2,1} is pairwise disjoint, then the group generated by reflections in their spines acts properly discontinuously on the entire space [7, 8, 10]. Thus spine reflections associated to disjoint crooked planes give rise to discrete subgroups of SO(3, 2). We will outline a way to construct such groups; see [4], for details.

Let S_{1}, S_{2} ⊂ Ein^{2,1} be a pair of spacelike circles that intersect in a point; conjugating if necessary, we may assume that this point is p_{∞}. Each circle S_{i}, i = 1, 2, is the projectivized nullcone of a subspace V_{i} ⊂ ℝ^{3,2} of type (2,1); V_{1} + V_{2} can be written as the direct sum

ℝv_{1} ⊕ ℝv_{2} ⊕ W,

where v_{1}, v_{2} are spacelike vectors and W = V_{1} ∩ V_{2} is of type (1,1). We call {S_{1}, S_{2}} an ultraparallel pair if v_{1}^{⊥} ∩ v_{2}^{⊥} is spacelike. Alternatively, we can define the pair to be ultraparallel if they are parallel to vectors u_{1}, u_{2} ∈ ℝ^{2,1} such that u_{1}^{⊥} ∩ u_{2}^{⊥} is a spacelike line in 𝔼^{2,1}.

Let S_{1}, S_{2} be an ultraparallel pair of spacelike circles in Ein^{2,1}. Denote by ι_{1} and ι_{2} the spine reflections fixing the respective circles. (Note that ι_{1} and ι_{2} are conjugate to s_{0}, since SO(3, 2) acts transitively on crooked surfaces.) Identifying the subgroup of SO(3, 2) fixing p_{0} and p_{∞} with the group of Lorentzian linear similarities

Sim(𝔼^{2,1}) = ℝ_{+} · O(2, 1),

then γ = ι_{2} ∘ ι_{1} has hyperbolic linear part—that is, it has three, distinct real eigenvalues. The proof of this fact and the following proposition may be found, for instance, in [4].

Proposition 9.1.1. Let S_{1} and S_{2} be an ultraparallel pair of spacelike circles as above. Then S_{1} and S_{2} are the spines of a pair of disjoint crooked planes, bounding a fundamental domain for ⟨γ⟩ in 𝔼^{2,1}.

Note that while ⟨γ⟩ acts freely and properly discontinuously on 𝔼^{2,1}, it fixes p_{∞} as well as two points on the ideal circle.

Next, let S_{i}, i = 1, 2, 3 be a triple of pairwise ultraparallel spacelike circles, all intersecting in p_{∞}, and let Γ = ⟨ι_{1}, ι_{2}, ι_{3}⟩ be the associated group of spine reflections. Then Γ contains an index-two free group generated by hyperbolic isometries of 𝔼^{2,1} (see [4]). Conversely, we have the following generalization of a well-known theorem in hyperbolic geometry.

Theorem 9.1.2. [4] Let Γ = ⟨γ_{1}, γ_{2}, γ_{3} | γ_{1}γ_{2}γ_{3} = Id⟩ be a subgroup of isometries of 𝔼^{2,1}, where each γ has hyperbolic linear part and such that their invariant lines are pairwise ultraparallel. Then there exist spine reflections ι_{i}, i = 1, 2, 3, such that γ_{1} = ι_{1}ι_{2}, γ_{2} = ι_{2}ι_{3} and γ_{3} = ι_{3}ι_{1}.

Note that Γ as above is discrete. Indeed, viewed as a group of affine isometries of 𝔼^{2,1}, its linear part G ≤ O(2, 1) acts on the hyperbolic plane and is generated by reflections in three ultraparallel lines. As mentioned before, if the spacelike circles are spines of pairwise disjoint crooked planes, then Γ acts properly discontinuously on the Minkowski patch. Applying this strategy, we obtain that the set of all properly discontinuous groups Γ, with linear part generated by three ultraparallel reflections, is non-empty and open [4].

Here is an example. For i = 1, 2, 3, let V_{i} ⊂ ℝ^{3,2} be the (2, 1)-subspace

V_{i} = \{[\begin{matrix}au_{i} + cp_{i} \\ a⟨u_{i}, p_{i}⟩ + b + c⟨p_{i}, p_{i}⟩ \\ c \end{matrix}] | a, b, c ∈ ℝ\} ,

where

u_{1} = [\sqrt{2} 0 1]^{†} u_{2} = [−\frac{\sqrt{2}}{2} \frac{\sqrt{6}}{2} 1]^{†}  u_{3} = [−\frac{\sqrt{2}}{2} −\frac{\sqrt{6}}{2} 1]^{†}
p_{1} = [0 \sqrt{2} 1]^{†} p_{2} = [−\frac{\sqrt{6}}{2} −\frac{\sqrt{2}}{2} 1]^{†} p_{3} = [\frac{\sqrt{6}}{2}  −\frac{\sqrt{2}}{2} 1]^{†} .

Then the projectivized nullcone of V_{i} is a spacelike circle—in fact, it corresponds to the spacelike geodesic in 𝔼^{2,1} passing through p_{i} and parallel to u_{i}. The crooked planes with vertex p_{i} and spine p + ℝu_{i}, respectively, are pairwise disjoint (one shows this using inequalities found in [10]).

9.2. Actions on photon space. Still in the same Minkowski patch as above, let G be a finitely generated discrete subgroup of O(2, 1) that is free and purely hyperbolic—that is, every nontrivial element is hyperbolic. Considered as a group of isometries of the hyperbolic plane, G is a convex cocompact free group. By Barbot [2],

Theorem 9.2.1. Let Γ be a subgroup of isometries of 𝔼^{2,1} with convex cocompact linear part. Then there is a pair of non-empty, Γ-invariant, open, convex sets Ω^{±} ⊂ 𝔼^{2,1} such that

• The action of Γ on Ω^{±} is free and proper;

• The quotient spaces Ω^{±}/Γ are globally hyperbolic;

• Each Ω^{±} is maximal among connected open domains satisfying these two properties;

• The only open domains satisfying all three properties above are Ω^{±}.

The notion of global hyperbolicity is central in General Relativity, see for example [3]. The global hyperbolicity of Ω^{±}/Γ implies that it is homeomorphic to the product (H^{2}/G) × ℝ. It also implies that no element of Γ preserves a null ray in Ω^{±}.

Let Γ be as in Theorem 9.2.1 and consider its action, for instance, on Ω^{+}. Since Ω^{+}/Γ is globally hyperbolic, it admits a Cauchy hypersurface, a spacelike surface S_{0} which meets every complete causal curve and with complement consisting of two connected components. The universal covering S̃_{0} is Γ-invariant. The subset Pho_{0}^{2,1} ⊂ Pho^{2,1} comprising photons which intersect S̃_{0} is open.

We claim that Γ acts freely and properly on Pho_{0}^{2,1}. Indeed, let K ≤ Pho_{0}^{2,1} be a compact set. Then K is contained in a product of compact subsets K_{1} × K_{2}, where K_{1} ⊂ S̃_{0} and K_{2} ⊂ S^{1}, the set of photon directions. The action of Γ restricts to a Riemannian action on S̃_{0}. Thus the set

{γ ∈ Γ | γ(K_{1}) ∩ K_{1} ≠ ∅}

is finite. As S̃_{0} is spacelike, it follows that {γ ∈ Γ | γ(K) ∩ K ≠ ∅} is finite too. Finally, global hyperbolicity of Ω^{+}/Γ implies that no photon intersecting Ω^{+} is invariant under the action of any element of Γ.

Corollary 9.2.2. There exists a non-empty open subset of Pho^{2,1} on which Γ acts freely and properly discontinuously.

9.3. Some questions. So far we have considered groups of transformations of Ein^{2,1} and Pho^{2,1} arising from discrete groups of Minkowski isometries. Specifically, we have focused on groups generated by spine reflections associated to spacelike circles intersecting in a point.

Question: Describe the action on Ein^{2,1} of a group generated by spine reflections corresponding to non-intersecting spacelike circles. In particular, determine the possible dynamics of such an action.

A related question is:

Question: What does a crooked surface look like when its spine does not pass through p_{∞}, or the lightcone at infinity altogether? Describe the action of the associated group of spine reflections.

More generally, we may wish to consider other involutions in the automorphism group of a crooked surface.

Question: Describe the action on Ein^{2,1} of a group generated by involutions, in terms of their associated crooked surfaces.

As for the action on photon space, here is a companion question to those asked in §7:

Question: Given a group generated by involutions, what is the maximal open subset of Pho^{2,1} on which the group acts properly discontinuously?

References

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Index

affine lightcone, 7 anti-de Sitter space, 9 anti-symplectic

involution, 25 attracting fixed point, 34

codimension-one, 34

Cartan decomposition, 35 causal

curve, 19 vector, 5

causally related, 19, 20 complex structure, 28 conformal inversion

Euclidean, 13 Lorentzian, 14

contact 1-form, 27 plane field, 27 structure, 27 vector field, 28

crooked

plane, 40 negatively-oriented, 41 positively-oriented, 41

surface, 42

dictionary, 29 discrete subgroup

of the first kind, 39 of the second kind, 39

dynamical quadruple, 34

Einstein hypersphere, 9, 25

flag

isotropic, 12 future

-conjugate point, 21 -oriented causal curve, 19 -oriented tangent vector, 18 -oriented vector, 18 causal, of a point, 19 lightcone, 18 of a point, 19

ideal sphere, 15 improper point, 15

spatial, 21 timelike, 21

Lagrangian Grassmannian, 22, 28 plane, 24

lightcone, 8 lightlike

vector, 5 Lorentzian vector space, 4

Maslov cycle, 28 index, 29

Minkowski patch, 8 negative, 8 positive, 8, 20

Minkowski space, 6

nullcone, 5

orientation, 18

parablolic

subgroup, 33 parabolic

subalgebra, 33 subgroup, 33

past -oriented causal curve, 19 -oriented tangent vector, 18 -oriented vector, 18 causal, of a point, 19 lightcone, 18 of a point, 19

photon, 8, 33 pointed photon, 12 polar decomposition, 35

repelling fixed point, 34

codimension-one, 34 root diagram

𝔰𝔭(4, ℝ), 30

sequence balanced distortion, 37 bounded distortion, 36 mixed distortion, 38 no-distortion, 36 unbalanced, 37 unbounded, 36

Siegel upper-half space, 26 similarity transformation

Euclidean, 13 Minkowski, 7

spacelike circle, 11 hypersphere, 9 vector, 5

stem, 40 stem configuration, 10, 43, 47 symplectic

involution, 25 plane, 25, 32 vector space, 23

symplectic basis, 29

time orientation, 17 timelike

circle, 11 curve, 19 vector, 5

Weyl chamber, 34 group, 33

wing, 40

1. Thierry Barbot, CNRS, UMR 5669, École Normale Supérieure de Lyon, 46 allée d’Italie, 69364 Lyon, France E-mail: Thierry.Barbot@umpa.ens-lyon.fr

2. Virginie Charette, Département de mathématiques, Université de Sherbrooke, Sherbrooke, Quebec, Canada E-mail: v.charette@usherbrooke.ca

3. Todd A. Drumm Department of Mathematics, University of Pennsylvania, Philadelphia, PA USA E-mail: tad@math.upenn.edu

4. William M. Goldman, Department of Mathematics, University of Maryland, College Park, MD 20742 USA E-mail: wmg@math.umd.edu

5. Karin Melnick, Department of Mathematics, Yale University, New Haven, CT 06520 USA E-mail: karin.melnick@yale.edu