Spinors and horospheres

Penrose and Rindler in [18] describe various aspects of relativity theory in terms of spinorial objects. The foundation of their spinorial description of spacetime is a construction associating objects of Minkowski space $\mathbb{R}^{1,3}$ to vectors $\kappa=(\xi,\eta)\in\mathbb{C}^{2}$, which they call spin vectors and which, for present purposes, we simply call spinors. To nonzero spinors, Penrose and Rindler associate a null flag, which is a point $p$ on the future light cone $L^{+}$, together with a 2-plane tangent to $L^{+}$ containing $p$, which behaves in a spinorial way. See Figure 1 (left).

On the other hand, Penner in [16] introduced a “decorated Teichmüller theory” which has since been highly developed (see e.g. [17]). A basic construction of this theory relates points on the future light cone $L^{+}$ in Minkowski space of one lower dimension $\mathbb{R}^{1,2}$, to horocycles in the hyperbolic plane $\mathbb{H}^{2}$, which sits in Minkowski space as the hyperboloid model, consisting of all points 1 unit in the future of the origin. See Figure 1 (right).

In this paper we observe that these two constructions can be combined and generalised, yielding the following theorem.

A decoration on a horosphere is a tangent parallel oriented line field, i.e. a choice of direction along the horosphere at each point which is invariant under parallel translation. Such decorations exist since horospheres in $\mathbb{H}^{3}$ are isometric to the Euclidean plane. A spin decoration is, roughly, a “spin lift” of such a decoration, where rotating the direction by $2\pi$ is not the identity, but rotation by $4\pi$ is; we define them precisely in Section 4. See Figure 2 (left).

The correspondence of Theorem 1 is expressed very simply in the upper half space model $\mathbb{U}$. Regarding the sphere at infinity $\partial\mathbb{H}^{3}$ of $\mathbb{H}^{3}$ as $\mathbb{C}\cup\{\infty\}$ in the usual way, the horosphere $H$ corresponding to $(\xi,\eta)$ has centre at $\xi/\eta$. If $\xi/\eta=\infty$ then $H$ is a horizontal plane in $\mathbb{U}$ at height $|\xi|^{2}$, and its decoration points in the direction $i\xi^{2}$. If $\xi/\eta\in\mathbb{C}$ then $H$ is a Euclidean sphere in $\mathbb{U}$ of Euclidean diameter $|\eta|^{-2}$, and its decoration at its highest point (“north pole”) points in the direction $i\eta^{-2}$. See Figure 2 (right). We prove this in Proposition 3.9.

As indicated, both sides of this correspondence have $SL(2,\mathbb{C})$-actions. The action on $\mathbb{C}^{2}$ is by the standard action of matrices on vectors. The action on horospheres is via the action of $PSL(2,\mathbb{C})$ as orientation-preserving isometries of $\mathbb{H}^{3}$; this action lifts to $SL(2,\mathbb{C})$ to preserve spin structures.

Penrose and Rindler in [18] define a complex-valued antisymmetric bilinear form $\{\cdot,\cdot\}$ on spinors, given by the $2\times 2$ determinant; it can also be regarded as the standard complex symplectic form on $\mathbb{C}^{2}$. On the other hand, given two horospheres $H_{1},H_{2}$, we may measure the signed distance $\rho$ from one to the other along their mutually perpendicular geodesic. As detailed in Section 4, we may also measure an angle $\theta$ between spin decorations, which is well defined modulo $4\pi$. We regard $d=\rho+i\theta$ as a complex distance between the two horospheres. See Figure 3.

The correspondence of Theorem 1 extends further to relate these structures, as follows.

In the 2-dimensional context, the distance between horocycles is just a real number $d$, and the quantity $\lambda=\exp(d/2)$ is known as a lambda length [16]. Thus, the standard bilinear form on spinors computes lambda lengths between corresponding spin-decorated horospheres.

Example.

Take $\kappa=(1,0)$ and $\omega=(0,1)$. In the upper half space model, $\kappa$ corresponds to a horosphere centred at $1/0=\infty$, so appears as a horizontal plane, at height $1$ and with decoration in the direction $i$. Similarly, the horosphere of $\omega$ has centre $0/1=0$, appearing as a sphere centred at $0$, with Euclidean diameter $1$, and direction at its highest point in the direction $i$. These two horospheres are tangent at the point $(0,0,1)$, where their decorations align. It turns out their spin directions also align, so $\rho=\theta=0$ and hence $\lambda=1=\{\kappa,\omega\}$.

Multiplying $\kappa$ by a complex number $re^{i\phi}$, with $r>0$ and $\phi$ real, its horosphere remains centred at $\infty$, but the horizontal plane moves to height $r^{2}$, translated upwards from its original position by hyperbolic distance $2\log r$, and its decoration is rotated by $2\phi$. From $\kappa$ to $\omega$ we then have $\rho=2\log r$ and $\theta=2\phi$, so $\lambda=\exp(\frac{1}{2}(2\log r+2\phi i))=re^{i\phi}=\{\kappa,\omega\}$.

Approach.

This paper proceeds through the constructions and proofs in a self-contained manner; we need to adapt and extend the constructions of Penrose–Rindler and Penner for our purposes. To give a rough overview, from a spinor $\kappa=(\xi,\eta)$, following Penrose–Rindler, we obtain points $(T,X,Y,Z)$ on the positive light cone $L^{+}$ in $\mathbb{R}^{1,3}$ via

$$\begin{pmatrix}\xi\\ \eta\end{pmatrix}\begin{pmatrix}\overline{\xi}&\overline{\eta}\end{pmatrix}=\frac{1}{2}\begin{pmatrix}T+Z&X+iY\\ X-iY&T-Z\end{pmatrix}$$

(1.1)

which uses the correspondence between Hermitian $2\times 2$ matrices and Minkowski space $\mathbb{R}^{1,3}$ given by Pauli matrices. The flag of $\kappa$ has flagpole along the ray of $(T,X,Y,Z)$ determined by (1.1), and 2-plane defined by the derivative of this map $\kappa\mapsto(T,X,Y,Z)$ in a certain direction depending on $\kappa$; we show this is a variation on the Penrose–Rindler construction. The correspondence between spinors and flags is $SL(2,\mathbb{C})$-equivariant, where $SL(2,\mathbb{C})$ acts on flags via its action on $\mathbb{R}^{1,3}$ in standard fashion as $SO(1,3)^{+}$. In Section 2 we describe these constructions precisely, along with explicit calculations, and details of $SL(2,\mathbb{C})$-equivariance, which we have not seen proved elsewhere in the literature.

From a point $p$ on the figure light cone $L^{+}$, Penner’s construction in [16] is to consider the affine plane in Minkowski space consisting of all $x$ satisfying

$$\langle x,p\rangle=1,$$

where $\langle\cdot,\cdot\rangle$ is the standard Lorentzian metric. This affine plane intersects the hyperboloid model of hyperbolic space $\mathbb{H}$ in a horosphere $H$. This construction works in any dimension, and in $\mathbb{R}^{1,3}$, we obtain an affine 3-plane which intersects the hyperboloid model of $\mathbb{H}^{3}$ in a 2-dimensional horosphere. We show that when $p$ arises from a spinor $\kappa$, this affine 3-plane contains an affine 2-plane parallel to the flag 2-plane of $\kappa$, and this affine 2-plane intersects the horosphere in an parallel oriented line field on $H$, yielding the picture of Figure 2 (left). Again, all constructions are $SL(2,\mathbb{C})$-equivariant. Precise details are given in Section 3.

Finally, all these constructions lift, in appropriate sense, to spin double covers, and remain $SL(2,\mathbb{C})$-equivariant, as we detail in Section 4.

Hyperbolic geometry applications.

These theorems have several applications and in this paper we consider some of them. In Section 5 we consider some applications to hyperbolic geometry.

Consider an ideal hyperbolic tetrahedron, i.e. with all vertices on $\partial\mathbb{H}^{3}$. Number the vertices $0,1,2,3$. We consider a spin decoration on the tetrahedron, consisting of a spin-decorated horosphere $H_{i}$ at each ideal vertex $i$. There is then a complex lambda length $\lambda_{ij}$ from $H_{i}$ to $H_{j}$. Each $\lambda_{ij}$ measures, in a certain sense, the distance between horospheres along each edge, along with the angle between them. See Figure 4.

This equation is similar to Ptolemy’s theorem relating the lengths of sides and diagonals of a cyclic quadrilaterals in classical Euclidean geometry, hence we call it a Ptolemy equation. Penner in [16] proved a corresponding Ptolemy equation in 2 dimensions: when an ideal quadrilateral in $\mathbb{H}^{2}$ has its vertices decorated with horocycles, the (real) lambda lengths of the edges and diagonals satisfy the same equation. Theorem 3 is a 3-dimensional generalisation showing that Ptolemy’s equation still holds, once we take horospheres to have spin decorations, and take lambda lengths to be complex. Roughly, 2-dimensional hyperbolic geometry corresponds to the case when spinors are real, i.e. lie in $\mathbb{R}^{2}$.

With the previous theorems in hand, the proof of Theorem 3 is not difficult. Indeed, the four spin-decorated horospheres correspond to four spinors in $\mathbb{C}^{2}$, which can be arranged into a $2\times 4$ matrix. The Ptolemy equation is then just the Plücker relation between $2\times 2$ determinants of a $2\times 4$ matrix. Alternatively, it can be seen as the relation

$$\varepsilon_{AB}\varepsilon_{CD}+\varepsilon_{BC}\varepsilon_{AD}+\varepsilon_{CA}\varepsilon_{BC}=0$$

satisfied by the spinor $\varepsilon$, as in [18] (e.g. eq. 2.5.21).

A related result is given in Proposition 5.3, where we show that the shape parameters of an ideal tetrahedron can be recovered from these six lambda lengths.

Truncations of ideal tetrahedra along horospheres arise naturally, for instance, in complete hyperbolic structures on 3-manifolds. In a forthcoming paper with Purcell [14] we show how Ptolemy equations can be used to describe hyperbolic structures on 3-manifolds, giving a directly hyperbolic-geometric version of the Ptolemy equations described by Garoufalidis–Thurston–Zickert [10] and enhanced Ptolemy variety of Zickert [23], in turn based on work of Fock–Goncharov [4].

Even in 2 dimensions, spinors provide a useful way to analyse the geometry of horocycles; we take the spinors to have real coordinates. In forthcoming work with Zymaris we apply this to circle packing theory and generalise a classical theorem of Descartes [15].

Indeed, when $\xi$ and $\eta$ are both integers then the horocycles obtained in the upper half plane model of $\mathbb{H}^{2}$ are the Ford circles, with their delightful relationships to Farey fractions, Diophantine approximation and continued fractions [9].

Cluster algebra applications.

In Section 6 we consider some applications to cluster algebras. We refer to [22] for an introduction to basic notions of cluster algebras.

We have already mentioned how 4 spinors arising from a spin-decorated ideal tetrahedron $\Delta$ can be arranged into a $2\times 4$ matrix. Considering those 4 spinors up to the common action of $SL(2,\mathbb{C})$ corresponds to considering such a $\Delta$ up to isometry. And considering appropriate $2\times 4$ matrices up to a left action by $2\times 2$ matrices is a standard description of a Grassmannian. Thus, the correspondences of the above theorems yield a relationship between hyperbolic geometry and Grassmannians.

In [7, sec. 12.2], Fomin–Zelevinsky described a geometric realisation of cluster algebras of type $A_{n}$ in terms of Grassmannians. Clusters in this case are in bijection with triangulations of an $(n+3)$-gon; two clusters are joined by an edge in the exchange graph if and only if the triangulations are related by a flip [3, 8]. Fomin–Zelevinsky showed that the cluster algebra is realised by $X(n+3)$, the affine cone over the Grassmannian $\operatorname{Gr}(2,n+3)$ in particular, the cluster algebra there denoted $\mathcal{A}_{\circ}$ in type $A_{n}$ is isomorphic to the $\mathbb{Z}$-form of the coordinate ring $\mathbb{C}[X(n+3)]$, with the cluster variables mapping to Plücker coordinates. This has since been generalised in various ways, for instance to other Grassmannians [20] and partial flag varieties [11].

On the other hand, work of Fock–Goncharov [4], Gekhtman–Shapiro–Vainshtein [12], Fomin–Shapiro–Thurston [5] and Fomin–Thurston [6] provides geometric realisations of cluster algebras arising from surfaces, in terms of the decorated Teichmüller space $\widetilde{T}(n+3)$ introduced by Penner [16], using lambda lengths. In the particular case of an $(n+3)$-gon, the lambda lengths of the diagonals provide cluster variables for the cluster algebra of type $A_{n}$ [6, examples 8.10, 16.1]. Fock–Goncharov in [4] also give numerous results relating Teichmüller spaces to higher algebraic structures, but as far as they relate to hyperbolic geometry and the results of this paper, they are in dimension 2. As mentioned earlier, the Ptolemy equations of Garoufalidis–Thurston–Zickert [10], which use variables provided by Fock–Goncharov, are given a 3-dimensional hyperbolic-geometric interpretation by our complex lambda lengths in forthcoming work with Purcell [14] .

In any case, there is thus a well-understood isomorphism between the cluster algebras arising from the affine cone $X(n+3)$ over the Grassmannian $\operatorname{Gr}(2,n+3)$, and from the decorated Teichmüler space $\widetilde{T}(n+3)$. In [6, remark 16.2] Fomin–Thurston note a connection between the underlying spaces.

The results of this paper further illuminate the situation, by giving a direct identification of the spaces underlying these cluster algebras, and extending them to 3 dimensions. The correspondence between spinors and spin-decorated horospheres naturally yields identifications of certain decorated Teichmüller spaces, and certain Grassmannian spaces, as follows.

In Section 6 we define all notions precisely and prove some properties about them, including these theorems.

The rough idea is simply that a collection of $n$ spinors $\kappa_{1},\ldots,\kappa_{n}$ describes the $n$ ideal vertices of an ideal $n$-gon (in 2 dimensions), or an ideal skew $n$-gon (in 3 dimensions); but on the other hand, we may place the $\kappa_{i}$ as the $n$ columns of a $2\times n$ matrix. The appropriate decorated Teichmüller space is then given by the certain (spin) isometry classes of such ideal (skew) $n$-gons, which is an orbit space of a $d$-tuple of spinors $(\kappa_{1},\ldots,\kappa_{d})$. The corresponding set of orbits of $2\times n$ matrices gives the affine cone on the appropriate Grassmannian.

It is not difficult to vary the conditions on $d$-gons or Grassmannians, and find identifications between diverse versions of decorated Teichmüller spaces, and corresponding diverse Grassmannian spaces.

The appearance of the positive Grassmannian here corresponds to the fact, proved in Proposition 6.9, that a sequence of horocycles with lambda lengths that are positive, in an appropriate sense, corresponds to the the centres of the horocycles being in order around $\partial\mathbb{H}^{2}$. Positivity of determinants and Grassmannians and their relationship to ordered or convex objects arises in a similar way in the physics of scattering amplitudes, see e.g. [1].

Acknowledgments.

The author thanks Varsha for assistance in the preparation of figures. He is supported by Australian Research Council grant DP210103136.

For us spinors are just elements of $\mathbb{C}^{2}$, which we regard as a complex symplectic vector space, with complex symplectic form denoted

$$\{\cdot,\cdot\}=d\xi\wedge d\eta$$

following [18]. We denote spinors by $\kappa=(\xi,\eta)$ or similar. Given $\kappa=(\xi,\eta)$ and $\kappa^{\prime}=(\xi^{\prime},\eta^{\prime})$ then

$$\{\kappa,\kappa^{\prime}\}=\xi\eta^{\prime}-\eta\xi^{\prime}=\det\begin{pmatrix}\xi&\xi^{\prime}\\ \eta&\eta^{\prime}\end{pmatrix}.$$

We write $\det(\kappa,\kappa^{\prime})$ for the above determinant. We denote by $\mathbb{C}_{*}^{2}$ the set of nonzero spinors.

For the purposes of linear algebra, we regard $\kappa$ as a column vector and write $\kappa^{T}$ for the corresponding row vector. The adjoint $\kappa^{*}=\overline{\kappa}^{T}$ is then a row vector.

We map spinors into the set $\mathcal{H}$ of Hermitian $2\times 2$ matrices, or equivalently into Minkowski space $\mathbb{R}^{1,3}$. We take $\mathbb{R}^{1,3}$ to have coordinates $(T,X,Y,Z)$ and metric $dT^{2}-dX^{2}-dY^{2}-dZ^{2}$, denoted $\langle\cdot,\cdot\rangle$. We observe $\mathcal{H}$ and $\mathbb{R}^{1,3}$ are isomorphic 4-dimensional real vector spaces and we identify them in a standard way (perhaps the constant is slightly unorthodox)

$$(T,X,Y,Z)\leftrightarrow\frac{1}{2}\begin{bmatrix}T+Z&X+iY\\ X-iY&T-Z\end{bmatrix}.$$

The right hand expression is $\frac{1}{2}\left(T+X\sigma_{X}+Y\sigma_{Y}+Z\sigma_{Z}\right)$, where the $\sigma_{\bullet}$ are the Pauli matrices. If a point $x=(T,X,Y,Z)\in\mathbb{R}^{1,3}$ corresponds to $S\in\mathcal{H}$ then we observe $\operatorname{Tr}S=T$ and $4\det S=\langle x,x\rangle$. The light cone $L=\{x\in\mathbb{R}^{1,3}\;\mid\;\langle x,x\rangle=0\}$ corresponds to $S$ with determinant zero, and the future light cone $L^{+}=\{x\in L\;\mid\;T>0\}$ corresponds to $S$ satisfying $\det S=0$ and $\operatorname{Tr}S>0$. We define the celestial sphere $\mathcal{S}^{+}$ to be the intersection of $L^{+}$ with the 3-plane $T=1$.

In other words,

$$\phi_{1}(\kappa)=\begin{pmatrix}\xi\\ \eta\end{pmatrix}\begin{pmatrix}\overline{\xi}&\overline{\eta}\end{pmatrix}=\begin{pmatrix}|\xi|^{2}&\xi\overline{\eta}\\ \overline{\xi}\eta&|\eta|^{2}\end{pmatrix}.$$

We observe that the image of $\phi_{1}$ has determinant zero, and its diagonal entries are $|\xi|^{2},|\eta|^{2}$, so that its trace is non-negative. Indeed it is not difficult to show $\phi_{1}(\mathbb{C}_{*}^{2})=L^{+}$. Thus $\phi_{1}$ maps a 4-(real)-dimensional domain onto a 3-dimensional image. The fibres are circles; it is not difficult to show that $\phi_{1}(\kappa)=\phi_{1}(\kappa^{\prime})$ iff $\kappa=e^{i\theta}\kappa^{\prime}$ for some real $\theta$. Indeed, on each 3-sphere in $\mathbb{C}^{2}$ given by $\kappa$ with $|\xi|^{2}+|\eta|^{2}$ fixed at some constant $c>0$, $\phi_{1}$ restricts to the Hopf fibration onto the 2-sphere in $L^{+}$ given by $T=c$. Thus $\phi_{1}$ is the cone on the Hopf fibration.

In order not to lose information, we extend $\phi_{1}$ to a map including tangent data. Given a tangent vector $\nu$ in the real tangent space $T_{\kappa}\mathbb{C}_{*}^{2}$, we write $D_{\kappa}\phi_{1}(\nu)$ for the derivative of $\phi_{1}$ at $\kappa$ in the direction $\nu$. Since, for real $t$,

$$\phi_{1}\left(\kappa+t\nu\right)=\left(\kappa+t\nu\right)\left(\kappa+t\nu\right)^{*}=\kappa\kappa^{*}+\left(\kappa\nu^{*}+\nu\kappa^{*}\right)t+\nu\nu^{*}t^{2}$$

we have

$$D_{\kappa}\phi_{1}(\nu)=\left.\frac{d}{dt}\phi_{1}\left(\kappa+t\nu\right)\right|_{t=0}=\kappa\nu^{*}+\nu\kappa^{*}.$$

(2.2)

In the abstract index notation of [18], this directional derivative is $\kappa^{A}\overline{\nu}^{A^{\prime}}+\nu^{A}\overline{\kappa}^{A^{\prime}}$. At each point $\kappa$ we will build a flag structure using the derivative in a certain direction $Z(\kappa)$.

Let us attempt to motivate this definition. Penrose–Rindler use a spinor $\tau^{A}$ forming a spin frame, or standard symplectic basis, with $\kappa^{A}$, i.e. so that $\{\kappa,\tau\}=\kappa_{A}\tau^{A}=1$. They then form a 2-plane defined by the bivector $K^{a}\wedge L^{b}$ where $K^{a}=\kappa^{A}\overline{\kappa}^{A^{\prime}}$ and $L^{a}=\kappa^{A}\overline{\tau}^{A^{\prime}}+\tau^{A}\overline{\kappa}^{A^{\prime}}$. These two vectors are our $\phi_{1}(\kappa)$ and $D_{\kappa}\phi_{1}(\tau)$. But the same oriented 2-plane is obtained using any positive multiple of such $\tau$, so we could equally fix $\kappa_{A}\tau^{A}$ simply to be positive real. Choosing $\tau$ to make $\kappa_{A}\tau^{A}$ negative real, or positive/negative imaginary, works also for our purposes. Our choice of $Z$ ensures $\{\kappa,Z(\kappa)\}=-i(|\xi|^{2}+|\eta|^{2})$ is negative imaginary. Though somewhat arbitrary, this works well for our purposes.

Another perspective on $Z$ is obtained by identifying $(\xi,\eta)\in\mathbb{C}^{2}$ with the quaternion $\xi+\eta j$. Then $Z\kappa=-k\kappa$. On the $S^{3}$ centred at the origin in $\mathbb{C}^{2}$ through $\kappa$, the tangent space at $\kappa$ has basis $i\kappa,j\kappa,k\kappa$. In the $i\kappa$ direction lies the fibre $e^{i\theta}\kappa$, and $\phi_{1}$ is constant; $Z\kappa$ is another tangent vector to this $S^{3}$.

In any case, (2.2) and Definition 2.3 immediately yield

$$D_{\kappa}\phi_{1}(Z\kappa)=\kappa\kappa^{T}J+J\overline{\kappa}\kappa^{*}.$$

(2.4)

We now define the type of flag structure we need.

Thus $p$ is a on flagpole $\mathbb{R}p$, which runs towards the future along the light cone; and the flag plane $V_{2}$ is a tangent plane to the light cone, with its relative orientation equivalent to choosing the half-plane to one side of $\mathbb{R}p$ or the other. Note that $V_{2}$ contains no timelike vectors, and $\mathbb{R}p$ generates the unique 1-dimensional lightlike subspace of $V_{2}$. The tangent space to $L^{+}$ at $p$ is defined by the equation $\langle x,p\rangle=0$, i.e. is the (Minkowski-)orthogonal complement $p^{\perp}$. Thus $\mathbb{R}p\subset V_{2}\subset p^{\perp}$.

Given linearly independent $p\in L^{+}$ and $v\in T_{p}L^{+}$, we denote by $[[p,v]]$ the flag given by $p$, the line $\mathbb{R}p$ oriented from the origin towards $p$, the plane $V_{2}$ spanned by $p$ and $v$, and the orientation on $V_{2}/\mathbb{R}p$ induced by $v$. We observe that two flags so given $[[p,v]]$, $[[p^{\prime},v^{\prime}]]$ are equal if and only if $p=p^{\prime}$ and there exist real $a,b,c$ such that $ap+bv+cv^{\prime}=0$, where $b,c$ (which are necessarily nonzero) have opposite sign.

Note that $\mathcal{F}$ is diffeomorphic to $UTS^{2}\times\mathbb{R}$, where $UTS^{2}$ is the unit tangent bundle of $S^{2}$: a point of $S^{2}$ describes a future-oriented ray in $L^{+}$, a unit tangent vector there describes a relatively oriented 2-plane, and the $\mathbb{R}$ factor fixes $p$ along the ray. Since $UTS^{2}\cong\mathbb{RP}^{3}$ we also have $\mathcal{F}\cong\mathbb{RP}^{3}\times\mathbb{R}$

Our version of Penrose–Rindler null flags can now be defined as the following map, upgrading $\phi_{1}$.

Thus the point $\phi_{1}(\kappa)$ yields the flagpole, and the derivative of $\phi_{1}$ in the $Z\kappa$ direction yields the relatively oriented flag plane. We verify that $D_{\kappa}\phi_{1}(Z\kappa)$ is (real-)linearly independent from $\phi_{1}(\kappa)$ using (2.4): if $a\kappa\kappa^{*}+b\left(\kappa\kappa^{T}J+J\overline{\kappa}\kappa^{*}\right)=0$ for some real $a,b$ then $\kappa\left(a\kappa^{*}+b\kappa^{T}J\right)=\left(J\overline{\kappa}\right)\left(-b\kappa^{*}\right)$; both sides of this equation being the product of a $2\times 1$ and $1\times 2$ matrix, the corresponding matrices must be proportional, say $\kappa=cJ\overline{\kappa}$ for some real $c$; in components then $\xi=ci\overline{\eta}$ and $\eta=-ci\overline{\xi}$, so $\xi=-c^{2}\xi$ and $\eta=-c^{2}\eta$, so that $\xi=\eta=0$, a contradiction.

The spaces $\mathbb{C}^{2}$, $\mathcal{H}\cong\mathbb{R}^{1,3}$ and $\mathcal{F}$ all have natural $SL(2,\mathbb{C})$ actions; in all cases we denote the action of $A\in SL(2,\mathbb{C})$ by a dot. An $A\in SL(2,\mathbb{C})$ acts on $\mathbb{C}^{2}$ by the defining representation, $A.\kappa=A\kappa$, yielding a symplectomorphism:

$$\{A\kappa,A\kappa^{\prime}\}=\det(A\kappa,A\kappa^{\prime})=\det A(\kappa,\kappa^{\prime})=\det(\kappa,\kappa^{\prime})=\{\kappa,\kappa^{\prime}\}$$

since $\det A=1$. The same $A$ acts on $S\in\mathcal{H}$ by $A.S=ASA^{*}$, which in $\mathbb{R}^{1,3}$ yields in the standard way the linear maps of $SO(1,3)^{+}$, i.e. those which preserve the Minkowski metric and space and time orientation. The action on $\mathbb{R}^{1,3}$ induces orientation-preserving actions on $L^{+}$ and planes in $\mathbb{R}^{1,3}$, yielding an action on $\mathcal{F}$, so that $A.[[p,v]]=[[A.p,A.v]]$. Essentially by definition $\phi_{1}$ is equivariant with respect to these actions,

$$\phi_{1}(A\kappa)=A\kappa\kappa^{*}A^{*}=A\phi_{1}(\kappa)A^{*}=A.\phi_{1}(\kappa),$$

and we have an equivariance property on its derivatives

$$A.D_{\kappa}\phi_{1}(\nu)=D_{A\kappa}\phi_{1}(A\nu)$$

since $A.(\kappa\nu^{*}+\nu\kappa^{*})=A\kappa\nu^{*}A^{*}+A\nu\kappa^{*}A^{*}=(A\kappa)(A\nu)^{*}+(A\nu)(A\kappa)^{*}$. We now show the equivariance property extends to $\Phi_{1}$; we have not seen a proof of this in the existing literature.

It is possible to express explicitly the linear dependence implied by the equality of the flags $\Phi_{1}(A\kappa)=[[\phi_{1}(A\kappa),D_{A\kappa}\phi_{1}(Z(A\kappa))]]$ and $A.\Phi_{1}(\kappa)=[[\phi_{1}(A\kappa),D_{A\kappa}\phi_{1}(A(Z\kappa))]]$: a direct computation verifies the (perhaps surprising) identity

$$\left(\kappa^{T}JA^{*}A\kappa+\kappa^{*}A^{*}AJ\overline{\kappa}\right)\phi_{1}(A\kappa)+\left(\kappa^{*}\kappa\right)\left[D_{A\kappa}\phi_{1}\left(Z\left(A\kappa\right)\right)\right]-\left(\kappa^{*}A^{*}A\kappa\right)\left[D_{A\kappa}\phi_{1}\left(A\left(Z\kappa\right)\right)\right]=0.$$

We can compute $\Phi_{1}$ completely explicitly.

The fact that $D_{\kappa}\phi_{1}(Z\kappa)$ has zero $T$-coordinate follows from $Z\kappa$ being tangent to the $S^{3}$ centred at the origin through $\kappa$, which maps under $\phi_{1}$ to the $S^{2}$ given by the intersection of $L^{+}$ with a plane at constant $T$.

We denote by $\mathbb{H}^{3}$ the hyperboloid model of hyperbolic 3-space

$$\mathbb{H}^{3}=\left\{x=(T,X,Y,Z)\in\mathbb{R}^{1,3}\;\mid\;\langle x,x\rangle=1,\;T>0\right\}$$

and by $\partial\mathbb{H}^{3}$ the boundary at infinity of $\mathbb{H}^{3}$. So $\partial\mathbb{H}^{3}\cong S^{2}$ and $\partial\mathbb{H}^{3}$ is naturally bijective with the celestial sphere $\mathcal{S}^{+}$.

Indeed, projectivising $L^{+}$ yields an the boundary at infinity $\partial\mathbb{H}^{3}\cong S^{2}$ and under this projectivisation, 2-planes tangent to $L^{+}$ containing a ray of $L^{+}$ correspond bijectively with tangent lines at points of $\partial\mathbb{H}^{3}$. Moreover, relatively oriented planes containing a ray of $L^{+}$ correspond bijectively with tangent directions at points of $\partial\mathbb{H}^{3}$.

The orientation-preserving isometry group $SO(1,3)^{+}$ of $\mathbb{H}^{3}$ acts transitively on the future light cone $L^{+}$, and indeed acts transitively on the tangent directions at points of $\partial\mathbb{H}^{3}$. Further, if we take an oriented flag consisting of a future-oriented line $R$ of $L^{+}$ and a relatively oriented 2-plane $\pi$ tangent to $L^{+}$, then there is an element of $SO(1,3)^{+}$ fixing $R$ (and its orientation) and $\pi$ (and its relative orientation), which sends any point on the ray to any other. Such an element is given by a hyperbolic translation along any geodesic with an endpoint at infinity corresponding to $R$. In other words, $SL(2,\mathbb{C})$ acts transitively on $\mathcal{F}$, and the action factors through $PSL(2,\mathbb{C})\cong SO(1,3)^{+}$.

Taking $\kappa=(e^{i\theta},0)$, we have $\phi_{1}(\kappa)=(1,0,0,1)$, which we denote $p_{0}$, and by Lemma 2.10, $\Phi_{1}(\kappa)$ is the flag with basepoint $p_{0}$ and 2-plane spanned by $p_{0}$ and $(0,-\sin 2\theta,\cos 2\theta,0)$. Thus as we multiply $\kappa$ by $e^{i\theta}$ to move through a fibre of $\phi_{1}$, the flag $\Phi_{1}(\kappa)$ rotates about a fixed pointed flagpole twice as fast. It follows that $\Phi_{1}$ takes the value of each such flag exactly twice.

Using the equivariance of $\Phi_{1}$ and the transitive action of $SL(2,\mathbb{C})$, the same applies for the flags based at any point on $L^{+}$. It follows that $\Phi_{1}$ is smooth, surjective and 2–1. Moreover, the stabiliser of a flag in $SO(1,3)^{+}$ is trivial, so that $PSL(2,\mathbb{C})$ acts freely and transitively on $\mathcal{F}$. Topologically $\Phi_{1}$ is a map $\mathbb{C}_{*}^{2}\cong S^{3}\times\mathbb{R}\longrightarrow\mathbb{RP}^{3}\times\mathbb{R}\cong\mathcal{F}$ which is a double cover.

We have now built the maps $\phi_{1}$ and $\Phi_{1}$ in the commutative diagram

$$\begin{array}[]{ccccc}\mathbb{C}_{*}^{2}&\stackrel{{\scriptstyle\Phi_{1}}}{{\longrightarrow}}&\mathcal{F}&\stackrel{{\scriptstyle\Phi_{2}}}{{\longrightarrow}}&\operatorname{Hor}^{D}\\ &\stackrel{{\scriptstyle\phi_{1}}}{{\searrow}}&\downarrow&&\downarrow\\ &&L^{+}&\stackrel{{\scriptstyle\phi_{2}}}{{\longrightarrow}}&\operatorname{Hor}\end{array}$$

where the downwards arrow $\mathcal{F}\longrightarrow L^{+}$ forgets all structure of a flag except its point on $L^{+}$. In this section we define the maps $\phi_{2},\Phi_{2}$, and the spaces $\operatorname{Hor},\operatorname{Hor}^{D}$, which involve horospheres and decorations.

Horospheres in the hyperboloid model $\mathbb{H}^{3}$ are given by the intersection of $\mathbb{H}^{3}$ with certain affine 3-planes in $\mathbb{R}^{1,3}$. Any affine 3-plane in $\mathbb{R}^{1,3}$ is given by $x\in\mathbb{R}^{1,3}$ satisfying an equation of the form $\langle x,n\rangle=c$, where $n$ is a (Minkowski-)normal vector to the plane and $c$ is a real constant. We call such an affine 3-plane lightlike if its normal $n$ is lightlike. We observe that a lightlike 3-plane can be defined by an equation $\langle x,p\rangle=c$ where $p\in L^{+}$; if $c>0$ then this plane intersects $\mathbb{H}^{3}$ in a horosphere, and if $c\leq 0$ the plane is disjoint from $\mathbb{H}^{3}$. Normalising such equations by requiring the constant to be $1$, i.e. $\langle x,p\rangle=1$, then gives a bijection between points $p\in L^{+}$ and horospheres. We denote the set of horospheres in $\mathbb{H}^{3}$ by $\operatorname{Hor}$.

Thus the map $\phi_{2}$ is a bijection. Indeed, it is a diffeomorphism: $\operatorname{Hor}\cong S^{2}\times\mathbb{R}$, with an $\mathbb{R}$-family of horospheres at each point at infinity in $S^{2}=\partial\mathbb{H}^{3}$. Any horosphere has a unique point at infinity in $\partial\mathbb{H}^{3}$, which we also call its centre. The map $\phi_{2}^{\partial}$ can be regarded as the projectivisation map $L^{+}\longrightarrow S^{2}$ or projection to the celestial sphere $L^{+}\longrightarrow\mathcal{S}^{+}$.

Note $SL(2,\mathbb{C})$ acts naturally on $\mathbb{H}^{3}$ (as on $L^{+}$ and $\mathbb{R}^{1,3}$) in the standard way, via linear maps of $SO(1,3)^{+}$, and hence also on $\partial\mathbb{H}^{3}$ and $\operatorname{Hor}$, and we again denote all actions via a dot. We observe an $A\in SL(2,\mathbb{C})$ sends the horosphere $\phi_{2}(p)$, defined by $\langle x,p\rangle=1$, to the horosphere $A.\phi_{2}(p)$ defined by $\langle A^{-1}x,p\rangle=1$. Since the action of $A$ preserves the Minkowski metric, this horosphere is also given by $\langle x,A.p\rangle=1$. In other words, $A.\phi_{2}(p)=\phi_{2}(A.p)$ so that $\phi_{2}$ is $SL(2,\mathbb{C})$-equivariant. Forgetting the horospheres and recording only points at infinity, similarly $\phi_{2}^{\partial}$ is $SL(2,\mathbb{C})$-equivariant.