Causal cells: spacetime polytopes with null hyperfaces by Neiman

This week I have been reading a paper about polyhedra and 4-polytopes in Minkowski spacetime – in particular, null polyhedra
with zero volume, and 4-polytopes that have such polyhedra as their hyperfaces.

A 3D projection of an tesseract performing an ...
A 3D projection of a tesseract performing an isoclinic rotation.


The paper presents the basic properties of several classes of null faced 4-polytopes: 4-simplices, tetrahedral diamonds and 4-parallelotopes. A most regular representative of each class is proposed.

The most-regular parallelotope is of particular interest: its edges, faces and hyperfaces are all congruent, and it features both null hyperplanes and null segments. A tiling of spacetime with copies of this polytope can be viewed alternatively as a lattice with null edges, such that each point is at the intersection of four light rays in a tetrahedral pattern. This construct may have relevance for discretizations of curved spacetime and for quantum gravity.

In this paper, the author studies the properties of some special 4-polytopes in spacetime. The main qualitative difference between spacetime and Euclidean space is the existence of null i.e. lightlike directions. So, there exist line segments with vanishing length, plane elements with vanishing area, and hyperplane elements with vanishing volume. 3d null hyperplane elements are especially interesting. In relativistic physics, null hypersurfaces play the role of causal boundaries between spacetime regions. They also function as characteristic surfaces for the differential equations of relativistic field theory.

Important examples of null hypersurfaces include the lightcone of an event and the event horizon of a black hole.

SVG version of

The prime example of a closed null hypersurface is a causal diamond – the intersection of two light cones originating from two timelike-separated points.


Null Hyperplanes 

Null 3d polyhedra or polyhedra with vanishing volume reside in null hyperplanes, such as the hyperplane t = z. let’s look at the geometry of these hyperplanes. The normal ℓμ to the hyperplane, ℓμ (1, 0, 0, 1) is a null vector, i.e. ℓμμ = 0.  As a result, it is also tangent to the hyperplane. It’s integral lines  form null geodesics. The hyperplane
is  foliated into light rays. All intervals within the hyperplane are spacelike, except the null intervals along the rays.

Null Polyhedra

In 3d Euclidean space, each area element has a normal vector n. When discussing polyhedra, it is convenient to define the norm of n to equal the area of the corresponding face. The orientation of the normals is chosen to be outgoing. Not every set of area normals {ni}
describes the faces of some polyhedron. For this to be true, the normals must sum up to zero:


This can be understood as the requirement that the flux of any constant vector field through the polyhedron vanishes. In loop quantum gravity, this condition encodes the local SO(3) rotation symmetry.


Null tetrahedra

The simplest null polyhedron is a tetrahedron. Up to reflections along the null axis, null tetrahedra come in two distinct types: (1,3) and (2,2). The pairs of numbers denote how many of the tetrahedron’s four faces are past-pointing and future- pointing, respectively.


Null-faced 4-simplices

Null-faced 4-simplices have hyperfaces which have zero volume. A 4-simplex has five tetrahedral hyperfaces, which in this case will be null tetrahedra,

The scalar products  ημν(i)μ(j)νof the null volume normals are directly related to the spacetime volume of the 4-simplex and to the areas of the 2d faces. To express the spacetime volume, we must choose a set of four volume normals ℓ(i)μ. The time-orientation of the
normals should be correlated with the past/future status of their hyperfaces.  Next, we construct a symmetric 4 × 4 matrix  Lij =(3!)²ημν(i)μ(j)ν of their scalar products. The diagonal elements of Lij are zero. Elements corresponding to past- future pairs ij are positive, while those for past-past and future-future pairs are negative.
The spacetime volume can then be found as:


The area of the face at the intersection of the i’th and j’th hyperplanes can be found as:


Can also define the 4-volume directly in terms of triangle areas:



Null parallelepipeds

The six faces of a null parallelepiped are spacelike parallelograms. There are three pairs of opposing faces, such that each pair is parallel and congruent. In a given pair of opposing faces, one is past-pointing, and the other future-pointing.



Tetrahedral diamonds

Beginning with an arbitrary spacelike tetrahedron, situated at t = 0 hyperplane, this is the base tetrahedron. For each of the base tetrahedron’s four faces,  the lightcross of two null hyperplanes orthogonal to it are drawn. The tetrahedral diamond is then defined by the convex hull of the intersections of these null hyperplanes.


The 4-volume of a tetrahedral diamond can be found as twice the volume of a 4-simplex, with the spacelike tetrahedron as its base and the inscribed radius r as its height. The result is:


where V is the base tetrahedron’s volume.


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Deformations of Polyhedra and Polygons by the Unitary Group by Livine

This week whilst preparing some calculations I have been looking at the paper ‘Deformations of Polyhedra and Polygons by the Unitary Group’. In the paper the author inspired by loop quantum gravity, the spinorial formalism and the structures of twisted geometry discusses the phase space of polyhedra in three dimensions and its quantization, which serve as basic building of the kinematical states of discrete geometry.

They show that the Grassmannian space U(N)/(U(N − 2) × SU(2)) is the space of framed convex polyhedra with N faces up to 3d rotations. The framing consists in the additional information of a U(1) phase per face. This provides an extension of the Kapovich-Milson phase space  for polyhedra with fixed number of faces and fixed areas for each face – see the post:

Polyhedra in loop quantum gravity

They describe the Grassmannian as the symplectic quotient C2N//SU(2), which provides canonical complex variables for the Poisson bracket. This construction allows a natural U(N) action on the space of polyhedra, which has two main features. First, U(N) transformations act non-trivially on polyhedra and change the area and shape of each individual face. Second, this action is cyclic: it allows us to go between any two polyhedra with fixed total area  – sum of the areas of the faces.

On quantization, the framed polyhedron phase space leads to the Hilbert space of SU(2) intertwiners, which is interpreted as the space of quantum polyhedra. By performing a canonical quantization from the complex variables of C2N//SU(2) all the classical features are automatically exported to the quantum level. Each face carries now a irreducible representation of SU(2) – a half-integer spin j, which defines the area of the face. Intertwiners are then SU(2)-invariant states in the tensor product of these irreducible representations. These intertwiners are the basic
building block of the spin network states of quantum geometry in loop quantum gravity.

The U(N) action on the space of intertwiners changes the spins of the faces and each Hilbert space for fixed total area defines an irreducible representation of the unitary group U(N). The U(N) action is cyclic and allows us to generate the whole Hilbert space from the action of U(N) transformation on the highest weight
vector. This construction provides coherent intertwiner states peaked on classical polyhedra.

At the classical level, we can use the U(N) structure of the space of polyhedra to compute the averages of polynomial observables over the ensemble of polyhedra and  to use the Itzykson-Zuber formula from matrix models  as a generating functional for these averages. It computes the integral over U(N) of the exponential of the matrix elements of a unitary matrix tensor its complex conjugate.

At the quantum level the character formula, giving the trace of unitary transformations either over the standard basis or the coherent intertwiner basis, provides an extension of the Itzykson-Zuber formula. It allows us in principle to generate the expectation values of all polynomial observables and so their spectrum.

This paper defines and describe the phase space of framed polyhedra, its parameterization in terms of spinor variables and the action of U(N) transformations. Then it shows  how to compute the averages and correlations of polynomial observables using group integrals over U(N) and the Itzykson-Zuber integral as a generating function. It discusses the quantum case, with the Hilbert space
of SU(2) intertwiners, coherent states and the character formula.

The paper also investigates polygons in two dimensions and shows that the unitary group is replaced by the orthogonal group and that
the Grassmannian Ø(N)/(Ø(N −2)×SO(2)) defines the phase space for framed polygons. It then discusses the issue of gluing such polygons together into a consistent 2d cellular decomposition, as a toy model for the gluing of framed polyhedra into 3d discrete manifolds. These constructions are relevant to quantum gravity in 2+1 and 3+1 dimensions, especially to discrete approaches based on a description of the geometry using glued polygons and polyhedra such as loop quantum gravity  and dynamical

The paper’s goal is to clarify how to parametrize the set of polygons or polyhedra and their deformations, and to introduce mathematical tools to compute the average and correlations of observables over the ensemble of polygons or polyhedra at the classical level and then the spectrum and expectation values of geometrical operators on
the space of quantum polygons or polyhedra at the quantum level.

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