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.

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.

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 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 ℓ^{μ} to the hyperplane, ℓ^{μ} (1, 0, 0, 1) is a null vector, i.e. ℓ_{μ}ℓ^{μ} = 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 {n_{i}}

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 η^{μν}ℓ^{(i)}_{μ}ℓ^{(j)}_{ν}of 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 L^{ij }=(3!)²η^{μν}ℓ^{(i)}_{μ}ℓ^{(j)}_{ν }of their scalar products. The diagonal elements of L^{ij } 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.

**Related posts**