The Geometry of a four-simplex by Wolfgang Wieland

This post looks at a small part of Wolfgang Wieland’s PhD thesis about the geometry of a four-simplex. In this the author  finds  that the Gauss law and the linear simplicity constraints together imply its geometricity. Geometricity  means that we can introduce a metric compatible with the fluxes, and speak  about the unique length of the bones bounding the triangles of the four simplex. This is the reconstruction, or shape-matching problem.



Lets define some terminology, and see what  a four-simplex actually looks like . Given its corners wieequ1 we can view the four-simplex as the set of points


So  5×4 = 20 numbers determining a four-simplex. If we identify any two four-simplices related by a Poincaré transformation, there are only ten of them left: Ten numbers define a four-simplex up to Poincaré transformations. A four-simplex contains several
sub-simplices: There are the corners, bones, triangles and tetrahedra. The bones are the geodesic lines connecting any two of the corners, they are given by the vectors:



Two bones b(kl) and b(lm) span an oriented triangle τij : If (ijklm) is an even permutation of (12345),  the pair (b(kl); b(lm)) have positive orientation in τij . The triangles τij , τik, τil, and τim bound a tetrahedron, we call it Ti, and vi is its volume, while nμ(i) denotes its normal. In the following we restrict ourselves to the case where nμ(i) is a future oriented timelike vector, i.e. nμ(i)nν(i) = -1, n0(i) > 0.

Assume the four-simplex be non-degenerate, which means
that its four-volume V should not vanish. This is the condition that:


Because of their prominent role in loop quantum gravity, we are most interested in the fluxes through the triangles. We compute them as the integrals:


Stoke’s theorem implies for any tetrahedron the fluxes sum up to zero:


This is nothing but Gauss’s law. If n (i) denotes the normal of the i-th tetrahedron we also have the condition:


which is the linear simplicity constraint that has appeared at several occasions in our theory as a reality condition on phase space.

Reconstruction of a four-simplex from fluxes

For every tetrahedron bounding a four-simplex, the four fluxes sum up to zero and lie perpendicular to the tetrahedron’s normal.  The opposite is also true – given five future oriented normals n (i), that span all of R4, and ten bivectors  subject to both the Gauss law and the linear simplicity constraint, there is a four-simplex Δ, such that the bivectors ∑αβ  (ij) are the fluxes through its triangles and the vectors n (i) are the normals of the tetrahedra. The resulting four-simplex is unique up to rigid translations and inversion at the origin.

The fluxes have the following form:


where a(ij) > 0 is the area of the triangle and Ξji = Ξij ∈ R is the rapidity between the two adjacent normals, i.e.:


Introduce some simplifying abbreviations. First of all we define the three dimensional volume elements and  the three-metric:


Now write the fluxes in the i-th tetrahedron as the spatial pseudo vectors:


The three-dimensional volume vi > 0 of the i-th tetrahedron is the wedge product of three bounding triangles:


This brings the volume formula into the form:


A similar formula exists for the four-volume V of the polytope, it is:


Substitute for the fluxes and  the volume formula turns into:


Combining the formulae for the three- and four-volume  get the following expression for the rapidity:


This implies for the ratio between any two volumina that:


Have defined the three-volume of the tetrahedra


and the four volume:


Now construct the underlying four-simplex. Substitute the  solution of the simplicity constraints into the Gauss’ law and get:



The last identity is the same as :


Now use assumption that any quadruple of normals be linearly independent in R4 – the non-degeneracy of the geometry), then all i must be proportional to the three-volume vi through a universal sign:


We can then also find a set of numbers  εi∈{-1,1} such that:


The numbers εi are unique up to a global sign. Equation (3.218) now have:


This condition is the four-dimensional analogue of Gauss’s law: wieequ205

The Minkowski theorem in four dimensions guarantees that there is a
corresponding four-simplex consisting of five tetrahedra with normals n (i) and volume vi. This four-simplex is unique up to rigid translations and reflection at the origin.

Now compute the fluxes through the triangles of the four-simplex.  The first step is to look at the bones. A bone b(il) connects the i-th corner with the l-th. From the perspective of the j-th tetrahedron, this bone follows the intersection of the triangles τjm and τjk.

Inserting the parametrisation of the fluxes, this gives:


Forming a triangle, the bones must close to zero:


This suggests the following ansatz for the bone b(il):


Find the value of  by demanding that their wedge product gives the fluxes through the triangles:


This fixes  to the value  ±3/2³ , and we can eventually write the bones in terms of the fluxes as:


This concludes the reconstruction.


The relation with loop gravity is the following.  Starting from the Plebanski two-form and the connection smeared over triangles and links. We integrate the Plebanski two-form over the triangles in the frame of the tetrahedron, obtaining:


where h(p →Ti) is a Lorentz holonomy mapping any point in the triangle τij towards the centre of the tetrahedron Ti.

Variables attached to different tetrahedra belong to different frames. To compare them, map into a common frame. This is the bulk holonomy from the centre of the four-simplex – the vertex to the centre of the i-th tetrahedron – the node. The resulting variables are in one-to-one correspondence with the fluxes and normals of this supplement, that is:


so we have the Gauss law together with the linear simplicity constraints impose the geometricity of the underlying four-simplex. In particular, the length l(ij) of each bone b(ij) turns into a unique function of the fluxes. This function is needed to explore the relation between loop quantum gravity and classical Regge calculus.


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