The tetrahedron and its Regge conjugate

This week I have been reading the PhD thesis ‘Single and collective dynamics of discretized geometries’ by Dimitri Marinelli. In this post I’ll look at a small portion about Regge calculus, the  tetrahedron and its Regge conjugate.

Regge Calculus is a dynamical theory of space-time introduced in 1961 by Regge as a discrete approximation for the Einstein theory of gravity. The basic idea is to replace a smooth space-time with a collection of simplices. The collective dynamics of these geometric objects is driven by the Regge action and the dynamical variables are their edge lengths – which play the role of the metric tensor of General Relativity. Simplices are the n-dimensional generalization of triangles and tetrahedra. Regge Calculus inspired and is at the base of almost all the present discretized models for a quantum theory of gravity for at least two reasons:

  • It is a discretized model, so it represents a possible atomistic system typical of quantum systems
  • There is a deep connection between the Regge action, the asymptotic of the 6j symbol and a path integral formulation of gravity.

Let’s see  how the Regge transformation acts on a tetrahedral shape. The formulas

reggeequ4.01

and the association between 6j symbol and an Euclidean tetrahedron tell us that any Regge transformation acts on four edges of a tetrahedron keeping a pair of opposite edges unchanged. The Regge-transformed tetrahedra is called `conjugate’.

Using the Ponzano-Regge formula for the 6j,

reggeequ2.18we can immediately say that the volume of a tetrahedron and that of a Regge transformed one must coincide.

thereom23

tetrahedron

The volume of a tetrahedron is also invariant under the Regge transformation of four consecutive edges.

The volume of a tetrahedron, being a function of six parameters, can be expressed in several ways. For the tetrahedron below:

tetrahedron with dihedral angleThe ‘orientated’ volume reads, 

reggeequ3.2.6

where AABC and AACD are respectively the areas of the triangles ABC and ACD, lAC is the length of the common edge and β is the dihedral angle between these two faces.

The importance of the Regge symmetry is that it constrains the shape dynamics of a single tetrahedron,  it relates different tetrahedra equating their quantum representations and it is the key tool to understand the classical motion of a four-bar linkage mechanical systems and its link to the the quantum dynamics of tetrahedra.

This thesis also contains a section on the Askey scheme which I’ll be following up in future posts:

askey scheme

 

 

 

 

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