Closure constraints for hyperbolic tetrahedra by Charles and Livine

This week I been continuing to look at non-euclidean tetrahedra as in the post:

In particular I have been looking the paper ‘Closure constraints for hyperbolic tetrahedra’ in this paper the authors  investigate the generalization of loop gravity‘s twisted geometries to a q-deformed gauge group. In the standard undeformed case, loop gravity is a formulation of general relativity as a diffeomorphism-invariant SU(2) gauge theory. Its classical states are graphs provided with algebraic data. In particular closure constraints at every node of the graph ensure their interpretation as twisted geometries. Dual to each node, one has a polyhedron embedded in at space R³. One then

glues them allowing for both curvature and torsion.

It was recently conjectured that q-deforming the gauge group SU(2) would allow to account for a non-vanishing cosmological constant Λ ≠ 0, and in particular that deforming the loop gravity phase space with real parameter q  would lead to a generalization of twisted geometries to a hyperbolic curvature.

The Flat tetrahedron

Closure constraints

closfig2

We define the outward vectors normal to the faces with their norm given by the face area:

closequ5and

closequ8

These normals satisfy the symmetric relation:

closequ9

This is the closure constraints for the flat tetrahedron.

The hyperbolic triangle 

The curved tetrahedron  lives in the 3d one-sheet hyperboloid defined as the submanifold of R³’¹ of timelike vectors satisfying:

closequ23

where κ is the curvature radius of the hyperboloid.

The hyperbolic triangle is  embedded in the 3-
hyperboloid H3  which is a coset of the Lorentz group:

closequ25Studying the closure constraints defining the hyperbolic triangle is a first step towards developing the closure constraints for the hyperbolic tetrahedron. In particular, it will lead us to introduce the SU(2) holonomy around the triangle, which will play the role of a group-valued normal vector to the triangle.

closfig4

The triangle is defined by three points a, b, c. They are defined by three translations from the hyperboloid origin  Ω= (1, 0, 0,0).  Introduce the three oriented edges of the triangle as 1 = (ab), 2 = (ac) and 3 = (bc). We define the translations along each edge:

closequ33

These three group elements obviously satisfy the following closure relation:

closequ34

Compact closure constraint for the hyperbolic tetrahedron

Consider the tetrahedron in hyperbolic space. It is defined by four points, a, b, c and d, defined by their respective triangular matrix la,b,c,d describing their position on the hyperboloid.

closfig6We want to find a closure constraint for this hyperbolic tetrahedron in terms of the hyperbolic triangle normals.

closfig7

We can construct the three SU(2) holonomies defining the normal rotations to the hyperbolic triangles:

closequ51

 

and closequ52

It satisfies a simple closure relation:

closequ53

From the point of view of differential geometry, the holonomies are the discrete counterparts of curvature and this closure relation is just the Bianchi identity.

Non-Compact closure constraint for the hyperbolic tetrahedron

Consider a hyperbolic tetrahedron formed by the 4 points a, b, c, d, with the four hyperbolic triangles (abc), (bcd), (cda) and (dab). Looking at one triangle, say (abc), its three vertices and defined by three translations la,b,c . The hyperbolic translation vectors along the three edges are SB(2,C) elements, which satisfy
the triangle closure constraint:

closequ63

Define the SB(2,C) normal to the triangle as:

closequ64

Choosing a definite path along the tetrahedron vertices

closfig10

say (abcd) following alphabetical and consider the SB(2,C) normal to the triangles following the path’s order:

closequ65

These SB(2,C) normals satisfy the following closure relation:

closequ66

Conclusion

In this paper the authors investigated the question of closure constraints for the hyperbolic tetrahedron in the context of loop
quantum gravity with a non-vanishing cosmological constant.

Their goal in doing this was as a first step towards interpreting
the deformed phase space structure for loop gravity on a given graph defined as discrete 3d hyperbolic geometries to be embedded in a 3 + 1-dimensional space-time.

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