Curvatures and discrete Gauss–Codazzi equation in (2 + 1)-dimensional loop quantum gravity

Abstract

We derive the Gauss-Codazzi equation in the holonomy and plane-angle representations and we use the result to write a Gauss-Codazzi equation for a discrete (2+1)-dimensional manifold, triangulated by isosceles tetrahedra. This allows us to write operators acting on spin network states in (2+1)dimensional loop quantum gravity, representing the 3-dimensional intrinsic, 2-dimensional intrinsic, and 2-dimensional extrinsic curvatures. II. GAUSS-CODAZZI EQUATION A. Standard Gauss-Codazzi equationThe Gauss-Codazzi equation relates the Riemann intrinsic curvatures of a manifold and its submanifold with the extrinsic curvature of the submanifold. We will briefly review the continuous (2+1)-dimensional Gauss-Codazzi equation in this section, but the formula will be valid in (n + 1)-dimension. arXiv:1503.05943v1 [gr-qc]

(2+1) Regge Calculus: Discrete Curvatures, Bianchi Identity, and Gauss-Codazzi Equation

“…The ADM formalism is based by an older theorem of Gauss, widely known by mathematicians as the Gauss-Codazzi relation, which describe the relation between curvatures of a manifold with its embeddded submanifold, or, in the language of general relativity, between spacetime and its hypersurface foliation. Works on canonical formulation of Regge calculus had been started by [37][38][39][40][41], and specifically, on the hypersurface foliation and Gauss-Codazzi equation in discrete geometry by [42,43], with the recent works by [44,45]. Our work is an attempt to clarify some parts of these previous results.…”

“…The following derivation will be based on our previous work [45]; e 0 in general will be a linear combination of coordinate basis vector in T p M :…”

“…The curvatures need to satisfy some geometrical relations, which are, the Bianchi identity and the Gauss-Codazzi relation. The discrete version of the first has been studied extensively, for instance [52][53][54], and the latter in [42,43,45], but the discrete geometrical interpretation of these relations remain unclear.…”

Statistical discrete geometry

“…10. For a detailed explanation about these angles, see [46,49,50]. Therefore, it is clear that formula (27) is purely intrinsic to the 2D surface, it does not depend on the embedding in R 3 , we only use R 3 to help us to derive relation (27) in an easy way.…”

Degrees of freedom in discrete geometry

“…Intrinsic curvatures as an emergent property. Given the definition of intrinsic curvature of discrete geometry, which is the deficit angle located on the hinge shared by several simplices [1,34,41], it is clear that curvature can only be defined in a system of coupled n-particles of space, in other words, we could think the intrinsic curvature as an emergent property of a many-body system, it is the measure of how strong is the 'interaction' among the 'particles' of space.…”