Measuring the scalar curvature with clocks and photons: Voronoi-Delaunay lattices in Regge calculus

The Riemann scalar curvature plays a central role in Einstein’s geometric theory of gravity. We describe a new geometric construction of this scalar curvature invariant at an event (vertex) in a discrete spacetime geometry. This allows one to constructively measure the scalar curvature using only clocks and photons. Given recent interest in discrete pre-geometric models of quantum gravity, we believe is it ever so important to reconstruct the curvature scalar with respect to a finite number of communicating observers. This derivation makes use of a new fundamental lattice cell built from elements inherited from both the original simplicial (Delaunay) spacetime and its circumcentric dual (Voronoi) lattice. The orthogonality properties between these two lattices yield an expression for the vertex-based scalar curvature which is strikingly similar to the corresponding hinge-based expression in Regge calculus (deficit angle per unit Voronoi dual area). In particular, we show that the scalar curvature is simply a vertex-based weighted average of deficits per weighted average of dual areas. PACS numbers: 83C27, 83C45 and 70G45 Submitted to: Class. Quantum Grav. The scalar curvature in Regge calculus 2 The Riemann scalar curvature invariant plays such a central role in Einstein’s still standard geometric theory of gravitation introduced in 1915, its centrality in the theory cannot be over emphasized. The extremum of this quantity over a proper 4volume of spacetime, yields a solution compatible with Einstein’s field equations. It is this scalar, so central to the Hilbert action, which yields the conservation of energymomentum (contracted Bianchi identities) when variations are done with respect to the diffeomorphic degrees of freedom of the spacetime geometry. This scalar also augments the Ricci tensor in coupling the non-gravitational fields and matter to the curvature of spacetime. It not only appears in its 4-dimensional form in the integrand Hilbert action principle of general relativity, it makes its presence felt in 3-dimensions as an “effective potential energy” in the ADM action. Given this curvature invariant’s pivotal role in the theory of general relativity, we believe it is important to understand how to locally construct this geometric object at a chosen event in an arbitrary curved spacetime. Given recent interest in discrete pre-geometric models of quantum gravity, it is ever so important to reconstruct the curvature scalar with respect to a finite number of observers and photons[1, 2]. Even though we do have familiar discrete representations of each of the twenty components of the Riemann curvature tensor in terms of geodesic deviation or parallel transport around closed loops[3, 4, 5], and apart from the sterile act of simply taking the trace of the Riemann tensor, we are not aware of such a chronometric construction of the scalar curvature. In this manuscript we provide such a discrete geometric description of this scalar curvature invariant utilizing the approach of Regge calculus[6, 7, 8], and the convergencein-mean of Regge calculus was rigorously demonstrated[9]. In the spirit of quantum mechanics and recent approaches to quantum gravity, our construction uses only clocks and photons local to an event on an observer’s world line. Furthermore, this construction is based on a finite number of observers (clocks) exchanging a finite amount of information via photon ranging and yields the scalar curvature naturally expressed in terms of Voronoi and Delaunay lattices[10]. It has been shown that these lattices naturally arrise in Regge calculus[11, 12, 13, 14, 15, 16, 17, 18, 19, 20].This constriction further emphasizes the fundamental role that Voronoi and Delaunay lattices have in the discrete representations of spacetime which is perhaps not so surprising given its preponderant role in self-evolving and interacting structures in nature[10]. In this analysis we introduce a new hybrid (half Voronoi, half Delaunay) simplex which we argue is fundamental to Regge calculus[20] and perhaps fundamental to any discrete representation of classical and quantum gravity. Consider the familiar simplical representation of the geometry of spacetime common in Regge calculus[6, 7]. Here the spacetime is composed of a countable number simplicies. Each 4-simplex is endowed with a flat Minkowski spacetime interior. This lattice is a 4-dimensional spacetime Delaunay lattice. By construction, the curvature in this lattice spacetime does not reside in its 4-simplicies, nor in its tetrahedra; however, the curvature is concentrated on each of its 2-dimensional triangle hinges, h. Each of these hinges is The scalar curvature in Regge calculus 3 the meeting place of three or more 4-simplicies. In the traditional description of Regge calculus, this hinge-based curvature is viewed as a conic singularity; however, it has been shown that the areas h∗ of the Voronoi lattice dual to the Delaunay simplicial lattice provides a natural area to distribute the curvature[20, 19]. The Voronoi lattice is constructed in the usual way by utilizing the circumcentric dual of the Delaunay lattice[10]. The key to our derivation of the Riemann-scalar curvature is the identification Ih ≡ Iv of the usual hinge-based expression the Regge calculus version of the Hilbert action principle [6, 20] with its corresponding vertex-based expression. We begin with the Hilbert action in a d-dimensional continuum spacetime, which is expressible as an integral of the Riemann scalar curvature over the proper d-volume of the spacetime. I = 1 16π ∫ RdVproper (1) On our lattice spacetime, and following the standard techniques of Regge calculus, we can approximate this action as a sum over the triangular hinges h. I ≈ Ih = 1 16π ∑