A Kirchhoff-like conservation law in Regge calculus

TL;DR

The paper tackles the long-standing challenge of coupling non-gravitational matter to Regge Calculus by deriving a discrete contracted Bianchi identity through the Cartan moment of rotation on the circumcentric dual lattice, embedding this within the boundary-of-a-boundary principle to obtain a Kirchhoff-like conservation law at each vertex. It introduces a Regge-Einstein tensor defined per edge $L$, via the relation $G_{LL} L^* = \tfrac{1}{2} \sum_{h\supset L} L \cot{\theta_{Lh}} \epsilon_h$, and shows how the contracted Bianchi identity emerges from summing dual 3-boundaries around a dual 4-volume, up to controllable ${O}(L^5)$ corrections due to non-commuting finite rotations. This framework enables an approximate vertex-based conservation of energy-momentum and provides a principled path to extend Regge Calculus to non-vacuum spacetimes by encoding stress-energy as a vertex-based, doubly-projected quantity along lattice edges. The results deepen the understanding of simplicial diffeomorphism structure and have potential implications for classical and quantum gravity scenarios where spacetime is fundamentally discrete.

Abstract

Simplicial lattices provide an elegant framework for discrete spacetimes. The inherent orthogonality between a simplicial lattice and its circumcentric dual yields an austere representation of spacetime which provides a conceptually simple form of Einstein's geometric theory of gravitation. A sufficient understanding of simplicial spacetimes has been demonstrated in the literature for spacetimes devoid of all non-gravitational sources. However, this understanding has not been adequately extended to non-vacuum spacetime models. Consequently, a deep understanding of the diffeomorphic structure of the discrete theory is lacking. Conservation laws and symmetry properties are attractive starting points for coupling matter with the lattice. We present a simplicial form of the contracted Bianchi identity which is based on the E. Cartan moment of rotation operator. This identity manifests itself in the conceptually-simple form of a Kirchhoff-like conservation law. This conservation law enables one to extend Regge Calculus to non-vacuum spacetimes and provides a deeper understanding of the simplicial diffeomorphism group.

Figures (4)

• Figure 1: The polyhedral boundary of a 4-polytope: This illustration shows the 2-dimensional projection of a typical 4-dimensional polytope, $V^*$, of the circumcentric dual (Voronoi) spacetime. It is dual to a vertex, $V$, of the simplicial (Delaunay) spacetime. This 4-polytope is bounded by six polygons (shown exploded off into the perimeter of the polytope). These 4-polytopes are ordinarily not simplexes nor are their bounding polyhedra. The orientation of $V^*$ induces an orientation on each of its polyhedral faces, $L^*$. The orientation of each polyhedron consequently induces an orientation on each its polygonal faces. However, each 2-face is shared by two polyhedra thereby inducing equal and opposite orientations on it. In this sense none of these polygonal faces are exposed and their orientations cancel. This is the origin of the BBP principle in its 2-3-4 dimensional form.
• Figure 2: BBP as a Geometric Identity: Here two adjoining faces of the 3-dimensional boundary of a 4-dimensional volume are depicted with their induced orientation. The orientation of the 2-dimensional area is seen to be opposite for the adjoining 3-volumes such that in the sum over the boundary of the boundary these areas cancel one another. Furthermore, if a vector $\vec{U}$ is parallel transported around the area adjoining the two 3-volumes, then the vector will ordinarily come back rotated. When the area is associated with the left 3-volume, the vector $\vec{U}$ comes back rotated as $\vec{U}^{'}$, but when the area is associated with the right 3-volume it will come back as $\vec{U}^{"}$. The rotation in both cases is in the same plane and rotated by the same amount but in opposite directions of rotation.
• Figure 3: Hinges and Moment Arm: In the simplicial lattice each edge is common to multiple hinges $h$ (left). The circumcentric 3-volume $L^*$ dual to edge $L$ has 2-dimensional boundaries dual to each of the hinges $h$ (right). The parallel transport of a vector around the perimeter of these dual areas will result in a net rotation by an angle equal to the deficit angle, $\epsilon_h$, associated with the hinge, $h$. The moment of rotation is given by a moment arm $P_{L}+dP_{Lh}$ wedge the rotation associated with the parallel transport around the dual area. However, the first term does not contribute as it is equal to zero by the ordinary Bianchi identity. On a given hinge, the effective moment arm is the vector from the edge to the center of rotation, i.e. the circumcenter of the hinge $C$, which has length $(1/2) L \cot{\theta_{Lh}}$.
• Figure 4: The Kirchhoff-like form of the contracted Bianchi identity in RC: On the left is an exploded view of the edges meeting a vertex $V$ and their dual 3-volumes, $L^*$. The first step towards the Kirchhoff-like conservation principle is constructing the total moment of rotation for each of the dual 3-volumes, $L^*$. On the right is a depiction of the flow of moment-of-rotation, or equivalently flow of stress-energy along each of the edges, $L$ meeting at the vertex $V$. The flow of moment-of-rotation entering or leaving vertex, $V$, is conserved to second order in the lattice spacing, $L$. Since the Einstein equations, and their RC equivalent, equates the moment-of-rotation with the stress-energy, this contracted Bianchi identity can be viewed as a circuit-like conservation law. Here the "wires" of the circuit are the edges of the simplicial lattice, and the "current" in each of the wires is the doubly-projected stress-energy tensor $T_{LL}$ along the given edge $\mathbf{L}$ emanating from vertex $V$.