Einstein-Bianchi system for smooth lattice general relativity. II. $3+1$ vacuum spacetimes

We will present a complete set of equations, in the form of an Einstein-Bianchi system, that describe the evolution of generic smooth lattices in spacetime. All 20 independent Riemann curvatures will be evolved in parallel with the leg lengths of the lattice. We will show that the evolution equations for the curvatures forms a hyperbolic system and that the associated constraints are preserved. This work is a generalization of our previous paper [L. Brewin, Phys. Rev. D 85, 124045 (2012)] on the Einstein-Bianchi system for the Schwarzschild spacetime to general $3+1$ vacuum spacetimes.

Figure 1

Here we show the evolution of one leg ($oa$) within one computational cell. Clearly the four vectors form a closed loop and thus $(Nn{)}_o\delta t^{\prime }+v_a^{\prime }=v_a+(Nn{)}_a)\delta t^{\prime }$ which leads directly to Eq. (5.6).

Figure 2

An example of the overlap, the shaded region, between a pair of computational cells. The central vertex of each computational cell is denoted by the large dots whereas the smaller dotes denote the vertices that define the boundary of the computation cells. These vertices are themselves the central vertices of other computational cells. In this two-dimensional example the overlap consists of just the pair of triangles. In 3 dimensions the over lap would consist of a closed loop of tetrahedra. In each case there is ample information available to obtain a coordinate transformation between the pair of local Riemann normal frames.