Einstein-Bianchi system for smooth lattice general relativity. I. The Schwarzschild spacetime

The second Bianchi identity can be recast as an evolution equation for the Riemann curvatures. Here we will report on such a system for a vacuum static spherically symmetric spacetime. This is the first of two papers. In the following paper we will extend the ideas developed here to general vacuum spacetimes. In this paper we will demonstrate our ideas on a Schwarzschild spacetime and give detailed numerical results. For suitable choices of lapse function we find that the system gives excellent results with long term stability.

Figure 1

The local structure of the lattice. The $(x,y,z)$ are a set of coordinates local to this set of legs. There is one such coordinate frame for each $L_{xx}$ along the lattice. These coordinates are never used in the computer code but help to define the metric in the neighborhood of $L_{xx}$.

Figure 2

A simple lattice for a Schwarzschild spacetime. This consists of two identical halves joined at the throat (denoted by the thick line). In our computer code we only store the right-hand half (plus a few nodes from the left-hand half to ensure reflection symmetry at the throat).

Figure 3

The Schwarzschild areal coordinate $r$ and the lattice $L_{xx}$ at the throat are related by $r(t)=2m{L}_{xx}(t)/L_{xx}(0)$. In geodesic slicing $r(t)$ at the throat is described by the parametric equations $r(t)=m(1+\cos \eta (t))$, $t(\eta )=m(\eta +\sin \eta )$. These equations allow us to plot the exact evolution of $r(t)$ (the smooth curve) against estimates from the lattice (solid points). The relative errors are seen to be very small and are dominated by the truncation errors in the Runge-Kutta scheme.

Figure 4

The curvature profiles for four choices of lapse function. Each figure shows the lapse function for $t=0$ to $t=100$ in steps of 10. The small diamond on each curve represents the location of the apparent horizon. There are only two curves visible in the algebraic slicing $N=20L_{xx}/(1+20L_{xx})$ due to the rapid rise in the curvatures as the slicing approaches the singularity.

Figure 5

As per Fig. 4 but this time we display the lapse profiles. Note how quickly the lapse collapse at the throat in the $1+\log $ slicing. This would likely cause serious underflow problems for $t\gtrsim 300$. Notice also the uniform spacing of the curves along the logarithmic axis for the maximal lapse. This show that the lapse collapses exponentially at the throat (a well known result for maximal slicing).

Figure 6

The Riemann curvature ${\mathcal{R}}_{xyxy}$ and the lapse function $N$ for the algebraic slicing with $N=1/(1+{\mathcal{R}}_{xyxy})$. The left pair of figures shows the evolution over the range $t=0$ to 100 in steps of 10, while the right pair cover the range $t=0$ to 1000 in steps of 100. This lapse does not appear (on this time scale) to exhibit an exponential collapse at the throat. The profiles for the algebraic slicings appear to propagate into the asymptotically flat regions far more rapidly than any of the differential slicings. This may be explain why an artificial viscosity was not needed for the algebraic slicings—they carry away any small numerical errors before they have chance to grow.

Figure 7

In the absence of gravitational radiation the area of the horizon should remain constant. It follows that the $L_{xx}$ should be constant on the horizon. Here we plot the fractional variation of $L_{xx}$ on the horizon. The irregular behavior of the plots for later times is due, in part, to the difficulty in accurately locating the horizon.

Figure 8

According to Eq. (6.9) the quantity $L_{xx}^3{\mathcal{R}}_{xyxy}$ should be conserved. In the top row we display the relative error, defined by $1-C(t)/C(0)$ with $C(t)=L_{xx}^3(t){\mathcal{R}}_{xyxy}(t)$, for two choices of lapse. The errors for the $1+\log $ slicing are much larger than those for the algebraic slicing which we attribute to the action of the artificial viscosity terms. In the bottom row we display the momentum constraint for the same pair of slicings. This shows a slow growth in the momentum constraint over time (judging by the peaks in the plots, the growth appears to be linear in time).

Figure 9

This shows an $xz$ section of the lattice obtained by cloning the original two-dimensional lattice. The data in the upper and lower cells are identical to that in the middle cell, this follows from spherical symmetry. The small squares denote the central vertices of each computational cell. The angle $\alpha$ can be computed using standard Euclidean trigonometry as described in the text.