Smooth lattice construction of the Oppenheimer-Snyder spacetime

We present test results for the smooth lattice method using an Oppenheimer-Snyder spacetime. The results are in excellent agreement with theory and numerical results from other authors.

Figure 1

In this figure we show how the lattice is constructed from two radial geodesics $G_0$ and $G_2$ and the series of interconnecting legs $L_{xx}$. The third geodesic $G_1$ lies midway between $G_0$ and $G_2$ is used to define the radial legs $L_{zz}$. The grey patch to the left represents (part) of the dustball. Note that the $L_{xx}$ legs have been drawn as straight lines as a reminder that these legs are geodesic segments.

Figure 2

In this figure we introduce most of the kinematical quantities on the lattice. The dust particle’s unit 4-velocity is $v^{\mu }$, while ${\gamma }^{\mu }$ is the drift vector, $n^{\mu }$ the unit normal to the Cauchy surface ${\Sigma }_t$, $N$ is the lapse function and $\delta \tau$ is the proper time measured along the dust particle’s trajectory.

Figure 3

In this diagram it is easy to see that $[m_{x\mu }{\tilde{\gamma }}^{\mu }]=d{L}_{xx}/dz$ where $\tilde{\gamma }$ is the unit vector parallel to $\gamma$. This result is used in Sec. 4 when deriving Eq. (9).

Figure 4

The $L_{xx}$ leg lengths for $t=0$ to $t=32$ in 20 steps (top, with $0<z<10$) and $t=0$ to $t=500$ in 10 steps (bottom, with $10<z<400$). This time the motion of the apparent horizon is much more noticeable than for the case of geodesic slicing.

Figure 5

A logarithmic plot of the lapse. The even gaps between the curves shows clearly that the collapse is exponential in time. Note the extreme value of the lapse at the origin for late times, of order ${10}^{-110}$.

Figure 6

Here we display the three-dimensional cell which we use to compute the energy density from the conservation Eq. (19).

Figure 7

The $L_{zz}$ leg lengths for $t=0$ to $t=32$ in 20 steps (top, with $0<z<10$) and $t=0$ to $t=500$ in 10 steps (bottom, with $10<z<400$). Notice the extreme change in $L_{zz}$ at the junction. The curves bunch together late in the evolution due to the exponential collapse of the lapse (see Fig. 5).

Figure 8

The Riemann curvature ${\mathcal{R}}_{xyxy}$.

Figure 9

The Riemann curvature ${\mathcal{R}}_{xzxz}$.

Figure 10

The extrinsic curvature ${\mathcal{K}}_{xx}$.

Figure 11

The Hamiltonian and momentum constraints. This shows a slowly growing error for later times.

Figure 12

The Hamiltonian and momentum constraints across the lattice at a fixed time with maximal slicing. The peak occurs near the junction and appears to vary as $1/n_{\infty }$.

Figure 13

The fractional errors for the lapse and the rest density across the lattice with maximal slicing. The large error in the coarsest lattice is probably due to having too few nodes. The finer lattice show much better errors but note that the lapse appears not to converge at the origin. This is due to the use of a finite outer boundary for the lapse.

Figure 14

The fractional errors for the radius and the central density for maximal slicing. The peak occurs around the time when the apparent horizon forms. The height of the peak for the finest resolution is limited by the location of the outer boundary. Doubling $z_{\infty }$ halves the height of the peak.

Figure 15

The lapse at the origin for the three models superimposed on the exact data of Petrich et al. (top) and for the single long-term model (bottom). This shows clearly that the lapse collapses exponentially.

Figure 16

The fractional change in the $L_{xx}$ on the horizon for the three models (top) and for the long-term integration (bottom) in maximal slicing). These errors should be zero by Hawking area theorem.