Wormholes and trumpets: Schwarzschild spacetime for the moving-puncture generation

We expand upon our previous analysis of numerical moving-puncture simulations of the Schwarzschild spacetime. We present a derivation of the family of analytic stationary $1+\log $ foliations of the Schwarzschild solution, and outline a transformation to isotropic coordinates. We discuss in detail the numerical evolution of standard Schwarzschild puncture data, and the new time-independent $1+\log $ data. Finally, we demonstrate that the moving-puncture method can locate the appropriate stationary geometry in a robust manner when a numerical code alternates between two forms of $1+\log $ slicing during a simulation.

Figure 1

Embedding diagram of a two-dimensional slice ($T=\mathrm{const}$, $\theta =\pi /2$) of the extended Schwarzschild solution. The distance to the rotation axis is $R$. A wormhole with a throat at $R=2M$ connects two asymptotically flat ends.

Figure 2

Embedding diagram of a two-dimensional slice ($t=\mathrm{const}$, $\theta =\pi /2$) of the maximal solution (17, 18, 19, 20). The distance to the rotation axis is $R$. In contrast to Fig. 1 there is only one asymptotically flat end. The other end is an infinitely long cylinder with radius $R_0=3M/2$.

Figure 3

Location of the throat, $R_0$, as a function of the coefficient $n$ in the slicing equation (36). In the limit $n\to \infty$ the stationary slice is maximal and the throat is at $R_0=1.5M$, indicated in the figure by a horizontal line. For small values of $n$, the throat approaches the singularity.

Figure 4

The Penrose diagram of the slices defined by the stationary solution of the $1+\log $ condition with $n=2$. Every slice approaches $i_L^{+}$ along the curve $R=R_0\approx 1.31M$ and spatial infinity $i_R^0$ along a curve of constant $T$. The slices are displayed in time steps of $4M$.

Figure 5

The lapse $\alpha$, $x$-component of the shift vector ${\beta }^x$, and $K$ for time-independent $1+\log $ data in isotropic coordinates. The data are shown along the $x$ axis.

Figure 6

A sketch of the method we use to obtain $T(r,t)$. The lines are drawn from actual data with $r_0=3.25M$ for the integration in time and $t=10M$ for the integration in space. Only a subset of the grid points is displayed.

Figure 7

The top two panels show the proper separation from the (outer) horizon versus the Schwarzschild coordinate $R$. The left panel shows the slices at $t=0$, 1, 2, $3M$, and the right panel shows the slice at $t=50M$. The final numerical slice terminates at $R\approx 1.31M$. The vertical line indicates the horizon at $R=2M$, and the six points represent $x/M=1/40$, $1/20$, $1/8$, 2, 5, 8 on each slice. The lower two panels show Schwarzschild $R$ versus numerical $r$ for the same points at the same times. The horizontal lines show $R=2M$ and $R=1.31M$.

Figure 8

Contour plots showing the behavior of $R(r,t)$ on the numerical grid for the first $20M$ of a simulation. The axes of the figures give $r$ and $t$ in units of $M$ for three different coordinate ranges. The contour labels and the colors indicate the value of $R$. We clearly see that $R$ changes rapidly for a given value of the $r$ coordinate for the first few $M$ of evolution before settling down to approximately stationary values at later time. Points with $r>M/2$ accelerate towards the black hole ($R$ decreases) before settling down again at some smaller value of $R$. As the slice moves towards the Schwarzschild singularity, the horizon, initially at $r=M/2$ and $R=2M$, splits into two copies, and values of $R<2M$, which are not present in the initial data, appear as time progresses. The initial slice covers the interior of the black hole from $R=2M$ to $R=+\infty$ for $r=M/2$ to $r=0$. This region is quickly squeezed towards zero extent in $r$, as the lines of constant $R$ in the lower left of the panels indicate. Note that for given $r$, the coordinate motion $R(r,t)$ is not monotonic in $t$, but $R(r,t)$ approaches its asymptotic value via a damped oscillation, see also Fig. 11.

Figure 9

Evolution of $R$ for fixed $r$ for the first $5M$ of an extremely high-resolution simulation. Left panel: the evolution of the third, fourth, and fifth closest points to the puncture, at $r=\{3,4,5\}M/8192$, are shown. We do not expect the closest two points to be reliable, due to finite-difference errors across the puncture, although they show similar behavior, shown in the right panel.

Figure 10

Time for grid points to pass the inner horizon, as a function of coordinate $r$. The innermost grid point is initially at $R=2031M$, and reaches the inner horizon in $T_{2M}=3.35M$.

Figure 11

The Schwarzschild coordinate $R$ of a point at $r=3M/32$ as a function of time. The point overshoots $R_0=1.31M$, indicated by a dashed horizontal line, but for $t>40M$ settles on a value slightly larger than $R_0$.

Figure 12

Time development of $K$ at the horizon, for simulations with initial lapse (a) $\alpha ={\psi }^{-2}$, and (b) $\alpha =1$.

Figure 13

Convergence of $K$ at the horizon, for simulations with initial lapse (a) $\alpha ={\psi }^{-2}$, and (b) $\alpha =1$. The $\alpha ={\psi }^{-2}$ simulations are not convergent at early times, while the $\alpha =1$ simulations lose clean convergence after about $50M$.

Figure 14

Logarithm of the deviation of $K$ at the horizon from the analytic value $K=0.066811$. The deviations are scaled assuming fourth-order convergence, and the results indicate that $K$ converges to the analytic value with fourth-order accuracy.

Figure 15

Penrose diagram produced from numerical data. We can clearly see the numerical slice retract from the second asymptotically flat end. The times shown are $t/M=0$, 0.25, 0.75, 1.25, 2.0, 2.5, 3.5, 8.

Figure 16

A close-up of the region near the cylinder at $R_0=1.31M$. Although the time has elapsed from $t_a\approx 4.5M$ to $t_b=13M$ (with non-negative lapse), $R$ at the innermost gridpoint has increased, $R_a<R_b$.

Figure 17

Penrose diagram of an evolution identical to that used for Fig. 15, except that in this case the shift is zero throughout the evolution and the data points are joined in the plot. We see that the numerical slices no longer retract from the second asymptotically flat end, and now penetrate $R_0$, and stay there. The times shown are the same as in Fig. 15.

Figure 18

The numerical data at Schwarzschild times $t=30$, 37.25, 40, $50M$, with the initial time chosen as $T_0=-40M$. The analytic solution, evaluated at the same times, is shown by a solid gray line. We see that the numerical points lie perfectly on the analytic solution.

Figure 19

The error in ${\tilde{\gamma }}_{xx}$ (i.e., ${\tilde{\gamma }}_{xx}-1$) after $50M$ of evolution of the time-independent trumpet puncture data. The errors for simulations at three resolutions are scaled assuming fourth-order convergence, and we indeed see that the errors converge to zero at fourth order.

Figure 20

The $L_2$ norm of the error in ${\tilde{\gamma }}_{xx}$ over the course of the evolution. The errors again converge to zero at fourth order.

Figure 21

The value of $K$ at the horizon, for a simulation where the slicing condition alternates between standard $1+\log $ and asymptotically maximal $1+\log $, Eqs. (21, 36). The dashed horizontal lines indicate the respective analytic values of $K$ at the horizon.

Figure 22

The coordinate location the horizon, for two simulations with an alternating slicing condition. The dashed line is for a simulation with the standard choice of $\eta =2/M$. The solid line is for a simulation with $\eta =0$. In this case, the horizon location almost returns to its original value when the solution returns to the $1+\log $ stationary geometry.