Where do moving punctures go?

TL;DR

The paper investigates how moving-puncture evolutions of Schwarzschild spacetime reshape the geometry of slicings, showing that the numerical slices detach from the second asymptotically flat end and settle on a cylinder at finite radius, with the conformal factor diverging as $1/\sqrt{r}$. It connects this stationary state to the $t \rightarrow \infty$ Estabrook maximal-slicing solution and demonstrates consistency between numerical evolutions and analytic maximal slices. The authors then construct time-independent, 'cylindrical' initial data by solving the Hamiltonian constraint with a tailored conformal factor, providing a rigorous testbed for numerical relativity codes and a potential route toward binary initial data in favorable coordinates. Overall, the work clarifies the gauge dynamics of moving punctures, demonstrates the robustness of the method, and offers practical tools for accuracy testing and improved initial-data construction in black-hole simulations.

Abstract

Currently the most popular method to evolve black-hole binaries is the ``moving puncture'' method. It has recently been shown that when puncture initial data for a Schwarzschild black hole are evolved using this method, the numerical slices quickly lose contact with the second asymptotically flat end, and end instead on a cylinder of finite Schwarzschild coordinate radius. These slices are stationary, meaning that their geometry does not evolve further. We will describe these results in the context of maximal slices, and present time-independent puncture-like data for the Schwarzschild spacetime.

Figures (6)

• Figure 1: The value of $K$ on the horizon $R = 2M$ as a function of time, for a moving-puncture evolution of Schwarzschild initial data. In this case the evolution reaches a stationary state.
• Figure 2: The Schwarzschild radial coordinate $R$ as a function of numerical distance $r$ for the initial data.
• Figure 3: The Schwarzschild radial coordinate $R$ as a function of numerical distance $r$ at times $T = 0,1M,2M,3M$ in the top-left, top-right, bottom-left and bottom-right figures. We see that the numerical slice loses contact with the second asymptotically flat end.
• Figure 4: The Schwarzschild radial coordinate $R$ at the puncture $r=0$ as a function of time. The end of the slice settles at $R = 3M/2$ after about 40$M$ of evoluution.
• Figure 5: The lapse function and $x$-component of the shift vector for time-independent puncture data for the Schwarzschild spacetime, based on the $t \rightarrow \infty$ limit of the Estabrook et al solution.