Analytical representation of a black hole puncture solution

The “moving-puncture” technique has led to dramatic advancements in the numerical simulations of binary black holes. Hannam et al. have recently demonstrated that, for suitable gauge conditions commonly employed in moving-puncture simulations, the evolution of a single black hole leads to a well-known, time-independent, maximal slicing of Schwarzschild spacetime. They construct the corresponding solution in isotropic coordinates numerically and demonstrate its usefulness, for example, for testing and calibrating numerical codes that employ moving-puncture techniques. In this brief report we point out that this solution can also be constructed analytically, making it even more useful as a test case for numerical codes.

Figure 1

The conformal factor $\psi$ as a function of isotropic radius $r$ (solid line). The short-dashed line shows the asymptotic value $\psi =\sqrt{3M/(2r)}$ in the limit $r\to 0$; the long-dashed line shows the asymptotic value $\psi =1+M/(2r)$ in the limit $r\to \infty$. The insert shows the same functions on a log-log scale.

Figure 2

The lapse $\alpha$ (top panel) and the shift ${\beta }^r$ (bottom panel) as a function of isotropic radius $r$ (Compare Fig. 5 in HHBGSO).