A single-domain spectral method for black hole puncture data

TL;DR

The paper develops a single-domain, pseudo-spectral method to compute puncture black hole initial data that satisfy the Hamiltonian constraint, using coordinate maps that smooth the punctures to achieve rapid convergence. It analyzes single- and two-puncture configurations, including the test-mass limit, and demonstrates exponential convergence near punctures but algebraic convergence at infinity due to logarithmic terms when momentum is nonzero. The authors compare binding energies in the test-mass limit to Schwarzschild and post-Newtonian results, finding a significant deviation at the ISCO and highlighting the influence of mass definitions and artificial radiation in puncture data. The work provides a high-accuracy, efficient framework for puncture initial data and sets the stage for improved mass definitions and domain decompositions in future studies.

Abstract

We calculate puncture initial data corresponding to both single and binary black hole solutions of the constraint equations by means of a pseudo-spectral method applied in a single spatial domain. Introducing appropriate coordinates, these methods exhibit rapid convergence of the conformal factor and lead to highly accurate solutions. As an application we investigate small mass ratios of binary black holes and compare these with the corresponding test mass limit that we obtain through a semi-analytical limiting procedure. In particular, we compare the binding energy of puncture data in this limit with that of a test particle in the Schwarzschild spacetime and find that it deviates by 50% from the Schwarzschild result at the innermost stable circular orbit of Schwarzschild, if the ADM mass at each puncture is used to define the local black hole masses.

Figures (6)

• Figure 1: For a single puncture with vanishing linear momentum parameter the spin $S^i=m^2 w\,\delta_1^i$ with $w=0.2$ has been chosen. The plot shows the relative global accuracy of the spectral method for expansion order $n_A=n_B=n$ compared to a reference solution with $n = 50$, see (\ref{['accuracy']}). For this axisymmetric example we have used $n_\varphi=4$.
• Figure 2: For a single puncture with vanishing spin parameter the linear momentum $P^i=mv\,\delta^i_1$ with $v=0.2$ has been chosen. The plot shows the relative global accuracy of the spectral method for expansion order $n_A=n_B=n$ compared to a reference solution with $n = 70$, see (\ref{['accuracy']}). For this axisymmetric example we have used $n_\varphi=4$.
• Figure 3: Several coordinate patches for the two puncture initial data problem. Shown are $(a)$ equidistant coordinate lines in the system of spectral coordinates $(A,B)$, as well as $(b)$ their images in prolate spheroidal coordinates $(\xi,\eta)$, $(c)$ in the coordinates $(X,R)$, and $(d)$ in cylindrical coordinates $(x,\rho)$. The punctures are indicated by bullets. The $(x=0)$-plane, several sections of the $x$-axis and their corresponding images in the other coordinate systems as well as spatial infinity given by $A=1$ are emphasized by thick lines.
• Figure 4: Two punctures with vanishing spins. The physical parameters are given by $m_+=m_-=b, P^i_\pm=\pm0.2\,b\,\delta^i_2$. For this plot we took $n_A=n_B=2n_\varphi=n$ and compared to a reference solution with $n = 70$. Apart from the global relative accuracy (see (\ref{['accuracy']})) taken over $6^3$ spatial points, the corresponding maximal deviations at infinity and at the punctures are shown. For small $n$, the error near the punctures is about ten times larger than the error at infinity, and the convergence rate is approximately exponential down to about $10^{-9}$. The error at infinity converges at roughly sixth algebraic order as expected, and for sufficiently large $n$ this becomes the dominant convergence rate.
• Figure 5: Example for a solution to the Hamiltonian constraint obtained with the spectral method and with a multigrid method on nested Cartesian grids. Shown is the regular part $u$ of the conformal factor for two punctures without spin and vanishing total linear momentum, which are located on the $x$-axis at $x=\pm 3M$. Results from the multigrid method are indicated by lines with markers. The panels on the left show the various levels of refinement combined into one line for the highest resolution (see text). The panels on the right show an enlargement of the region near one of the punctures for three resolutions of the multigrid method. In all panels the result for the single-domain spectral method with $n_A=n_B= 40$ and $n_\varphi=20$ is shown as a solid line without markers. Note that on this scale the methods agree well both far away and close to the punctures.