Initial data for general relativistic simulations of multiple electrically charged black holes with linear and angular momenta

Gabriele Bozzola and Vasileios Paschalidis

Phys. Rev. D 99, 104044 – Published 17 May 2019

Abstract

A general relativistic, stationary, and axisymmetric black hole in a four-dimensional asymptotically flat spacetime is fully determined by its mass, angular momentum, and electric charge. The expectation that astrophysically relevant black holes do not posses charge has resulted in a limited number of investigations of moving and charged black holes in the dynamical, strong-field gravitational (and electromagnetic) regime, in which numerical studies are necessary. Apart from having a theoretical interest, the advent of multimessenger astronomy with gravitational waves offers new ways to think about charged black holes. In this work, we initiate an exploration of charged binary black holes by generating valid initial data for general relativistic simulations of black hole systems that have generic electric charge and linear and angular momenta. We develop our initial data formalism within the framework of the conformal transverse-traceless (Bowen-York) technique using the puncture approach and apply the theory of isolated horizons to attribute physical parameters (mass, charge, and angular momentum) to each hole. We implemented our formalism in the case of a binary system by modifying the publicly available TwoPunctures and QuasiLocalMeasures codes. We demonstrate that our code can recover existing solutions and that it has excellent self-convergence properties for a generic configuration of two black holes.

Received 4 March 2019

DOI:https://doi.org/10.1103/PhysRevD.99.104044

Physics Subject Headings (PhySH)

Classical black holes General relativity equations & solutions

Numerical relativity

Gravitation, Cosmology & Astrophysics

Authors & Affiliations

Gabriele Bozzola^*

Department of Astronomy, University of Arizona, Tucson, Arizona 85721, USA

Vasileios Paschalidis^†

Departments of Astronomy and Physics, University of Arizona, Tucson, Arizona 85721, USA

^*gabrielebozzola@email.arizona.edu
^†vpaschal@email.arizona.edu

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Issue

Vol. 99, Iss. 10 — 15 May 2019

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Images

Figure 1
$u$ along the different coordinate axes (solid line for the $x$ axis and dotted and dashed linnes for the $y$ and $z$ axes, respectively) for two punctures with equal mass and opposite charge $Q_{1} = - Q_{2} = 0.5 M$ located on the $x$ axis at $\pm 2 M$ . This configuration is generated with a spectral grid resolution $n_{A} = n_{B} = n_{φ} = 64$ . We graphically compared our solution to the solution in Ref. [86] and found that the curves shown here perfectly match the solution of Ref. [86]. The horizons have areal radius $R_{S_{1}} = R_{S_{2}} = 0.387 M$ as defined by Eq. (23). The vertical dotted lines represent the coordinate radius of the horizons as found by AHFinderDirect. We note that the black hole horizons in binary black holes are generally nonspherical; see, e.g., Ref. [123].
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Figure 2
$u$ along the coordinate axes for a single puncture with charge $Q = 0.5 M$ rotating around the $z$ axis with angular momentum $S^{z} = 0.5 M^{2}$ . The plot corresponds to spectral grid resolution $n_{A} = n_{B} = n_{ϕ} = 64$ . The horizon has areal radius $R_{S} = 0.433 M$ . The different styles of curves have the same meanings as in Fig. 1.
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Figure 3
$u$ along the coordinate axes for a single puncture with charge $Q = 0.5 M$ with linear momentum $P^{z} = 0.5 M$ . The plot corresponds to spectral grid resolution $n_{A} = n_{B} = n_{ϕ} = 64$ . The horizon has areal radius $R_{S} = 0.421 M$ . The different styles of curves have the same meanings as in Fig. 1.
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Figure 4
Electric-field lines on the $x - z$ plane for two charged punctures. The first (left) black hole has charge $Q_{1} = - 0.3 M$ , linear momentum $P_{1}^{x} = - 0.5 M$ , and spin angular momentum $S_{1}^{z} = 0.5 M^{2}$ . The second (right) black hole has $Q_{2} = 0.5 M$ , linear momentum $P_{2}^{z} = - 0.5 M$ and spin angular momentum $S_{2}^{z} = 0.5 M^{2}$ . The black disks depict the apparent horizon of each black hole, which set the scale in the plot. This is the test case used in the self-convergence test reported in Fig. 6.
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Figure 5
$u$ along the coordinate axes for two punctures with charge $Q_{1} = - 0.3 M$ and $Q_{2} = 0.5 M$ . The first black hole has linear momentum $P_{1}^{x} = - 0.5 M$ and spin angular momentum $S_{1}^{z} = 0.5 M^{2}$ . The second black hole has linear momentum $P_{2}^{z} = - 0.5 M$ and spin angular momentum $S_{2}^{z} = 0.5 M^{2}$ . The electric-field lines are reported in Fig. 4. The plot corresponds to spectral grid resolution $n_{A} = n_{B} = n_{ϕ} = 64$ . This system is used for the self-convergence test in Fig. 6. The horizons have radii $R_{S_{1}} = 0.412 M$ and $R_{S_{2}} = 0.373$ and quasilocal masses $M_{S_{1}} = 1.187 M$ and $M_{S_{2}} = 1.202 M$ . The different styles of curves have the same meanings as in Fig. 1.
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Figure 6
Convergence properties of the algorithm measured by computing the maximum relative error on $u ∥ Δ_{n}^{64} u ∥_{\infty}^{T}$ over the test set $T$ (formed by points at distances $1 M$ , $2 M$ , $5 M$ , $10 M$ , $100 M$ , and $1000 M$ on the different coordinate axes). See Fig. 4 for the geometric setup and black hole parameters used for this test. The dashed line shows that the code is approximately sixth-order convergent. All the other physical properties (such as the ADM mass and momenta and the horizon quantities) inherit this excellent convergence behavior from $u$ .
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