Generation of Initial Data for General-Relativistic Simulations of Charged Black Holes

  • Gabriele BozzolaEmail author
  • Vasileios Paschalidis
Part of the Tutorials, Schools, and Workshops in the Mathematical Sciences book series (TSWMS)


Einstein–Maxwell theory is a description of electromagnetism and gravity from first principles that has yielded several interesting results on black holes with electromagnetic fields. However, since astrophysically-relevant black holes are believed to have negligible electric charge, the theory has been mostly confined within the realm of theoretical investigation. In addition to this, this theory has been studied mostly with analytical tools, which is why the vast majority of available results are restricted to spacetimes endowed with some degree of symmetry (e.g., stationarity and axisymmetry). Consequently, dynamical solutions (where no symmetry is assumed), for which the numerical approach is the only feasible one, represent a largely unexplored territory.

In this paper we present our efforts towards dynamical solutions of the coupled Einstein–Maxwell equations. As a first step, we solve numerically the constraint equations to generate valid initial data for dynamical general-relativistic simulations of generic configurations of black holes that possess electric charge, linear and angular momenta. The initial data are constructed with the conformal transverse-traceless approach, and the black holes are described as punctures within a modified Bowen–York framework. The attribution of physical parameters (mass, charge, and momenta) to the holes is performed by adopting the theory of isolated horizons. We implement our new formalism numerically and find it both to be in agreement with previous results and to show good convergence properties.


General relativity Numerical relativity Einstein–Maxwell Black holes 3 + 1 Decomposition 


  1. 1.
    M. Alcubierre, J.C. Degollado, M. Salgado, Einstein-Maxwell system in 3 + 1 form and initial data for multiple charged black holes. Phys. Rev. D 80(10), 104022 (2009). arXiv: 0907.1151 [gr-qc]
  2. 2.
    M. Ansorg, B. Brügmann, W. Tichy, A single-domain spectral method for black hole puncture data. Phys. Rev. D 70, 064011 (2004). Eprint: arXiv:gr-qc/0404056
  3. 3.
    A. Ashtekar et al., Generic isolated horizons and their applications. Phys. Rev. Lett. 85, 3564–3567 (2000). Eprint: gr-qc/0006006 ADSMathSciNetCrossRefGoogle Scholar
  4. 4.
    T.W. Baumgarte, S.L. Shapiro, Numerical Relativity: Solving Einstein’s Equations on the Computer (Cambridge University Press, Cambridge, 2010)CrossRefGoogle Scholar
  5. 5.
    J.M. Bowen, J.W. York Jr, Time asymmetric initial data for black holes and black hole collisions. Phys. Rev. D 21, 2047–2056 (1980). ADSCrossRefGoogle Scholar
  6. 6.
    G. Bozzola, V. Paschalidis, Initial data for general relativistic simulations of multiple electrically charged black holes with linear and angular momenta. Phys. Rev. D 99(10), 104044.
  7. 7.
    W. Israel, G.A. Wilson, A class of stationary electromagnetic vacuum fields. J. Math. Phys. 13, 865–867 (1972). ADSCrossRefGoogle Scholar
  8. 8.
    S.D. Majumdar, A class of exact solutions of Einstein’s field equations. Phys. Rev. 72, 390–398 (1947). ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    E.T. Newman et al., Metric of a rotating, charged mass. J. Math. Phys. 6, 918–919 (1965). ADSMathSciNetCrossRefGoogle Scholar
  10. 10.
    A. Papapetrou, A static solution of the equations of the gravitational field for an arbitary charge-distribution, in Proceedings of the Royal Irish Academy. Section A: Mathematical and Physical Sciences, vol. 51 (1945), pp. 191–204. ISSN: 00358975,
  11. 11.
    The LIGO Scientific Collaboration and the Virgo Collaboration, GWTC-1: A gravitational-wave transient catalog of compact binary mergers observed by LIGO and Virgo during the first and second observing runs. arXiv e-prints (2018). arXiv: 1811.12907 [astro-ph.HE] Google Scholar
  12. 12.
    J.W. York Jr, Kinematics and dynamics of general relativity, in Sources of Gravitational Radiation, ed. by L.L. Smarr (Cambridge University Press, Cambridge and New York, 1979), pp. 83–126Google Scholar

Copyright information

© Springer Nature Switzerland AG 2019

Authors and Affiliations

  1. 1.Department of Astronomy and Steward ObservatoryUniversity of ArizonaTucsonUSA
  2. 2.Departments of Astronomy and PhysicsUniversity of ArizonaTucsonUSA

Personalised recommendations