After providing some theoretical and practical background, we have demonstrated a novel method for constructing initial data for single boosted black holes. We begin with Schwarzschild spacetime written in Kerr-Schild coordinates and apply a series of coordinate transformations. These transformations include
1. a Lorentz boost, 2. trumpet slicing,
3. a radial rescaling to remove the singularity from the coordinate patch (and therefore numerical simu- lation domain), and
Analytically, this data contains no junk radiation. Our analysis showed that – when this data is im- plemented in a simulation – discretization and gauge condition mis-match lead to non-zero junk radiation, though it is several orders of magnitude smaller than the junk radiation in Bowen-York initial data. This analysis relied on developing a thorough understanding of how to physically interpret the Weyl scalars when the black hole puncture is not near the origin of the tetrad. We saw that both types of initial data exhibit a growth of a background trend in ψ4 which does not represent a physical gravitational wave and agrees
well with our calculated prediction from the offset. The boosted-trumpet data also yields more immediately consistent black hole speeds than Bowen-York does.
We built initial data for a binary black hole system by superposing two copies of the boosted-trumpet initial data for a single black hole; this data approximately satisfies the constraint equations. We saw that the numerical asymmetries present in the initial data grew to the point where the black holes’ trajectories were qualitatively different. We saw a similar growth in asymmetry for the Bowen-York black hole, indicating that attempting to evolve a perfectly symmetric binary may be difficult due to chaos in the non-linear evolution equations.
Finally, we looked at how boosted-trumpet initial data can be expanded into new areas. Using the approximate binary data could provide a helpful starting point for re-solving the constraint equations to obtain initial data which retains the benefits we found for a single boosted-trumpet black hole over a Bowen- York black hole while also satisfying the constraints. The application of rotation matricies to boost the black holes in any direction and using Kerr spacetime to obtain spinning boosted-trumpet black holes were discussed.
REFERENCES [1] Einstein toolkit homepage. https://einsteintoolkit.org/. [2] Gnu scientific library. http://www.gnu.org/software/gsl/.
[3] McLachlan Thorn Homepage. https://www.cct.lsu.edu/∼eschnett/McLachlan/.
[4] Multipole on einstein toolkit homepage. http://einsteintoolkit.org/documentation/ThornDoc/ EinsteinAnalysis/Multipole/.
[5] Simulation tools homepage. http://simulationtools.org/.
[6] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al. Observation of Gravitational Waves from a Binary Black Hole Merger. Physical Review Letters, 116(6):061102, February 2016.
[7] B. P. Abbott, R. Abbott, T. D. Abbott, M. R. Abernathy, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al. Tests of general relativity with gw150914. Phys. Rev. Lett., 116:221101, May 2016.
[8] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al. Gw170814: A three-detector observation of gravitational waves from a binary black hole coalescence. Phys. Rev. Lett., 119:141101, Oct 2017.
[9] B. P. Abbott, R. Abbott, T. D. Abbott, F. Acernese, K. Ackley, C. Adams, T. Adams, P. Addesso, R. X. Adhikari, and et al. Gw170817: Observation of gravitational waves from a binary neutron star inspiral. Phys. Rev. Lett., 119:161101, Oct 2017.
[10] Abbott, B. P. et al. (The LIGO Scientific Collaboration and VIRGO Collaboration). Directly comparing gw150914 with numerical solutions of einstein’s equations for binary black hole coalescence. Phys. Rev. D, 94:064035, Sep 2016.
[11] Abbott, B. P. et al. (The LIGO Scientific Collaboration and VIRGO Collaboration). GW151226: Observation of Gravitational Waves from a 22-Solar-Mass Binary Black Hole Coalescence. Phys. Rev. Lett., 116:241103, Jun 2016.
[12] Abbott, B. P. et al. (The LIGO Scientific Collaboration and VIRGO Collaboration). GW170104: Obser- vation of a 50-Solar-Mass Binary Black Hole Coalescence at Redshift 0.2. Phys. Rev. Lett., 118:221101, Jun 2017.
[13] M. Alcubierre. Introduction to 3+1 Numerical Relativity. Oxford University Press, UK, 2008.
[14] Daniela Alic, Carles Bona-Casas, Carles Bona, Luciano Rezzolla, and Carlos Palenzuela. Conformal and covariant formulation of the z4 system with constraint-violation damping.Phys. Rev. D, 85:064040, Mar 2012.
[15] W. Ames. Numerical Methods for Partial Differential Equations. Academic Press, Inc., 1977.
[16] Marcus Ansorg, Bernd Br¨ugmann, and Wolfgang Tichy. Single-domain spectral method for black hole puncture data. Phys. Rev. D, 70:064011, Sep 2004.
[17] Richard L. Arnowitt, Stanley Deser, and Charles W. Misner. The Dynamics of general relativity. Gen. Rel. Grav., 40:1997–2027, 2008.
[18] Abhay Ashtekar and Badri Krishnan. Dynamical horizons and their properties.Phys. Rev. D, 68:104030, Nov 2003.
[19] John Baker, Manuela Campanelli, and Carlos O. Lousto. The lazarus project: A pragmatic approach to binary black hole evolutions. Phys. Rev. D, 65:044001, Jan 2002.
[20] T. W. Baumgarte and S. L. Shapiro. Numerical Relativity: Solving Einstein’s Equations on the Com- puter. Cambridge University Press, June 2010.
[21] Eloisa Bentivegna. Solving the Einstein constraints in periodic spaces with a multigrid approach.Class. Quant. Grav., 31:035004, 2014.
[22] L. Blanchet. Gravitational Radiation from Post-Newtonian Sources and Inspiralling Compact Binaries. Living Reviews in Relativity, 17:2, February 2014.
[23] Jeffrey M. Bowen and James W. York. Time-asymmetric initial data for black holes and black-hole collisions. Phys. Rev. D, 21:2047–2056, Apr 1980.
[24] Katy Clough, Pau Figueras, Hal Finkel, Markus Kunesch, Eugene A Lim, and Saran Tunyasuvunakool. Grchombo : Numerical relativity with adaptive mesh refinement. Classical and Quantum Gravity, 32(24):245011, 2015.
[25] Kenneth A. Dennison and Thomas W. Baumgarte. A Simple Family of Analytical Trumpet Slices of the Schwarzschild Spacetime. Class.Quant.Grav., 31:117001, 2014.
[26] Kenneth A. Dennison, Thomas W. Baumgarte, and Pedro J. Montero. Trumpet Slices in Kerr Space- times. Phys. Rev. Lett., 113:261101, Dec 2014.
[27] Olaf Dreyer, Badri Krishnan, Deirdre Shoemaker, and Erik Schnetter. Introduction to isolated horizons in numerical relativity. Phys. Rev. D, 67:024018, Jan 2003.
[28] C. R. Evans. A method for numerical relativity: Simulation of axisymmetric gravitational collapse and gravitational radiation generation. PhD thesis, Univ. Texas at Austin, 1984.
[29] J. N. Goldberg, A. J. Macfarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan. Spins spherical harmonics and . Journal of Mathematical Physics, 8(11), 1967.
[30] Mark Hannam, Sascha Husa, Frank Ohme, Bernd Br¨ugmann, and Niall ´O Murchadha. Wormholes and trumpets: Schwarzschild spacetime for the moving-puncture generation. Phys. Rev. D, 78:064020, Sep 2008.
[31] R. A. Hulse and J. H. Taylor. Discovery of a pulsar in a binary system. The Astrophysical Journal, 195(2):L51–L53, 1975.
[32] Frank Loffler, Joshua Faber, Eloisa Bentivegna, Tanja Bode, Peter Diener, et al. The Einstein Toolkit: A Community Computational Infrastructure for Relativistic Astrophysics. Class.Quant.Grav., 29:115001, 2012.
[33] Geoffrey Lovelace, Mark A. Scheel, and B´ela Szil´agyi. Simulating merging binary black holes with nearly extremal spins. Phys. Rev. D, 83:024010, Jan 2011.
[34] Richard A. Matzner, Mijan F. Huq, and Deirdre Shoemaker. Initial data and coordinates for multiple black hole systems. Phys. Rev. D, 59:024015, Dec 1998.
[35] C.W. Misner, K.S. Thorne, and J.A. Wheeler. Gravitation. Freeman, San Francisco, CA, U.S.A., 1973. [36] Hiroyuki Nakano, Yosef Zlochower, Carlos O. Lousto, and Manuela Campanelli. Intermediate-mass-ratio black hole binaries. ii. modeling trajectories and gravitational waveforms.Phys. Rev. D, 84:124006, Dec 2011.
[37] Andrea Nerozzi, Christopher Beetle, Marco Bruni, Lior M. Burko, and Denis Pollney. Towards a wave- extraction method for numerical relativity. ii. the quasi-kinnersley frame. Phys. Rev. D, 72:024014, Jul 2005.
[38] E. Newman and R. Penrose. An Approach to Gravitational Radiation by a Method of Spin Coefficients. Journal of Mathematical Physics, 3:566–578, May 1962.
[39] E. Newman and R. Penrose. Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients. Journal of Mathematical Physics, 4:998–998, July 1963.
[40] Niall ´O Murchadha and James W. York. Gravitational energy. Phys. Rev. D, 10:2345–2357, Oct 1974. [41] Vasileios Paschalidis, Zachariah B. Etienne, Roman Gold, and Stuart L. Shapiro. An efficient spectral
interpolation routine for the TwoPunctures code. 2013.
[42] Eric Poisson, Adam Pound, and Ian Vega. The motion of point particles in curved spacetime. Living Rev. Rel., 14:7, 2011.
[43] Eric Poisson and Clifford M. Will. Gravity: Newtonian, Post-Newtonian, Relativistic. Cambridge University Press, Cambridge, UK, 1st edition, 2014.
[44] William H. Press, Saul A. Teukolsky, William T. Vetterling, and Brian P. Flannery.Numerical Recipes in C: The Art of Scientific Computing. Cambridge University Press, Cambridge, UK, 2nd edition edition, 1993.
[45] Christian Reisswig and Denis Pollney. Notes on the integration of numerical relativity waveforms. Classical and Quantum Gravity, 28(19):195015, 2011.
[46] I. Ruchlin, J. Healy, C. O. Lousto, and Y. Zlochower. New Puncture Initial Data for Black-Hole Binaries: High Spins and High Boosts. ArXiv e-prints, October 2014.
[47] Erik Schnetter, Scott H Hawley, and Ian Hawke. Evolutions in 3d numerical relativity using fixed mesh refinement. Classical and Quantum Gravity, 21(6):1465, 2004.
[48] Erik Schnetter, Badri Krishnan, and Florian Beyer. Introduction to dynamical horizons in numerical relativity. Phys. Rev. D, 74:024028, Jul 2006.
[49] Ulrich Sperhake. Binary black-hole evolutions of excision and puncture data. Phys. Rev. D, 76:104015, Nov 2007.
[50] Ulrich Sperhake, Vitor Cardoso, Frans Pretorius, Emanuele Berti, and Jos´e A. Gonz´alez. High-energy collision of two black holes. Phys. Rev. Lett., 101:161101, Oct 2008.
[51] Jonathan Thornburg. Finding apparent horizons in numerical relativity.Physical Review D, 54(8):4899– 4918, October 15 1996.
[52] Jonathan Thornburg. A fast apparent-horizon finder for 3-dimensional Cartesian grids in numerical relativity. Classical and Quantum Gravity, 21(2):743–766, January 21 2004. gr-qc/0306056.
[53] Petr Tsatsin and Pedro Marronetti. Initial data for neutron star binaries with arbitrary spins. Phys. Rev. D, 88:064060, Sep 2013.
[54] T A Wagner, S Schlamminger, J H Gundlach, and E G Adelberger. Torsion-balance tests of the weak equivalence principle. Classical and Quantum Gravity, 29(18):184002, 2012.
[55] Clifford M. Will. The confrontation between general relativity and experiment. Living Reviews in Relativity, 17(1):4, 2014.
[56] J. W. York. Kinematics and dynamics of general relativity. In Sources of gravitational radiation. Cambridge University Press, 1978.
[57] J. W. York and Tsvi Piran. Initial value problem and beyond. InSpacetime and Geometry: The Alfred Schild Lectures. University of Texas Press, 1982.