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Gravitational wave luminosity and net momentum flux in head-on mergers of black holes: Radiative patterns and mode mixing

Rafael Fernandes Aranha^1,*, Ivano Damião Soares^2,†, and Eduardo Valentino Tonini^3,‡

¹Universidade do Estado do Rio de Janeiro—UERJ, 20550-900 Rio de Janeiro, Brazil
²Centro Brasileiro de Pesquisas Físicas—CBPF/MCTI, 22290-180 Rio de Janeiro, Brazil
³Instituto Federal do Espírito Santo—IFES, 29040-333 Vitória, Brazil

^*rafael.aranha@uerj.br
^†ivano@cbpf.br
^‡tonini@cefetes.br

Phys. Rev. D 94, 064017 – Published 8 September, 2016

DOI: https://doi.org/10.1103/PhysRevD.94.064017

Abstract

We show that gravitational wave radiative patterns from a point test particle falling radially into a Schwarzschild black hole, as derived by Davis, Ruffini, Press and Price [M. Davis et al., Phys. Rev. Lett. 27, 1466 (1971).], are present in the nonlinear regime of head-on mergers of black holes. We use the Bondi-Sachs characteristic formulation and express the gravitational wave luminosity and the net momentum flux in terms of the news functions. We then evaluate the ( $- 2$ )-spin-weighted $ℓ$ -multipole decomposition of these quantities via exact expressions valid in the nonlinear regime and defined at future null infinity. Our treatment is made in the realm of Robinson-Trautman dynamics, with characteristic initial data corresponding to the head-on merger of two black holes. We consider mass ratios in the range $0.01 \leq α \leq 1$ . We obtain the exponential decay with $ℓ$ of the total energy contributed by each multipole $ℓ$ , with an accurate linear correlation in the log-linear plot of the points up to $α ≃ 0.7$ . Above this mass ratio the contribution of the odd modes to the energy decreases faster than that of the even modes, leading to the breaking of the linear correlation; for $α = 1$ the energy in all odd modes is zero. The dominant contribution to the total radiated energy comes from the quadrupole mode $ℓ = 2$ corresponding, for instance, to about $≃ 84 %$ for small mass ratios up to $≃ 99.8 %$ for the limit case $α = 1$ . The total rescaled radiated energy $E_{W}^{total} / m_{0} α^{2}$ decreases linearly with decreasing $α$ , yielding for the point particle limit $α \to 0$ the value $≃ 0.0484$ , about 5 times larger than the result of Davis et al. [1]. The mode decomposition of the net momentum flux and of the associated gravitational wave impulses results in an adjacent-even-odd mode-mixing pattern. We obtain that the impulses contributed by each $(ℓ, ℓ + 1)$ mixed mode also accurately satisfy the exponential decay with $ℓ$ , for the whole mass ratio domain considered, $0.01 \leq α < 1$ . The (2, 3) mode contributions to the total impulses are dominant. The mode-mixing effect can also be seen in the decomposition of the net kick velocity imparted to the system by the gravitational wave emission. The mixed mode impulses reach a maximum at $α ≃ 0.7$ ; for $α > 0.7$ the impulses decrease and are zero in the equal mass case, due to the decrease to zero of the odd modes of the news functions.

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References (52)

M. Davis, R. Ruffini, W. H. Press, and R. H. Price, Phys. Rev. Lett. 27, 1466 (1971).
B. Abbott et al. (LIGO Scientific Collaboration), Rep. Prog. Phys. 72, 076901 (2009).
G. M. Harry and LIGO Scientific Collaboration, Classical Quantum Gravity 27, 084006 (2010).
T. Accadia, F. Acernese, F. Antonucci, P. Astone, G. Ballardin et al., Classical Quantum Gravity 28, 114002 (2011).
K. Somiya (KAGRA Collaboration), Classical Quantum Gravity 29, 124007 (2012).
J. G. Baker, J. Centrella, D. Choi, M. Koppitz, R. van Meter, and M. C. Miller, Astrophys. J. 653, L93 (2006).
D. Merrit, M. Milosavljević, M. Favata, S. S. Hughes, and D. E. Holz, Astrophys. J. 607, L9 (2004).
M. Favata, S. A. Hughes, and D. E. Holz, Astrophys. J. 607, L5 (2004).
L. Blanchet, Living Rev. Relativ. 9, 4 (2006).
F. Pretorius, Phys. Rev. Lett. 95, 121101 (2005).
J. G. Baker, J. Centrella, D. I. Choi, M. Koppitz, and J. van Meter, Phys. Rev. Lett. 96, 111102 (2006).
M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Phys. Rev. Lett. 96, 111101 (2006).
J. A. González, U. Sperhake, B. Brügmann, M. Hannam, and S. Husa, Phys. Rev. Lett. 98, 091101 (2007).
J. A. González, U. Sperhake, and B. Brügmann, Phys. Rev. D 79, 124006 (2009).
C. F. Sopuerta, N. Yunes, and P. Laguna, Phys. Rev. D 74, 124010 (2006); 75, 069903(E) (2007).
A. Le Tiec, L. Blanchet, and C. M. Will, Classical Quantum Gravity 27, 012001 (2010).
C. O. Lousto, H. Nakano, Y. Zlochower, and M. Campanelli, Phys. Rev. Lett. 104, 211101 (2010); H. Nakano, Y. Zlochower, C. O. Lousto, and M. Campanelli, Phys. Rev. D 84, 124006 (2011); C. O. Lousto and Y. Zlochower, Phys. Rev. Lett. 106, 041101 (2011).
S. Bernuzzi and A. Nagar, Phys. Rev. D 81, 084056 (2010).
A. Nagar, T. Damour, and A. Tartaglia, Classical Quantum Gravity 24, S109 (2007).
S. Bernuzzi, A. Nagar, and A. Zenginoglu, Phys. Rev. D 83, 064010 (2011).
T. Damour and A. Nagar, Phys. Rev. D 76, 064028 (2007).
S. Bernuzzi, A. Nagar, and A. Zenginoglu, Phys. Rev. D 84, 084026 (2011).
V. Moncrief, Astrophys. J. 238, 333 (1980).
C. O. Lousto and R. H. Price, Phys. Rev. D 69, 087503 (2004).
H. Bondi, M. G. J. van der Berg, and A. W. K. Metzner, Proc. R. Soc. A 269, 21 (1962).
R. K. Sachs, Proc. R. Soc. A 270, 103 (1962).
R. F. Aranha, I. Damião Soares, and E. V. Tonini, Classical Quantum Gravity 30, 025014 (2013).
I. Robinson and A. Trautman, Phys. Rev. Lett. 4, 431 (1960).
I. Robinson and A. Trautman, Proc. R. Soc. A 265, 463 (1962).
R. F. Aranha, I. Damião Soares, and E. V. Tonini, Phys. Rev. D 81, 104005 (2010).
S. Bernuzzi, A. Nagar, and A. Zenginoğlu, Phys. Rev. D 84, 084026 (2011).
R. F. Aranha, I. Damião Soares, and E. V. Tonini, Phys. Rev. D 82, 104033 (2010).
L. Rezzolla, R. P. Macedo, and J. L. Jaramillo, Phys. Rev. Lett. 104, 221101 (2010).
J. L. Jaramillo, R. P. Macedo, P. Moesta, and L. Rezzolla, Phys. Rev. D 85, 084030 (2012).
E. T. Newman and R. Penrose, J. Math. Phys. (N.Y.) 7, 863 (1966).
J. N. Goldberg, A. J. MacFarlane, E. T. Newman, F. Rohrlich, and E. C. G. Sudarshan, J. Math. Phys. (N.Y.) 8, 2155 (1967).
P. Chruściel, Commun. Math. Phys. 137, 289 (1991); P. Chruściel and D. B. Singleton, 147, 137 (1992).
T. W. Baumgarte and S. L. Shapiro, Numerical Relativity: Solving Einstein’s Equations in the Computer (Cambridge University Press, Cambridge, England, 2010).
A. Z. Petrov, Sci. Nat. Kazan State University 114, 55 (1954); reprinted in Gen. Relativ. Gravit. 32, 1665 (2000).
F. A. E. Pirani, Introduction to Gravitational Radiation Theory, in Lectures on General Relativity, Brandeis Summer Institute in Theoretical Physics (Prentice-Hall, New Jersey, 1964), Vol. 1.
R. Sachs, Proc. R. Soc. Edinburgh, Sect. A 264, 309 (1961); E. T. Newman and R. Penrose, J. Math. Phys. (N.Y.) 3, 566 (1962).
E. T. Newman and R. Penrose, J. Math. Phys. (N.Y.) 3, 566 (1962).
R. K. Sachs, Phys. Rev. 128, 2851 (1962).
G. Arfken, Mathematical Methods for Physicists (Academic Press, New York, 1968), Sec. II.14.
R. A. D’Inverno and J. Stachel, J. Math. Phys. (N.Y.) 19, 2447 (1978); R. A. D’Inverno and J. Smallwood, Phys. Rev. D 22, 1233 (1980).
J. A. Vickers, in Approaches to Numerical Relativity, edited by R. A. D’Inverno (Cambridge University Press, Cambridge, England, 1992), p. 59.
C. A. J. Fletcher, Computational Galerkin Methods (Springer, Berlin, 1984).
R. F. Aranha, I. Damião Soares, and E. V. Tonini, Phys. Rev. D 85, 024003 (2012).
V. Moncrief, Ann. Phys. (Berlin) 88, 323 (1974).
C. O. Lousto and R. H. Price, Phys. Rev. D 56, 6439 (1997).
E. Mitsou, Phys. Rev. D 83, 044039 (2011).
L. Smarr, Sources of Gravitational Radiation, edited by L. Smarr (Cambridge University Press, Cambridge, England, 1978), p. 245.

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Gravitational wave luminosity and net momentum flux in head-on mergers of black holes: Radiative patterns and mode mixing

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