The emergence of gravitational wave science: 100 years of development of mathematical theory, detectors, numerical algorithms, and data analysis tools

Holst, Michael; Sarbach, Olivier; Tiglio, Manuel; Vallisneri, Michele

doi:10.1090/bull/1544

The emergence of gravitational wave science: 100 years of development of mathematical theory, detectors, numerical algorithms, and data analysis tools
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by Michael Holst, Olivier Sarbach, Manuel Tiglio and Michele Vallisneri PDF

Bull. Amer. Math. Soc. 53 (2016), 513-554 Request permission

Abstract:

On September 14, 2015, the newly upgraded Laser Interferometer Gravitational-wave Observatory (LIGO) recorded a loud gravitational-wave (GW) signal, emitted a billion light-years away by a coalescing binary of two stellar-mass black holes. The detection was announced in February 2016, in time for the hundredth anniversary of Einstein’s prediction of GWs within the theory of general relativity (GR). The signal represents the first direct detection of GWs, the first observation of a black-hole binary, and the first test of GR in its strong-field, high-velocity, nonlinear regime. In the remainder of its first observing run, LIGO observed two more signals from black-hole binaries, one moderately loud, another at the boundary of statistical significance. The detections mark the end of a decades-long quest and the beginning of GW astronomy: finally, we are able to probe the unseen, electromagnetically dark Universe by listening to it. In this article, we present a short historical overview of GW science: this young discipline combines GR, arguably the crowning achievement of classical physics, with record-setting, ultra-low-noise laser interferometry, and with some of the most powerful developments in the theory of differential geometry, partial differential equations, high-performance computation, numerical analysis, signal processing, statistical inference, and data science. Our emphasis is on the synergy between these disciplines and how mathematics, broadly understood, has historically played, and continues to play, a crucial role in the development of GW science. We focus on black holes, which are very pure mathematical solutions of Einstein’s gravitational-field equations that are nevertheless realized in Nature and that provided the first observed signals.

References

The website for the Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY, http://ccrg,rit.edu/GW150914
The website for the Gravitational Wave Physics and Astronomy Center, California State University, Fullerton, CA, http://physics.fullerton.edu/gwpac; image appears in arXiv:1607.05377 (20 July 2016)
The website for the IBM archives, http://www-03.ibm.com/ibm/history/exhibits/mainframe/mainframe_PP7090.html
A Center for Scientific and Engineering Supercomputing (proposal), http://www.ncsa.illinois.edu/20years/timeline/documents/blackproposal.pdf
Gravitational waves detected 100 years after Einstein’s prediction, https://www.nsf.gov//news/news_summ.jsp?cntn_id=137628
Report of the panel on Large Scale Computing in Science and Engineering, http://www.pnl.gov/scales/docs/las_report1982.pdf
The role of gravitation in physics. Report from the 1957 Chapel Hill conference, http://www.edition-open-access.de/sources/5/toc.html
The International Society on General Relativity & Gravitation Conferences, http://ares.jsc.nasa.gov/HumanExplore/Exploration/EXlibrary/docs/ApolloCat/Part1/LSG.htm
LIGO Magazine, http://www.ligo.org/magazine/LIGO-magazine-issue-8.pdf, 2016.
Spectral Einstein Code, http://www.black-holes.org/SpEC.html
J. Abadie et al. Predictions for the rates of compact binary coalescences observable by ground-based gravitational-wave detectors. Classical and Quantum Gravity 27 (2010), no. 17, 173001. iopscience.iop.org/article/10.1088/0264-9381/27/17/173001/meta
B. Abbott et al. GW151226: Observation of Gravitational Waves from a $22$-Solar-Mass Binary Black Hole Coalescence. Phys. Rev. Lett., 116 (2016), no. 24, 241103.
B. P. Abbott et al., The Rate of Binary Black Hole Mergers Inferred from Advanced LIGO Observations Surrounding GW150914. arXiv:1602.03842 (11 February 2016) and arXiv: 1602.08342v2 (13 June 2016).
B. P. Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016), no. 6, 061102.
A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. Gürsel, S. Kawamura, F. J. Raab, D. Shoemaker, L. Sievers, R. E. Spero, K. S. Thorne, R. E. Vogt, R. Weiss, S. E. Whitcomb, and M. E. Zucker, LIGO: The laser interferometer gravitational-wave observatory, Science 256 (1992), no. 5055, 325–333.
B. Allen et al., FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries, Phys. Rev. D 85 (2012), no. 12, 122006.
B. Allen and J. D. Romano, Detecting a stochastic background of gravitational radiation: Signal processing strategies and sensitivities, Phys. Rev. D 59 (1999), no. 10, 102001.
Lars Andersson and Piotr T. Chruściel, Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary conditions”, Dissertationes Math. (Rozprawy Mat.) 355 (1996), 100. MR 1405962
N. Arnaud, M. Barsuglia, M.-A. Bizouard, V. Brisson, F. Cavalier, M. Davier, P. Hello, S. Kreckelbergh, and E. K. Porter, Elliptical tiling method to generate a $2$-dimensional set of templates for gravitational wave search, Phys. Rev. D 67 (2003), no. 10, 102003.
R. Arnowitt, S. Deser, and C. W. Misner, Dynamical structure and definition of energy in general relativity, Phys. Rev. (2) 116 (1959), 1322–1330. MR 113667
J. G. Baker, J. Centrella, D.-I. Choi, M. Koppitz, and J. van Meter, Gravitational wave extraction from an inspiraling configuration of merging black holes, Phys. Rev. Lett. 96 (2006), no. 11, 111102.
Robert Bartnik and Jim Isenberg, The constraint equations, The Einstein equations and the large scale behavior of gravitational fields, Birkhäuser, Basel, 2004, pp. 1–38. MR 2098912
Thomas W. Baumgarte, Niall Ó Murchadha, and Harald P. Pfeiffer, Einstein constraints: uniqueness and nonuniqueness in the conformal thin sandwich approach, Phys. Rev. D 75 (2007), no. 4, 044009, 9. MR 2304419, DOI 10.1103/PhysRevD.75.044009
T. W. Baumgarte and S. L. Shapiro, Numerical Relativity: Solving Einstein’s Equations on the Computer, Cambridge University Press, 2010.
A. Behzadan and M. Holst Rough solutions of the Einstein constraint equations on asymptotically flat manifolds without near-CMC conditions, arXiv:1504.04661 [gr-qc].
D. Bernstein and M. Holst, “A 3D finite element solver for the initial-value problem”, in Proceedings of the Eighteenth Texas Symposium on Relativistic Astrophysics and Cosmology, December 16-20, 1996, Chicago, Illinois, A. Olinto, J. A. Frieman, and D. N. Schramm, editors, World Scientific, Singapore, 1998.
Emanuele Berti, Vitor Cardoso, and Andrei O. Starinets, Quasinormal modes of black holes and black branes, Classical Quantum Gravity 26 (2009), no. 16, 163001, 108. MR 2529208, DOI 10.1088/0264-9381/26/16/163001
J. Blackman, S. E. Field, C. R. Galley, B. Szilgyi, M. A. Scheel, M. Tiglio, and D. A. Hemberger, Fast and accurate prediction of numerical relativity waveforms from binary black hole coalescences using surrogate models, Phys. Rev. Lett. 115 (2015), no. 12, 121102.
L. Blanchet, Gravitational radiation from post-Newtonian sources and inspiralling compact binaries, Living Reviews in Relativity 17 (2014), no. 2. DOI 10.12942/lrr-2014-2.
J. Bowen and J. York, Time asymmetric initial data for black holes and black hole collisions, Phys. Rev. D, 21:2047–2051, 1980.
D. Brill, personal communication.
Othmar Brodbeck, Simonetta Frittelli, Peter Hübner, and Oscar A. Reula, Einstein’s equations with asymptotically stable constraint propagation, J. Math. Phys. 40 (1999), no. 2, 909–923. MR 1674255, DOI 10.1063/1.532694
J. David Brown, Puncture evolution of Schwarzschild black holes, Phys. Rev. D 77 (2008), no. 4, 044018, 5. MR 2421233, DOI 10.1103/PhysRevD.77.044018
A. Buonanno and T. Damour, Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D (3) 59 (1999), no. 8, 084006, 24. MR 1699287, DOI 10.1103/PhysRevD.59.084006
M. Campanelli, C. O. Lousto, P. Marronetti, and Y. Zlochower, Accurate evolutions of orbiting black-hole binaries without excision, Phys. Rev. Lett. 96 (2006), no. 11, 111101.
P. Canizares, S. E. Field, J. Gair, V. Raymond, R. Smith, and M. Tiglio, Accelerated gravitational-wave parameter estimation with reduced order modeling, Phys. Rev. Lett. 114 (2015), no. 7, 071104. DOI 10.1103/PhysRevLett.114.071104.
J. G. Charney, R. Fjörtoft, and J. von Neumann, Numerical integration of the barotropic vorticity equation, Tellus 2 (1950), 237–254. MR 42799, DOI 10.3402/tellusa.v2i4.8607
M. W. Choptuik, Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett. 70 (1993), no. 1, 9–12.
M. W. Choptuik, L. Lehner, and F. Pretorius, Probing strong field gravity through numerical simulations, arXiv: 1502.06853 [gr-qc]
Yvonne Choquet-Bruhat, Einstein constraints on compact $n$-dimensional manifolds, Classical Quantum Gravity 21 (2004), no. 3, S127–S151. A spacetime safari: essays in honour of Vincent Moncrief. MR 2053003, DOI 10.1088/0264-9381/21/3/009
Yvonne Choquet-Bruhat, General relativity and the Einstein equations, Oxford Mathematical Monographs, Oxford University Press, Oxford, 2009. MR 2473363
Y. Choquet-Bruhat, J. Isenberg, and J. W. York, Jr., Einstein constraints on asymptotically Euclidean manifolds, Phys. Rev. D 61 (2000), 084034.
N. Christensen and R. Meyer, Markov chain Monte Carlo methods for Bayesian gravitational radiation data analysis, Phys. Rev. D 58 (1998), no. 8, 082001.
Demetrios Christodoulou, The formation of black holes in general relativity, EMS Monographs in Mathematics, European Mathematical Society (EMS), Zürich, 2009. MR 2488976, DOI 10.4171/068
Demetrios Christodoulou and Sergiu Klainerman, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series, vol. 41, Princeton University Press, Princeton, NJ, 1993. MR 1316662
Piotr T. Chruściel and Erwann Delay, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, Mém. Soc. Math. Fr. (N.S.) 94 (2003), vi+103 (English, with English and French summaries). MR 2031583, DOI 10.24033/msmf.407
Piotr T. Chruściel, Gregory J. Galloway, and Daniel Pollack, Mathematical general relativity: a sampler, Bull. Amer. Math. Soc. (N.S.) 47 (2010), no. 4, 567–638. MR 2721040, DOI 10.1090/S0273-0979-2010-01304-5
P. Chruściel and R. Gicquaud, Bifurcating solutions of the Lichnerowicz equation, arXiv:1506.00101 [gr-qc].
P. T. Chruściel, J. L. Costa, and M. Heusler, Stationary black holes: Uniqueness and beyond, Living Reviews in Relativity 15 (2012), no 7. DOI 10.12942/lrr-2012-7.
Piotr T. Chruściel, James Isenberg, and Daniel Pollack, Initial data engineering, Comm. Math. Phys. 257 (2005), no. 1, 29–42. MR 2163567, DOI 10.1007/s00220-005-1345-2
Piotr T. Chruściel and Rafe Mazzeo, Initial data sets with ends of cylindrical type: I. The Lichnerowicz equation, Ann. Henri Poincaré 16 (2015), no. 5, 1231–1266. MR 3324104, DOI 10.1007/s00023-014-0339-z
Piotr T. Chruściel, Rafe Mazzeo, and Samuel Pocchiola, Initial data sets with ends of cylindrical type: II. The vector constraint equation, Adv. Theor. Math. Phys. 17 (2013), no. 4, 829–865. MR 3262517
L. Cohen. Time-frequency distributions—a review. Proceedings of the IEEE 77 (1989), no. 7, 941–981.
H. Collins. Gravity’s Shadow: The Search for Gravitational Waves. University of Chicago Press, 2010.
Gregory B. Cook, Initial data for axisymmetric black-hole collisions, Phys. Rev. D (3) 44 (1991), no. 10, 2983–3000. MR 1138889, DOI 10.1103/PhysRevD.44.2983
Gregory B. Cook, Initial data for numerical relativity, Living Rev. Relativ. 3 (2000), 2000-5, 53. MR 1799071, DOI 10.12942/lrr-2000-5
Gregory B. Cook and Saul A. Teukolsky, Numerical relativity: challenges for computational science, Acta numerica, 1999, Acta Numer., vol. 8, Cambridge Univ. Press, Cambridge, 1999, pp. 1–45. MR 1819642, DOI 10.1017/S0962492900002889
Leo Corry, Jürgen Renn, and John Stachel, Belated decision in the Hilbert-Einstein priority dispute, Science 278 (1997), no. 5341, 1270–1273. MR 1487067, DOI 10.1126/science.278.5341.1270
Justin Corvino, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys. 214 (2000), no. 1, 137–189. MR 1794269, DOI 10.1007/PL00005533
Justin Corvino and Daniel Pollack, Scalar curvature and the Einstein constraint equations, Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM), vol. 20, Int. Press, Somerville, MA, 2011, pp. 145–188. MR 2906924
Justin Corvino and Richard M. Schoen, On the asymptotics for the vacuum Einstein constraint equations, J. Differential Geom. 73 (2006), no. 2, 185–217. MR 2225517
M. Dafermos, G. Holzegel, and I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, arXiv:1601.06467 [gr-qc], 2016.
M. Dafermos, I. Rodnianski, and Y. Shlapentokh-Rothman, Decay for solutions of the wave equation on Kerr exterior spacetimes III: The full subextremal case $|a| < M$, arXiv:1402.7034 [gr-qc], 2014.
Mattias Dahl, Romain Gicquaud, and Emmanuel Humbert, A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method, Duke Math. J. 161 (2012), no. 14, 2669–2697. MR 2993137, DOI 10.1215/00127094-1813182
Sergio Dain, Trapped surfaces as boundaries for the constraint equations, Classical Quantum Gravity 21 (2004), no. 2, 555–573. MR 2030884, DOI 10.1088/0264-9381/21/2/017
Sergio Dain, Generalized Korn’s inequality and conformal Killing vectors, Calc. Var. Partial Differential Equations 25 (2006), no. 4, 535–540. MR 2214623, DOI 10.1007/s00526-005-0371-4
Sergio Dain, José Luis Jaramillo, and Badri Krishnan, Existence of initial data containing isolated black holes, Phys. Rev. D (3) 71 (2005), no. 6, 064003, 11. MR 2138833, DOI 10.1103/PhysRevD.71.064003
G. Darmois, Memorial des sciences mathematique: fascicule XXV: les equations de la gravitation einsteinienne, Gauthier-Villars, 1927.
James Dilts, The Einstein constraint equations on compact manifolds with boundary, Classical Quantum Gravity 31 (2014), no. 12, 125009, 27. MR 3216438, DOI 10.1088/0264-9381/31/12/125009
J. Dilts, M. Holst, and D. Maxwell, Analytic and numerical bifurcation analysis of the conformal formulation of the Einstein constraint equations, Preprint.
James Dilts, Jim Isenberg, Rafe Mazzeo, and Caleb Meier, Non-CMC solutions of the Einstein constraint equations on asymptotically Euclidean manifolds, Classical Quantum Gravity 31 (2014), no. 6, 065001, 10. MR 3176057, DOI 10.1088/0264-9381/31/6/065001
A. Einstein, Approximative Integration of the Field Equations of Gravitation, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916 (1916), 688–696.
A. Einstein, Über Gravitationswellen, Sitzungsberichte der Königlich Preußischen Akademie der Wissenschaften (Berlin), Seite 154-167, 1918.
A. Einstein and N. Rosen, On gravitational waves, Journal of the Franklin Institute 223 (1937), no. 1, 43–54.
K. R. Eppley, “The Numerical Evolution of the Collision of Two Black Holes”, PhD thesis, Princeton University, Princeton, New Jersey, 1975.
Arthur E. Fischer, Jerrold E. Marsden, and Vincent Moncrief, The structure of the space of solutions of Einstein’s equations. I. One Killing field, Ann. Inst. H. Poincaré Sect. A (N.S.) 33 (1980), no. 2, 147–194. MR 605194
E. E. Flanagan, Sensitivity of the Laser Interferometer Gravitational Wave Observatory to a stochastic background, and its dependence on the detector orientations, Phys. Rev. D 48 (1993), no. 6, 2389–2407.
Y. Fourès-Bruhat, Théorème d’existence pour certains systèmes d’équations aux dérivées partielles non linéaires, Acta Math. 88 (1952), 141–225 (French). MR 53338, DOI 10.1007/BF02392131
Jörg Frauendiener, Conformal infinity, Living Rev. Relativ. 7 (2004), 2004-1, 82. MR 2037619, DOI 10.12942/lrr-2004-1
Helmut Friedrich, On the hyperbolicity of Einstein’s and other gauge field equations, Comm. Math. Phys. 100 (1985), no. 4, 525–543. MR 806251
Helmut Friedrich and Gabriel Nagy, The initial boundary value problem for Einstein’s vacuum field equation, Comm. Math. Phys. 201 (1999), no. 3, 619–655. MR 1685892, DOI 10.1007/s002200050571
C. L. Fryer and K. C. B. New, Gravitational waves from gravitational collapse, Living Reviews in Relativity 14 (2011), no. 1. DOI 10.12942/lrr-2011-1.
C. J. Geyer, Markov Chain Monte Carlo maximum likelihood, 1991.
W. R. Gilks, S. Richardson, and D. J. Spiegelhalter (eds.), Markov chain Monte Carlo in practice, Interdisciplinary Statistics, Chapman & Hall, London, 1996. MR 1397966, DOI 10.1007/978-1-4899-4485-6
R. Gicquaud and Q. A. Ngo, A new point of view on the solutions to the Einstein constraint equations with arbitrary mean curvature and small TTtensor, Classical and Quantum Gravity 31 (2014), no. 19, 1–16.
R. Gicquaud and T. C. Nguyen, Solutions to the Einstein-scalar field constraint equations with small TT-sensor, Classical and Quantum Gravity 55 (2015), no. 2, 1–18.
Peter J. Green, Krzysztof Łatuszyński, Marcelo Pereyra, and Christian P. Robert, Bayesian computation: a summary of the current state, and samples backwards and forwards, Stat. Comput. 25 (2015), no. 4, 835–862. MR 3360496, DOI 10.1007/s11222-015-9574-5
P. C. Gregory, Bayesian logical data analysis for the physical sciences, Cambridge University Press, Cambridge, 2005. A comparative approach with Mathematica$^\circledR$ support. MR 2152425, DOI 10.1017/CBO9780511791277
Carsten Gundlach, Gioel Calabrese, Ian Hinder, and José M. Martín-García, Constraint damping in the Z4 formulation and harmonic gauge, Classical Quantum Gravity 22 (2005), no. 17, 3767–3773. MR 2168553, DOI 10.1088/0264-9381/22/17/025
Bertil Gustafsson, Heinz-Otto Kreiss, and Arne Sundström, Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp. 26 (1972), 649–686. MR 341888, DOI 10.1090/S0025-5718-1972-0341888-3
Susan G. Hahn, Stability criteria for difference schemes, Comm. Pure Appl. Math. 11 (1958), 243–255. MR 97886, DOI 10.1002/cpa.3160110207
Susan G. Hahn and Richard W. Lindquist, The two-body problem in geometrodynamics, Ann. Physics 29 (1964), 304–331. MR 177779, DOI 10.1016/0003-4916(64)90223-4
S. W. Hawking and W. Israel (eds.), Three hundred years of gravitation, 2nd ed., Cambridge University Press, Cambridge, 1989. MR 1024406
D. Hilbert, Die Grundlagen der Physik, Nachr. Ges. Wiss. Göttingen Math. Phys. KL., 1916, pp. 395–407.
M. Holst, Adaptive numerical treatment of elliptic systems on manifolds, Adv. Comput. Math. 15 (2001), no. 1-4, 139–191 (2002). A posteriori error estimation and adaptive computational methods. MR 1887732, DOI 10.1023/A:1014246117321
M. Holst and V. Kungurtsev, Numerical bifurcation analysis of conformal formulations of the Einstein constraints, Phys. Rev. D 84 (2011), no. 12, 124038(1)–124038(8).
Michael Holst, Lee Lindblom, Robert Owen, Harald P. Pfeiffer, Mark A. Scheel, and Lawrence E. Kidder, Optimal constraint projection for hyperbolic evolution systems, Phys. Rev. D (3) 70 (2004), no. 8, 084017, 17. MR 2117121, DOI 10.1103/PhysRevD.70.084017
M. Holst and C. Meier, Non-uniqueness of solutions to the conformal formulation. arXiv:1210.2156 [gr-qc].
Michael Holst and Caleb Meier, Non-CMC solutions to the Einstein constraint equations on asymptotically Euclidean manifolds with apparent horizon boundaries, Classical Quantum Gravity 32 (2015), no. 2, 025006, 28. MR 3291778, DOI 10.1088/0264-9381/32/2/025006
M. Holst, C. Meier, and G. Tsogtgerel, Non-CMC solutions of the Einstein constraint equations on compact manifolds with apparent horizon boundaries, arXiv:1310.2302 [gr-qc].
M. Holst, G. Nagy, and G. Tsogtgerel, Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics, Phys. Rev. Lett. 100 (2008), no. 16, 161101, 4. MR 2403263, DOI 10.1103/PhysRevLett.100.161101
Michael Holst, Gabriel Nagy, and Gantumur Tsogtgerel, Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions, Comm. Math. Phys. 288 (2009), no. 2, 547–613. MR 2500992, DOI 10.1007/s00220-009-0743-2
Michael Holst and Gantumur Tsogtgerel, The Lichnerowicz equation on compact manifolds with boundary, Classical Quantum Gravity 30 (2013), no. 20, 205011, 31. MR 3117005, DOI 10.1088/0264-9381/30/20/205011
J. Hough, J. R. Pugh, R. Bland, and R. W. P. Drever, Search for continuous gravitational radiation, Nature 254 (1975), 498–501. DOI 10.1038/254498a0.
R. Hulse and J. Taylor, Discovery of a pulsar in a binary system, Astrophys. J. 195 (1975), L51–L53.
James Isenberg, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity 12 (1995), no. 9, 2249–2274. MR 1353772
James Isenberg and Vincent Moncrief, A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical Quantum Gravity 13 (1996), no. 7, 1819–1847. MR 1400943, DOI 10.1088/0264-9381/13/7/015
E. T. Jaynes, Probability theory, Cambridge University Press, Cambridge, 2003. The logic of science; Edited and with a foreword by G. Larry Bretthorst. MR 1992316, DOI 10.1017/CBO9780511790423
Bernard S. Kay and Robert M. Wald, Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation $2$-sphere, Classical Quantum Gravity 4 (1987), no. 4, 893–898. MR 895907
Daniel Kennefick, Traveling at the speed of thought, Princeton University Press, Princeton, NJ, 2007. Einstein and the quest for gravitational waves. MR 2313291, DOI 10.1515/9781400882748
Lawrence E. Kidder, Mark A. Scheel, and Saul A. Teukolsky, Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations, Phys. Rev. D (3) 64 (2001), no. 6, 064017, 13. MR 1857419, DOI 10.1103/PhysRevD.64.064017
S. Klimenko, G. Vedovato, M. Drago, F. Salemi, V. Tiwari, G. A. Prodi, C. Lazzaro, K. Ackley, S. Tiwari, C. F. Da Silva, and G. Mitselmakher, Method for detection and reconstruction of gravitational wave transients with networks of advanced detectors, Phys. Rev. D 93 (2016), no. 4, 042004.
H.-O. Kreiss, O. Reula, O. Sarbach, and J. Winicour, Boundary conditions for coupled quasilinear wave equations with application to isolated systems, Comm. Math. Phys. 289 (2009), no. 3, 1099–1129. MR 2511662, DOI 10.1007/s00220-009-0788-2
H.-O. Kreiss and J. Winicour, Problems which are well posed in a generalized sense with applications to the Einstein equations, Classical Quantum Gravity 23 (2006), no. 16, S405–S420. MR 2254281, DOI 10.1088/0264-9381/23/16/S07
A. Lichernowicz, Sur l’intégration des équations d’Einstein, J. Math. Pures Appl. 23 (1944), 26–63.
A. Lichnerowicz and G. Darmois, Théories relativistes de la gravitation et de l’électromagnétisme: relativité générale et théories unitaires, par A. Lichnerowicz,... Préface du Pr G. Georges Darmois, Barnéoud frères et Cie, 1955.
Hans Lindblad and Igor Rodnianski, Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys. 256 (2005), no. 1, 43–110. MR 2134337, DOI 10.1007/s00220-004-1281-6
Thomas J. Loredo, Bayesian astrostatistics: a backward look to the future, Astrostatistical challenges for the new astronomy, Springer Ser. Astrostatistics, Springer, New York, 2013, pp. 15–40. MR 3051143, DOI 10.1007/978-1-4614-3508-2_{2}
M. Maggiore, Gravitational wave experiments and early universe cosmology, Physics Reports 331 (2000), no. 6, 283–367.
M. Maggiore, Gravitational waves, Oxford University Press, 2008.
R. Matzner, “Moving black holes, long-lived black holes and boundary conditions: Status of the binary black hole grand challenge”, Matters of Gravity. The newsletter of the Topical Group in Gravitation by the American Physical Society 11 (1998), 13–16.
D. Maxwell, Initial data in general relativity described by expansion, conformal deformation and drift, arXiv:1407.1467 [gr-qc].
David Maxwell, Solutions of the Einstein constraint equations with apparent horizon boundaries, Comm. Math. Phys. 253 (2005), no. 3, 561–583. MR 2116728, DOI 10.1007/s00220-004-1237-x
David Maxwell, Rough solutions of the Einstein constraint equations, J. Reine Angew. Math. 590 (2006), 1–29. MR 2208126, DOI 10.1515/CRELLE.2006.001
David Maxwell, A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, Math. Res. Lett. 16 (2009), no. 4, 627–645. MR 2525029, DOI 10.4310/MRL.2009.v16.n4.a6
David Maxwell, A model problem for conformal parameterizations of the Einstein constraint equations, Comm. Math. Phys. 302 (2011), no. 3, 697–736. MR 2774166, DOI 10.1007/s00220-011-1187-z
David Maxwell, The conformal method and the conformal thin-sandwich method are the same, Classical Quantum Gravity 31 (2014), no. 14, 145006, 34. MR 3233274, DOI 10.1088/0264-9381/31/14/145006
D. Maxwell, Conformal parameterizations of slices of flat Kasner spacetimes, Annales Henri Poincaré 16 (2015), no. 12, 2919–2954. DOI 10.1007/s00023-014-0386-5.
Lorenzo Mazzieri, Generalized gluing for Einstein constraint equations, Calc. Var. Partial Differential Equations 34 (2009), no. 4, 453–473. MR 2476420, DOI 10.1007/s00526-008-0191-4
C. Messenger, R. Prix, and M. A. Papa, Random template banks and relaxed lattice coverings, Phys. Rev. D 79 (2009), no. 10, 104017.
N. Metropolis, J. Howlett, and Gian-Carlo Rota (eds.), A history of computing in the twentieth century, Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London, 1980. A collection of essays. MR 584927
N. Metropolis, A. W. Rosenbluth, M. N. Rosenbluth, A. H. Teller, and E. Teller, Equation of state calculations by fast computing machines, J. Chem. Phys. 21 (1953), no. 6, 1087–1092. DOI 10-1063/1.1699114.
T.-C. Nguyen, Nonexistence and nonuniqueness results for solutions to the vacuum Einstein conformal constraint equations, arXiv:1507.01081 [math.AP].
Niall O’Murchadha and James W. York Jr., Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds, J. Mathematical Phys. 14 (1973), 1551–1557. MR 332094, DOI 10.1063/1.1666225
B. J. Owen, Search templates for gravitational waves from inspiraling binaries: Choice of template spacing, Phys. Rev. D 53 (1996), 6749–6761.
H. Pfeiffer, “The initial value problem in numerical relativity”, in Proceedings of Miami Waves 2004: Conference on Geometric Analysis, Nonlinear Wave Equations and General Relativity, 4–10 January 2004, Coral Gables, FL, FIZ Karlsruhe, Germany, 2004.
Harald P. Pfeiffer and James W. York Jr., Uniqueness and nonuniqueness in the Einstein constraints, Phys. Rev. Lett. 95 (2005), no. 9, 091101, 4. MR 2167142, DOI 10.1103/PhysRevLett.95.091101
Bruno Premoselli, Effective multiplicity for the Einstein-scalar field Lichnerowicz equation, Calc. Var. Partial Differential Equations 53 (2015), no. 1-2, 29–64. MR 3336312, DOI 10.1007/s00526-014-0740-y
Frans Pretorius, Evolution of binary black-hole spacetimes, Phys. Rev. Lett. 95 (2005), no. 12, 121101, 4. MR 2169088, DOI 10.1103/PhysRevLett.95.121101
R. Prix, Template-based searches for gravitational waves: efficient lattice covering of flat parameter spaces, Classical and Quantum Gravity 24 (2007), S481–S490.
Tullio Regge and John A. Wheeler, Stability of a Schwarzschild singularity, Phys. Rev. (2) 108 (1957), 1063–1069. MR 91832
T. Regimbau, The astrophysical gravitational wave stochastic background, Research in Astronomy and Astrophysics 11 (2011), no. 4, 369–390.
O. Sarbach and M. Tiglio, Continuum and discrete initial-boundary-value problems and Einstein’s field equations, Living Reviews in Relativity 15 (2012), no. 9, 194 pp.
Tilman Sauer, The relativity of discovery: Hilbert’s first note on the foundations of physics, Arch. Hist. Exact Sci. 53 (1999), no. 6, 529–575. MR 1672944
L. Scharf and C. Demeure, Statistical Signal Processing: Detection, Estimation, and Time Series Analysis, Addison-Wesley Series in Electrical and Computer Engineering. Addison-Wesley Publishing Company, Boston, 1991.
E. Sejdi, I. Djurovi, and J. Jiang, Time frequency feature representation using energy concentration: An overview of recent advances, Digital Signal Processing 19 (2009), no. 1, 153–183.
John Skilling, Nested sampling for general Bayesian computation, Bayesian Anal. 1 (2006), no. 4, 833–859. MR 2282208, DOI 10.1214/06-BA127
L. Smarr, Space-times generated by computers: Black holes with gravitational radiation, Annals of the New York Academy of Sciences 302 (1977), no. 1, 569–604.
J. H. Taylor and J. M. Weisberg, Further experimental tests of relativistic gravity using the binary pulsar PSR $1913 + 16$, Aptrophys J. 345 (1989), 434–450.
The LIGO Scientific Collaboration and the Virgo Collaboration. Observing gravitational-wave transient GW150914 with minimal assumptions, Phys. Rev. D 93 (2016), no. 12, 122004; arXiv: 1602.03843 [gr-qc]
The LIGO Scientific Collaboration and the Virgo Collaboration. Tests of general relativity with GW150914. Phys. Rev. Lett. 116 (2016), no. 22, 221101. arXiv: 1602.03841 [gr-qc]
The LIGO Scientific Collaboration and the Virgo Collaboration, GW150914: First results from the search for binary black hole coalescence with Advanced LIGO, Phys. Rev. D 93 (2016), no. 12, 122003. arXiv: 1602.03839 [gr-qc]
The LIGO Scientific Collaboration and the Virgo Collaboration, Properties of the binary black hole merger GW150914. Phys. Rev. Lett. 116 (2016), no. 24, 241102. arXiv: 1602.03840 [gr-qc]
George L. Turin, An introduction to matched filters, Trans. IRE IT-6 (1960), 311–329. MR 0115847, DOI 10.1109/tit.1960.1057571
M. Vallisneri, Use and abuse of the Fisher information matrix in the assessment of gravitational-wave parameter-estimation prospects, Phys. Rev. D 77 (2008), no. 4, 042001. arXiv: gr-qc/0703086
M. Vallisneri, Beyond the Fisher-matrix formalism: Exact sampling distributions of the maximum-likelihood estimator in gravitational-wave parameter estimation, Phys. Rev. Lett. 107 (2011), no. 19, 191104.
J. Veitch, V. Raymond, B. Farr, W. Farr, P. Graff, S. Vitale, B. Aylott, K. Blackburn, N. Christensen, M. Coughlin, W. Del Pozzo, F. Feroz, J. Gair, C.-J. Haster, V. Kalogera, T. Littenberg, I. Mandel, R. O’Shaughnessy, M. Pitkin, C. Rodriguez, C. Röver, T. Sidery, R. Smith, M. Van Der Sluys, A. Vecchio, W. Vousden, and L. Wade, Parameter estimation for compact binaries with ground-based gravitational-wave observations using the LALInference software library, Phys. Rev. D 91 (2015), no. 4, 042003.
L. A. Wainstein and V. D. Zubakov, Extraction of signals from noise, International Series in Applied Mathematics, Prentice-Hall, Inc., Englewood Cliffs, N.J., 1962. Translated from the Russian by Richard A. Silverman. MR 0142420
D. M. Walsh, Non-uniqueness in conformal formulations of the Einstein constraints, Classical Quantum Gravity 24 (2007), no. 8, 1911–1925. MR 2311452, DOI 10.1088/0264-9381/24/8/002
J. Weber, Evidence for discovery of gravitational radiation, Phys. Rev. Lett. 22 (1969), no. 24, 1320–1324.
R. Weiss, Electromagnetically Coupled Broadband Gravitational Antenna, Quarterly Progress Report of the MIT Research Laboratory of Electronics, 54(105), 1972.
Bernard F. Whiting, Mode stability of the Kerr black hole, J. Math. Phys. 30 (1989), no. 6, 1301–1305. MR 995773, DOI 10.1063/1.528308
C. M. Will, The confrontation between general relativity and experiment, Living Reviews in Relativity 9 (2006), no. 3. DOI 10.12942/lrr-2006-3
James W. York Jr., Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity, J. Mathematical Phys. 14 (1973), 456–464. MR 329562, DOI 10.1063/1.1666338