×

The emergence of gravitational wave science: 100 years of development of mathematical theory, detectors, numerical algorithms, and data analysis tools. (English) Zbl 1350.83009

Summary: 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.

MSC:

83C35 Gravitational waves
83-02 Research exposition (monographs, survey articles) pertaining to relativity and gravitational theory
83-03 History of relativity and gravitational theory
85-02 Research exposition (monographs, survey articles) pertaining to astronomy and astrophysics
85-03 History of astronomy and astrophysics
01A60 History of mathematics in the 20th century
01A61 History of mathematics in the 21st century
62P35 Applications of statistics to physics
83F05 Relativistic cosmology
83C57 Black holes
35Q75 PDEs in connection with relativity and gravitational theory
53Z05 Applications of differential geometry to physics
65T60 Numerical methods for wavelets
68U10 Computing methodologies for image processing
83-08 Computational methods for problems pertaining to relativity and gravitational theory

Software:

SG; SpEC; FINDCHIRP
PDFBibTeX XMLCite
Full Text: DOI arXiv

References:

[1] 1 The website for the Center for Computational Relativity and Gravitation, Rochester Institute of Technology, Rochester, NY, http://ccrg,rit.edu/GW150914
[2] 2 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)
[3] IBM7090 The website for the IBM archives, http://www-03.ibm.com/ibm/history/exhibits/mainframe/mainframe_PP7090.html
[4] BlackProposal A Center for Scientific and Engineering Supercomputing (proposal), http://www.ncsa.illinois.edu/20years/timeline/documents/blackproposal.pdf
[5] NSF-LIGO-press Gravitational waves detected 100 years after Einstein’s prediction, https://www.nsf.gov//news/news_summ.jsp?cntn_id=137628
[6] LaxReport Report of the panel on Large Scale Computing in Science and Engineering, http://www.pnl.gov/scales/docs/las_report1982.pdf
[7] ChapelHill The role of gravitation in physics. Report from the 1957 Chapel Hill conference, http://www.edition-open-access.de/sources/5/toc.html
[8] gw-moon The International Society on General Relativity & Gravitation Conferences, http://ares.jsc.nasa.gov/HumanExplore/Exploration/EXlibrary/docs/ApolloCat/Part1/LSG.htm
[9] LIGO-mag8 LIGO Magazine, http://www.ligo.org/magazine/LIGO-magazine-issue-8.pdf, 2016.
[10] specweb Spectral Einstein Code, http://www.black-holes.org/SpEC.html
[11] Abadie:2010cf 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
[12] Abbott:2016nmj 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.
[13] 2016arXiv160203842A 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).
[14] Abbott:2016blz B. P. Abbott et al., Observation of Gravitational Waves from a Binary Black Hole Merger, Phys. Rev. Lett. 116 (2016), no. 6, 061102.
[15] Abramovici325 A. Abramovici, W. E. Althouse, R. W. P. Drever, Y. G\"ursel, 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.
[16] PhysRevD.85.122006 B. Allen et al., FINDCHIRP: An algorithm for detection of gravitational waves from inspiraling compact binaries, Phys. Rev. D 85 (2012), no. 12, 122006.
[17] PhysRevD.59.102001 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.
[18] Andersson, Lars; Chru{\'s}ciel, Piotr T., Solutions of the constraint equations in general relativity satisfying “hyperboloidal boundary conditions”, Dissertationes Math. (Rozprawy Mat.), 355, 100 pp. (1996) · Zbl 0873.35101
[19] 2003PhRvD..67j2003A 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.
[20] Arnowitt, R.; Deser, S.; Misner, C. W., Dynamical structure and definition of energy in general relativity., Phys. Rev. (2), 116, 1322-1330 (1959) · Zbl 0092.20704
[21] Baker:2005vv 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.
[22] Bartnik, Robert; Isenberg, Jim, “The constraint equations”. The Einstein equations and the large scale behavior of gravitational fields, 1-38 (2004), Birkh\"auser, Basel · Zbl 1073.83009
[23] Baumgarte, Thomas W.; \'OMurchadha, Niall; Pfeiffer, Harald P., Einstein constraints: uniqueness and nonuniqueness in the conformal thin sandwich approach, Phys. Rev. D, 75, 4, 044009, 9 pp. (2007) · doi:10.1103/PhysRevD.75.044009
[24] NR:Book T. W. Baumgarte and S. L. Shapiro, Numerical Relativity: Solving Einstein’s Equations on the Computer, Cambridge University Press, 2010. · Zbl 1198.83001
[25] BeHo14a 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].
[26] BeHo96 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.
[27] Berti, Emanuele; Cardoso, Vitor; Starinets, Andrei O., TOPICAL REVIEW:Quasinormal modes of black holes and black branes, Classical and Quantum Gravity, 26, 16, 163001, 108 pp. (2009) · Zbl 1173.83001 · doi:10.1088/0264-9381/26/16/163001
[28] Blackman:2015pia 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.
[29] 2014LRR....17....2B 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. · Zbl 1316.83003
[30] BoY80 J. Bowen and J. York, Time asymmetric initial data for black holes and black hole collisions, Phys. Rev. D, 21:2047-2051, 1980.
[31] BrillStevens D. Brill, personal communication.
[32] Brodbeck, Othmar; Frittelli, Simonetta; H{\`“u}bner, Peter; Reula, Oscar A., Einstein”s equations with asymptotically stable constraint propagation, J. Math. Phys., 40, 2, 909-923 (1999) · Zbl 0946.83046 · doi:10.1063/1.532694
[33] Brown, J. David, Puncture evolution of Schwarzschild black holes, Phys. Rev. D, 77, 4, 044018, 5 pp. (2008) · doi:10.1103/PhysRevD.77.044018
[34] Buonanno, A.; Damour, T., Effective one-body approach to general relativistic two-body dynamics, Phys. Rev. D (3), 59, 8, 084006, 24 pp. (1999) · doi:10.1103/PhysRevD.59.084006
[35] Campanelli:2005dd 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. · Zbl 1219.83101
[36] Canizares:2014fya 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.
[37] Charney, J. G.; Fj{\"o}rtoft, R.; von Neumann, J., Numerical integration of the barotropic vorticity equation, Tellus, 2, 237-254 (1950)
[38] Choptuik:1992jv M. W. Choptuik, Universality and scaling in gravitational collapse of a massless scalar field, Phys. Rev. Lett. 70 (1993), no. 1, 9-12.
[39] Choptuik:2015mma M. W. Choptuik, L. Lehner, and F. Pretorius, Probing strong field gravity through numerical simulations, arXiv: 1502.06853 [gr-qc]
[40] Choquet-Bruhat, Yvonne, Einstein constraints on compact \(n\)-dimensional manifolds. A space time safari: Essays in honour of Vincent Moncrief, Classical and Quantum Gravity, 21, 3, S127-S151 (2004) · Zbl 1040.83004 · doi:10.1088/0264-9381/21/3/009
[41] Choquet-Bruhat, Yvonne, General relativity and the Einstein equations, Oxford Mathematical Monographs, xxvi+785 pp. (2009), Oxford University Press, Oxford · Zbl 1157.83002
[42] CBIY00 Y. Choquet-Bruhat, J. Isenberg, and J. W. York, Jr., Einstein constraints on asymptotically Euclidean manifolds, Phys. Rev. D 61 (2000), 084034.
[43] 1998PhRvD..58h2001C N. Christensen and R. Meyer, Markov chain Monte Carlo methods for Bayesian gravitational radiation data analysis, Phys.Rev.D 58 (1998), no. 8, 082001.
[44] Christodoulou, Demetrios, The formation of black holes in general relativity, EMS Monographs in Mathematics, x+589 pp. (2009), European Mathematical Society (EMS), Z\"urich · Zbl 1197.83004 · doi:10.4171/068
[45] Christodoulou, Demetrios; Klainerman, Sergiu, The global nonlinear stability of the Minkowski space, Princeton Mathematical Series 41, x+514 pp. (1993), Princeton University Press, Princeton, NJ · Zbl 0827.53055
[46] Chru{\'s}ciel, Piotr T.; Delay, Erwann, On mapping properties of the general relativistic constraints operator in weighted function spaces, with applications, M\'em. Soc. Math. Fr. (N.S.), 94, vi+103 pp. (2003) · Zbl 1058.83007
[47] Chru{\'s}ciel, Piotr T.; Galloway, Gregory J.; Pollack, Daniel, Mathematical general relativity: a sampler, Bull. Amer. Math. Soc. (N.S.), 47, 4, 567-638 (2010) · Zbl 1205.83002 · doi:10.1090/S0273-0979-2010-01304-5
[48] ChruscielGicquad:2015 P. Chru\'sciel and R. Gicquaud, Bifurcating solutions of the Lichnerowicz equation, arXiv:1506.00101 [gr-qc].
[49] lrr-2012-7 P. T. Chru\'sciel, 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. · Zbl 1316.83023
[50] Chru{\'s}ciel, Piotr T.; Isenberg, James; Pollack, Daniel, Initial data engineering, Comm. Math. Phys., 257, 1, 29-42 (2005) · Zbl 1080.83002 · doi:10.1007/s00220-005-1345-2
[51] Chru{\'s}ciel, Piotr T.; Mazzeo, Rafe, Initial data sets with ends of cylindrical type: I. The Lichnerowicz equation, Ann. Henri Poincar\'e, 16, 5, 1231-1266 (2015) · Zbl 1314.83011 · doi:10.1007/s00023-014-0339-z
[52] Chru{\'s}ciel, Piotr T.; Mazzeo, Rafe; Pocchiola, Samuel, Initial data sets with ends of cylindrical type: II. The vector constraint equation, Adv. Theor. Math. Phys., 17, 4, 829-865 (2013) · Zbl 1291.83023
[53] Cohen1989 L. Cohen. Time-frequency distributions-a review. Proceedings of the IEEE 77 (1989), no. 7, 941-981.
[54] collins2010gravity H. Collins. Gravity’s Shadow: The Search for Gravitational Waves. University of Chicago Press, 2010.
[55] Cook, Gregory B., Initial data for axisymmetric black-hole collisions, Phys. Rev. D (3), 44, 10, 2983-3000 (1991) · doi:10.1103/PhysRevD.44.2983
[56] Cook, Gregory B., Initial data for numerical relativity, Living Reviews in Relativity, 3, 2000-5, 53 pp. (electronic) (2000) · Zbl 1024.83001 · doi:10.12942/lrr-2000-5
[57] Cook, Gregory B.; Teukolsky, Saul A., Numerical relativity: challenges for computational science. Acta Numerica, 8, 1-45 (1999), Cambridge Univ. Press, Cambridge · Zbl 0948.83005 · doi:10.1017/S0962492900002889
[58] Corry, Leo; Renn, J{\"u}rgen; Stachel, John, Belated decision in the Hilbert-Einstein priority dispute, Science, 278, 5341, 1270-1273 (1997) · Zbl 1226.83001 · doi:10.1126/science.278.5341.1270
[59] Corvino, Justin, Scalar curvature deformation and a gluing construction for the Einstein constraint equations, Comm. Math. Phys., 214, 1, 137-189 (2000) · Zbl 1031.53064 · doi:10.1007/PL00005533
[60] Corvino, Justin; Pollack, Daniel, Scalar curvature and the Einstein constraint equations. Surveys in geometric analysis and relativity, Adv. Lect. Math. (ALM) 20, 145-188 (2011), Int. Press, Somerville, MA · Zbl 1268.53048
[61] Corvino, Justin; Schoen, Richard M., On the asymptotics for the vacuum Einstein constraint equations, J. Differential Geom., 73, 2, 185-217 (2006) · Zbl 1122.58016
[62] Dafermos:2016uzj M. Dafermos, G. Holzegel, and I. Rodnianski, The linear stability of the Schwarzschild solution to gravitational perturbations, arXiv:1601.06467 [gr-qc], 2016. · Zbl 1419.83023
[63] Dafermos:2014cua 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. · Zbl 1347.83002
[64] Dahl, Mattias; Gicquaud, Romain; Humbert, Emmanuel, A limit equation associated to the solvability of the vacuum Einstein constraint equations by using the conformal method, Duke Math. J., 161, 14, 2669-2697 (2012) · Zbl 1258.53037 · doi:10.1215/00127094-1813182
[65] Dain, Sergio, Trapped surfaces as boundaries for the constraint equations, Classical and Quantum Gravity, 21, 2, 555-573 (2004) · Zbl 1050.83019 · doi:10.1088/0264-9381/21/2/017
[66] Dain, Sergio, Generalized Korn’s inequality and conformal Killing vectors, Calc. Var. Partial Differential Equations, 25, 4, 535-540 (2006) · Zbl 1091.35097 · doi:10.1007/s00526-005-0371-4
[67] Dain, Sergio; Jaramillo, Jos{\'e} Luis; Krishnan, Badri, Existence of initial data containing isolated black holes, Phys. Rev. D (3), 71, 6, 064003, 11 pp. (2005) · doi:10.1103/PhysRevD.71.064003
[68] darmois1927memorial G. Darmois, Memorial des sciences mathematique: fascicule XXV: les equations de la gravitation einsteinienne, Gauthier-Villars, 1927.
[69] Dilts, James, The Einstein constraint equations on compact manifolds with boundary, Classical and Quantum Gravity, 31, 12, 125009, 27 pp. (2014) · Zbl 1298.83015 · doi:10.1088/0264-9381/31/12/125009
[70] DiltsHolstMaxwell15 J. Dilts, M. Holst, and D. Maxwell, Analytic and numerical bifurcation analysis of the conformal formulation of the Einstein constraint equations, Preprint.
[71] Dilts, James; Isenberg, Jim; Mazzeo, Rafe; Meier, Caleb, Non-CMC solutions of the Einstein constraint equations on asymptotically Euclidean manifolds, Classical and Quantum Gravity, 31, 6, 065001, 10 pp. (2014) · Zbl 1292.83009 · doi:10.1088/0264-9381/31/6/065001
[72] Einstein:1916cc A. Einstein, Approximative Integration of the Field Equations of Gravitation, Sitzungsber. Preuss. Akad. Wiss. Berlin (Math. Phys.) 1916 (1916), 688-696. · JFM 46.1293.02
[73] 1918SPAW.......154E A. Einstein, \`“Uber Gravitationswellen, Sitzungsberichte der K\'”oniglich Preuischen Akademie der Wissenschaften (Berlin), Seite 154-167, 1918. · JFM 46.1295.02
[74] Einstein193743 A. Einstein and N. Rosen, On gravitational waves, Journal of the Franklin Institute 223 (1937), no. 1, 43-54. · JFM 63.1259.03
[75] eppley75 K. R. Eppley, “The Numerical Evolution of the Collision of Two Black Holes”, PhD thesis, Princeton University, Princeton, New Jersey, 1975.
[76] Fischer, Arthur E.; Marsden, Jerrold E.; Moncrief, Vincent, The structure of the space of solutions of Einstein’s equations. I. One Killing field, Ann. Inst. H. Poincar\'e Sect. A (N.S.), 33, 2, 147-194 (1980) · Zbl 0454.53044
[77] PhysRevD.48.2389 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.
[78] Four{\`“e}s-Bruhat, Y., Th\'”eor\`“eme d”existence pour certains syst\`“emes d”\'equations aux d\'eriv\'ees partielles non lin\'eaires, Acta Math., 88, 141-225 (1952) · Zbl 0049.19201
[79] Frauendiener, J{\"o}rg, Conformal infinity, Living Reviews in Relativity, 7, 2004-1, 82 pp. (electronic) (2004) · Zbl 1070.83006
[80] Friedrich, Helmut, On the hyperbolicity of Einstein’s and other gauge field equations, Comm. Math. Phys., 100, 4, 525-543 (1985) · Zbl 0588.35058
[81] Friedrich, Helmut; Nagy, Gabriel, The initial boundary value problem for Einstein’s vacuum field equation, Comm. Math. Phys., 201, 3, 619-655 (1999) · Zbl 0947.83007 · doi:10.1007/s002200050571
[82] 2011LRR....14....1F 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. · Zbl 1316.83026
[83] Geyer1991 C. J. Geyer, Markov Chain Monte Carlo maximum likelihood, 1991.
[84] W. R. Gilks; S. Richardson; D. J. Spiegelhalter{,} eds., Markov chain Monte Carlo in practice, Interdisciplinary Statistics, xviii+486 pp. (1996), Chapman & Hall, London · Zbl 0832.00018 · doi:10.1007/978-1-4899-4485-6
[85] GiNg14a 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 \textupTTtensor, Classical and Quantum Gravity 31 (2014), no.19, 1-16.
[86] GiNg15a R. Gicquaud and T. C. Nguyen, Solutions to the Einstein-scalar field constraint equations with small \textupTT-sensor, Classical and Quantum Gravity 55 (2015), no.2, 1-18.
[87] Green, Peter J.; \L atuszy{\'n}ski, Krzysztof; Pereyra, Marcelo; Robert, Christian P., Bayesian computation: a summary of the current state, and samples backwards and forwards, Stat. Comput., 25, 4, 835-862 (2015) · Zbl 1331.62017 · doi:10.1007/s11222-015-9574-5
[88] Gregory, P. C., Bayesian logical data analysis for the physical sciences. A comparative approach with {\it Mathematica}\(^\circledR\) support, xviii+468 pp. (2005), Cambridge University Press, Cambridge · Zbl 1069.62109 · doi:10.1017/CBO9780511791277
[89] Gundlach, Carsten; Calabrese, Gioel; Hinder, Ian; Mart{\'{\i }}n-Garc{\'{\i }}a, Jos{\'e} M., Constraint damping in the Z4 formulation and harmonic gauge, Classical and Quantum Gravity, 22, 17, 3767-3773 (2005) · Zbl 1154.83302 · doi:10.1088/0264-9381/22/17/025
[90] Gustafsson, Bertil; Kreiss, Heinz-Otto; Sundstr{\"o}m, Arne, Stability theory of difference approximations for mixed initial boundary value problems. II, Math. Comp., 26, 649-686 (1972) · Zbl 0293.65076
[91] Hahn, Susan G., Stability criteria for difference schemes, Comm. Pure Appl. Math., 11, no. 2{,} 243-255 (1958) · Zbl 0082.12301
[92] Hahn, Susan G.; Lindquist, Richard W., The two-body problem in geometrodynamics, Ann. Physics, 29, 304-331 (1964) · Zbl 0127.43505
[93] S. W. Hawking; W. Israel{,} eds., Three hundred years of gravitation, xiv+690 pp. (1989), Philosophiae Naturalis, Principia Mathematica, Cambridge University Press, Cambridge
[94] Hilbert D. Hilbert, Die Grundlagen der Physik, Nachr. Ges. Wiss. G\"ottingen Math. Phys. KL., 1916, pp. 395-407. · JFM 45.1111.01
[95] Holst, M., Adaptive numerical treatment of elliptic systems on manifolds, A posteriori error estimation and adaptive computational methodsAdv. Comput. Math., 15, 1-4, 139-191 (2002) (2001) · Zbl 0997.65134 · doi:10.1023/A:1014246117321
[96] HoKu09a 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).
[97] Holst, Michael; Lindblom, Lee; Owen, Robert; Pfeiffer, Harald P.; Scheel, Mark A.; Kidder, Lawrence E., Optimal constraint projection for hyperbolic evolution systems, Phys. Rev. D (3), 70, 8, 084017, 17 pp. (2004) · doi:10.1103/PhysRevD.70.084017
[98] HoMe12a M. Holst and C. Meier, Non-uniqueness of solutions to the conformal formulation. arXiv:1210.2156 [gr-qc].
[99] Holst, Michael; Meier, Caleb, Non-CMC solutions to the Einstein constraint equations on asymptotically Euclidean manifolds with apparent horizon boundaries, Classical and Quantum Gravity, 32, 2, 025006, 28 pp. (2015) · Zbl 1307.83002 · doi:10.1088/0264-9381/32/2/025006
[100] HMT13a 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].
[101] Holst, M.; Nagy, G.; Tsogtgerel, G., Far-from-constant mean curvature solutions of Einstein’s constraint equations with positive Yamabe metrics, Phys. Rev. Lett., 100, 16, 161101, 4 pp. (2008) · Zbl 1228.83015 · doi:10.1103/PhysRevLett.100.161101
[102] Holst, Michael; Nagy, Gabriel; Tsogtgerel, Gantumur, Rough solutions of the Einstein constraints on closed manifolds without near-CMC conditions, Comm. Math. Phys., 288, 2, 547-613 (2009) · Zbl 1175.83010 · doi:10.1007/s00220-009-0743-2
[103] Holst, Michael; Tsogtgerel, Gantumur, The Lichnerowicz equation on compact manifolds with boundary, Classical and Quantum Gravity, 30, 20, 205011, 31 pp. (2013) · Zbl 1276.83007 · doi:10.1088/0264-9381/30/20/205011
[104] Hough1975 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.
[105] Hulse:1974eb R. Hulse and J. Taylor, Discovery of a pulsar in a binary system, Astrophys. J. 195 (1975), L51-L53.
[106] Isenberg, James, Constant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical and Quantum Gravity, 12, 9, 2249-2274 (1995) · Zbl 0840.53056
[107] Isenberg, James; Moncrief, Vincent, A set of nonconstant mean curvature solutions of the Einstein constraint equations on closed manifolds, Classical and Quantum Gravity, 13, 7, 1819-1847 (1996) · Zbl 0860.53056 · doi:10.1088/0264-9381/13/7/015
[108] Jaynes, E. T., Probability theory, xxx+727 pp. (2003), The logic of science; Edited and with a foreword by G. Larry Bretthorst, Cambridge University Press, Cambridge · Zbl 1045.62001 · doi:10.1017/CBO9780511790423
[109] Kay, Bernard S.; Wald, Robert M., Linear stability of Schwarzschild under perturbations which are nonvanishing on the bifurcation \(2\)-sphere, Classical and Quantum Gravity, 4, 4, 893-898 (1987) · Zbl 0647.53065
[110] Kennefick, Daniel, Traveling at the speed of thought: Einstein and the quest for gravitational waves, xiv+319 pp. (2007), Princeton University Press, Princeton, NJ · Zbl 1120.01013 · doi:10.1515/9781400882748
[111] Kidder, Lawrence E.; Scheel, Mark A.; Teukolsky, Saul A., Extending the lifetime of 3D black hole computations with a new hyperbolic system of evolution equations, Phys. Rev. D (3), 64, 6, 064017, 13 pp. (2001) · doi:10.1103/PhysRevD.64.064017
[112] 2016PhRvD..93d2004K 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.
[113] Kreiss, H.-O.; Reula, O.; Sarbach, O.; Winicour, J., Boundary conditions for coupled quasilinear wave equations with application to isolated systems, Comm. Math. Phys., 289, 3, 1099-1129 (2009) · Zbl 1172.35077 · doi:10.1007/s00220-009-0788-2
[114] Kreiss, H.-O.; Winicour, J., Problems which are well posed in a generalized sense with applications to the Einstein equations, Classical and Quantum Gravity, 23, 16, S405-S420 (2006) · Zbl 1191.83010 · doi:10.1088/0264-9381/23/16/S07
[115] L44 A. Lichernowicz, Sur l’int\'egration des \'equations d’Einstein, J. Math. Pures Appl. 23 (1944), 26-63.
[116] lichnerowicz1955theories A. Lichnerowicz and G. Darmois, Th\'eories relativistes de la gravitation et de l’\'electromagn\'etisme: relativit\'e g\'en\'erale et th\'eories unitaires, par A. Lichnerowicz,... Pr\'eface du Pr G. Georges Darmois, Barn\'eoud fr\`eres et Cie, 1955.
[117] Lindblad, Hans; Rodnianski, Igor, Global existence for the Einstein vacuum equations in wave coordinates, Comm. Math. Phys., 256, 1, 43-110 (2005) · Zbl 1081.83003 · doi:10.1007/s00220-004-1281-6
[118] Loredo, Thomas J., Bayesian astrostatistics: a backward look to the future. Astrostatistical challenges for the new astronomy, Springer Ser. Astrostatistics, 15-40 (2013), Springer, New York · doi:10.1007/978-1-4614-3508-2\_2
[119] Maggiore2000 M. Maggiore, Gravitational wave experiments and early universe cosmology, Physics Reports 331 (2000), no. 6, 283-367.
[120] Maggiore2008 M. Maggiore, Gravitational waves, Oxford University Press, 2008.
[121] CGA-MoG 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.
[122] Maxwell:2014c D. Maxwell, Initial data in general relativity described by expansion, conformal deformation and drift, arXiv:1407.1467 [gr-qc].
[123] Maxwell, David, Solutions of the Einstein constraint equations with apparent horizon boundaries, Comm. Math. Phys., 253, 3, 561-583 (2005) · Zbl 1065.83011 · doi:10.1007/s00220-004-1237-x
[124] Maxwell, David, Rough solutions of the Einstein constraint equations, J. Reine Angew. Math., 590, 1-29 (2006) · Zbl 1088.83004 · doi:10.1515/CRELLE.2006.001
[125] Maxwell, David, A class of solutions of the vacuum Einstein constraint equations with freely specified mean curvature, Math. Res. Lett., 16, 4, 627-645 (2009) · Zbl 1187.83022 · doi:10.4310/MRL.2009.v16.n4.a6
[126] Maxwell, David, A model problem for conformal parameterizations of the Einstein constraint equations, Comm. Math. Phys., 302, 3, 697-736 (2011) · Zbl 1215.53064 · doi:10.1007/s00220-011-1187-z
[127] Maxwell, David, The conformal method and the conformal thin-sandwich method are the same, Classical and Quantum Gravity, 31, 14, 145006, 34 pp. (2014) · Zbl 1295.83015 · doi:10.1088/0264-9381/31/14/145006
[128] Maxwell:2014b D. Maxwell, Conformal parameterizations of slices of flat Kasner spacetimes, Annales Henri Poincar\'e 16 (2015), no. 12, 2919-2954. DOI 10.1007/s00023-014-0386-5. · Zbl 1330.83006
[129] Mazzieri, Lorenzo, Generalized gluing for Einstein constraint equations, Calc. Var. Partial Differential Equations, 34, 4, 453-473 (2009) · doi:10.1007/s00526-008-0191-4
[130] 2009PhRvD..79j4017M C. Messenger, R. Prix, and M. A. Papa, Random template banks and relaxed lattice coverings, Phys.Rev.D 79 (2009), no. 10, 104017.
[131] N. Metropolis; J. Howlett; Gian-Carlo Rota{,} eds., A history of computing in the twentieth century. A collection of essays, xix+659 pp. (1980), Academic Press, Inc. [Harcourt Brace Jovanovich, Publishers], New York-London · Zbl 0456.68001
[132] Metropolis1953 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. · Zbl 1431.65006
[133] Nguyen:2015 T.-C. Nguyen, Nonexistence and nonuniqueness results for solutions to the vacuum Einstein conformal constraint equations, arXiv:1507.01081 [math.AP].
[134] O’Murchadha, Niall; York, James W., Jr., Existence and uniqueness of solutions of the Hamiltonian constraint of general relativity on compact manifolds, J. Math. Phys., 14, 1551-1557 (1973) · Zbl 0281.53031
[135] 1996PhRvD..53.6749O B. J. Owen, Search templates for gravitational waves from inspiraling binaries: Choice of template spacing, Phys.Rev.D 53 (1996), 6749-6761.
[136] Pfei04 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.
[137] Pfeiffer, Harald P.; York, James W., Jr., Uniqueness and nonuniqueness in the Einstein constraints, Phys. Rev. Lett., 95, 9, 091101, 4 pp. (2005) · doi:10.1103/PhysRevLett.95.091101
[138] Premoselli, Bruno, Effective multiplicity for the Einstein-scalar field Lichnerowicz equation, Calc. Var. Partial Differential Equations, 53, 1-2, 29-64 (2015) · Zbl 1321.83013 · doi:10.1007/s00526-014-0740-y
[139] Pretorius, Frans, Evolution of binary black-hole spacetimes, Phys. Rev. Lett., 95, 12, 121101, 4 pp. (2005) · doi:10.1103/PhysRevLett.95.121101
[140] 2007CQGra..24S.481P R. Prix, Template-based searches for gravitational waves: efficient lattice covering of flat parameter spaces, Classical and Quantum Gravity 24 (2007), S481-S490.
[141] Regge, Tullio; Wheeler, John A., Stability of a Schwarzschild singularity, Phys. Rev. (2), 108, 1063-1069 (1957) · Zbl 0079.41902
[142] Regimbau2011 T. Regimbau, The astrophysical gravitational wave stochastic background, Research in Astronomy and Astrophysics 11 (2011), no. 4, 369-390.
[143] Sarbach:2012pr 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. · Zbl 1320.83012
[144] Sauer, Tilman, The relativity of discovery: Hilbert’s first note on the foundations of physics, Arch. Hist. Exact Sci., 53, 6, 529-575 (1999) · Zbl 0926.01004
[145] Scharf1991 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.
[146] Sejdic2009 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.
[147] Skilling, John, Nested sampling for general Bayesian computation, Bayesian Anal., 1, 4, 833-859 (electronic) (2006) · Zbl 1332.62374 · doi:10.1214/06-BA127
[148] NYAS:NYAS569 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.
[149] 1989ApJ...345..434T 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.
[150] 2016arXiv160203843T 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]
[151] 2016arXiv160203841T 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]
[152] 2016arXiv160203839T 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]
[153] 2016arXiv160203840T 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]
[154] Turin, George L., An introduction to matched filters, Trans. IRE, IT-6, 311-329 (1960)
[155] 2008PhRvD..77d2001V 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
[156] 2011PhRvL.107s1104V 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.
[157] 2015PhRvD..91d2003V 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. R\"over, 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.
[158] Wainstein, L. A.; Zubakov, V. D., Extraction of signals from noise, Translated from the Russian by Richard A. Silverman. International Series in Applied Mathematics, xii+382 pp. (1962), Prentice-Hall, Inc., Englewood Cliffs, N.J.
[159] Walsh, D. M., Non-uniqueness in conformal formulations of the Einstein constraints, Classical and Quantum Gravity, 24, 8, 1911-1925 (2007) · Zbl 1113.83008 · doi:10.1088/0264-9381/24/8/002
[160] Weber:1969bz J. Weber, Evidence for discovery of gravitational radiation, Phys. Rev. Lett. 22 (1969), no. 24, 1320-1324.
[161] Weiss R. Weiss, Electromagnetically Coupled Broadband Gravitational Antenna, Quarterly Progress Report of the MIT Research Laboratory of Electronics, 54(105), 1972. · Zbl 1518.83021
[162] Whiting, Bernard F., Mode stability of the Kerr black hole, J. Math. Phys., 30, 6, 1301-1305 (1989) · Zbl 0689.53041 · doi:10.1063/1.528308
[163] lrr-2006-3 C. M. Will, The confrontation between general relativity and experiment, Living Reviews in Relativity 9 (2006), no. 3. DOI 10.12942/lrr-2006-3 · Zbl 1316.83020
[164] York, James W., Jr., Conformally invariant orthogonal decomposition of symmetric tensors on Riemannian manifolds and the initial-value problem of general relativity, J. Math. Phys., 14, 456-464 (1973) · Zbl 0259.53014
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.