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  • Print publication year: 2015
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7 - Probing Strong-Field Gravity Through Numerical Simulations

from Part Three - Gravity is Geometry, after all

Summary

This chapter describes what has been learned about the dynamical, strong-field regime of general relativity via numerical methods. There is no rigorous way to identify this regime, in particular since notions of energies, velocities, length and timescales are observer- dependent at best, and at worst are not well-defined locally or even globally. Loosely speaking, however, dynamical strong-field phenomena exhibit the following properties: there is at least one region of spacetime of characteristic size R containing energy E where the compactness 2GE/(c4R) approaches unity, local velocities approach the speed of light c, and luminosities (of gravitational or matter fields) can approach the Planck luminosity c5 /G. A less physical characterization, though one better suited to classifying solutions, involves spacetimes where even in “well-adapted” coordinates the non-linearities of the field equations are strongly manifest. In many of the cases where these conditions are met, numerical methods are the only option available to solve the Einstein field equations, and such scenarios are the subject of this chapter.

Mirroring trends in the growth and efficacy of computation, numerical solutions have had greatest impact on the field in the decades following the 1987 volume [1] celebrating the 300th anniversary of Newton's Principia. However, several pioneering studies laying the foundation for subsequent advances were undertaken before this, and they are briefly reviewed in Section 7.1 below. Though this review focuses on the physics that has been gleaned from computational solutions, there are some unique challenges in numerical evolution of the Einstein equations; these as well as the basic computational strategies that are currently dominant in numerical relativity are discussed in Section 7.2. As important as computational science has become in uncovering details of solutions too complex to model analytically, it is a rare moment when qualitatively new physics is uncovered. The standout example in general relativity is the discovery of critical phenomena in gravitation collapse (Section 7.3.1); another noteworthy example is the formation of so-called spikes in the approach to cosmological singularities (Section 7.3.7).

[1] Israel, W., and Hawking, S. W. 1987. Three hundred years of gravitation. Cambridge: Cambridge University Press.
[2] Hahn, S. G., and Lindquist, R. W. 1964. Ann. Phys., 29, 304–331.
[3] Misner, C. W. 1960. Phys. Rev., 118, 1110–1111.
[4] Smarr, L. L. 1975. The structure of general relativity with a numerical illustration: the collision of two black holes. Ph.D. thesis, University of Texas, Austin, Austin, Texas.
[5] Smarr, L. L. 1979. Basic concepts in finite differencing of partial differential equations. Page 139 of: Smarr, L. L. (ed), Sources of gravitational radiation. Cambridge: Cambridge University Press.
[6] Eppley, K. 1975. The numerical evolution of the collision of two black holes. Ph.D. thesis, Princeton University, Princeton, New Jersey.
[7] Smarr, L., et al. 1976. Phys. Rev. D, 14, 2443–2452.
[8] Anninos, P., et al. 1993. Phys. Rev. Lett., 71, 2851–2854.
[9] Brügmann, B. 1999. Int. J. Mod. Phys. D, 8, 85–100.
[10] Pretorius, F. 2005. Phys. Rev. Lett., 95, 121101.
[11] Campanelli, M., et al. 2006. Phys. Rev. Lett., 96, 111101.
[12] Baker, J. G., et al. 2006. Phys. Rev. Lett., 96, 111102.
[13] Pretorius, F. 2009. Binary black hole coalescence. Pages 305–369 of: Colpi, M. et al. (eds), Physics of relativistic objects in compact binaries: from birth to coalescence. Heidelberg: Springer.
[14] May, M. M., and White, R. H. 1966. Phys. Rev., 141, 1232–1241.
[15] Wilson, J. R. 1971. Astrophys. J., 163, 209.
[16] Wilson, J. R. 1979. A numerical method for relativistic hydrodynamics. Pages 423–445 of: Smarr, L. L. (ed), Sources of gravitational radiation. Cambridge: Cambridge University Press.
[17] Shapiro, S. L., and Teukolsky, S. A. 1980. Astrophys. J., 235, 199–215.
[18] Stark, R. F., and Piran, T. 1985. Phys. Rev. Lett., 55, 891–894. Erratum: ibid. 56, 97 (1986).
[19] Nakamura, T. 1981. Prog. Theor. Phys., 65, 1876–1890.
[20] Nakamura, T. 1983. Prog. Theor. Phys., 70, 1144–1147.
[21] Evans, C. R. 1986. An approach for calculating axisymmetric gravitational collapse. Pages 3-39 of: Centrella, J. M. (ed), Dynamical spacetimes and numerical relativity. Cambridge: Cambridge University Press.
[22] Shapiro, S. L., and Teukolsky, S. A. 1991. Phys. Rev. Lett., 66, 994–997.
[23] Thorne, K. S. 1972. Nonspherical gravitational collapse: a short review. Page 231 of: Klauder, J. (ed), Magic without magic: John Archibald Wheeler. San Francisco, CA: Freeman.
[24] Piran, T. 1980. J. Comp. Phys., 35, 254–283.
[25] Cook, G. B. 2000. Living Rev. Rel., 3.
[26] Gourgoulhon, E. 2007. J. Phys.: Conf. Ser., 91, 012001.
[27] Pfeiffer, H. P. 2005. Initial data for black hole evolutions. Ph.D. thesis, Cornell University, Ithaca, New York.
[28] York Jr., J. W. 1979. Kinematics and dynamics of general relativity. Pages 83–126 of: Smarr, L. L. (ed), Sources of gravitational radiation. Cambridge: Cambridge University Press.
[29] York Jr., J. W., and Piran, T. 1982. The initial value problem and beyond. Pages 147–176 of: Matzner, R. A., and Shepley, L. C. (eds), Spacetime and geometry: the Alfred Schild lectures. Austin, TX: University of Texas Press.
[30] Arnowitt, R., Deser, S., and Misner, C. W. 1962. The dynamics of general relativity. Pages 227–265 of: Witten, L. (ed), Gravitation: an introduction to current research. NewYork: Wiley.
[31] Bowen, J. M., and York Jr., J. W. 1980. Phys. Rev. D, 21, 2047–2056.
[32] Brandt, S. R., and Brügmann, B. 1997. Phys. Rev. Lett., 78, 3606–3609.
[33] Thornburg, J. 1987. Class. Quant. Grav., 4, 1119–1131.
[34] Centrella, J. M. 1980. Phys. Rev. D, 21, 2776–2784.
[35] Centrella, J. M., and Wilson, J. R. 1984. Astrophys. J. Suppl. Ser., 54, 229–249.
[36] Anninos, P., Centrella, J. M., and Matzner, R. A. 1991. Phys. Rev. D, 43, 1808.
[37] Kurki-Suonio, H., Laguna, P., and Matzner, R. A. 1993. Phys. Rev. D, 48, 3611–3624.
[38] Berger, B. K., and Moncrief, V. 1993. Phys. Rev. D, 48, 4676–4687.
[39] Kaup, D. J. 1968. Phys. Rev., 172, 1331–1342.
[40] Ruffini, R., and Bonazzola, S. 1969. Phys. Rev., 187, 1767–1783.
[41] Colpi, M., Shapiro, S. L., and Wasserman, I. 1986. Phys. Rev. Lett., 57, 2485–2488.
[42] Liebling, S. L., and Palenzuela, C. 2012. Living Rev. Rel., 15, 6.
[43] Seidel, E., and Suen, W.-M. 1992. Phys. Rev. Lett., 69, 1845–1848.
[44] Bona, C., and Massó, J. 1993. A vacuum fully relativistic 3D numerical code. Pages 258–264 of: d'Inverno, R. A. (ed), Approaches to numerical relativity. Cambridge: Cambridge University Press.
[45] Cook, G. B., et al. 1998. Phys.Rev. Lett., 80, 2512–2516.
[46] Abrahams, A. M., et al. 1998. Phys. Rev. Lett., 80, 1812–1815.
[47] Gómez, R., et al. 1998. Phys. Rev. Lett., 80, 3915–3918.
[48] Lehner, L. 2001. Class. Quant. Grav., 18, R25–R86.
[49] Bona, C., and Palenzuela, C. (eds). 2005. Elements of numerical relativity. Lecture Notes in Physics, vol. 673. Berlin/Heidelberg: Springer.
[50] Alcubierre, M. 2008. Introduction to 3+1 numerical relativity. Oxford: Oxford University Press.
[51] Baumgarte, T. W., and Shapiro, S. L. 2010. Numerical relativity: solving Einstein's equations on the computer. Cambridge: Cambridge University Press.
[52] Gourgoulhon, E. (ed). 2012. 3+1 Formalism in general relativity. Lecture Notes in Physics, vol. 846. Berlin: Springer.
[53] Kreiss, H.-O., and Ortiz, O. E. 2002. Some mathematical and numerical questions connected with first and second order time-dependent systems of partial differential equations. Pages 359–370 of: Frauendiener, J., and Friedrich, H. (eds), The conformal structure of space-time. Lecture Notes in Physics, vol. 604. Berlin: Springer.
[54] Gustafsson, B., Kreiss, H.-O., and Oliger, J. 1995. Time dependent problems and difference methods. New York: Wiley.
[55] Sarbach, O., and Tiglio, M. 2012. Living Rev. Rel., 15, 9.
[56] Friedrich, H., and Rendall, A. D. 2000. The Cauchy problem for the Einstein equations. Pages 127–223 of: Schmidt, B. G. (ed), Einstein's field equations and their physical Lecture Notes in Physics, vol. 540. Berlin: Springer.
[57] Reula, O. A. 1998. Living Rev. Rel., 1.
[58] Alic, D., et al. 2012. Phys. Rev. D, 85, 064040.
[59] Nakamura, T., Oohara, K.-I., and Kojima, Y. 1987. Prog. Theor. Phys. Suppl., 90, 1–218.
[60] Shibata, M., and Nakamura, T. 1995. Phys. Rev. D, 52, 5428–5444.
[61] Baumgarte, T. W., and Shapiro, S. L. 1998. Phys. Rev. D, 59, 024007.
[62] Renn, J., and Sauer, T. 1999. Heuristics and mathematical representation in Einstein's search for a gravitational field equation. Page 87 of: Goenner, H., Renn, J., Ritter, J., and Sauer, T. (eds), The expanding worlds of general relativity. Basel: Birkhäuser.
[63] Lindblom, L., et al. 2006. Class. Quant. Grav., 23, S447–S462.
[64] Garfinkle, D. 2002. Phys. Rev. D, 65, 044029.
[65] Gundlach, C., et al. 2005. Class. Quant. Grav., 22, 3767–3774.
[66] Brodbeck, O., et al. 1999. J. Math. Phys., 40, 909–923.
[67] Palenzuela, C., Lehner, L., and Yoshida, S. 2010. Phys. Rev. D, 81, 084007.
[68] Headrick, M., Kitchen, S., and Wiseman, T. 2010. Class. Quant. Grav., 27, 035002.
[69] Hannam, M. D. et al. 2007. Phys. Rev. Lett., 99, 241102.
[70] Hannam, M. D. et al. 2008. Phys. Rev. D, 78, 064020.
[71] Winicour, J. 1998. Living Rev. Rel., 1.
[72] Gómez, R., et al. 1998. Phys. Rev. D, 57, 4778–4788.
[73] Bishop, N. T., et al. 1997. Phys.Rev. D, 54, 6153–6165.
[74] Reisswig, C., et al. 2010. Class. Quant. Grav., 27, 075014.
[75] Brady, P. R., and Smith, J. D. 1995. Phys. Rev. Lett., 75, 1256–1259.
[76] Chesler, P. M., and Yaffe, L. G., 2013. arXiv:1309.1439.
[77] Friedrich, H. 2002. Conformal Einstein evolution. Pages 1–50 of: Frauendiener, J., and Friedrich, H. (eds), The conformal structure of space-time. Lecture Notes in Physics, vol. 604. Berlin: Springer.
[78] Frauendiener, J. 1998. Phys. Rev. D, 58, 064002.
[79] Husa, S. 2002. Problems and successes in the numerical approach to the conformal field equations. Pages 239–260 of: Frauendiener, J., and Friedrich, H. (eds), The conformal structure ofspace-time. Lecture Notes in Physics, vol. 604. Berlin: Springer.
[80] Berger, M. J., and Oliger, J. 1984. J. Comput. Phys., 53, 484.
[81] Choptuik, M. W. 1989. Experiences with an adaptive mesh refinement algorithm in numerical relativity. In: Evans, C. R., Finn, L. S., and Hobill, D. W. (eds), Frontiers in numerical relativity. Cambridge: Cambridge University Press.
[82] Lehner, L., Liebling, S. L., and Reula, O. A. 2006. Class. Quant. Grav., 23, S421–S446.
[83] Boyd, J. P. 1989. Chebyshev and Fourier spectral methods. New York: Springer-Verlag.
[84] Grandclement, P., and Novak, J. 2009. Living Rev. Rel., 12.
[85] Szilágyi, B., Lindblom, L., and Scheel, M. A. 2009. Phys. Rev. D, 80, 124010.
[86] Font, J. A. 2008. Living Rev. Rel., 11, 7.
[87] LeVeque, R. J. 1992. Numerical methods for conservation laws. Basel: Birkhäuser Verlag.
[88] Cardoso, V., Gualtieri, L., Herdeiro, C., and Sperhake, U. 2014. Exploring new physics frontiers through numerical relativity. To appear in Living Reviews in Relativity.
[89] Evans, C. R., and Coleman, J. S. 1994. Phys. Rev. Lett., 72, 1782–1785.
[90] Koike, T., Hara, T., and Adachi, S. 1995. Phys. Rev. Lett., 74, 5170–5173.
[91] Maison, D. 1996. Phys. Lett., B366, 82–84.
[92] Gundlach, C., and Martin-Garcia, J. M. 2007. Living Rev. Rel., 10, 5.
[93] Gundlach, C. 1998. Adv. Theor. Math. Phys., 2, 1–49.
[94] Choptuik, M. W. 1993. Phys. Rev. Lett., 70, 9–12.
[95] Gundlach, C. 1997. Phys. Rev. D, 55, 695–713.
[96] Hod, S., and Piran, T. 1997. Phys. Rev. D, 55, 3485–3496.
[97] Martin-Garcia, J. M., and Gundlach, C. 1999. Phys. Rev. D, 59, 064031.
[98] Choptuik, M. W., Hirschmann, E. W., Liebling, S. L., and Pretorius, F. 2003. Phys. Rev. D, 68, 044007.
[99] Healy, J., and Laguna, P., 2013. arXiv:1310.1955.
[100] Brady, P. R., Chambers, C. M., and Goncalves, S. M. 1997. Phys. Rev. D, 56, 6057–6061.
[101] Seidel, E., and Suen, W. 1991. Phys. Rev. Lett., 66, 1659–1662.
[102] Hawley, S. H., and Choptuik, M. W. 2000. Phys. Rev. D, 62, 104024.
[103] Husain, V., and Olivier, M. 2001. Class. Quant. Grav., 18, L1–L10.
[104] Pretorius, F., and Choptuik, M. W. 2000. Phys. Rev. D, 62, 124012.
[105] Bizon, P., and Rostworowski, A. 2011. Phys. Rev. Lett., 107, 031102.
[106] Garfinkle, D. 2001. Phys. Rev. D, 63, 044007.
[107] Abrahams, A. M., and Evans, C. R. 1993. Phys. Rev. Lett., 70, 2980–2983.
[108] Sorkin, E. 2011. Class. Quant. Grav., 28, 025011.
[109] Garfinkle, D., and Duncan, G. C. 1998. Phys.Rev. D, 58, 064024.
[110] Bizon, P., Chmaj, T., and Schmidt, B. G. 2005. Phys. Rev. Lett., 95, 071102.
[111] Bizon, P., Chmaj, T., and Schmidt, B. G. 2006. Phys. Rev. Lett., 97, 131101.
[112] Bizon, P., et al. 2005. Phys. Rev. D, 72, 121502.
[113] Szybka, S. J., and Chmaj, T. 2008. Phys. Rev. Lett., 100, 101102.
[114] Gundlach, C. 1998. Phys. Rev. D, 57, 7080–7083.
[115] Cahill, M. E., and Taub, A. H. 1971. Commun. Math. Phys., 21, 1–40.
[116] Noble, S. C. 2003. A numerical study of relativistic fluid collapse. Ph.D. thesis, The University of British Columbia, Vancouver, British Columbia. [arXiv:gr-qc/0310116].
[117] Neilsen, D. W., and Choptuik, M. W. 2000. Class. Quant. Grav., 17, 761–782.
[118] Noble, S. C., and Choptuik, M. W. 2008. Phys. Rev. D, 78, 064059.
[119] Novak, J. 2001. Astron. Astrophys., 376, 606–613.
[120] Niemeyer, J. C., and Jedamzik, K. 1998. Phys. Rev. Lett., 80, 5481–5484.
[121] Jin, K.-J., and Suen, W.-M. 2007. Phys. Rev. Lett., 98, 131101.
[122] Kellermann, T., Rezzolla, L., and Radice, D. 2010. Class. Quant. Grav., 27, 235016.
[123] Radice, D., Rezzolla, L., and Kellermann, T. 2010. Class. Quant. Grav., 27, 235015.
[124] Wan, M.-B. 2011. Class. Quant. Grav., 28, 155002.
[125] Liebling, S. L., et al. 2010. Phys. Rev. D, 81, 124023.
[126] Choptuik, M. W., Chmaj, T., and Bizon, P. 1996. Phys. Rev. Lett., 77, 424–427.
[127] Bartnik, R., and McKinnon, J. 1988. Phys. Rev. Lett., 61, 141–144.
[128] Choptuik, M. W., Hirschmann, E. W., and Marsa, R. L. 1999. Phys. Rev. D, 60, 124011.
[129] Liebling, S. L., and Choptuik, M. W. 1996. Phys. Rev. Lett., 77, 1424–1427.
[130] Lechner, C., et al. 2002. Phys. Rev. D, 65, 081501.
[131] Andreasson, H., and Rein, G. 2006. Class. Quant. Grav., 23, 3659–3678.
[132] Olabarrieta, I., and Choptuik, M. W. 2002. Phys. Rev. D, 65, 024007.
[133] Rein, G., Rendall, A. D., and Schaeffer, J. 1998. Phys. Rev. D, 58, 044007.
[134] Martin-Garcia, J. M., and Gundlach, C. 2002. Phys. Rev. D, 65, 084026.
[135] Centrella, J., et al. 2010. Ann. Rev. Nucl. Particle Sci., 60, 75–100.
[136] Hinder, I., et al. 2013. Class. Quant. Grav., 31, 025012.
[137] Ajith, P., et al. 2011. Phys. Rev. Lett., 106, 241101.
[138] Pan, Y., et al., 2013. arXiv:1307.6232.
[139] Damour, T., Nagar, A., and Bernuzzi, S. 2013. Phys. Rev. D, 87, 084035.
[140] Buonanno, A., Cook, G. B., and Pretorius, F. 2007. Phys. Rev. D, 75, 124018.
[141] Bishop, N. T., et al. 1997. Phys. Rev. D, 56, 6298–6309.
[142] Chu, T., Pfeiffer, H. P., and Cohen, M. I. 2011. Phys. Rev. D, 83, 104018.
[143] Gonzalez, J. A. et al. 2007. Phys. Rev. Lett., 98, 091101.
[144] Baker, J. G., et al. 2006. Astrophys. J., 653, L93–L96.
[145] Herrmann, F., et al. 2007. Class. Quant. Grav., 24, 33.
[146] Berti, E. et al. 2007. Phys. Rev. D, 76, 064034.
[147] Campanelli, M., Lousto, C., and Zlochower, Y. 2006. Phys. Rev. D, 74, 041501.
[148] Hemberger, D. A., et al. 2013. Phys. Rev. D, 88, 064014.
[149] Schmidt, P., et al. 2011. Phys. Rev. D, 84, 024046.
[150] Boyle, M., Owen, R., and Pfeiffer, H. P. 2011. Phys. Rev. D, 84, 124011.
[151] O'Shaughnessy, R., et al. 2011. Phys. Rev. D, 84, 124002.
[152] Campanelli, M., et al. 2007. Astrophys. J. Lett., 659, L5–L8.
[153] González, J. A., et al. 2007. Phys. Rev. Lett., 98, 231101.
[154] Lousto, C. O., and Zlochower, Y. 2011. Phys. Rev. Lett., 107, 231102.
[155] Komossa, S. 2012. Adv. Astron., 2012.
[156] Schnittman, J. D. 2013. Class. Quant. Grav., 30, 244007.
[157] Komossa, S., Zhou, H., and Lu, H. 2008. Astrophys. J., 678, L81–L84.
[158] Lippai, Z., Frei, Z., and Haiman, Z. 2008. Astrophys. J., 676, L5–L8.
[159] Milosavljevic, M., and Phinney, E. 2005. Astrophys. J., 622, L93–L96.
[160] Loeb, A. 2007. Phys. Rev. Lett., 99, 041103.
[161] Noble, S. C., et al. 2012. Astrophys. J., 755, 51.
[162] Farris, B. D., et al. 2012. Phys. Rev. Lett., 109, 221102.
[163] Stone, N., and Loeb, A. 2011. Mon. Not. Roy. Astr. Soc., 412, 75–80.
[164] Palenzuela, C., Lehner, L., and Leibling, S. L. 2010. Science, 329, 927–930.
[165] East, W. E., McWilliams, S. T., Levin, J., and Pretorius, F. 2013. Phys. Rev. D, 87, 043004.
[166] Pretorius, F., and Khurana, D. 2007. Class. Quant. Grav., 24, S83–S108.
[167] Healy, J., Levin, J., and Shoemaker, D. 2009. Phys. Rev. Lett., 103, 131101.
[168] Gold, R., and Bruegmann, B. 2013. Phys. Rev. D, 88, 064051.
[169] Metzger, B., and Berger, E. 2012. Astrophys. J., 746, 48.
[170] Piran, T., Nakar, E., and Rosswog, S. 2013. Mon. Not. R. Astron. Soc., 430, 2121–2136.
[171] Shibata, M., and Uryu, K. 2000. Phys. Rev. D, 61, 064001.
[172] Nakamura, T., and Oohara, K.-I. 1999. A way to 3D numerical relativity – coalescing binary neutron stars. arXiv:gr-qc/9812054.
[173] Read, J. S. et al. 2013. Phys.Rev. D, 88, 044042.
[174] Lackey, B. D., et al., 2013. arXiv:1303.6298.
[175] Tsang, D., et al. 2012. Phys. Rev. Lett., 108, 011102.
[176] Hotokezaka, K., et al. 2011. Phys. Rev. D, 83, 124008.
[177] Anderson, M., et al. 2008. Phys. Rev. Lett., 100, 191101.
[178] Sekiguchi, Y., et al., 2012. arXiv:1206.5927.
[179] Kaplan, J., et al. 2013. Phys. Rev. D, 88, 064009.
[180] Rezzolla, L. et al. 2010. Class. Quant. Grav., 27, 114105.
[181] Tanvir, N. R., et al. 2013. Nature, 500, 547–549.
[182] Berger, E., Fong, W., and Chornock, R. 2013. Astrophys. J. Lett., 774, L23.
[183] Hinderer, T., et al. 2010. Phys. Rev. D, 81, 123016.
[184] Markakis, C., et al. 2009. J. Phys.: Conf. Ser., 189, 012024.
[185] Sekiguchi, Y., et al. 2011. Phys. Rev. Lett., 107, 051102.
[186] Lehner, L., et al. 2012. Phys. Rev. D, 86, 104035.
[187]Kyutoku, K., Ioka, K., and Shibata, M. 2014. Mon. Not. Roy. Astron. Soc., 437, L6. [arXiv:1209.5747].
[188] Palenzuela, C., et al. 2013. Phys. Rev. Lett., 111, 061105.
[189] Hotokezaka, K., et al. 2013. Phys. Rev. D, 88, 044026.
[190] Kyutoku, K., Ioka, K., and Shibata, M. 2013. Phys. Rev. D, 88, 041503.
[191] Chawla, S., et al. 2010. Phys. Rev. Lett., 105, 111101.
[192] Foucart, F., et al. 2013. Phys. Rev. D, 87, 084006.
[193] Foucart, F. 2012. Phys. Rev. D, 86, 124007.
[194] Hansen, B. M., and Lyutikov, M. 2001. Mon. Not. Roy. Astron. Soc., 322, 695.
[195] McWilliams, S. T., and Levin, J. 2011. Astrophys. J., 742, 90.
[196] Paschalidis, V., Etienne, Z. B., and Shapiro, S. L. R.Phys. Rev. D, 88, 021504.
[197] Lackey, B. D., et al. 2012. Phys. Rev. D, 85, 044061.
[198] Stephens, B. C., East, W. E., and Pretorius, F. 2011. Astrophys. J. Lett., 737, L5.
[199] Gold, R., et al. 2012. Phys. Rev. D, 86, 121501.
[200] Tsang, D. 2013. Astrophys. J., 777, 103.
[201] Duez, M. D. 2010. Class. Quant. Grav., 27, 114002.
[202] Pfeiffer, H. P. 2012. Class. Quant. Grav., 29, 124004.
[203] Faber, J. A., and Rasio, F. A. 2012. Living Rev. Rel., 15, 8.
[204] Ott, C. D. 2009. Class. and Quant. Grav., 26, 063001.
[205] Janka, H.-T., et al. 2007. Phys. Reports, 442, 38–74.
[206] Burrows, A., et al. 2007. Phys. Reports, 442, 23–37.
[207] Ott, C. D., et al. 2013. Astrophys. J., 768, 115.
[208] Dimmelmeier, H., Font, J. A., and Müller, E. 2002. Astron. Astrophys., 388, 917–935.
[209] Obergaulinger, M., et al. 2006. Astron. Astrophys., 457, 209–222.
[210] Müller, B., Janka, H.-T., and Dimmelmeier, H. 2010. Astrophys. J. Suppl. Ser., 189, 104–133.
[211] Wongwathanarat, A., Janka, H.-T., and Mueller, E. 2013. Astron. Astrophys., 552, A126.
[212] Ott, C. D., et al. 2011. Phys. Rev. Lett., 106, 161103.
[213] Penrose, R. 1966. General relativistic energy flux and elementary optics. In: Hoffman, B. (ed), Perspectives in geometry and relativity. Indiana University Press.
[214] Aichelburg, P., and Sexl, R. 1971. Gen. Rel. Grav., 2, 303–312.
[215] Arkani-Hamed, N., Dimopoulos, S., and Dvali, G. 1998. Phys. Lett. B, 429, 263–272.
[216] Randall, L., and Sundrum, R. 1999. Phys. Rev. Lett., 83, 3370–3373.
[217] Giddings, S. B., and Thomas, S. D. 2002. Phys. Rev. D, 65, 056010.
[218] Feng, J. L., and Shapere, A. D. 2002. Phys. Rev. Lett., 88, 021303.
[219] Chatrchyan, S., et al. [CMS Collaboration]. 2013. JHEP, 1307, 178.
[220] Aad, G., et al. [ATLAS Collaboration]. 2013. Phys. Rev. D, 88, 072001.
[221] de los Heros, C. 2007. ArXiv Astrophysics e-prints.
[222] Thorne, K. S. 1972. Nonspherical gravitational collapse – a short review. Page 231 of: Klauder, J. (ed), Magic without magic: John Archibald Wheeler. San Francisco, CA: Freeman.
[223] Sperhake, U. et al. 2008. Phys. Rev. Lett., 101, 161101.
[224] Shibata, M., Okawa, H., and Yamamoto, T. 2008. Phys. Rev. D, 78, 101501.
[225] Sperhake, U., et al. 2009. Phys. Rev. Lett., 103, 131102.
[226] Sperhake, U., et al. 2011. Phys. Rev. D, 83, 024037.
[227] Sperhake, U., Berti, E., Cardoso, V., and Pretorius, F. 2013. Phys. Rev. Lett., 111, 041101.
[228] Okawa, H., Nakao, K.-i., and Shibata, M. 2011. Phys. Rev. D, 83, 121501.
[229] D'eath, P. D., and Payne, P. N. 1992. Phys. Rev. D, 46, 694–701.
[230] Eardley, D. M., and Giddings, S. B. 2002. Phys. Rev. D, 66, 044011.
[231] Berti, E. et al. 2010. Phys. Rev. D, 81, 104048.
[232] Gundlach, C., et al. 2012. Phys. Rev. D, 86, 084022.
[233] Gralla, S. E., Harte, A. I., and Wald, R. M. 2010. Phys. Rev. D, 81, 104012.
[234] Choptuik, M. W., and Pretorius, F. 2010. Phys. Rev. Lett., 104, 111101.
[235] East, W. E., and Pretorius, F. 2013. Phys. Rev. Lett., 110, 101101.
[236] Rezzolla, L., and Takami, K. 2013. Class. Quant. Grav., 30, 012001.
[237] Kaloper, N., and Terning, J. 2008. Int. J. Mod. Phys., D17, 665–672.
[238] Galley, C. R., and Porto, R. A. 2013. JHEP, 1311, 096.
[239] Gal'tsov, D., Spirin, P., and Tomaras, T. N. 2013. JHEP, 1301, 087.
[240] Grumiller, D., and Romatschke, P. 2008. JHEP, 0808, 027.
[241] Casalderrey-Solana, J., Heller, M. P., Mateos, D., and van der Schee, W. 2013. Phys. Rev. Lett., 111, 181601.
[242] Cardoso, V. et al. 2012. Class. Quant. Grav., 29, 244001. [arXiv:1201.5118].
[243] Antoniadis, I., et al. 1998. Phys. Lett. B, 436, 257–263.
[244] Baumann, D., 2009. arXiv:0907.5424.
[245] Lehners, J.-L. 2008. Phys. Rept., 465, 223–263.
[246] Carlip, S. 2005. Living Rev. Rel., 8, 1.
[247] Gegenberg, J., and Kunstatter, G., 2009. arXiv:0902.0292.
[248] Maldacena, J. M. 1998. Adv. Theor. Math. Phys., 2, 231–252.
[249] Aharony, O., et al. 2000. Phys. Rept., 323, 183–386.
[250] Horowitz, G. T., 2012. Black holes in higher dimensions. Cambridge: Cambrdige University Press.
[251] Emparan, R., and Reall, H. S. 2008. Living Rev. Rel., 11, 6.
[252] Reall, H. S., 2012. arXiv:1210.1402.
[253] Emparan, R., and Myers, R. C. 2003. JHEP, 0309, 025.
[254] Lehner, L., and Pretorius, F. 2010. Phys. Rev. Lett., 105, 101102.
[255] Shibata, M., and Yoshino, H. 2010. Phys. Rev. D, 81, 021501.
[256] Shibata, M., and Yoshino, H. 2010. Phys. Rev. D, 81, 104035.
[257] Gregory, R., and Laflamme, R. 1993. Phys. Rev. Lett., 70, 2837–2840.
[258] Eggers, J. 1993. Phys. Rev. Lett, 71, 3458.
[259] Sorkin, E. 2004. Phys. Rev. Lett., 93, 031601.
[260] Figueras, P., Murata, K., and Reall, H. S. 2012. JHEP, 1211, 071.
[261] Dias, O. J., et al. 2009. Phys. Rev. D, 80, 111701.
[262] Figueras, P., and Wiseman, T. 2011. Phys. Rev. Lett., 107, 081101.
[263] Tanaka, T. 2003. Prog. Theor. Phys. Suppl., 148, 307–316.
[264] Emparan, R., Fabbri, A., and Kaloper, N. 2002. JHEP, 0208, 043.
[265] Son, D. T., and Starinets, A. O. 2007. Ann. Rev. Nucl. Part. Sci., 57, 95–118.
[266] DeWolfe, O., et al., 2013. arXiv:1304.7794.
[267] Chesler, P. M., and Yaffe, L. G. 2009. Phys. Rev. Lett., 102, 211601.
[268] Chesler, P. M., and Teaney, D., 2011. arXiv:1112.6196.
[269] Chesler, P. M., and Yaffe, L. G. 2011. Phys. Rev. Lett., 106, 021601.
[270] Bjorken, J. 1983. Phys. Rev. D, 27, 140–151.
[271] Luzum, M., and Romatschke, P. 2008. Phys. Rev. C, 78, 034915.
[272] Buchel, A., Lehner, L., and Myers, R. C. 2012. JHEP, 1208, 049.
[273] Buchel, A., et al. 2013. JHEP, 1305, 067.
[274] Buchel, A., Myers, R. C., and van Niekerk, A. 2013. Phys. Rev. Lett., 111, 201602.
[275] Hubeny, V. E., Minwalla, S., and Rangamani, M., 2011. arXiv:1107.5780.
[276] Van Raamsdonk, M. 2008. JHEP, 0805, 106.
[277] Carrasco, F., et al. 2012. Phys. Rev. D, 86, 126006.
[278] Adams, A., Chesler, P. M., and Liu, H., 2013. arXiv:1307.7267.
[279] Bantilan, H., Pretorius, F., and Gubser, S. S. 2012. Phys. Rev. D, 85, 084038.
[280] Green, S. R., Carrasco, F., and Lehner, L., 2013. arXiv:1309.7940.
[281] Berger, B. K. 2002. Living Rev. Rel., 5,1.
[282] Belinskii, V., Lifshitz, E., and Khalatnikov, I. 1972. Zh. Eksp. Teor. Fiz., 62, 1606–1613.
[283] Barrow, J. D., and Tipler, F. J. 1979. Phys. Rept., 56, 371–W2.
[284] Berger, B., et al. 1998. Mod. Phys. Lett., A13, 1565–1574.
[285] Garfinkle, D. 2004. Phys. Rev. Lett., 93, 161101.
[286] Lim, W. C., et al. 2009. Phys. Rev. D, 79, 123526.
[287] Poisson, E., and Israel, W. 1990. Phys. Rev. D, 41, 1796–1809.
[288] Ori, A., and Flanagan, É. É. 1996. Phys. Rev. D, 53, 1754.
[289] Dafermos, M., 2012. arXiv:1201.1797.
[290] Christodoulou, D., and Klainerman, S. 1993. The global nonlinear stability of the Minkowski space. Princeton, NJ: Princeton University Press.
[291] Friedrich, H. 1986. J. Geom. Phys., 3, 101–117.
[292] Dias, O. J., Horowitz, G. T., and Santos, J. E. 2012. Class. Quant. Grav., 29, 194002.
[293] Buchel, A., Liebling, S. L., and Lehner, L. 2013. Phys. Rev. D, 87, 123006.
[294] Dias, O. J., et al. 2012. Class. Quant. Grav., 29, 235019.
[295] Maliborski, M., and Rostworowski, A., 2013. arXiv:1303.3186.
[296] Callan, C. G., et al. 1992. Phys. Rev. D, 45, 1005–1009.
[297] Ashtekar, A., Pretorius, F., and Ramazanoglu, F. M. 2011. Phys. Rev. Lett., 106, 161303.
[298] Ashtekar, A., Pretorius, F., and Ramazanoglu, F. M. 2011. Phys.Rev. D, 83, 044040.
[299] Ramazanoglu, F. M., and Pretorius, F. 2010. Class. Quant. Grav., 27, 245027.
[300] Ashtekar, A., Taveras, V., and Varadarajan, M. 2008. Phys. Rev. Lett., 100, 211302.
[301] Almheiri, A., et al. 2013. JHEP, 1302, 062.
[302] Guth, A. H. 2007. J. Phys., A40, 6811–6826.
[303] Aguirre, A., and Johnson, M. C. 2011. Rept. Prog. Phys., 74, 074901.
[304] Kleban, M. 2011. Class. Quant. Grav., 28, 204008.
[305] Johnson, M. C., Peiris, H. V., and Lehner, L. 2012. Phys. Rev. D, 85, 083516.
[306] Wainwright, C. L., et al., 2013. arXiv:1312.1357.
[307] Buchert, T., and Räsänen, S. 2012. Ann. Rev. Nucl. Particle Sci., 62, 57–79.
[308] Zhao, X., and Mathews, G. J. 2011. Phys. Rev. D, 83, 023524.
[309] Yoo, C.-M., Okawa, H., and Nakao, K.-i. 2013. Phys. Rev. Lett., 111, 161102.
[310] Yoo, C.-M., and Okawa, H. 2014. ArXiv e-prints.
[311] East, W. E., Ramazanoglu, F. M., and Pretorius, F. 2014. Phys. Rev. D, 89, 061503.
[312] Barausse, E., et al. 2013. Phys. Rev. D, 87, 081506.
[313] Chesler, P. M., and Yaffe, L. G. 2015. arXiv:1501.04644.