Abstract
The geometry of twisted null geodesic congruences in gravitational plane wave spacetimes is explored, with special focus on homogeneous plane waves. The rôle of twist in the relation of the Rosen coordinates adapted to a null congruence with the fundamental Brinkmann coordinates is explained and a generalised form of the Rosen metric describing a gravitational plane wave is derived. The Killing vectors and isometry algebra of homogeneous plane waves (HPWs) are described in both Brinkmann and twisted Rosen form and used to demonstrate the coset space structure of HPWs. The van Vleck-Morette determinant for twisted congruences is evaluated in both Brinkmann and Rosen descriptions. The twisted null congruences of the Ozsváth-Schücking, ‘anti-Mach’ plane wave are investigated in detail. These developments provide the necessary geometric toolkit for future investigations of the rôle of twist in loop effects in quantum field theory in curved spacetime, where gravitational plane waves arise generically as Penrose limits; in string theory, where they are important as string backgrounds; and potentially in the detection of gravitational waves in astronomy.
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References
R. Penrose, A remarkable property of plane waves in general relativity, Rev. Mod. Phys. 37 (1965) 215 [INSPIRE].
R. Penrose, Any space-time has a plane wave as a limit, in Differential geometry and relativity: a volume in honour of André Lichnerowicz on his 60th birthday, Springer, Dordrecht The Netherlands, (1976), pg. 271.
M. Blau, D. Frank and S. Weiss, Fermi coordinates and Penrose limits, Class. Quant. Grav. 23 (2006) 3993 [hep-th/0603109] [INSPIRE].
T.J. Hollowood and G.M. Shore, The refractive index of curved spacetime: the fate of causality in QED, Nucl. Phys. B 795 (2008) 138 [arXiv:0707.2303] [INSPIRE].
T.J. Hollowood and G.M. Shore, The causal structure of QED in curved spacetime: analyticity and the refractive index, JHEP 12 (2008) 091 [arXiv:0806.1019] [INSPIRE].
T.J. Hollowood, G.M. Shore and R.J. Stanley, The refractive index of curved spacetime II: QED, Penrose limits and black holes, JHEP 08 (2009) 089 [arXiv:0905.0771] [INSPIRE].
T.J. Hollowood and G.M. Shore, The effect of gravitational tidal forces on renormalized quantum fields, JHEP 02 (2012) 120 [arXiv:1111.3174] [INSPIRE].
T.J. Hollowood and G.M. Shore, Causality violation, gravitational shockwaves and UV completion, JHEP 03 (2016) 129 [arXiv:1512.04952] [INSPIRE].
T.J. Hollowood and G.M. Shore, Causality, renormalizability and ultra-high energy gravitational scattering, J. Phys. A 49 (2016) 215401 [arXiv:1601.06989] [INSPIRE].
J.I. McDonald and G.M. Shore, Radiatively-induced gravitational leptogenesis, Phys. Lett. B 751 (2015) 469 [arXiv:1508.04119] [INSPIRE].
J.I. McDonald and G.M. Shore, Leptogenesis from loop effects in curved spacetime, JHEP 04 (2016) 030 [arXiv:1512.02238] [INSPIRE].
J.I. McDonald and G.M. Shore, Leptogenesis and gravity: baryon asymmetry without decays, Phys. Lett. B 766 (2017) 162 [arXiv:1604.08213] [INSPIRE].
G. Papadopoulos, J.G. Russo and A.A. Tseytlin, Solvable model of strings in a time dependent plane wave background, Class. Quant. Grav. 20 (2003) 969 [hep-th/0211289] [INSPIRE].
M. Blau, M. O’Loughlin, G. Papadopoulos and A.A. Tseytlin, Solvable models of strings in homogeneous plane wave backgrounds, Nucl. Phys. B 673 (2003) 57 [hep-th/0304198] [INSPIRE].
M. Blau and M. O’Loughlin, Homogeneous plane waves, Nucl. Phys. B 654 (2003) 135 [hep-th/0212135] [INSPIRE].
C. Duval, G.W. Gibbons, P.A. Horvathy and P.-M. Zhang, Carroll symmetry of plane gravitational waves, Class. Quant. Grav. 34 (2017) 175003 [arXiv:1702.08284] [INSPIRE].
P.-M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, The memory effect for plane gravitational waves, Phys. Lett. B 772 (2017) 743 [arXiv:1704.05997] [INSPIRE].
P.-M. Zhang, C. Duval, G.W. Gibbons and P.A. Horvathy, Soft gravitons & the memory effect for plane gravitational waves, arXiv:1705.01378 [INSPIRE].
M. Blau, J.M. Figueroa-O’Farrill, C. Hull and G. Papadopoulos, A new maximally supersymmetric background of IIB superstring theory, JHEP 01 (2002) 047 [hep-th/0110242] [INSPIRE].
M. Blau and S. Weiss, Penrose limits versus string expansions, Class. Quant. Grav. 25 (2008) 125014 [arXiv:0710.3480] [INSPIRE].
M. Blau, M. Borunda, M. O’Loughlin and G. Papadopoulos, Penrose limits and space-time singularities, Class. Quant. Grav. 21 (2004) L43 [hep-th/0312029] [INSPIRE].
M. Blau, M. Borunda, M. O’Loughlin and G. Papadopoulos, The universality of Penrose limits near space-time singularities, JHEP 07 (2004) 068 [hep-th/0403252] [INSPIRE].
M. Cahen and N. Wallach, Lorentzian symmetric spaces, Bull. Amer. Math. Soc. 76 (1970) 585.
I. Ozsváth and E. Schücking, An anti-Mach metric, in Recent Developments in General Relativity, Pergamon Press, Oxford U.K., (1962), pg. 339.
D. Sarma, M. Patgiri and F.U. Ahmed, Causality violation in plane wave spacetimes, arXiv:1203.6173 [INSPIRE].
D. Sarma, M. Patgiri and F.U. Ahmed, Pure radiation metric with stable closed timelike curves, Gen. Rel. Grav. 46 (2014) 1633 [INSPIRE].
D. Marolf and S.F. Ross, Plane waves: to infinity and beyond!, Class. Quant. Grav. 19 (2002) 6289 [hep-th/0208197] [INSPIRE].
H. Stephani, D. Kramer, M.A.H. MacCallum, C. Hoenselaers and E. Herlt, Exact solutions of Einstein’s field equations, 2nd edition, Cambridge University Press, Cambridge U.K., (2003) [INSPIRE].
J.B. Griffiths and J. Podolsky, Exact space-times in Einstein’s general relativity, 1st edition, Cambridge University Press, Cambridge U.K., (2009) [INSPIRE].
J.B. Griffiths, Colliding plane waves in general relativity, Clarendon Press, Oxford U.K., (1991) [INSPIRE].
G.W. Gibbons, Quantized fields propagating in plane wave space-times, Commun. Math. Phys. 45 (1975) 191 [INSPIRE].
A.I. Harte, Strong lensing, plane gravitational waves and transient flashes, Class. Quant. Grav. 30 (2013) 075011 [arXiv:1210.1449] [INSPIRE].
A.I. Harte, Optics in a nonlinear gravitational plane wave, Class. Quant. Grav. 32 (2015) 175017 [arXiv:1502.03658] [INSPIRE].
S. Chandrasekhar, The mathematical theory of black holes, Clarendon Press, Oxford U.K., (1985) [INSPIRE].
D.G. Boulware and L.S. Brown, Symmetric space scalar field theory, Annals Phys. 138 (1982) 392 [INSPIRE].
G.M. Shore, Geometry of supersymmetric σ models, Nucl. Phys. B 320 (1989) 202 [INSPIRE].
G.M. Shore, Symmetry restoration and the background field method in gauge theories, Annals Phys. 137 (1981) 262 [INSPIRE].
J.H. Van Vleck, The correspondence principle in the statistical interpretation of quantum mechanics, Proc. Nat. Acad. Sci. 14 (1928) 178 [INSPIRE].
W. Pauli, Selected topics in field quantization, in Pauli lectures 6, MIT Press, U.S.A., (1973), pg. 161.
C. Morette, On the definition and approximation of Feynman’s path integrals, Phys. Rev. 81 (1951) 848 [INSPIRE].
L. Van Hove, Sur certaines représentations unitaires d’un groupe infini de transformations (in French), in Academie Royale de Belgique, classe des sciences, no. 1618, Tome XXVI, Fascicule 6, Belgium, (1951) [INSPIRE].
M. Visser, Van Vleck determinants: geodesic focusing and defocusing in Lorentzian space-times, Phys. Rev. D 47 (1993) 2395 [hep-th/9303020] [INSPIRE].
E. Poisson, A. Pound and I. Vega, The motion of point particles in curved spacetime, Living Rev. Rel. 14 (2011) 7 [arXiv:1102.0529] [INSPIRE].
P. Nurowski and J. Tafel, New algebraically special solutions of the Einstein-Maxwell equations, Class. Quant. Grav. 9 (1992) 2069 [INSPIRE].
E.T. Newman, Maxwell fields and shear free null geodesic congruences, Class. Quant. Grav. 21 (2004) 3197 [gr-qc/0402056] [INSPIRE].
W. Davidson, On twisting pure radiation and Einstein-Maxwell fields, Adv. Studies Theor. Phys. 5 (2011) 315.
Virgo and LIGO Scientific collaborations, B.P. Abbott et al., Observation of gravitational waves from a binary black hole merger, Phys. Rev. Lett. 116 (2016) 061102 [arXiv:1602.03837] [INSPIRE].
Virgo and LIGO Scientific collaborations, B.P. Abbott et al., GW 151226: observation of gravitational waves from a 22-solar-mass binary black hole coalescence, Phys. Rev. Lett. 116 (2016) 241103 [arXiv:1606.04855] [INSPIRE].
P. Amaro-Seoane et al., Low-frequency gravitational-wave science with eLISA/NGO, Class. Quant. Grav. 29 (2012) 124016 [arXiv:1202.0839] [INSPIRE].
A. Loeb and D. Maoz, Using atomic clocks to detect gravitational waves, arXiv:1501.00996 [INSPIRE].
S. Kolkowitz, I. Pikovski, N. Langellier, M.D. Lukin, R.L. Walsworth and J. Ye, Gravitational wave detection with optical lattice atomic clocks, Phys. Rev. D 94 (2016) 124043 [arXiv:1606.01859] [INSPIRE].
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Shore, G.M. A new twist on the geometry of gravitational plane waves. J. High Energ. Phys. 2017, 39 (2017). https://doi.org/10.1007/JHEP09(2017)039
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DOI: https://doi.org/10.1007/JHEP09(2017)039