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Asymptotically Flat Spacetimes

  • Geoffrey Compère
Chapter
Part of the Lecture Notes in Physics book series (LNP, volume 952)

Abstract

For the next lecture, we consider four-dimensional asymptotically flat spacetimes, which are the solutions of General Relativity with localised energy-momentum sources. We will start with a review of the work of Penrose on the conformal compactification of asymptotically flat spacetimes in order to get a global view on the asymptotic structure. We will then concentrate on the properties of radiative fields by reviewing the work of van der Burg, Bondi, Metzner and Sachs of 1962. One may think at first that the group of asymptotic symmetries of radiative spacetimes is the Poincaré group, but a larger group appears, the BMS group which contains so-called supertranslations. Additional symmetries, known as superrotations, also play a role and we shall briefly discuss them too. Finally, we will give some comments on the scattering problem in General Relativity. We will show that the extended asymptotic group gives conserved quantities once junction conditions are fixed at spatial infinity

References

  1. 1.
    G. Barnich, C. Troessaert, Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited. Phys. Rev. Lett. 105, 111103 (2010). arXiv:0909.2617 [gr-qc]; http://dx.doi.org/10.1103/PhysRevLett.105.111103
  2. 2.
    G. Barnich, C. Troessaert, Aspects of the BMS/CFT correspondence. J. High Energy Phys. 1005, 062 (2010). arXiv:1001.1541 [hep-th]; http://dx.doi.org/10.1007/JHEP05(2010)062
  3. 3.
    G. Barnich, C. Troessaert, BMS charge algebra. J. High Energy Phys. 12, 105 (2011). arXiv:1106.0213 [hep-th]; http://dx.doi.org/10.1007/JHEP12(2011)105
  4. 4.
    L. Bieri, P. Chen, S.-T. Yau, Null asymptotics of solutions of the Einstein-Maxwell equations in general relativity and gravitational radiation. Adv. Theor. Math. Phys. 15(4), 1085–1113 (2011). arXiv:1011.2267 [math.DG]; http://dx.doi.org/10.4310/ATMP.2011.v15.n4.a5 MathSciNetCrossRefGoogle Scholar
  5. 5.
    L. Blanchet, T. Damour, Tail transported temporal correlations in the dynamics of a gravitating system. Phys. Rev. D37, 1410 (1988). http://dx.doi.org/10.1103/PhysRevD.37.1410 ADSGoogle Scholar
  6. 6.
    L. Blanchet, T. Damour, Hereditary effects in gravitational radiation. Phys. Rev. D46, 4304–4319 (1992). http://dx.doi.org/10.1103/PhysRevD.46.4304 ADSMathSciNetGoogle Scholar
  7. 7.
    H. Bondi, M.G.J. van der Burg, A.W.K. Metzner, Gravitational waves in general relativity. 7. Waves from axisymmetric isolated systems. Proc. R. Soc. Lond. A269, 21–52 (1962). http://dx.doi.org/10.1098/rspa.1962.0161
  8. 8.
    R. Bousso, M. Porrati, Soft hair as a soft wig. Class. Quant. Grav. 34(20), 204001 (2017). arXiv:1706.00436 [hep-th]; http://dx.doi.org/10.1088/1361-6382/aa8be2 ADSMathSciNetCrossRefGoogle Scholar
  9. 9.
    F. Cachazo, A. Strominger, Evidence for a new soft graviton theorem (2014, unpublished). arXiv:1404.4091 [hep-th]
  10. 10.
    M. Campiglia, A. Laddha, Asymptotic symmetries and subleading soft graviton theorem. Phys. Rev. D90(12), 124028 (2014). arXiv:1408.2228 [hep-th]; http://dx.doi.org/10.1103/PhysRevD.90.124028
  11. 11.
    M. Campiglia, A. Laddha, New symmetries for the Gravitational S-matrix. J. High Energy Phys. 04, 076 (2015). arXiv:1502.02318 [hep-th]; http://dx.doi.org/10.1007/JHEP04(2015)076
  12. 12.
    D. Christodoulou, Nonlinear nature of gravitation and gravitational wave experiments. Phys. Rev. Lett. 67, 1486–1489 (1991). http://dx.doi.org/10.1103/PhysRevLett.67.1486 ADSMathSciNetCrossRefGoogle Scholar
  13. 13.
    D. Christodoulou, S. Klainerman, The Global nonlinear stability of the Minkowski space. Princeton Legacy Library, 1993Google Scholar
  14. 14.
    G. Compère, A. Fiorucci, Asymptotically flat spacetimes with BMS3 symmetry. Class. Quant. Grav. 34(20), 204002 (2017). arXiv:1705.06217 [hep-th]; http://dx.doi.org/10.1088/1361-6382/aa8aad ADSMathSciNetCrossRefGoogle Scholar
  15. 15.
    G. Compère, J. Long, Vacua of the gravitational field. J. High Energy Phys. 07, 137 (2016). arXiv:1601.04958 [hep-th]; http://dx.doi.org/10.1007/JHEP07(2016)137
  16. 16.
    G. Compère, J. Long, Classical static final state of collapse with supertranslation memory. Class. Quant. Grav. 33(19), 195001 (2016). arXiv:1602.05197 [gr-qc]; http://dx.doi.org/10.1088/0264-9381/33/19/195001 ADSMathSciNetCrossRefGoogle Scholar
  17. 17.
    G. Compère, A. Fiorucci, R. Ruzziconi, Superboost transitions, refraction memory and super-Lorentz charge algebra. J. High Energy Phys. 2018, 200 (2018). https://doi.org/10.1007/JHEP11(2018)200 ADSCrossRefGoogle Scholar
  18. 18.
    J. de Boer, S.N. Solodukhin, A holographic reduction of Minkowski space-time. Nucl. Phys. B665, 545–593 (2003). arXiv:hep-th/0303006 [hep-th]; http://dx.doi.org/10.1016/S0550-3213(03)00494-2
  19. 19.
    E.E. Flanagan, D.A. Nichols, Conserved charges of the extended Bondi-Metzner-Sachs algebra. Phys. Rev. D95(4), 044002 (2017). arXiv:1510.03386 [hep-th]; http://dx.doi.org/10.1103/PhysRevD.95.044002
  20. 20.
    S.W. Hawking, M.J. Perry, A. Strominger, Soft hair on black holes. Phys. Rev. Lett. 116(23), 231301 (2016). arXiv:1601.00921 [hep-th]; http://dx.doi.org/10.1103/PhysRevLett.116.231301
  21. 21.
    T. He, V. Lysov, P. Mitra, A. Strominger, BMS supertranslations and Weinberg’s soft graviton theorem. J. High Energy Phys. 05, 151 (2015) . arXiv:1401.7026 [hep-th]. http://dx.doi.org/10.1007/JHEP05(2015)151
  22. 22.
    M. Henneaux, C. Troessaert, BMS group at spatial infinity: the Hamiltonian (ADM) approach. J. High Energy Phys. 03, 147 (2018). arXiv:1801.03718 [gr-qc]. http://dx.doi.org/10.1007/JHEP03(2018)147
  23. 23.
    P.D. Lasky, E. Thrane, Y. Levin, J. Blackman, Y. Chen, Detecting gravitational-wave memory with LIGO: implications of GW150914. Phys. Rev. Lett. 117(6), 061102 (2016). arXiv:1605.01415 [astro-ph.HE]; http://dx.doi.org/10.1103/PhysRevLett.117.061102
  24. 24.
    M. Mirbabayi, M. Porrati, Dressed hard states and black hole soft hair. Phys. Rev. Lett. 117(21), 211301 (2016). arXiv:1607.03120 [hep-th]. http://dx.doi.org/10.1103/PhysRevLett.117.211301
  25. 25.
    D.A. Nichols, Spin memory effect for compact binaries in the post-Newtonian approximation. Phys. Rev. D95(8), 084048 (2017). arXiv:1702.03300 [gr-qc]; http://dx.doi.org/10.1103/PhysRevD.95.084048
  26. 26.
    S. Pasterski, A. Strominger, A. Zhiboedov, New gravitational memories. J. High Energy Phys. 12, 053 (2016). arXiv:1502.06120 [hep-th]; http://dx.doi.org/10.1007/JHEP12(2016)053
  27. 27.
    R. Penrose, The geometry of impulsive gravitational waves, in General Relativity: Papers in Honour of J.L. Synge, ed. by L. O’Raifeartaigh (Clarendon Press, Oxford, 1972), pp. 101–115Google Scholar
  28. 28.
    R.K. Sachs, Gravitational waves in general relativity. 8. Waves in asymptotically flat space-times. Proc. R. Soc. Lond. A270, 103–126 (1962). http://dx.doi.org/10.1098/rspa.1962.0206
  29. 29.
    A. Strominger, On BMS invariance of gravitational scattering. J. High Energy Phys. 07, 152 (2014). arXiv:1312.2229 [hep-th]; http://dx.doi.org/10.1007/JHEP07(2014)152
  30. 30.
    A. Strominger, Black hole information revisited (2017, unpublished). arXiv:1706.07143 [hep-th]
  31. 31.
    A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory (Princeton University Press, Princeton, 2018). ISBN 978-0-691-17950-6CrossRefGoogle Scholar
  32. 32.
    A. Strominger, A. Zhiboedov, Gravitational memory, BMS supertranslations and soft theorems. J. High Energy Phys. 01, 086 (2016). arXiv:1411.5745 [hep-th]; http://dx.doi.org/10.1007/JHEP01(2016)086
  33. 33.
    A. Strominger, A. Zhiboedov, Superrotations and black hole pair creation. Class. Quant. Grav. 34(6), 064002 (2017). arXiv:1610.00639 [hep-th]; http://dx.doi.org/10.1088/1361-6382/aa5b5f ADSMathSciNetCrossRefGoogle Scholar
  34. 34.
    K.S. Thorne, Gravitational-wave bursts with memory: the Christodoulou effect. Phys. Rev. D45(2), 520–524 (1992) . http://dx.doi.org/10.1103/PhysRevD.45.520 ADSMathSciNetGoogle Scholar
  35. 35.
    C. Troessaert, The BMS4 algebra at spatial infinity. arXiv:1704.06223 [hep-th]
  36. 36.
    S. Weinberg, Infrared photons and gravitons. Phys. Rev. 140, B516–B524 (1965) . http://dx.doi.org/10.1103/PhysRev.140.B516 ADSMathSciNetCrossRefGoogle Scholar
  37. 37.
    Y.B. Zel’dovich, A.G. Polnarev, Radiation of gravitational waves by a cluster of superdense stars. Sov. Astron. 18, 17 (1974)ADSGoogle Scholar

Copyright information

© The Author(s), under exclusive license to Springer Nature Switzerland AG 2019

Authors and Affiliations

  • Geoffrey Compère
    • 1
  1. 1.Physique théorique mathématiqueUniversité Libre de BruxellesBrusselsBelgium

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