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
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
Strictly speaking it implies that Ψ = 0 up to the lowest l = 0, 1 spherical harmonics. However, these harmonics are exactly zero modes of the differential operator ε C(A D B) D C Ψ defined in C AB. Therefore we can set them to zero.
- 2.
Similarly to the construction of the AdS 3 phase space of stationary field configurations with boundary fields turned on that was described in (2.57) or for the asymptotically flat analogue (2.73), one can construct gravitational vacua and Schwarzschild black holes that carry a supertranslation field, see [15, 16].
- 3.
- 4.
The attentive reader might already notice that such an observable is not clearly the difference between a quantity in the initial and final state. Instead \(\int _{u_i}^{u_f} du \: \Psi \) is non-local in retarded time! Therefore, though it bears analogy with the displacement memory effect, the spin memory effect is not (yet?) proven to be a memory effect at the first place! For a discussion of memory effects associated with Diff(S 2) symmetries, see [17].
- 5.
Shortcut: just formally replace u →−v everywhere while leaving r unchanged.
References
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
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
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
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
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
L. Blanchet, T. Damour, Hereditary effects in gravitational radiation. Phys. Rev. D46, 4304–4319 (1992). http://dx.doi.org/10.1103/PhysRevD.46.4304
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
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
F. Cachazo, A. Strominger, Evidence for a new soft graviton theorem (2014, unpublished). arXiv:1404.4091 [hep-th]
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
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
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
D. Christodoulou, S. Klainerman, The Global nonlinear stability of the Minkowski space. Princeton Legacy Library, 1993
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
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
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
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
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
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
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
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
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
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
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
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
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
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. 101–115
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
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
A. Strominger, Black hole information revisited (2017, unpublished). arXiv:1706.07143 [hep-th]
A. Strominger, Lectures on the Infrared Structure of Gravity and Gauge Theory (Princeton University Press, Princeton, 2018). ISBN 978-0-691-17950-6
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
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
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
C. Troessaert, The BMS4 algebra at spatial infinity. arXiv:1704.06223 [hep-th]
S. Weinberg, Infrared photons and gravitons. Phys. Rev. 140, B516–B524 (1965) . http://dx.doi.org/10.1103/PhysRev.140.B516
Y.B. Zel’dovich, A.G. Polnarev, Radiation of gravitational waves by a cluster of superdense stars. Sov. Astron. 18, 17 (1974)
Author information
Authors and Affiliations
Rights and permissions
Copyright information
© 2019 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Compère, G. (2019). Asymptotically Flat Spacetimes. In: Advanced Lectures on General Relativity. Lecture Notes in Physics, vol 952. Springer, Cham. https://doi.org/10.1007/978-3-030-04260-8_3
Download citation
DOI: https://doi.org/10.1007/978-3-030-04260-8_3
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-04259-2
Online ISBN: 978-3-030-04260-8
eBook Packages: Physics and AstronomyPhysics and Astronomy (R0)