The Asymptotic Structure of Gravity at Spatial Infinity in Four Spacetime Dimensions

Marc Henneaux ; Cédric Troessaert

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Modern problems of mathematical and theoretical physics, Tome 309 (2020), pp. 141-164.

Résumé

A review of our results on the asymptotic structure of gravity at spatial infinity in four spacetime dimensions is given. Finiteness of the action and integrability of the asymptotic Lorentz boost generators are key criteria that we implement through appropriate boundary conditions. These conditions are “twisted parity conditions,” expressing that the leading order of the asymptotic fields obeys strict parity conditions under the sphere antipodal map up to an improper gauge transformation. The asymptotic symmetries are shown to form the infinite-dimensional Bondi–Metzner–Sachs group, which has a nontrivial action. The charges and their algebra are worked out. The presentation aims at being self-contained and at possessing a pedagogical component.

Bibliographie

[1] Arnowitt R., Deser S., Misner C.W., “The dynamics of general relativity”, Gravitation: An introduction to current research, Ch. 7, ed. by L. Witten, J. Wiley and Sons, New York, 1962, 227–264 ; Gen. Relativ. Gravitation, 40:9 (2008), 1997–2027 ; arXiv: gr-qc/0405109 | MR | MR | Zbl

[2] Ashtekar A., Bombelli L., Reula O., “The covariant phase space of asymptotically flat gravitational fields”, Mechanics, analysis and geometry: 200 years after Lagrange, North-Holland, Amsterdam, 1991, 417–450 | MR

[3] Ashtekar A., Hansen R.O., “A unified treatment of null and spatial infinity in general relativity. I: Universal structure, asymptotic symmetries, and conserved quantities at spatial infinity”, J. Math. Phys., 19:7 (1978), 1542–1566 | MR

[4] Beig R., “Integration of Einstein's equations near spatial infinity”, Proc. R. Soc. London A, 391 (1984), 295–304 | MR | Zbl

[5] Beig R., Schmidt B.G., “Einstein's equations near spatial infinity”, Commun. Math. Phys., 87:1 (1982), 65–80 | MR | Zbl

[6] Benguria R., Cordero P., Teitelboim C., “Aspects of the Hamiltonian dynamics of interacting gravitational gauge and Higgs fields with applications to spherical symmetry”, Nucl. Phys. B, 122:1 (1977), 61–99

[7] Bondi H., van der Burg M.G.J., Metzner A.W.K., “Gravitational waves in general relativity. VII: Waves from axi-symmetric isolated systems”, Proc. R. Soc. London A, 269 (1962), 21–52 | MR | Zbl

[8] Bousso R., Porrati M., “Soft hair as a soft wig”, Classical Quantum Gravity, 34:20 (2017), 204001 ; arXiv: 1706.00436 | MR | Zbl

[9] Brown J.D., Henneaux M., “Central charges in the canonical realization of asymptotic symmetries: An example from three dimensional gravity”, Commun. Math. Phys., 104:2 (1986), 207–226 | MR | Zbl

[10] Bunster C., Portugues R., Cnockaert S., Henneaux M., “Monopoles for gravitation and for higher spin fields”, Phys. Rev. D, 73:10 (2006), 105014 ; arXiv: hep-th/0601222 | MR

[11] Christodoulou D., Klainerman S., The global nonlinear stability of the Minkowski space, Princeton Math. Ser., 41, Princeton Univ. Press, Princeton, NJ, 1993 | MR | Zbl

[12] Dirac P.A.M., “The theory of gravitation in Hamiltonian form”, Proc. R. Soc. London A, 246 (1958), 333–343 | MR | Zbl

[13] Friedrich H., “Gravitational fields near space-like and null infinity”, J. Geom. Phys., 24:2 (1998), 83–163 | MR

[14] Friedrich H., Kánnár J., “Bondi-type systems near spacelike infinity and the calculation of the Newman–Penrose constants”, J. Math. Phys., 41:4 (2000), 2195–2232 ; arXiv: gr-qc/9910077 | MR | Zbl

[15] Friedrich H., Kánnár J., “Calculating asymptotic quantities near space-like and null infinity from Cauchy data”, Ann. Phys., 9:3–5 (2000), 321–330 ; arXiv: gr-qc/9911103 | MR | Zbl

[16] Giddings S.B., Gravitational dressing, soft charges, and perturbative gravitational splitting, E-print, 2019, arXiv: 1903.06160 | MR

[17] Henneaux M., Teitelboim C., “Asymptotically anti-de Sitter spaces”, Commun. Math. Phys., 98:3 (1985), 391–424 | MR | Zbl

[18] Henneaux M., Troessaert C., “BMS group at spatial infinity: The Hamiltonian (ADM) approach”, J. High Energy Phys., 2018:03 (2018), 147 ; arXiv: 1801.03718 | MR | Zbl

[19] Henneaux M., Troessaert C., “Asymptotic symmetries of electromagnetism at spatial infinity”, J. High Energy Phys., 2018:05 (2018), 137 ; arXiv: 1803.10194 [hep-th] | MR | Zbl

[20] Henneaux M., Troessaert C., “Hamiltonian structure and asymptotic symmetries of the Einstein–Maxwell system at spatial infinity”, J. High Energy Phys., 2018:07 (2018), 171 ; arXiv: 1805.11288 | MR | Zbl

[21] Henneaux M., Troessaert C., “Asymptotic structure of a massless scalar field and its dual two-form field at spatial infinity”, J. High Energy Phys., 2019:05 (2019), 147 ; arXiv: 1812.07445 | MR | Zbl

[22] Henneaux M., Troessaert C., “Asymptotic structure of electromagnetism in higher spacetime dimensions”, Phys. Rev. D, 99:12 (2019), 125006 ; arXiv: 1903.04437 | MR | Zbl

[23] Herberthson M., Ludvigsen M., “A relationship between future and past null infinity”, Gen. Relativ. Gravitation, 24:11 (1992), 1185–1193 | MR | Zbl

[24] Misner C.W., “The flatter regions of Newman, Unti, and Tamburino's Generalized Schwarzschild Space”, J. Math. Phys., 4:7 (1963), 924–937 | MR

[25] Regge T., Teitelboim C., “Role of surface integrals in the Hamiltonian formulation of general relativity”, Ann. Phys., 88:1 (1974), 286–318 | MR | Zbl

[26] Sachs R.K., “Gravitational waves in general relativity. VIII: Waves in asymptotically flat space–time”, Proc. R. Soc. London A, 270 (1962), 103–126 | MR | Zbl

[27] Sachs R., “Asymptotic symmetries in gravitational theory”, Phys. Rev., 128:6 (1962), 2851–2864 | MR | Zbl

[28] Slavnov A.A., “Gauge invariant infrared regularization for non-abelian fields”, Phys. Lett. B, 98:1–2 (1981), 57–58 | MR

[29] Strominger A., Lectures on the infrared structure of gravity and gauge theory, E-print, 2017, arXiv: 1703.05448 | MR

[30] Troessaert C., “The BMS4 algebra at spatial infinity”, Classical Quantum Gravity, 35:7 (2018), 074003 ; arXiv: 1704.06223 | MR | Zbl