supertranslations and superrotations
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Supertranslations and superrotations Geoffrey Compre Universit - PowerPoint PPT Presentation

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Supertranslations and superrotations Geoffrey Compre Universit Libre de Bruxelles (ULB) Workshop on Topics in Three Dimensional Gravity, Trieste, March 22, 2016 G. Compre (ULB) 1 / 40 Credits Vacua of the gravitational field,

  1. Supertranslations and superrotations Geoffrey Compère Université Libre de Bruxelles (ULB) Workshop on Topics in Three Dimensional Gravity, Trieste, March 22, 2016 G. Compère (ULB) 1 / 40

  2. Credits “Vacua of the gravitational field”, G.C., J. Long, arXiv :1601.04958 “Classical static final state of collapse with supertranslation memory, G.C., J. Long, arXiv :1602.05197 with inspiration from “Aspects of the BMS/CFT correspondence”, G. Barnich, C. Troessaert, arXiv :1001.1541 “Gravitational Memory, BMS Supertranslations and Soft Theorems, A. Strominger, A. Zhiboedov, arXiv :1411.5745 G. Compère (ULB) 2 / 40

  3. On this talk : Fascinating properties and algebra of symmetries of asymptotically flat spacetimes 4 d ñ 3 d ñ 4 d . Many lessons can be drawn from 3 d to help understand 4 d physics. Interplay between various concepts : asymptotic symmetries, gravitational memory, holography, black holes Tackle classical problems : gravitational collapse, cosmic censorship, black hole information paradox G. Compère (ULB) 3 / 40

  4. Asymptotically flat spacetimes No black hole in 3 d Einstein-positive matter theory. [Ida, 2000] G. Compère (ULB) 4 / 40

  5. Preambule : the BMS 3 and BMS 4 groups The space of solutions to Einstein gravity with “reasonable” asymptotically flat boundary conditions can be expanded close to null infinity in a fixed gauge. ´ du 2 ´ 2 dudr ` r 2 d 2 Ω ` . . . “ ds 2 ´ dv 2 ` 2 dvdr ` r 2 d 2 Ω antipodal ` . . . “ The group of diffeomorphisms which preserve the form of the asymptotic metric, mapping one metric to another but preserving the gauge, are associated with finite and non-trivial canonical charges is the asymptotic symmetry group. Using “reasonable” boundary conditions, the asymptotic symmetry group was found to be the BMS 4 group in 4 d [Bondi, van der Burg, Metzner, 1962] [Sachs, 1962] and the BMS 3 group in 3 d [Ashtekar, Bicak, Schmidt, 1996] G. Compère (ULB) 5 / 40

  6. What reasonable boundary conditions may mean ? 4 d Admit Kerr, gravitational waves and electromagnetic fields Positive energy Allow to describe memory effects [Zeldovich, Polnarev, 1974] [Christodoulou, 1991] Allow to describe a semi-classical S-matrix which obeys all known theorems [Weinberg, 1965] [Cachazo, Strominger, 2014] Allow for small perturbations to decay (non-linear stability) [Christodoulou, Klainerman, 1993] 3 d Admit “appropriate” matter fields Positive energy Flat region can be embedded in AdS 3 G. Compère (ULB) 6 / 40

  7. A translation in Minkowski spacetime p t , x , y , z q B z p t , r , θ, φ q cos θ B r ´ 1 r sin θ B θ p u , r , θ, φ q , retarded time u “ t ´ r ´ cos θ B u ` cos θ B r ´ 1 r sin θ B θ G. Compère (ULB) 7 / 40

  8. The bms 4 algebra bms 4 » so p 3 , 1 q i Supertranslations Supertranslations are either translations or pure supertranslations. Pure supertranslations are (abelian) “higher harmonic angle-dependent translations” T p θ, φ qB u ` 1 2 ∇ 2 T B r ´ 1 1 r pB θ T B θ ` B φ T B φ q ` . . . sin 2 θ The solutions to ∇ 2 p ∇ 2 ` 2 q T “ 0 are the translations. Those are the ℓ “ 0 and ℓ “ 1 spherical harmonics, T “ 1, T “ cos θ , T “ sin θ cos φ , T “ sin θ sin φ . What are supertranslations in the bulk ? G. Compère (ULB) 8 / 40

  9. The extended bms 4 algebra [Barnich, Troessaert, 2010] bms 4 » Superrotations ˚ i Supertranslations ˚ where Vir ˚ ‘ Vir ˚ , Superrotations ˚ » Supertranslations ˚ » Regular supert. ‘ Meromorphic supert. The Lorentz subalgebra so p 3 , 1 q » sl p 2 , R q ‘ sl p 2 , R q Ă Vir ˚ ‘ Vir ˚ is generated by global conformal transformations on the sphere. The rest of the algebra has generators which contain meromorphic functions, with poles on S 2 . G. Compère (ULB) 9 / 40

  10. The extended bms 4 algebra : comments The algebra is not realized as asymptotic symmetry algebra, at least in the standard sense : The Kerr black hole has infinite meromorphic supertranslation momenta. [Barnich, Troessaert, 2010] Minkowski acted upon with a finite superrotation diffeomorphism has negative energy. [G.C., Long, 2016] The superrotations still have a role to play : Superrotation charges are finite and can be non-trivial [Barnich, Troessaert, 2011] [Flanagan, Nichols, 2015] [G.C., Long, 2016] The subleading soft graviton theorem has been related to the Ward identity of the superrotation algebra [Kapec, Lysov, Pasterski, Strominger, 2014] [Campiglia, Laddha, 2015] G. Compère (ULB) 10 / 40

  11. The bms 3 algebra In 3 d : Poincaré » so p 2 , 1 q i R 3 . The Poincaré algebra is i r R m , R n s “ p m ´ n q R m ` n , i r R m , T n s “ p m ´ n q T m ` n , i r T m , T n s “ m , n “ ´ 1 , 0 , 1 . 0 , 1+2 Translations T 0 “ B t ; T 1 ` T ´ 1 “ B x , i p T 1 ´ T ´ 1 q “ B y 1+2 Lorentz transformations R 0 “ B φ ; R 1 ` R ´ 1 , i p R 1 ´ R ´ 1 q The algebra can be promoted as an asymptotic symmetry algebra of asymptotically flat spacetimes, for n P Z : » Superrotations p R n q i Supertranslations p T n q bms 3 z » i u p 1 q Virasoro [Ashtekar, Bicak, Schmidt, 1996] [Barnich, G.C., 2007] The BMS 3 group is Diff p S 1 q ˙ Vect p S 1 q [Barnich, Oblak, 2014] . G. Compère (ULB) 11 / 40

  12. The bms 3 algebra : comments Limit from Brown-Henneaux In large ℓ Ñ 8 limit, AdS 3 Ñ Mink 3 . The exact symmetries are contracted as so p 2 , 2 q Ñ iso p 2 , 1 q . The asymptotic symmetries with Brown-Henneaux/Dirichlet boundary conditions are contracted as Vir ‘ Vir Ñ Superrotations i Supertranslations [Barnich, G.C., 2007] Isomorphism The bms 3 algebra is also isomorphic to the infinite-dimensional extension of the 2 d Galilean conformal algebra. [Bagchi, Gopakumar, 2009] G. Compère (ULB) 12 / 40

  13. 4 d supertranslations and memories After the passage of either gravitational waves or null matter between two detectors placed in the asymptotic null region, the detectors generically acquire a finite relative displacement and a finite time shift. This is the memory effect . Historically, it is refered to as the linear memory effect for null matter [Zeldovich, Polnarev, 1974] and the non-linear memory or Christodoulou effect for gravitational waves [Christodoulou, 1991] . Memory effects follow from the existence of the supertranslation field C p θ, φ q which is effectively shifted by a supertranslation after the passage of radiation as [Geroch, Winicour, 1981] δ T C p θ, φ q “ T p θ, φ q . Memory effects are a 2.5PN General Relativity effect. [Damour, Blanchet, 1988] Memory effects cannot be detected by LIGO. G. Compère (ULB) 13 / 40

  14. More precisely, supertranslation memories follow from an angle-dependent energy conservation law deduced from Einstein’s equations integrated over a finite retarded time interval of I ` : [Strominger, Zhiboedov, 2014] ż u 1 4 ∇ 2 p ∇ 2 ` 2 qp C | u 2 ´ C | u 1 q “ m | u 2 ´ m | u 1 ` ´ 1 duT uu , u 2 4 N zz N zz ` 4 π G lim T uu ” 1 r Ñ8 r r 2 T matter s . uu The supertranslation shift can be constructed from the radiation flux history. It allows to compute the shift of the geodesic deviation vector s A , A “ θ, φ s A | u 2 ´ s A | u 1 „ 1 r B A B B p C | u 2 ´ C | u 1 q s B This is a classical effect of Einstein gravity, O p � 0 q . G. Compère (ULB) 14 / 40

  15. What is the supertranslation field in the bulk ? In 3 d , part of the answer is the phase space of analytic solutions to vacuum Einstein gravity with Dirichlet boundary conditions : [Barnich, Troessaert, 2010] ´ ¯ ds 2 “ Θ p φ q du 2 ´ 2 dudr ` 2 Ξ p φ q ` u 2 B φ Θ p φ q dud φ ` r 2 d φ 2 . The transformation laws of Θ p φ q under bms 3 is δ T , R Θ “ R B φ Θ ` 2 B φ R Θ ´ 2 B 3 φ R This is the coadjoint representation of the Virasoro algebra. We deduce that Θ p φ q is the superrotation field itself plus a zero mode. The zero mode is the mass (a conical defect). In order to concentrate on the supertranslation field, we set Θ “ ´ 1 p no conical defect q . This sets to the supertranslation charge to 0 (rest frame). G. Compère (ULB) 15 / 40

  16. The transformation law of Ξ p φ q under a supertranslation is then “ ´B φ T ´ B 3 δ T Ξ φ T . We deduce that Ξ p φ q is a composite field in terms of the supertranslation field C p φ q plus a zero mode Ξ p φ q “ 4 GJ ´ B φ p 1 ` B 2 φ q C , δ T C “ T . The zero mode is attributed to the spin of a massless particle. It creates a dislocation responsible for closed timelike curves. So we set J “ 0. The metric becomes ds 2 “ ´ du 2 ´ 2 dud p r ` C p φ q ` B 2 φ C p φ qq ` r 2 d φ 2 . G. Compère (ULB) 16 / 40

  17. ds 2 “ ´ du 2 ´ 2 dud p r ` C p φ q ` B 2 φ C p φ qq ` r 2 d φ 2 . We switch to static coordinates ρ “ r ` B 2 φ C p φ q ` C p φ q ´ C p 0 q , t “ u ` ρ . The shift of C by its zero mode ensures that the space coordinate ρ is not affected by time shifts. The metric becomes [G.C., Long, 2016] ds 2 “ ´ dt 2 ` d ρ 2 ` p ρ ´ ρ SH p φ qq 2 d φ 2 . In the rest frame, supertranslations only act spatially, except the zero mode which is a time translation. Coordinates break down at the supertranslation horizon ρ “ ρ SH p C q ” B 2 φ C p φ q ` C p φ q ´ C p 0 q . G. Compère (ULB) 17 / 40

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