Topics in asymptotically flat gravity in 3 and 4 dimensions Glenn - PowerPoint PPT Presentation

Higher Spin Gravity The Erwin Schrödinger International Institute for Mathematical Physics April 13, 2012 Topics in asymptotically flat gravity in 3 and 4 dimensions Glenn Barnich Physique théorique et mathématique Université Libre de Bruxelles & International Solvay Institutes

Overview 3d AdS gravity in BMS gauge 3d flat gravity as a modified Penrose limit G. B., A. Gomberoff, and H. Gonzalez, “The flat limit of three dimensional asymptotically anti-de Sitter spacetimes.” to appear. “Higher spin like” infinite-dimensional extension of 3d flat gravity G. B., A. Garbarz, G. Giribet, and M. Leston, “A Chern-Simons action for the Virasoro algebra.” in preparation. 4d flat gravity, null infinity: symmetries & charges G. B., C. Troessaert, “Symmetries of asymptotically flat 4 dimensional spacetimes at null infinity revisited,” Phys. Rev. Lett. 105 (2010) 111103, 0909.2617 . “Aspects of the BMS/CFT correspondence,” JHEP 05 (2010) 062, 1001.1541 . “BMS charge algebra,” JHEP 1112 (2011) 105, 1106.0213 . G. B., P.-H. Lambert, “A note on the Newman-Unti group,” 1102.0589 .

FG gauge AdS3 � ⇥ Fefferman-Graham gauge l 2 0 Λ = − 1 r 2 g µ ν = l 2 0 g AB 2d metric γ AB ( x C ) + O (1) g AB = r 2 ¯ t, φ r flat metric on the cylinder existence of general solution Ξ ++ = Ξ ++ ( x + ) , Ξ −− = Ξ −− ( x − ) integration “constants” g AB dx A dx B = − ( r 2 + l 4 r 2 Ξ ++ Ξ −− ) dx + dx − + l 2 Ξ ++ ( dx + ) 2 + l 2 Ξ −− ( dx − ) 2 , Ξ ±± = 2 G ( M ± J M = − 1 BTZ black hole AdS3 space l ) 8 G, J = 0 cannot be taken naively in these coordinates

AdS3 BMS gauge conformal boundary for flat case: null infinity BMS gauge same gauge for asymptotically flat and AdS3 spacetimes ds 2 = − du 2 − 2 dudr + r 2 d φ 2 Minkowski AdS3 fall-offs

AdS3 Asymptotic symmetries L ξ g rr = 0 = L ξ g r φ , L ξ g φφ = 0 , asymptotic symmetries L ξ g ur = O ( r − 1 ) , L ξ g u φ = O (1) , L ξ g uu = O (1) general (exact) solution linear representation of conformal algebra in bulk spacetime modified bracket

AdS3 Solution space and conformal properties general solution to EOM asymptotic symmetries transform solutions into solutions conformal transformation properties

AdS3 Charge algebra surface charge generators Dirac bracket algebra modes

AdS3 Charge fields on the plane conventional normalization: mapping to the plane:

Penrose limit Generalities action scaling most general solution solution with flat space solution Penrose rescaling of coordinates limit : null orbifold

Penrose limit modified scaling alternative scaling well defined limit if limiting metric

3d flat Charge algebra contraction appropriate combination for the limit Virasoro algebra contracts to ∪ iso (2 , 1) relation to AdS 3 so (2 , 2) → iso (2 , 1) similar to contraction between Virasoro factor: centrally non extended superrotations

3d flat Charge fields normalized fields Minkowski background

AdS3 & 3d flat Zero modes zero mode solutions in both cases M M black holes cosmological solutions J J angular defects angular defects angular excess angular excess (a) (b)

3d flat CS extension Chern-Simons for Virasoro 3d flat gravity, Chern-Simons formulation invariant metric always exists for Virasoro CS co-adjoint vs adjoint extension of 3d flat gravity

3d flat CS extension Deformations cosmological constant same inner product invariance of metric implies completely skew, extended theory Jacobi implies invariant under co-adjoint action no such tensor, AdS deformation does not survive extension “exotic deformation” survives on its own related to to be studied further: asymptotics, boundary theory, solutions, 1-loop effects ...

BMS4/CFT2 Asymptotically flat spacetimes  − e − 2 β  0 0 BMS ansatz g µ ν = − e − 2 β − V r e − 2 β − U B e − 2 β   − U A e − 2 β g AB 0 ⇢ θ , φ ζ = cot θ x A = r u 2 e i φ ζ , ¯ ζ   − 1 − 1 0 0 − 1 0 0 0   η µ ν = Minkowski   u = t − r r 2 0 0 0   r 2 sin 2 θ 0 0 0 g AB dx A dx B = r 2 ¯ γ AB dx A dx B + O ( r ) 0 γ AB dx A dx B = d θ 2 + sin 2 θ d φ 2 γ AB = e 2 ϕ Sachs: unit sphere 0 γ AB ¯ ζ = e i φ cot θ γ AB dx A dx B = e 2 e Riemann sphere ϕ d ζ d ¯ 2 , ζ ¯ ζ ) = 1 d θ 2 + sin 2 θ d φ 2 = P − 2 d ζ d ¯ P ( ζ , ¯ 2(1 + ζ ¯ ζ ) , ζ , ϕ = ϕ − ln P ˜ det g AB = r 4 4 e 4 e determinant condition ϕ V/r = − 1 U A = O ( r − 2 ) , ¯ β = O ( r − 2 ) , R + O ( r − 1 ) fall-off conditions 2

BMS4/CFT2 Asymptotic symmetries L ξ g AB g AB = 0 , asymptotic symmetries L ξ g rr = 0 , L ξ g rA = 0 , L ξ g ur = O ( r − 2 ) , L ξ g uu = O ( r − 1 ) L ξ g uA = O (1) , L ξ g AB = O ( r ) , ⇤ u ϕ + 1 T + 1 ξ u = f, � ˙ ⇒ f = e ϕ � ⇥ du ⇥ e � ϕ ψ f = f ˙ , 2 ψ ⇐ ⌅ 2 ⇤ ⇧ ⇥ ξ A = Y A + I A , I A = − f ,B 0 general solution dr � ( e 2 β g AB ) , r ξ r = − 1 D A ξ A − f ,B U B + 2 f ∂ u ϕ ) , 2 r ( ¯ D A Y A ψ = ¯ ⌅ ⇥ Y A = Y A ( x B ) conformal Killing vectors of the sphere T = T ( x B ) generators for supertranslations spacetime vectors with modified bracket algebra bms 4 form linear representation of [( Y 1 , T 1 ) , ( Y 2 , T 2 )] = ( � Y , � T ) Sachs 1962 � Y A Y B 1 ∂ B Y A 2 − Y B 1 ∂ B Y A = 2 , 2 ∂ A T 1 + 1 � Y A 1 ∂ A T 2 − Y A 2 ( T 1 ∂ A Y A 2 − T 2 ∂ A Y A = 1 ) T SL (2 , C ) / Z 2 ' SO (3 , 1) standard GR choice: restrict to globally well-defined transformations Y A generators of Lorentz algebra

BMS4/CFT2 New proposal CFT choice : allow for meromorphic functions on the Riemann sphere Y ζ = Y ζ ( ζ ) , ζ = Y ¯ ζ (¯ ¯ solution to conformal Killing equation ζ ) Y l n = − ζ n +1 ∂ ζ n +1 ∂ ¯ l n = − ¯ superrotations ∂ζ , ζ , n ∈ Z ∂ ¯ generators supertranslations T m,n = ζ m ¯ ζ n , m, n ∈ Z commutation relations [¯ l m , ¯ l n ] = ( m − n )¯ [ l m , ¯ [ l m , l n ] = ( m − n ) l m + n , l n ] = 0 , l m + n , [ l l , T m,n ] = ( l + 1 l l , T m,n ] = ( l + 1 [¯ − m ) T m + l,n , − n ) T m,n + l . 2 2 Poincaré subalgebra ¯ l − 1 , ¯ l 0 , ¯ l − 1 , l 0 , l 1 , l 1 , T 0 , 0 , T 1 , 0 , T 0 , 1 , T 1 , 1 ,

BMS4/CFT2 solution space u dependence fixed through evolution equation integration “constants plays no role asymptotically free data free u dependence news tensor unit sphere

Conformal properties BMS4/CFT2 bms4 transformations Interpretation and consequences: work in progress field dependent Schwarzian derivative: Lie algebra Lie algebroid →

Charge algebra BMS4/CFT2 asymptotic charge : non integrable due to the news / Q ξ [ δ X , X ] = δ ( Q s [ X ]) + Θ s [ δ X , X ] , δ Proposal : “Dirac” bracket { Q s 1 , Q s 2 } ∗ [ X ] = ( − δ s 2 ) Q s 1 [ X ] + Θ s 2 [ − δ s 1 X , X ] . if one can integrate by parts Proposition : { Q s 1 , Q s 2 } ∗ = Q [ s 1 ,s 2 ] + K s 1 ,s 2 , generalized cocycle condition K [ s 1 ,s 2 ] ,s 3 − δ s 3 K s 1 ,s 2 + cyclic (1 , 2 , 3) = 0 .

Charges for Kerr black hole BMS4/CFT2 Q T m,n , 0 [ X Kerr ] = 2 M I m,n = 1 1 Z ζ n . ζ ζ m ¯ d 2 Ω supertranslations : G I m,n , 1 + ζ ¯ 4 π Z 1 I ( m ) = 1 dµ (1 + µ ) m I m,n = δ m n I ( m ) 4 (1 − µ ) m − 1 − 1 Q T =1 ,Y =0 [ X Kerr ] = M G , divergences for proper supertranslations ! iaM Q 0 ,l m [ X Kerr ] = − δ m superrotations : 2 G . 0 Q T =0 ,Y φ =1 ,Y θ =0 [ X Kerr ] = − Ma ∂ φ = − i ( l 0 − ¯ l 0 ) G

BMS4/CFT2 Central charges for Kerr black hole central charges : K (0 ,l m ) , (0 ,l n ) [ X Kerr ] = 0 = K (0 , ¯ l n ) [ X Kerr ] = K (0 ,l m ) , (0 , ¯ l n ) [ X Kerr ] , l m ) , (0 , ¯ K (0 ,l l ) , ( T m,n , 0) [ X Kerr ] = a l ( l − 1)( l + 1) J m + l,n , 16 G l l ) , ( T m,n , 0) [ X Kerr ] = a l ( l − 1)( l + 1) K (0 , ¯ J m,n + l , 16 G Z 1 dµ (1 + µ ) m − 3 2 J m,n = δ m n J ( m ) J ( m ) = 2 2 , (1 − µ ) m + 1 − 1 form in-line with extremal Kerr/CFT correspondence, but divergences ! problem: one cannot integrate by parts if there are poles there are no poles for but then way out (i) define the analog of charge fields to regularize the divergences in the charges Green’s function for (ii) take correctly into account the boundary contributions to correct the central charges

Conclusions and perspectives BMS4/CFT2 4d gravity is dual to an extended conformal field theory to be done: particles as UIRREPS for bms 4 scattering theory between and Penrose, Les Houches 1963

Conclusions and perspectives BMS4/CFT2 angular momentum problem in GR: Geroch, Asymptotic structure of spacetime, 1977 Lorentz = Poincaré /translation 4 conditions needed to fix rotations Lorentz = bms4(old)/supertranslations infinite # conditions needed to fix rotations bms4(new)/supertranslations = Virasoro infinite # conditions needed to fix infinite # of superrotations