BMS current algebra and central extension

Transcription

1 Recent Developments in General Relativity The Hebrew University of Jerusalem, -3 May 07 BMS current algebra and central extension Glenn Barnich Physique théorique et mathématique Université libre de Bruxelles & International Solvay Institutes

2 Overview BMS symmetry Would-be conserved BMS current algebra The field dependent central charge Cardyology at null infinity In collaboration with C. Troessaert

3 Introduction Bondi mass loss due to gravitational radiation : non-linear GR effect that was important to settle the controversy on the existence of gravitational waves D. Kennefick King's College and the story of how gravitational waves became real

4 The set-up retarded time Penrose & Rindler Vol II complex coordinates on i e cot celestial sphere d sin d P d d S PS (, ) ( )

5 BMS symmetry The paper

6 BMS symmetry The algebra Poincaré algebra GR choice: globally welldefined quantities CFT choice: allow for poles sl(, ) ST Lorentz generators as globally well-defined conformal Killing vectors fields of celestial sphere [ l, l ] ( m n) l m n mn [ l, l ] ( m n) l m n mn [ l, l ] 0 m n [ l, P ] ( m k) P [ l, P ] ( ml) P [ P, P ] 0 m k, l mk, l m k, l k, ml kl op ln n superrotations k, l k l S P P supertranslations Poincaré subalgebra l, l0, l, l, l0, l, P, P, P, P,,,,

7 BMS current algebra Currents u J f ( ) ( ð ð( )) c.c. [( ) ] GPS mass aspect angular momentum aspect P Y, P Y S S u T (, ) ð f PS T conformal Killing vectors supertranslation generators ð 0 i Weyl tensor 0 0, asymptotic part of shear & news information on TT polarizations of gravity waves

8 BMS current algebra Action on fields 3 [ f ð ð ð ð ] ð 0 0 u f [ f ð ðð ] ð 0 0 u 3 3 [ f ð ð ð ð ] ð f u 3 [ f ð ðð ð ] 3ðf u transformations of fields involve inhomogeneous terms Minkowski vacuum breaks BMS invariance

9 BMS current algebra The formula J ( ) J K L a a a a [ ab] [, ], b x a ( u,, ) local formula works with poles breaking due to news [ ] u 0 0 ( ) f c.c. 8 GPS field dependent central extension K f ð ( ) c.c. [( ) ] u 0, 8 GPS vanishes when there are no poles/superrotations

10 BMS current algebra Non-conservation current non-conservation for, u ð ð ( ) u u u u, u u charges when there are no poles Q S d u d du Q 0 0 d [ c.c.] 8 G S Bondi mass loss formula for u d du Q 8 G 0 0 d [ u S 0 c.c.]

11 Central extension WZ consistency condition in QFT the Adler-Bardeen anomaly satisfies the Wess-Zumino consistency condition a a a b c a A DC C C C f bc Tr F d 3 0,5 H 0,5,4 d H,4,3 d H 0 0 4, 5 d C 5 Tr C 0 H Tr 0 fermionic generators Y ( ) T C(, ), u d d K dud K dud K, 3, d H 0 3, 4,0 d H 0 4,0 0

12 Central extension Extended algebroid structure functions [ e, e ] f ( ) e R f f f i [ i ] [ ] i [ e, f ( )] R( ) i f R [ ir f R ] i j j Lie algebra over functions f ( ) e [, ] ( f ) e i 0 R[ i ] [ f ] extended algebroid [ e, e ] f ( ) e ( ) Z needs all spatial boundary terms to vanish

13 Central extension Getting numbers PR 0 3, ( ) c.c. [( ) ] K d d f Y from the conformal dimensions : n n u u k, l kl, n n3 k l ( u,, ) ( ) ( u) admit Laurent series (delta function singularities), integrals as residues K u ( m )( n ) [ n ( n ) m ( m )] l m 0, ln mn, K u, ( m lm ln )( n )[ m ( m ) n ( n )] 0 0 m, n m, n K 0 l, P m k, l ( ) m k, l m m K P, P 0 k, l o, p

14 Central extension Cardyology at null infinity? But 0 0 for Kerr black hole transform Scri to a cylinder times a line by a finite superrotation L e 0 k l n n n 0 L L 0 u ( u,, ) ( ) [( u ) k, l ( u) e e ] ( ) ( nu n) L L 4 finite shift! thermal circle iu ~ iu Work in progress.

15 BMS current algebras Technical details Asymptotic symmetries Conserved currents and charges NP formalism & solution space Finite BMS transformations

16 Asymptotic symmetries Gauge fixation Main idea : asymptotic symmetries = residual gauge symmetries BMS ansatz 0 e 0 V g e e U e r A 0 U e g B u r null coordinate A x AB,,, d- gauge conditions g uu 0 g ua determinant condition ( d ) det g AB r det AB fix diffeomorphism invariance in d dimensions AB dx dx e d A B d conformal to metric on unit d- sphere

17 Asymptotic symmetries Residual gauge transformations (weak) fall-off conditions V o U o o r r A B A B g ABdx dx r ABdx dx o( r ) A (), (), ( ), residual symmetries leave this class of spacetimes invariant exact conditions g g g rr ra AB 0 0 g AB 0 u f A A AB y B f dre g r r r B u B ( DB B U ) d A A A B fix r dependence up to integration functions f f ( u, x ) y y ( u, x ) asymptotic conditions g g g g ur ua uu AB o() o r ( ) o r ( ) o r ( ) A A y Y u f e [ T due ], 0 A DY A fix u dependence up to integration functions conformal Killing equation d- sphere T B A A B T ( x ) Y Y ( x ) d Y AB AB

18 Asymptotic symmetries Lie algebroids Metric dependence of bulk asymptotic Killing vectors ( xg, ) requires modified Lie bracket [, ] M [, ] g g leads to representation of asymptotic symmetry algebra in the bulk spacetime Particular example of a Lie algebroid

19 Asymptotic symmetries Weyl transformations or Motivations to keep the conformal factor ( u,, ) P P( u,, ) arbitrary in A B ABdx dx e d ( PP) d d ) because one can (general solution to Einstein s equation is known for this case) ) finite left-over ambiguity in geometric definition of asymptotic flatness through conformal compactification 3) solution space manifestly contains Robinson-Trautman waves ds Hdu dudr r P d d Inclusion of Weyl transformations Gauge symmetry of dual theory, P P replace A DY A in asymptotic Killing vectors bms4 Weyl

20 Current algebra Newman-Penrose formalism first order Cartan formulation 6 G 4 S[ abc, ea ] d x er ab [ ab] [ ab] spin coefficients c e c covariant derivative ð s s ( s s ), ð s s PP P PP ( P s s ) s [ð,ð] s R s conformal Killing vectors P Y, P Y ð 0 ð transformation law ð ð hð hð, s w s w [ ] ( hh, ) (, )

21 Current algebra Asymptotic solution space asymptotic solution space free data ( u, r,, ) r Or ( ) ( u,, ) 0 0 ( )( u,, ) ( u,, ) Pu (,, ) free u dependence evolution equations ( u 5 ) ð ( u 4 ) ð ( u 3 3 ) ð on-shell constraints news tensor 0 ln P P R 0 0 ln P u ð( ) ( u 3 ) ð ð ð ð ð ( u 4 )

22 Current algebra Transformation of free data BMS & Weyl transformations P P(, ) u ð ð f P T (, ) S ( Y, Y, T, ) (field dependent) inhomogeneous pieces, Schwarzian derivatives Strominger: soft gravitons = Goldstone modes for these transformations

23 Current algebra Motivation/Global symmetries Interpretation requires charges, canonical generators for the transformations + Dirac bracket algebra Problem: some ADM type charges diverge because of poles on the sphere Local non integrated version of Ward identities x j ( x) j ( y) X ( z) i ( x y) j ( y) X ( z) i ( x z) j ( y) X ( z) Q Q [ Q, Q ] Q Q classical version : i n Q dh jq Q d x i L dh ( Q jq j[ Q, Q ] TQ, Q ) 0 TQ Q, 0 j j T d K n Q Q [ Q, Q ] Q, Q H Q, Q T n Q, Q dh ~0 central extension highly constrained trivial Noether current, Belinfante ambiguities n [ KQ, Q ] H ( dh ) Classification [ j] [ Q] may be field dependent cocycle condition Q K Q, Q K 3 [ Q, Q ], Q cyclic (,,3) 0 3 i i R ( f ) T ~ 0 i T 0

24 Current algebra Gauge symmetries/holography gauge symmetries trivial Noether current i i i i f R ( f ) R f R f L S f R f d x i n ( )( ) i Classification n dh k 0 [ ] [ ] k i f R ( f ) 0 no solution in full GR, in linearized GR solutions classified by Kvf of background constructive k f i [ ] ( ) S i dx f ADM-type charges conservation in time and in the bulk asymptotic case x A ( u, r, x ) r integrability? k k ( d x) f [ ] n f k J, k J conservation? [ ur] u [ Ar ] A f f f current of lower dimensional theory x a A ( u, x )

25 Finite transformations BMS and Weyl group integrate BMS Lie algebra group finite transformations of solution space Residual gauge symmetries : find the local Lorentz transformations + diffeomorphisms that leave NPU solution space invariant How do they act on solution space? ( ( ), ( ), (, ), E( u,, ) E ie ) finite superrotations, supertranslations, complex Weyl rescalings R I determine,, ER u ' u '( u, ) (, ) dv( PP) u ˆ 0 Weyl invariant time coordinate u u( u,, ) dv( PP) ( v,, ) P( u,, ) P( u,, ) e 0 E ER u( u,, ) dve u( u,, ) J u( u,, ) (, ) u uˆ [ ] J NB: simple formulas when P 0 u u P standard BMS group when P is fixed

26 Finite transformations Action on solution space For the Riemann sphere P=

27 Finite transformations From the Riemann sphere to arbitrary P Solve evolution equation in terms of integrations functions Bondi mass aspect apply a pure complex rescaling generate solution for arbitrary P from P=

28 Finite transformations Schwarzian derivatives transformation of the Weyl invariant quantities