Fourth Aegean Summer School: Black Holes Mytilene, Island of Lesvos September 18, 2007

Transcription

1 Fourth Aegean Summer School: Black Holes Mytilene, Island of Lesvos September 18, 2007

2 Central extensions in flat spacetimes Duality & Thermodynamics of BH dyons New classical central extension in asymptotically flat spacetimes at null infinity with G. Compère, CQG 24 (2007) F15 Thermodynamics of black hole dyons with duality invariant extended double potential formalism with A. Gomberoff arxiv: [hep-th]

3 Central extensions: Generalized conserved charges: p=1: { Surface charges & generalized Killing vectors dω n p 0 ω n p ω n p + dη n p 1 + tn p, conserved currents associated with global symmetries irreducible gauge theories (no 2,3-forms): n p Hchar (d) = 0 f or p! 3 n 2 i Hchar (d) f α such that Rα (f α ) = 0 charges: Qf [φs ] =! S n 2 kfn 2 [φs ] tn p 0

4 Central extensions: Surface charges & generalized Killing vectors Examples semi-simple YM theory: δ ɛ A a µ = D µ ɛ a = 0 = ɛ a = 0 δ ɛ A µ = µ ɛ = 0 = ɛ = cte EM: k n 2 = F electric charge Q = S n 2 F GR: δ ξ g µν = L ξ g µν = 0 = ξ µ = 0 linearized gravity: δ ξ h µν = L ξ ḡ µν = 0 = ξ µ Killing vector of ḡ µν Q ξ = S n 2 k ξ [h, ḡ] Abbott & Deser Nucl. Phys. B195 (1982) 76

5 Central extensions: Algebra & asymptotics global symmetry L ξ ḡ µν = 0 = δ 1 ξh µν = L ξ h µν ḡ µν = η µν Poincaré invariance of Pauli-Fierz theory Algebra: Surface charges form a representation of the algebra of Killing vectors {Q ξ1, Q ξ2 } := δ 1 ξ 1 Q ξ2 = Q [ξ1,ξ 2 ] full GR, asymptotics g µν = ḡ µν + O( replace 1 r χ µν ) h µν = g µν ḡ µν at boundary charges r Q ξ = k ξ [g ḡ, ḡ] S

6 Central extensions: Algebra & asymptotics new feature: asymptotic Killing vectors L ξ ḡ µν 0 to leading order that preserve the fall-off conditions 1 L ξ g µν = O( r χ ) µν suitable tuning of fall-off conditions on metrics and asymptotic Killing vectors: centrally extended charge representation of algebra of asymptotic Killing vectors {Q ξ1, Q ξ2 } := δ ξ1 Q ξ2 = Q [ξ1,ξ 2 ] + K ξ1,ξ 2 K ξ1,ξ 2 = k ξ2 [L ξ1 ḡ, ḡ] S NB: central extension vanishes for exact symmetries of the background

7 Central extensions: Asymptotically ADS spacetimes non trivial asymptotic Kvf= conformal Kvf of flat boundary metric n>3: so(n 1, 2) only exact Killing vectors of AdS, no central extension n=3: pseudo-conformal algebra in 2 dimensions, 2 copies of Wit algebra charges algebra: 2 copies of Virasoro Brown & Henneaux CMP 104 (1986) 207 similar results in de Sitter spacetimes at timelike infinity

8 Central extensions: Asymptotically flat spacetimes conformal boundary in asymptotically flat spacetimes: null infinity bms n Y A (θ A ) T (θ A ) conformal Kvf of n-2 sphere supertranslations, arbitrary function on n-2 sphere

9 Central extensions: Asymptotically flat spacetimes ξ = [ξ, ξ ] algebra: semi-direct product with abelian ideal i n 2 n>4: so(n 1, 1) i n 2 n=4: conformal algebra in 2d so(3, 1) i 2 Bondi-Metzner-Sachs (1962)

10 Central extensions: Asymptotically flat spacetimes n=3: no restriction on Y (θ) 1 copy of Wit algebra acting on i 1 iso(2, 1) Ashtekar et al. Phys. Rev. D55 (1997) 669 charge algebra: relation to AdS 3 similar to contraction between so(2, 2) iso(2, 1) L ± m = 1 2 ( lp ±m ± J ±m ) l

11 Magnetic charge as a surface charge? Magnetic charge: d F = 0 F = db What about dyons? Action principle?

12 Thermodynamics for RN RN dyon: infer thermodynamics for parameter variations from purely electric case using duality Problem: excluded in action based derivations of 1st law for arbitrary variations because of string singularity and absence of magnetic potential

13 Double potential formalism key: existence of 1st order formalism that makes invariance of action under duality rotations manifest in curved space: Deser & Teitelboim Phys. Rev. D 13 (1976) 1592

14 Double potential formalism independent rederivation : Schwarz & Sen Nucl. Phys. B411 (1994) 35

15 Applications to black hole duality complicated proof of duality between electric and magnetic BH partition function through semi-classical evaluation of Euclidean path integral Z(β, P ) vs Z(β, φ H ) Hawking & Ross, Phys. Rev. D52 (1995) 5685 additional Legendre transformation needed to compare

16 Applications to black hole duality duality symmetric formulation:

17 Applications to black hole duality action principle appropriate for fixed charges duality: BH duality: Deser, Henneaux, Teitelboim Phys. Rev. D 55 (1997) 826

18 Extended formulation New construction: dynamical longitudinal fields and non spurious scalar potentials A a µ (A µ, Z µ ) C a (C, Y ) B a ( B, E) B a = A a + C a external sources: action principle: µ j aµ = 0 I M [A a µ, C a ; j aµ ] = 1 2 d 4 x [ ɛ ab ( Ba ( 0 A b A b 0) C a A b C a A b ) B a B a + 2ɛ ab A a µj bµ], Maxwell s equations: { A a 0 : B a 2 C a = j 0a C a : 2 C a = ɛ ab ( 0 A b 2 A b 0) A a : ɛ ab 0 B b + B a = ɛ ab j b.

19 Extended formulation point particle dyon at origin: j aµ (x) = 4πQ a δ µ 0 δ3 (x) A a = ɛab Q b r dt, C a = Qa r gauge invariance : δ ɛ A a µ = µ ɛ a, δ ɛ C a = 0 no string singularity! spectrum: additional pure gauge degrees of freedom quartet besides longitudinal and temporal photon 2 reducibility parameters: magnetic charge is surface integral, no longer a topologically charge

20 Extended formulation curved space: B ai = [ijk] j A a k + I M [A a µ, C a, g ij, N, N i ] = 1 8π g N i C a d 4 x [ (B ai + g N i C a )ɛ ab ( 0 A b i i A b 0) N g B i ab a i + ɛ ab [ijk]n i B aj B bk] RN dyon: ds 2 = N 2 dt 2 + N 2 dr 2 + r 2 (dθ 2 + sin 2 θdφ 2 ), N = 1 2M + Q2 + P 2 r r 2, A a = ɛab Q b r BH thermodynamics in grand canonical ensemble dt, C a = Qa r variables: z A (g ij, π ij, Z i, A i, Y, C) λ a (N, N i, A 0, Z 0 ) ɛ a (ɛ, ɛ i, λ, µ) surface integrals: δ(γ α ɛ α ) = δz A δ(γ α ɛ α ) δz A i k T [0i] ɛ [δz A ] Regge & Teitelboim, Ann. Phys. 88 (1974) 286 reducibility parameters for RN: ɛ α = λ α = (N, 0, 0, A a 0) Hamiltonian EOM: { δz B = i k T [0i] ż A = σ AB δ(γ α λ α ) γ α = 0 λ = 0

21 Extended formulation Stokes theorem: 1 16π S d n 1 x i k T [0i] λ = 1 16π S r+ d n 1 x i k T [0i] λ. explicitly: k T [0i] ɛ [δz A ] = kɛ grav[0i] [δg ij, δπ ij ] + kɛ mat[0i] [δz A ]. ( ɛ kɛ mat[0i] [δz A ] = 4 [ijk](e j δz k + B j δa k ) ɛ g N (E i δy + B i δc) matter part: gg ɛ i (B k δz k E k δa k ) + B i ɛ k δz k E i ɛ k il δa k N [ljk]ɛj (B k δy E k δc) 1 g ) 2 N (λ i δy i λδy µ i δc + i µδc). both magnetic and electric contributions! first law: δm = κ 8π δa + φ HδQ + ψ H δp

22 Conclusion new classical central extension in asymptotically flat 3D spacetimes at null infinity contraction of the ADS case construction of an explicitly duality invariant version of electromagnetism through addition of pure gauge degrees of freedom enhanced gauge invariance static dyon described by Coulomb fields without string singularities electric and magnetic charges are surface integrals applications in the context of thermodynamics of BH dyons

23 Dynamical sources & Dirac strings duality requires both electric and magnetic sources action principle for charged point particles: Dirac string monopole field described by singular potential replaced by thin solenoid with no magnetic charge

24 Dynamical sources & Dirac strings + dynamical string Felsager Goddard & Olive action principle: NB: asymmetric treatment of both types of sources

25 Dynamical sources & Dirac strings dynamical dyons with Dirac strings: generalizes Dirac s action Deser, Henneaux, Teitelboim, Gomberoff Nucl. Phys. B520 (1998) 179

26 Extended formulation dynamical point particle dyons need strings for Lorentz force law: j aµ (x) = qn a δ 4 (x z n )dz n µ I P [z n] µ = n Γ n n total action: I M [A a µ, C a, y µ n] = 1 2 I M [A a µ, C a, y µ n] + I I[A a µ, z µ n] + I P [z µ n] d 4 x { ɛ ab [ ( B a + C a )( 0 A b A b 0 + α b ) Γ n dz µ ndz nµ A a 0β b β a α b β a 2 0β b ] B a Ba }, I I[A a µ, z n] µ = d 4 x ɛ ab (A a 0j 0b + 1 A 2 a j b). y µ n(σ n, τ n ), y µ n(0, τ n ) = z µ n(τ n ) α a = n q a n δ 4 (x y n ) 1 Σ n 2 d y n dy n, β a = n q a n Σ n δ 4 (x y n )dy 0 n dy n, β a = j a0, α a 0 β a = j a. variation with respect to z n µ gives Lorentz force law standard veto: string attached to dyon n cannot cross any other dyon leads to standard quantization condition: ēg eḡ = 2πn