Scaling symmetry and the generalized Smarr relation

Transcription

1 Scaling symmetry and the generalized Smarr relation Park, Sang-A Yonsei Univ. Jan. 13, 2016 The 10th Asian Winter OIST 1 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 1/10

2 Talk Based on works collaboration with B.Ahn, S.Hyun, J.Jeong, K.Kim, S.Yi Scaling symmetry and scalar hairy Lifshitz black holes JHEP 10:105 (2015) Scaling symmetry and scalar hairy rotating AdS 3 black holes to appear in PRD Holography without Counter Terms also related with, Quasi-local charges and asymptotic symmetry generators JHEP 1406:151 (2014) Quasi-local conserved charges and holography Phys. Rev. D 90: (2014) Frame-independent holographic conserved charges Phys. Rev. D 91: (2015) which are generalizations of the off-shell formalism for the conserved charges. [Kim, Kulkarni, Yi 13 2 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 2/10

3 Introduction BH Thermodynamics and Smarr relation Basic physical quantities of black holes in gravity Conserved charges The 1st law of black hole thermodynamics δm = T H δs BH + Ω H δj + Φ H δq Black hole entropy is also the Noether charge [Wald 93 κ 2π δs BH = 1 dx µν δk µν (ξ H ) The Smarr relation for charged Kerr sol. [Smarr 73 Scaling symmetry gives Smarr relation H M = 2T H S + Ω H J + Φ H Q 3 dim Einstein gravity with an minimally coupled scalar hair in AAdS geometry. [Banados, Theisen 05 We generalize this formulation. 3 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 3/10

4 Scaling symmetry of the reduced action Ansatz metric : ds 2 = e 2A(r) f(r)dt 2 + dr2 f(r) + r2 dx 2 D 2 for matter fields : A µ = A µ (r), φ = φ(r), Reduced action I red = 1 δi red = 1 drdx L(r, Ψ), Ψ (A, f, A µ, φ, ) drdx [E Ψ δψ + Θ r (δψ), r Consider the transformation, δ σ Ψ = σ(ωψ rψ ), s.t. δ σi red = 1 drdx S r, S r = rl(r, Ψ) [ Noether charge : C = C(r) 1 8G Θ(δ σ Ψ) S 4 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 4/10

5 Scaling symmetry of the reduced action Ansatz metric : ds 2 = e 2A(r) f(r)dt 2 + dr2 f(r) + r2 dx 2 D 2 for matter fields : A µ = A µ (r), φ = φ(r), Reduced action I red = 1 δi red = 1 drdx L(r, Ψ), Ψ (A, f, A µ, φ, ) drdx [E Ψ δψ + Θ r (δψ), r Consider the transformation, δ σ Ψ = σ(ωψ rψ ), s.t. δ σi red = 1 drdx S r, S r = rl(r, Ψ) [ Noether charge : C = C(r) 1 8G Θ(δ σ Ψ) S Meaning of C? 4 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 4/10

6 Smarr relation horizon C(r H ) Scaling symmetry infinity C(r! 1) compare Entropy Mass Smarr relation T H S BH = 1 Z dx µ K µ ( H ) 16 G T HS H M = 1 BH Z= 1 dx µν K µν (ξ H) H dx µ K µ ( T )-2 [µ 16 G T ( )+ p -ga 1 M = δm = 1 ( ) ds dx µν δk µν (ξ T ) 2ξ [µ T Θν (δψ) 5 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 5/10 1/1

7 Application on Lifshitz BH arxiv: Model : NMG coupled with a scalar field in 3 dim L [g, φ = η [R 2Λ + 1 (R m 2 µν R µν 3 ) 8 R2 1 2 ( φ)2 α 2 Rφ2 V (φ) C = η ( [e A 1 + λ e A 8G 4m 2 r (Z e A f ) + η α ) 2 φ2 (2f rf ) ( ) + λz + e A 2rfλ + η r 2 fφ 2 + e A 2m 2 r Z(2Z 2 rz ) T H S BH = C = (1 + z)m 2. Model : Einstein-Maxwell-dilaton gravity in D dim L [g, ϕ, A = R 2Λ 1 ( ) 2 1 ϕ 2 4 eλϕ F 2 (D + z 2)(D + z 3), Λ = 2 [ C = rd 3 e A ) (D 2) (2f rf r 2 fϕ 2 (D 2) 8G 8G q a (D 2)T H S BH = C + D 2 8G q a(r H) = (D + z 1)M 6 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 6/10

8 Modify: including spatial dependences Ansatz metric : ds 2 = e 2A(r,x) f(r, x)dt 2 + dr2 f(r,x) + r2 dx 2 D 2 for matter fields : A µ = A µ (r, x), φ = φ(r, x), Consider the transformation, ) δ σ Ψ = σ(ωψ rψ ), δ σ Ψ,i = σ ((ω + 1)Ψ,i r(ψ,i ), δ σ α n = σω n α n, which gives δ σ I red = 1 ( drdx E Ψ δ σ Ψ + a Θ a (δ σ Ψ) + δi [ ) red δi red δσ, i Ψ + δ σ α n. δ i Ψ δα n n Charge function C(r) C(r H) = r r H dr dx [ D 2 i=1 δi red δ i Ψ iψ n δi red ω nα n δα n on-shell 7 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 7/10

9 Application to rotating BH arxiv: For rotating BH with one Killing vector: ds 2 = f(r, y)e 2A(r,y) dt 2 + dr2 f(r,y) +r2 (dθ (r, y)dt) 2, φ = φ(r, y), y Ω H t θ ξ K = + t ΩH ξt + ΩHξR θ Conserved charges using asymptotic Killing vectors [ M = 1 ds dx µν δk µν (ξ T ) 2ξ [µ T Θν + ga µν J = [ K 1 µν (ξ R ) + ds ga µν dx µν Smarr-like relation M = 1 2 T HS H + Ω H J 1 32πG [ δired drdy δ Ψ Ψ on-shell issues in integrability 1st law still invariant thermodynamic stability of hairy BH 8 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 8/10

10 Application to holographic CMT models arxiv: Thermodynamics of dual system using grand potential for homogeneous system and expression of c W ϵ µq T H S c(r H ) = (D 2)T H S BH we obtain the expression for the renormalized on-shell action T H I ren on-shell = W = M µq 1 D 2 c(rh) [ = M µq 1 D 2 lim c(r) + r r without concerning counter terms. r H dr ( D 2 i=1 δi red δ i Ψ iψ + n ω nα n δi red δα n ) on-shell 9 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 9/10

11 Summary We obtained charge for scaling symmetry from reduced action. This gives: Smarr relation for asymptotically Lifshitz planar-bh [ consistent with known results. T H S = D + z 2 D 2 M Smarr-like relation for rotating BH with scalar hair [ M = 1 2 T HS H + Ω H J 1 [ δired drdy 32πG δ Ψ Ψ on-shell describing thermodynamic stability. Smarr-like relation for CMT model [ giving renormalized on-shell action. T H I ren on-shell = M µq 1 D 2 c(r H) 10 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 10/10

12 Thank you for listening! 10 / 10 Sang-A Park Yonsei Univ. Scaling symmetry and the generalized Smarr relation 10/10