Backreaction effects of matter coupled higher derivative gravity

Transcription

1 Backreaction effects of matter coupled higher derivative gravity Lata Kh Joshi (Based on arxiv: , work done with Ramadevi) Indian Institute of Technology, Bombay DAE-HEP, Dec 09, 2014 Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

2 Introduction Introduction and Motivation Strongly coupled field theories Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

3 Introduction Introduction and Motivation Strongly coupled field theories Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

4 Introduction Introduction and Motivation Strongly coupled field theories AdS CFT Correspondence Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

5 Introduction and Motivation AdS CFT correspondence AdS 5 /CFT 4 [Maldacena, Adv.Theor.Math.Phys.2: ,1998] Type IIB string theory on AdS 5 S 5 N = 4 SYM theory. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

6 Introduction and Motivation AdS CFT correspondence AdS 5 /CFT 4 [Maldacena, Adv.Theor.Math.Phys.2: ,1998] Type IIB string theory on AdS 5 S 5 N = 4 SYM theory. A correspondence to relate theories of two very different kind. One in gravity and other in field theory. Strong coupling Weak coupling AdS CFT Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

7 Introduction and Motivation AdS CFT correspondence AdS 5 /CFT 4 [Maldacena, Adv.Theor.Math.Phys.2: ,1998] Type IIB string theory on AdS 5 S 5 N = 4 SYM theory. A correspondence to relate theories of two very different kind. One in gravity and other in field theory. Strong coupling Weak coupling AdS CFT Specific limit: Fluid-Gravity Correspondence Hydrodynamics: Large length scales in field theories. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

8 Introduction and Motivation To locate transition temperature precise behavior of η/s with T is needed. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

9 Introduction and Motivation To locate transition temperature precise behavior of η/s with T is needed. Figure: arxiv: The figure shows expectation taking analogy with H 2 O and He plots. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

10 Using AdS/CFT Calculation of the correlator? Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

11 Using AdS/CFT Calculation of the correlator? Use AdS-CFT correspondence e i M φ 0Õ = e is cl [φ] Õ is an operator in the boundary; φ field in the bulk, with φ 0 value at boundary. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

12 Using AdS/CFT Calculation of the correlator? Use AdS-CFT correspondence e i M φ 0Õ = e is cl [φ] Õ is an operator in the boundary; φ field in the bulk, with φ 0 value at boundary. Giving, Õ(x 1 )Õ(x 2 ) = i 2 δ2 δφ 0 (x 1 )δφ 0 (x 2 ) eis cl [φ 0 ] at φ 0 0 Note: We need to know the background geometry for the above calculations. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

13 Background Solutions Background solutions in gravity Einstein gravity with higher derivatives S = d D x g ( R 2Λ + c 1 R 2 + c 2 R µν R µν + c 3 R µνρσ R µνρσ) (1) Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

14 Background Solutions Background solutions in gravity Einstein gravity with higher derivatives S = d D x g ( R 2Λ + c 1 R 2 + c 2 R µν R µν + c 3 R µνρσ R µνρσ) (1) (D 1)(D 2) with Λ = 2 ds 2 = 1 (( f (r)dt 2 + ( d x 2) ) + dr ) 2 r 2 f (r) (2) (3) where, f (r) = 1 r D 1 + δ + κr 2(D 1) δ and κ being dependent on D, c 1, c 2 and c 3 (arxiv: ) Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

15 Matter coupling Background Solutions To know the properties of QGP, it becomes necessary to consider dilaton field in the gravity Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

16 Matter coupling Background Solutions To know the properties of QGP, it becomes necessary to consider dilaton field in the gravity Consider the action S = 1 2κ 2 D d D x[r 4 D 2 ( Φ(r))2 V (Φ(r))] ; V (Φ(r)) = 2Λe αφ(r) (4) The already known solution to this action shows effects of matter on the background metric Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

17 Background Solutions It is interesting to look upon the case of matter coupled higher derivative gravity S = 1 2κ 2 d D x(r 4 D D 2 ( Φ(r))2 V (Φ(r)) + βg(φ(r))r µνσρ R µνσρ ) (5) with V (Φ) = 2Λe αφ ; G(Φ) = e γφ. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

18 Background Solutions It is interesting to look upon the case of matter coupled higher derivative gravity S = 1 2κ 2 d D x(r 4 D D 2 ( Φ(r))2 V (Φ(r)) + βg(φ(r))r µνσρ R µνσρ ) (5) with V (Φ) = 2Λe αφ ; G(Φ) = e γφ. Varying the action with respect to metric field and scalar field gives, R µν 1 2 g µνr = T matter µν + T hd µν (6) 8 D 2 Φ(r) = Φ(βG(Φ(r))R µνσρ R µνσρ V (Φ(r))) (7) where,t µν = 1 g gl M g µν (8) Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

19 Background Solutions Choosing an ansatz is the most crucial part in solving the above equation. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

20 Background Solutions Choosing an ansatz is the most crucial part in solving the above equation. To find the solution up to O(β), we choose the ansatz as: ds 2 = r 2a (1 r c(β,r) )dt 2 dr 2 + r 2a (1 r c(β,r) )r 4 + r 2a d x 2 Φ(r) = mlog(r) + βφ 1 (r) (9) where c[β, r] is chosen as: c[r] = c + Log(1 βκ(r)) Log(r) (10) Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

21 Background Solutions The first order correction κ(r), is obtained as: 6(( 1) D +1)m (D 4)(D 1)16D D 2 (g r) ad+1 (D 4) 4 + (D 2) ((D 2) 2 α 2 + 2Λr m 2 (α+γ) 16(D 1)) (D 2) (r mγ+a(d 1) ( (D 2) 2 α (3(D 2)α + 4(D 3)γ) 16(D 4)(D 3) ) ((D 2) 2 αγ + 16(D 1)) r mα ( 4(D 1)(D 2) 2 α 2 + 8(D 3)(D 2) 2 αγ + 64(D 4) ) ((D 2) 2 α(α + 2γ) + 16(D 1)) ), Horizon gets a linear order correction as: r h = 1 + βr 1. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

22 Shear viscosity Shear viscosity to entropy ratio Kubo formula for viscosity Low energy and low momentum limit of retarded Green s function of stress tensor in CFT. 1 η = lim ω 0 ω ImG xy;xy(ω, R k = 0) (11) Translate the calculation of the correlator to a holographic one Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

23 Shear viscosity Shear viscosity to entropy ratio Kubo formula for viscosity Low energy and low momentum limit of retarded Green s function of stress tensor in CFT. 1 η = lim ω 0 ω ImG xy;xy(ω, R k = 0) (11) Translate the calculation of the correlator to a holographic one perturbation in the xy components of the metric. g xy = gxy 0 + ɛh xy (r, x) = gxy 0 [ + ɛg ii φ(r, x), x = (t, ] x ) with φ(r, x) = 1 (2π) 4 d 4 ke ik.x φ(r, k), [k = (ω, k ), k.x = k µ x µ] Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

24 Entropy density Shear viscosity to entropy ratio Wald formula S = 2π Σ d D 2 x δl h ɛ µν ɛ ρσ (12) δr µνρσ 1 ɛ µν ɛ µν = 2 2 ɛ µν = ξ µ η ν ξ ν η µ The L above is chosen by writing the action as S action d 5 x gl Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

25 Shear viscosity to entropy ratio Backreaction effects on η and s For the matter coupled higher derivative action the shear viscosity and entropy density is [ 1 η = 2κ 2 rh P 4β(D 2)2 αγ ( α 2 (D 2) 2 16(D 1) ) ] r0 M D (16 + α 2 (D 2) 2 ) 2 (13) [ 1 s = 4πrh P 128πβ(D 4) ( α 2 (D 2) 2 16(D 1) ) ] r0 N (16 + α 2 (D 2) 2 ) 2 (14) 2κ 2 D Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

26 Shear viscosity to entropy ratio Backreaction effects on η and s For the matter coupled higher derivative action the shear viscosity and entropy density is [ 1 η = 2κ 2 rh P 4β(D 2)2 αγ ( α 2 (D 2) 2 16(D 1) ) ] r0 M D (16 + α 2 (D 2) 2 ) 2 (13) [ 1 s = 4πrh P 128πβ(D 4) ( α 2 (D 2) 2 16(D 1) ) ] r0 N (16 + α 2 (D 2) 2 ) 2 (14) 2κ 2 D Giving, η s = 1 4π [1 4β ( 16(D 1) + (D 2) 2 α 2) ( 8(D 4) + (D 2) 2 αγ ) ] (16 + (D 2) 2 α 2 ) 2 (15). Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

27 Summary Shear viscosity to entropy ratio Found the first order correction to the background metric for matter coupled higher derivative AdS gravity. Horizon gets a linear order correction due to matter presence. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

28 Summary Shear viscosity to entropy ratio Found the first order correction to the background metric for matter coupled higher derivative AdS gravity. Horizon gets a linear order correction due to matter presence. Shear viscosity and Entropy density get linear order correction via the correction in horizon However, the ratio η/s remains unaffected due to the backreaction in the metric. Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

29 Summary Shear viscosity to entropy ratio Found the first order correction to the background metric for matter coupled higher derivative AdS gravity. Horizon gets a linear order correction due to matter presence. Shear viscosity and Entropy density get linear order correction via the correction in horizon However, the ratio η/s remains unaffected due to the backreaction in the metric. Puzzle: Backreaction to the matter solution still remains unknown Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

30 Summary Shear viscosity to entropy ratio Found the first order correction to the background metric for matter coupled higher derivative AdS gravity. Horizon gets a linear order correction due to matter presence. Shear viscosity and Entropy density get linear order correction via the correction in horizon However, the ratio η/s remains unaffected due to the backreaction in the metric. Puzzle: Backreaction to the matter solution still remains unknown Thank You Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

31 Shear viscosity to entropy ratio Anti de Sitter space Solution to Einstein equation with negative cosmological constant. Space of Lorentzian signature (, +, +..+), but of constant negative curvature On boundary d 1 x0 2 + i=1 x 2 i x 2 d+1 = R2 Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

32 Shear viscosity to entropy ratio R µν 1 2 g µνr g µνλ = 0 (16) Thus constant negative curvature (d + 1) = R = Λ d 1 (17) R µν = 1 d 1 Λg µν (18) Parametrize, d(d 1) Λ = L 2 d(d + 1) = R = L 2 (19) Back Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

33 Shear viscosity to entropy ratio Conformal Field Theory Guage field theory with enhanced symmetries. Poincare symmetry + dilatations + special conformal symmetry Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

34 Shear viscosity to entropy ratio Conformal Field Theory Guage field theory with enhanced symmetries. Poincare symmetry + dilatations + special conformal symmetry Dilatations: x µ λx µ special conformal: Translation+Inversion+Translation Under special conformal: x µ = x µ + 2x µ b.x x 2 b µ Back Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14

35 Shear viscosity to entropy ratio Translation:P µ = i µ (20) Dilatations:D = ix µ µ (21) Rotations:L µν = i(x µ ν x ν µ ) (22) SCT:G µ = 2ix µ (x. ) + ix 2 µ (23) Back Lata Joshi (IITB) Matter coupled higher derivative gravity DAE-HEP / 14