Anisotropic expansion, second order hydrodynamics and holographic dual
Priyanka Priyadarshini Pruseth ([email protected]) Utkal University, Bhubaneswar-751004 (in collaboration with Prof. S. Mahapatra)
XXIV DAE-BRNS high energy physics symposium (14-18 December-2020) Plan of Talk
Introduction Bjorken’s hydrodynamics and Kasner spacetime Energy-momentum tensor in second order hydrodynamics A proposal for the gravity dual of anisotropic expansion Summary Introduction
One of the fundamental question in the field of high energy physics is to understand the properties of matter at extreme density and temperature in the first few microseconds after the big bang. Such a state of matter is known as Quark-Gluon-Plasma (QGP) state where the quarks and the gluons are in deconfined state. Lot of progress has been made in understanding the properties and various aspects of the evolution of strongly coupled QGP through the heavy ion collision experiments at RHIC as well as in LHC. AdS/CFT correspondence has provided a very important tool for studying dynamics of the strongly coupled QGP. Perfect local equilibrium system is described by ideal fluid dynamics. For small departures from equilibrium, the system is described by dissipative fluid dynamics. Introduction
Navier-stokes equation is the fundamental equation in the non- relativistic viscous hydrodynamics. However, in relativistic Navier-Stokes equation, causality is violated in the first order theory. The second order gradient terms are included to preserve causality in the relativistic hydrodynamics equations. The gravity dual description of the expansion of the strongly coupled QGP has been very useful in understanding various aspects of both the boundary and bulk theory. Fefferman-Graham coordinates (FG) have been very useful for the gravity dual description. However, the regularity of the dual geometry in late time regime has been an important issue. This problem has been addressed by working with Eddington-Finkelstein type coordinates. We consider the viscous hydrodynamics including second order gradient expansion terms and study the anisotropic three dimensional expansion of the fluid with Kasner spacetime as the local rest frame (LRF). Bjorken’s hydrodynamics and Kasner spacetime
The Minkowski metric in terms of proper time τ and rapidity y is given by
2 2 2 2 2 2 ds = −(dτ) + τ dy + (dX2) + (dX3) (1)
Generalizating the one dim. expansion to 3-dim. expansion of the plasma, we consider Kasner space-time as the LRF of the fluid. The metric is
2 2 2a 2 2b 2 2c 2 ds = −(dτ) + τ (dx1) + τ (dx2) + τ (dx3) (2)
x1, x2, x3 are the comoving coordinates. a, b, c are constants and are known as Kasner parameters. The Kasner parameters satisfy the conditions,
a + b + c = 1, a2 + b2 + c2 = 1 (3)
Kasner metric is an exact solution of vacuum Einstein’s equation and it describes a homogeneous and anisotropic expansion of the Universe. Bjorken’s hydrodynamics and Kasner spacetime
µν By using conservation law ∇µν T = 0 (where µ, ν = τ, x1, x2, x3),we obtain (a + b + c) a b c ∂ T + T + τ −2a T + τ −2b T + τ −2c T = 0 (4) τ ττ τ ττ τ x1x1 τ x2x2 τ x3x3 µ and from conformal invariance Tµ = 0 , we get 1 1 1 −T + T + T + T = 0 (5) ττ τ 2a x1x1 τ 2b x2x2 τ 2c x3x3 The energy momentum tensor in relativistic viscous hydrodynamics is given by
T µν = uµuν + P∆µν + Πµν (6)
where uµ, and P are fluid velocity, the energy density and pressure respectively. ∆µν = g µν + uµuν Energy-momentum tensor in second order hydrodynamics
1 Πµν = −ησµν + ητ hDσµνi + σµν (∇ · u) + κ R<µν> − 2u Rα<µν>βu π 3 α β <µ ν>λ <µ ν>λ <µ ν>λ + λ1σ λσ + λ2σ λΩ + λ3Ω λΩ (7)
where η=shear viscosity, τπ=relaxation time and κ, λ1, λ2, λ3 are the other transport coefficients. For first order gradient expansion, the energy-momentum tensor in Kasner space-time is obtained as,
(τ) 0 0 0 2η (P − 3τ (3a − 1)) 0 0 0 τ 2a T µν = 2η (P − 3τ (3b − 1)) 0 0 0 τ 2b (P − 2η (3c − 1)) 0 0 0 3τ τ 2c (8) Energy-momentum tensor in second order hydrodynamics
The corresponding equation of state and conservation condition are respectively found to be, P = (9) 3 d 4 4η + − = 0 (10) dτ 3τ 3τ 2 Including second order gradient expansion, the energy-momentum tensor in Kasner space-time is obtained as, T 00 = (τ) P(τ) 2η(3a − 1) 4ητ (3a − 1) 4λ (9a2 − 6a − 1) 2κ(1 − a)a T 11 = − − π + 1 + τ 2a 3τ 2a+1 9τ 2a+2 9τ 2a+2 τ 2a+2 P(τ) 2η(3b − 1) 4ητ (3b − 1) 4λ (9b2 − 6b − 1) 2κ(1 − b)b T 22 = − − π + 1 + τ 2b 3τ 2b+1 9τ 2b+2 9τ 2b+2 τ 2b+2 P(τ) 2η(3c − 1) 4ητ (3c − 1) 4λ (9c2 − 6c − 1) 2κ(1 − c)c T 33 = − − π + 1 + τ 2c 3τ 2c+1 9τ 2c+2 9τ 2c+2 τ 2c+2 (11) Energy-momentum tensor in second order hydrodynamics
The equation of state and the conservation law become: P = (12) 3
d 4 4η 8ητ 2κ(−1 + a3 + b3 + c3) + = + π + dτ 3τ 3τ 2 9τ 3 τ 3 λ (−7 + 9(a3 + b3 + c3)) − 1 (13) 9τ 3 From the conformal invariance of the fluid, the proper-time dependence of the transport coefficients are given by
3/4 −1/4 1/2 1/2 0 0 η = 0 η0 , τπ = τπ , λ1 = 0 λ1 , κ = 0 κ0 0 0 0 0 (14) 0 0 where 0, η0, τπ, λ1, κ0 are constants. Energy-momentum tensor in second order hydrodynamics
The solution of the equation for energy density (τ) is obtained as,
2 0 (τ) −4/3 η0 3η0 λ1 3 3 3 = τ − 2 2 + + (−7 + 9a + 9b + 9c ) 0 τ 2 3 2η τ 0 3κ − 0 π − 0 (−1 + a3 + b3 + c3) τ −8/3 + ... (15) 3 2
Denoting the term in the above square bracket as
3η2 λ0 2η τ 0 3κ 0 + 1 (−7 + 9a3 + 9b3 + 9c3) − 0 π − 0 (−1 + a3 + b3 + c3) = ˜(2) 2 3 3 2 0 (16) The solution for (τ) can be rewritten as,
(τ) −4/3 −2 (2) −8/3 = τ − 2η0τ + ˜0 τ + ..., (17) 0 Energy-momentum tensor in second order hydrodynamics
From Stefan-Boltzmann’s law ( ∝ T 4) , we obtain the proper time dependence of the temperature T as, 1 η 3κ (−1 + a3 + b3 + c3) T (τ) = 1/4 − 0 + 0 0 τ 1/3 2τ 8τ 5/3 λ0(−7 + 9a3 + 9b3 + 9c3) η τ 0 + 1 − 0 π + ... (18) 12τ 5/3 6τ 5/3 From thermodynamic relation dE + PdV = TdS , the entropy as a function of τ is obtained as 3η 3η2 η τ 0 S(τ) = 3/4 1 − 0 τ −2/3 + 0 τ −4/3 − 0 π τ −4/3 0 2 4 2 λ0(−7 + 9a3 + 9b3 + 9c3) + 1 τ −4/3 4 9κ (−1 + a3 + b3 + c3) − 0 τ −4/3 + 0(τ −2) (19) 8 In the limit a=1, b=0 and c=0 the above expressions reduce to the one dimensional expansion case. A proposal for the gravity dual of anisotropic expansion
The five dimensional asymptotically AdS metric in Fefferman-Graham (FG) coordinates is given by
g dx µdx ν + dz2 ds2 = µν (20) z2
gµν is the four-dimensional metric which is expanded with respect to z as
(0) 2 (2) 4 (4) 6 (6) gµν (τ, z) = gµν (τ) + z gµν (τ) + z gµν (τ) + z gµν (τ) + .... (21)
The higher order terms in the expansion of g can be obtained by solving the five dimensional bulk Einstein’s equation with a negative cosmological constant recursively. 1 R − G R − 6G = 0 (22) MN 2 MN MN A proposal for the gravity dual of anisotropic expansion
We use Eddington-Finkelstein type coordinates and propose the following parametrization for the dual geometry in the three dimensional expansion case with Kasner space-time as the local rest frame of the fluid as follows
1 2 ds2 = −r 2Pdτ 2 + 2dτdr + r 2τ 2ae2Q−2R 1 + dx 2 uτ 2/3 1 2 2b R 2 2 2c R 2 + r τ e dx2 + r τ e dx3 (23)
where the new variable u is defined as u = rτ 1/3. With the boundary conditions P → 1, Q → 0 and R → 0 as r → ∞ the 5D bulk metric becomes
2 2 2 2a 2 2b 2 2c 2 ds |r→∞ = r −(dτ) + τ (dx1) + τ (dx2) + τ (dx3) + 2dτdr (24) A proposal for the gravity dual of anisotropic expansion
The parameters P, Q, R are expanded in powers of τ −2/3 as
−2/3 −4/3 P(τ, u) = P0(u) + P1(u)τ + P2(u)τ + ... −2/3 −4/3 Q(τ, u) = Q0(u) + Q1(u)τ + Q2(u)τ + ... −2/3 −4/3 R(τ, u) = R0(u) + R1(u)τ + R2(u)τ + ... (25)
where Pn, Qn and Rn are obtained by solving the 5D Einstein’s equation order by order in late time regime with the boundary conditions. The zeroth order solution is given by
w 4 P (u) = 1 − , Q = 0, R = 0 (26) 0 u4 0 0 where, w is a constant. A proposal for the gravity dual of anisotropic expansion
The corresponding Kretschmann scalar is obtained as
72w 8 32(a + b + c) R Rµναβ = 40 + + µναβ u8 uτ 2/3 8(ab + bc + ca) 16(a2 + b2 + c2) + + (27) u2τ 4/3 u2τ 4/3
After putting Kasner conditions the Kretschmann scalar is given by,
9w 8 R Rµναβ = 8 5 + + O(τ −2/3) (28) µναβ u8
which is regular except for the physical singularity at u = 0. This matches with the first order expansion result. Summary
1 We have studied the three dimensional anisotropic expansion of a conformal fluid by using Kasner space-time as the local rest frame of the fluid as an example of time dependent AdS/CFT correspondence. 2 Considering second order gradient expansion terms we have obtained the expressions for energy density, temperature and the components of the energy momentum tensor and entropy density in terms of Kasner parameters in the late time regime. 3 We have made a proposal for the 5-dimensional dual geometry with the boundary metric as the Kasner space-time in Eddington- Finkelstein coordinates in the large proper time approximation. The Kretschmann scalar for the zeroth order solution is found to be regular except at the physical singularity. References
1 R. A. Janik and R. Peschanski, “Asymptotic perfect fluid dynamics as a consequence of AdS/CFT,” Phys. Rev. D73 (2006) 045013, arXiv:hep-th/0512162 2 S. Sin, S. Nakamura and S. Kim, “Elliptic Flow, Kasner Universe and Holographic Dual of RHIC Fireball, ” arXiv:hep-th/0610113. 3 R. Baier, P. Romatschke, D. T. Son, A. O. Starinets, M. A. Stephanov, “Relativistic viscous hydrodynamics, conformal invariance, and holography” ,JHEP0804:100,2008 , (arXiv:0712.2451 [hep-th]) 4 S. Kinoshita, S. Mukohyama, S. Nakamura, K. Oda, “A Holographic Dual of Bjorken Flow”, Prog. Theor. Phys. 121 (2009) , (arXiv:0807.3797 [hep-th]) 5 P. P. Pruseth, S. Mahapatra, “Anisotropic expansion, second order hydrodynamics and holographic dual ”, (arXiv:2008.13699 [hep-th]) Thank you