A Holographic Dual of Bjorken Flow Shinji Mukohyama
A Holographic Dual of Bjorken Flow Shinji Mukohyama Institute for Physics and Mathematics of the Universe (IPMU) University of Tokyo with S. Kinoshita, S. Nakamura and K. Oda, to appear
Then and now • 2005 : the centenary of Special Relativity • 2006 -2007: the centenary of Yukawa & Tomonaga • Now • 2015 : the centenary of General Relativity Our dream now: Unification of Quantum Theory & General Relativity
Why are black holes important? • Cosmology & black holes will play as important a role in the development of quantum gravity as blackbody radiation & hydrogen atoms did in the history of quantum mechanics. • Black hole thermodynamics involves gravity GN quantum mechanics statistical mechanics k. B
Black brane to holography: Ad. S/CFT • Near horizon geometry of a BPS black brane is Ad. S 5 x. S 5. • Light d. o. f. (open string massless states) on the corresponding stack of D-branes is described by a N=4 4 D SYM theory. • Maldacena conjecture (Ad. S/CFT correspondence): There must be a relation between gravity in asymptotically Ad. S 5 x. S 5 spacetime and the N=4 4 D SYM theory.
String theory & real world • String cosmology • String phenomenology • Gauge/gravity (Ad. S/CFT) duality understanding gauge theory from gravity viewpoint • In this talk we consider a 5 D dynamical black brane to understand 4 D QGP.
RHIC (relativistic heavy ion collider) http: //www. bnl. gov/RHIC/heavy_ion. htm
Bjorken flow • Colliding pancakes “big bang” • QGP after “big bang” • Toy model 1) hydrodynamics 2) infinite pancakes 3) boost invariance
Bjorken flow • t = t cosh y, x 1 = t sinh y ds 2 = -dt 2 + dx 12 + dx 22 + dx 32 ds 2 = -dt 2 + t 2 dy 2 + dx 22 + dx 32 • Boost invariance t as time • Traceless & conservation eqs Tmu =
2 nd order hydrodynamics • Tmn = e umun + P qmn + Pmn umum = -1, qmn = gmn + umun , Pmm = 0 • Expand Pmn up to 2 nd derivatives of um Pmn = - hsmn + ht. P [ uls<mn>; l + smnul; l/3] + l 1 sl<msn>l + l 2 sl<m. Wn>l + l 3 Wl<m. Wn>l smn = 2 u<m; n> , Wmn = um; n - un; m ( A<mn> = qmaqnb(Aab+Aba)/2 - qmnqab Aab/3 ) • Assume the scaling (according to dimension)
Hydro solution • Solve the traceless & conservation equations, assuming that transport coefficients are small (derivative expansion). • Result Ttt = e 0 t-4/3 [ 1 - 2 h 0 t-2/3 + c 2 t-4/3 + … ] t-2 Tyy = e 0 t-4/3 [ 1/3 - 2 h 0 t-2/3 + (5/3)c 2 t-4/3 + … ] Txx = e 0 t-4/3 [ 1/3 - (1/3)c 2 t-4/3 + … ] c 2 = (3/2) h 02 + (2/3)(l 102 - h 0 t. P 0) • Coefficients h 0 and c 2 are undetermined.
Dual geometry • Microphysics is needed to determine h 0 and c 2. • However, a strongly coupled field theory is not easy to analyze. • Use the Ad. S/CFT correspondence as a tool. • Bjorken flow of large Nc N=4 SYM is dual to an asymptotically Ad. S 5 x. S 5 geometry. singularity EH center boundary AH
Dual geometry: ansatz • Ad. S-Sch : inconsistent with boundary metric • Our ansatz A, B, C : functions of ( t+ , r ) remaining gauge freedom: r r + f(t+) • Boundary metric @ if A 1, B 0, C 0 boundary condition
Boundary Tmn • ADM-like decomposition • Einstein + Gibbons-Hawking + counter terms • Brown-York’s prescription d=4 : • Scaling & limit Tmn from hydro
Late time expansion • Motivated by the hydro solution, let us expand metric functions by t-2/3. • Solve the Einstein equation order by order. • Result (in each order) i) The solution includes 3 integration constants. One is gauge d. o. f. but two are physical. ii) One of the two physical integration constants is determined by the boundary metric. iii) The last integration constant is related to the transport coefficient and determined by the regularity of the apparent horizon.
Summary • We proposed a dynamical and anisotropic 5 D brack brane as a holographic dual of Bjorken flow of strongly coupled large Nc N=4 4 D SYM theory plasma. • We performed a late-time expansion in Eddington. Finkelstein coordinates. • The dual geometry is regular on and outside an apparent horizon at all orders in the late-time expansion, provided that transport coefficients are chosen appropriately. • Conversely, the regularity of the dual geometry determines transport coefficients. • An example of time-dependent Ad. S/CFT.
- Slides: 26