Enstrophy and AdS4 Black Branes: Avant-Garde Methods for QFT and Gravity
"Explore the avant-garde methods of using enstrophy and AdS4 black branes in the realms of quantum field theory and gravity, shedding new light on relativistic hydrodynamics and its correspondence with gravity. Dive into the manifestation of enstrophy in the gravity dual, unraveling the interplay between fluid dynamics and gravitational phenomena."
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Enstrophy and AdS4Black Branes Avant-garde methods for QFT and Gravity, Nazareth, 2019 Raja Marjieh In collaboration with: Prof. Amos Yarom and Natascia Pinzani-Fokeeva
Outline Introduction Relativistic Hydrodynamics and Enstrophy The Fluid/Gravity Correspondence AdS4Black Branes Summary
Introduction For 2+1 dimensional flows there is a special quantity that is conserved at large Reynolds number. vorticity2 Enstrophy plays an important role in turbulent phenomena.
Introduction Boosted AdS?+1black branes are dual to (conformal) relativistic hydrodynamics in ? dimensions. [Bhattacharyya, Hubeny, Minwalla, Rangamani (2008)] [Haack, Yarom (2008)] Natural Question: What is the manifestation of enstrophy in the gravity dual?
Relativistic Hydrodynamics Universal IR effective description of many body systems. Conserved currents of the hydrodynamic theory are given using local temperature ?, velocity field ??, chemical potential ?. Derivative corrections suppressed by ??? For a neutral fluid, Energy density Pressure
Relativistic Hydrodynamics and Enstrophy The hydrodynamic equations of motion are conservation laws. For a neutral fluid energy-momentum conservation The (relativistic) enstrophy current is given by Entropy density
Enstrophy and Symmetry In ? = 2 + 1 and in the absence of dissipation (on-shell) Noether theorem look for underlying symmetry. Ideal hydrodynamics can be described as a sigma model. Target space Physical space of fluid ???? Worldvolume Labels fluid elements ? ?
Enstrophy and Symmetry Fluid data (??,?) are specified by a worldvolume vector ??(?). The hydrodynamic effective action is given by where
Enstrophy and Symmetry Under a variation of the action ?? For ?? to be a genuine (off-shell) symmetry we must have
Enstrophy and Symmetry In the conformal and flat space limit But symmetry of the action Indeed, the EOM ?????= 0 are invariant under symmetry of the equations of motion.
Enstrophy and Symmetry In fact, the invariance under follows from diff. invariance and that (on-shell) If ??is a solution then it is invariant along the integral curves of ??.
Fluid/Gravity Correspondence A map between asymptotically AdS solutions in ? = ? + 1 and conformal relativistic hydrodynamics in ? dimensions. The starting point is a boosted black brane in ? dimensions where
Fluid/Gravity Correspondence The parameter ? Hawking temperature. We then let ??,? vary slowly ??? ,? ? Solve Einstein eq. perturbatively in transverse derivatives ??. The constraint equations are the fluid EOM ?????= 0.
AdS4and Enstrophy For ? = 4 we can again construct the enstrophy current. Resulting ???is not ideal ??is approximately conserved To linear order in derivatives ? = ?2+ ? ?2. ? ? 0,? ? ?? Thus, ??can be easily lifted to 4d by noting that for ??
AdS4and Enstrophy ?locally) One can uplift ??into ??such that (take ??= ?? For r-v geodesics ??= ??,??,0,0 + ? ? Conserved charge
Summary We have recast enstrophy conservation as a Noether current. We identified the symmetry transformation associated with it at the level of the equations of motion. Using the Fluid/Gravity correspondence we mapped this fluid symmetry into geometrical properties of the dual description.
