Motivation and overview From gravity to fluid The end From Gravity to Fluid Yu Tian ( ✵ è ) 1 1 College of Physical Sciences, Graduate University of Chinese Academy of Sciences ( ✲ ý Ñ ❢ ❜ ✔ ✈ ✤ ❜ ✐ ✝ Ñ ❢❢ ❜ ) YITP 2012 Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview From gravity to fluid The end Outline Motivation and overview 1 Holography (bulk/boundary correspondence) In equilibrium: thermodynamics and phase transition In non-equilibrium: transportation and entropy production From gravity to fluid 2 The gravity/fluid case Non-relativistic long-wavelength expansion on an arbitrary cutoff surface Incompressible Navier-Stokes equations from Petrov-like condition Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production Outline Motivation and overview 1 Holography (bulk/boundary correspondence) In equilibrium: thermodynamics and phase transition In non-equilibrium: transportation and entropy production From gravity to fluid 2 The gravity/fluid case Non-relativistic long-wavelength expansion on an arbitrary cutoff surface Incompressible Navier-Stokes equations from Petrov-like condition Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production Holography: a brief introduction Early (rough) ideas of holography G. ’t Hooft, [gr-qc/9310026]. L. Susskind, J. Math. Phys. 36, 6377 (1995). A more precise prescription: AdS/CFT J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998). Basic principle (Euclidean): � Z B d + 1 [ ¯ D ψ exp ( − I CFT [ ¯ φ ] = φ , ψ ]) � � � Z B d + 1 [ ¯ φ + δ ¯ φ ] = Z B d + 1 [ ¯ S d δ ¯ φ ] exp φ O φ CFT Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production Holography: a brief introduction Early (rough) ideas of holography G. ’t Hooft, [gr-qc/9310026]. L. Susskind, J. Math. Phys. 36, 6377 (1995). A more precise prescription: AdS/CFT J. Maldacena, Adv. Theor. Math. Phys. 2, 231 (1998). E. Witten, Adv. Theor. Math. Phys. 2, 253 (1998). Basic principle (Euclidean): � Z B d + 1 [ ¯ D ψ exp ( − I CFT [ ¯ φ ] = φ , ψ ]) � � � Z B d + 1 [ ¯ φ + δ ¯ φ ] = Z B d + 1 [ ¯ S d δ ¯ φ ] exp φ O φ CFT Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production Holography: a brief introduction More info from superstring theory Classical limit ↔ Large N c limit Weak coupling ↔ Strong coupling Generalization: bulk/boundary correspondence AdS/QCD( r s ✴ ③ ), AdS/CMT, HEE, gravity/fluid, . . . � Z bulk [ ¯ D ψ exp ( − I FT [ ¯ φ ] = φ , ψ ]) � � � Z bulk [ ¯ φ + δ ¯ φ ] = Z bulk [ ¯ bdry δ ¯ φ ] exp φ O φ FT Basic dictionary: φ | bdry ↔ Non-dynamical field ¯ φ Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production Holography: a brief introduction More info from superstring theory Classical limit ↔ Large N c limit Weak coupling ↔ Strong coupling Generalization: bulk/boundary correspondence AdS/QCD( r s ✴ ③ ), AdS/CMT, HEE, gravity/fluid, . . . � Z bulk [ ¯ D ψ exp ( − I FT [ ¯ φ ] = φ , ψ ]) � � � Z bulk [ ¯ φ + δ ¯ φ ] = Z bulk [ ¯ bdry δ ¯ φ ] exp φ O φ FT Basic dictionary: φ | bdry ↔ Non-dynamical field ¯ φ Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production The bulk/boundary correspondence Under the classical approximation of the bulk gravity, Z bulk [ ¯ φ ] → exp ( − I bulk [ ¯ φ ]) = ⇒ � exp ( − I bulk [ ¯ D ψ exp ( − I FT [ ¯ φ ]) = φ , ψ ]) with I bulk [ ¯ φ ] the on-shell action (Hamilton’s principal function). Variation with respect to ¯ φ gives − δ I bulk [ ¯ O φ = − δ I FT [ ¯ φ ] φ , ψ ] � � φ ( x ) = O φ ( x ) FT , δ ¯ δ ¯ φ ( x ) Further variations give the correlations of O φ on the boundary. Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production The bulk/boundary correspondence Under the classical approximation of the bulk gravity, Z bulk [ ¯ φ ] → exp ( − I bulk [ ¯ φ ]) = ⇒ � exp ( − I bulk [ ¯ D ψ exp ( − I FT [ ¯ φ ]) = φ , ψ ]) with I bulk [ ¯ φ ] the on-shell action (Hamilton’s principal function). Variation with respect to ¯ φ gives − δ I bulk [ ¯ O φ = − δ I FT [ ¯ φ ] φ , ψ ] � � φ ( x ) = O φ ( x ) FT , δ ¯ δ ¯ φ ( x ) Further variations give the correlations of O φ on the boundary. Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production The bulk/boundary correspondence Under the classical approximation of the bulk gravity, Z bulk [ ¯ φ ] → exp ( − I bulk [ ¯ φ ]) = ⇒ � exp ( − I bulk [ ¯ D ψ exp ( − I FT [ ¯ φ ]) = φ , ψ ]) with I bulk [ ¯ φ ] the on-shell action (Hamilton’s principal function). Variation with respect to ¯ φ gives − δ I bulk [ ¯ O φ = − δ I FT [ ¯ φ ] φ , ψ ] � � φ ( x ) = O φ ( x ) FT , δ ¯ δ ¯ φ ( x ) Further variations give the correlations of O φ on the boundary. Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production The bulk/boundary correspondence Important examples (with n µ the unit normal of the boundary) Fields Bulk Boundary − n µ F µ a | bdry Current � J a � Electromagnetic Brown-York t ab | bdry T ab � � Gravitational Stress tensor Additional dictionary Black holes ↔ Thermal field theory Local Hawking temperature ↔ Temperature Holographic renormalization group (RG) flow Position of the boundary ↔ Energy scale Black hole horizon ↔ IR limit Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production The bulk/boundary correspondence Important examples (with n µ the unit normal of the boundary) Fields Bulk Boundary − n µ F µ a | bdry Current � J a � Electromagnetic Brown-York t ab | bdry T ab � � Gravitational Stress tensor Additional dictionary Black holes ↔ Thermal field theory Local Hawking temperature ↔ Temperature Holographic renormalization group (RG) flow Position of the boundary ↔ Energy scale Black hole horizon ↔ IR limit Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production The bulk/boundary correspondence Important examples (with n µ the unit normal of the boundary) Fields Bulk Boundary − n µ F µ a | bdry Current � J a � Electromagnetic Brown-York t ab | bdry T ab � � Gravitational Stress tensor Additional dictionary Black holes ↔ Thermal field theory Local Hawking temperature ↔ Temperature Holographic renormalization group (RG) flow Position of the boundary ↔ Energy scale Black hole horizon ↔ IR limit Yu Tian ( ✵ è ) From Gravity to Fluid
Motivation and overview Holography (bulk/boundary correspondence) From gravity to fluid In equilibrium: thermodynamics and phase transition The end In non-equilibrium: transportation and entropy production Outline Motivation and overview 1 Holography (bulk/boundary correspondence) In equilibrium: thermodynamics and phase transition In non-equilibrium: transportation and entropy production From gravity to fluid 2 The gravity/fluid case Non-relativistic long-wavelength expansion on an arbitrary cutoff surface Incompressible Navier-Stokes equations from Petrov-like condition Yu Tian ( ✵ è ) From Gravity to Fluid
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