Holographic Description of Lifshitz Fluids and Black Holes
The study explores the correspondence between black holes and fluids in the context of hyperscaling-violating Lifshitz hydrodynamics. It delves into Lifshitz spacetimes, their holographic descriptions, and the fluid/gravity correspondence for Lifshitz spacetimes. The Einstein-Maxwell-Dilaton model and first-order solutions are discussed, along with the stress-energy tensor for Lifshitz fluid, providing insights into the behavior and properties of fluids in non-relativistic gravity backgrounds.
Download Presentation
Please find below an Image/Link to download the presentation.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.If you encounter any issues during the download, it is possible that the publisher has removed the file from their server.
You are allowed to download the files provided on this website for personal or commercial use, subject to the condition that they are used lawfully. All files are the property of their respective owners.
The content on the website is provided AS IS for your information and personal use only. It may not be sold, licensed, or shared on other websites without obtaining consent from the author.
E N D
Presentation Transcript
Hyperscaling-violating Lifshitz hydrodynamics from black holes Yoshinori Matsuo National Taiwan University In collaboration with Elias Kiritsis arXiv:1508.02494 arXiv:1611.nnnnn
Introduction The holography gives the correspondences between black holes and fluids. [Policastro-Son-Starinets, 02] [Bhattacharyya-Hubeny-Minwalla-Rangamani, 07] [Son-Starinets, 02] Lifshitz spacetimes give holographic description of the Lifshitz scaling invariant theories. [Son, 08] [Balasubramanian-McGreevy, 08] ??2= ?2???2+ ?2???2+??2 [Kachru-Liu-Mulligan, 08] [Taylor, 08] ?2 Boundary theories in the Lifshitz spacetimes have the Newton-Cartan geometry, which is a non-relativistic gravity, as a background. [Christensen-Hartong-Obers-Rollier, 14] [Hartong-Kiritsis-Obers, 14, 14, 15] Here we would like to see that Fluid/gravity correspondence for the Lifshitz spacetime give a holographic description of fluids in Newton-Cartan background.
Fluid/Gravity correspondence In CFT side, matters behaves as a fluid at finite temperature. In AdS side, black holes appear at finite temperature. Fluids in CFT corresponds to Black holes in AdS, at finite temperature. For fluids Energy density (~ temperature) depends on position ?? Horizon radius ( temperature) of BH depends on ?? introduce (??-dependent) boost to BH geometry Fluids have flow The modified black hole solution describes fluids in dual field theory We generalize the Fluid/Gravity correspondence to asymptotically Lifshitz spacetime.
Einstein-Maxwell-Dilaton model The action 16?? ??+1? ? ? 2 1 1 4e???2 1 2??2 ? = This model has the Lifshitz spacetime as a solution ?2= 2? 1 ? 1 ? + ? 1 ? + ? 2 2 2 ? 1 ? + ? 1 ??2= ?2???2+??2 ?2???2 ?2+ = ? ???= ??2(1 ?) ? = ???+? 1?? ??2= The metric has Lifshitz scaling symmetry ? ??? ? ? 1? ?? ? ??
First order solution (for ? = 4, ? = 2) We modify the asymptotically Lifshitz black hole solution and introduce the correction terms such that it is a solution of EOM. 2 ??2= ?4? ? ??2+ 2? ?? ?? + ?2??? ??(?)?? +2 3?2????(?)??2 ?2? ? ?????? ??(?)?? ? = ?(?) ?5? ? 1 3?3????? ??? ??(?)?? ?? ?(?)?2?? + ??(?) ??? ??(?)?? 5(?) ?5 ? ? = 1 ?0 ???= ?(?)?2(1 ?) 3 ?3 ?0 ?(?5 ?0 ???= ????+ ???? 2 ? ? = ?? 3??????? 5) It must satisfy the following constraints 0 = ???0+ ?????0+1 3?0???? 0 = ??? + ????? ????? 5 0 = ????+ ??????+ ??????+ ????0
Stress-energy tensor for Lifshitz fluid The stress-energy tensor is calculated from the solution by using GKPW as 0= ?= + ? ??+ ??????+ ???? ?0 ?0 ?= ???? ???? 0= 0 ?? ?? The energy density, pressure and charge density are ? =? + 3 ? 1 16??? ? 3 ?0? 16???0?+3 16???0?+3 ? = ? = = 4? The shear viscosity and thermal conductivity are 1 1 3 8?(? 1)?0?+1 ? = 16???0 ? = Bulk viscosity vanishes. Energy density and pressure satisfy the Lifshitz scaling condition ? = ? 1 ?
Non-relativistic fluid equations Constraints for parameters ?0, ??, ?, etc. give fluid equations 0 = ?? + ???? + + ? ???? 1 2??????? ?????? 0 = ??? ?????? + ?? ?? 0 = ??? + ????? Ordinary fluid equations are 0 = ?? + ???? + + ? ???? 1 2??????? ?????? 0 = ??? + ?????+ ??????? ?????? + ????? 0 = ??? + ????? Energy conservation (1stline) and continuity equation (3rdline) agree. But the Navier-Stokes does not agree. The difference can be explained by using the Newton-Cartan theory.
Newton-Cartan geometry Non-relativistic theory of gravity Described by Newton-Cartan data ??, ??, ??, ?? ??: metric on the time slice ??: time direction ??: gauge field of Newton gravity ??: velocity of space In relativistic theory, stress-energy tensor ???is associated to metric ???. Associated operators to Newton-Cartan data are Energy density (flow): ? Viscous stress tensor: ??? Mass current: ?? Momentum density: ?? The Newton-Cartan theory has Milne boost symmetry ? ? + ??????? 1 2?2? ?? ??+ ?? where ????= 0
Fluid equations and Newton-Cartan theory In our case, velocity of space is not ??= (1,0) but (1,??) By using the Milne boost, ? ? = ? + ????? 1 2?2?? ??= 1,0 (1,??) We identify ?? with ? = ? + ????? 1 2?2?? . Then, the Navier-Stokes equation from Lifshitz black hole solution is ??? ?????? = ???? is expressed as ??? + ?????+ ??????? ?????? = ????? This is standard Navier-Stokes equation with external force from ?? . Therefore, ?? in black hole solution should be identified with the gauge field in Newton-Cartan theory ? .
Hyperscaling-violation in geometry To consider hyperscaling-violation, we consider the following action 1 ? ? 2 e ?? 1 4e???2 1 16?? ??+1? 2??2 ? = Lifshitz spacetime solution with hyperscaling-violation ? 1 ?2???2+??2 2? (? 1) ? 1 ?? ? ?2???2 ??2= ? ?2+ ? 2? 2 ???= ? (? 1) ? 1 ?? 2?1 ? = ? ? ??+? 1 ??? The additional factor in the metric breaks the Lifshitz scaling symmetry
Coordinate redefinition The black hole geometry is given by 2? 3 3 ?? ? 2+ ? ?2= ? ? 1 ?2????2+ 2?? 1???? + ?2??? ???? ? ?+3 ? ? = 1 ?0 ??+3 ? For hydrodynamics, we introduce ??-dependence to ?0, ?? and ?, etc. ?(?) appears also in the overall factor of the metric The background on boundary becomes non-flat In order to have flat background, we take ? ?? ? 3 3 ??? Then, thermodynamic variables are also rescaled, for example, temperature is ? =? + 3 ? 4? ? 3 3 ??0? ?
First law of thermodynamics Energy ?, entropy ? and (particle number) charge ? ? ? = ? =3 ? ?: Volume of space 3(3 ?)?0?+3 ?? 16??? 1 4??0 ? 1 16???? 3 ?? ? = ?? = ? = ?? = From first law of thermodynamics ?? = ??? ??? + ??? Temperature ?, pressure ? and chemical potential ? are estimated ? ? ? ? =? + 3 ? 3 ? 1?1+ 3 3 ??0?+3 ? ? = 3(3 ?)?0? ? 4? ? 1 ? ? 3(3 ?)?0?+3 ? ? = ? 16?? 3 Hyperscaling-violation gives non-zero chemical potential
Hyperscaling-violation and scaling dimensions Scaling dimensions in Lifshitz scaling theory in ?-dimensional spacetime ??= 1 : energy density ? = ? ?: entropy density = ? + (? 1) ? = ? 1 Due to the coordinate redefinition, the scaling dimensions are modified to ? ? ??= 1 + ? = ? + ? 1 ? 1 ? ? = ? 1 ? = ? + ? 1 ? 1? The parameter ? is nothing but the hyperscaling-violation exponent. The ward identity for the Lifshitz scaling symmetry is modified as ? ? = ? 1 ? ? ? 1
Stress-energy tensor and fluid equation The stress-energy tensor is calculated as 0= ?= + ? ??+ ??????+ ???? ?0 ?0 ?= ???? ???? 0= 0 ?? ?? The transport coefficients are given by ? 1 1 8 ? 1 ?? 3 ? 3 3 ??0?+1 ? ? = 0 ? = 16???0 ? = The bulk viscosity is zero even with the hyperscaling-violation Fluid equations are given by 0 = ?? + ???? + + ? ???? 1 2??????? ?????? 0 = ??? ???? ?????? + ?? ?? 0 = ??? + ?????
Zero bulk viscosity and dimensional reduction The Lifshitz spacetime with hyperscaling-violation can be obtained by dimensional reduction of the Lifshitz spacetime without hyperscaling-violation. Lifshitz black hole with hyperscaling-violation Non-relativistic fluid with zero bulk viscosity correspondence Dimensional reduction Dimensional reduction ? correspondence Lifshitz black hole without hyperscaling-violation Non-relativistic fluid with zero bulk viscosity
Zero bulk viscosity and dimensional reduction However, naively, dimensional reduction of fluid with zero bulk viscosity gives fluid with non-zero (positive) bulk viscosity. ????+ ???? 2 ????????= ????+ ???? 2 3???????+? 3 ??????? 3? (shear and bulk viscosity are the coefficients of traceless and trace part of ????+ ????) Why Lifshitz black hole with hyperscaling-violation corresponds to fluid with zero bulk viscosity? Volume of compactified directions is not constant but depends on the position of the other directions. In this case, expansion appears in the following combination = ???? ? 1??? + ????? For ? = ?, charge conservation for ? gives = 0. For ? = 1, = ???? is expansion and ? is bulk viscosity.
Summary We have studied fluid/gravity correspondence for Lifshitz spacetime Energy conservation and continuity equations agree with those for ordinary non-relativistic fluids. By identifying the gauge field as that in the Newton-Cartan theory, the Navier-Stokes equation (from black hole solution) agrees with that for the standard fluids. We have also considered fluid/gravity correspondence for the Lifshitz black hole geometry with hyperscaling-violation. We have introduced the coordinate redefinition such that the background on the boundary is flat. The thermodynamic relations implies non-zero chemical potential. The bulk viscosity is zero even with the hyperscaling-violation. Zero bulk viscosity is consistent with the dimensional reduction with ??-dependent radius.
