Home >Documents >Numerical Relativity & AdS/CFT Correspondence - · PDF fileNumerical Relativity &...

Numerical Relativity & AdS/CFT Correspondence - · PDF fileNumerical Relativity &...

Date post:16-Aug-2018
Category:
View:215 times
Download:0 times
Share this document with a friend
Transcript:
  • Numerical Relativity& AdS/CFT Correspondence

    Christian Ecker

    Einstein Toolkit EU School & WorkshopTrento

    June 16, 2016

    Based on work with Daniel Grumiller, Stefan Stricker 1506.02658 (JHEP);& Wilke van der Schee, Philipp Stanzer (16XX.XXXXX)

  • Quark-gluon plasma in heavy ion collisions

    Quark-gluon plasma (QGP) is a deconfined phase of quarks and gluonsproduced in heavy ion collision (HIC) experiments at RHIC and LHC.

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 1/17

  • Why AdS/CFT?The QGP produced in HIC's behaves like a strongly coupled liquid rather than a weakly coupled gas.

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 2/17

    Perturbative QCD? Lattice QCD?Not at strong

    coupling!Not for dynamics!

  • AdS/CFT correspondence

    AdS/CFT correspondence:

    Type IIB string theory on AdS5 x S5 is equivalent to N =4 super symmetric SU(NC) Yang-Mills theory in 4D.

    [Maldacena 97]

    Supergravity limit:

    Strongly coupled large NC N =4 SU(NC ) SYM theory is equivalent to classical supergravity on AdS5.

    Boundary:4-dim. CFT

    Bulk:5-dim. GR

    z

    Strategy: Use N=4 SYM as toymodel for QCD in the strongly coupled regime. Build a gravity model dual to HICs, like colliding gravitational shock waves. Switch on the computer and solve the 5-dim. gravity problem numerically. Use the holographic dictionary to compute observables in the 4 dim. field

    theory form those gravity result.

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 3/17

  • Solving the Einstein equations on asymptotically AdS

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 4/17

    Eddington-Finkelstein gauge (light-like slicing) decouples the EE into a nested set of ODEs. (method of characteristics)

    BC's: boundary metric is 4-dim Minkovski = background metric of the boundary QFT

    IC's: two gravitational shock waves in AdS = Lorentz contracted nuclei in the QFT

    The time evolution is done with a 4th order Runge-Kutta method (RK4).

    These ODEs can be efficiently solved with a spectral method.

    We want to solve the 5 dim. vacuum Einstein equations (EE) with negative cosmological constant .

    AdS is not globally hyperbolic need IC's & BC's to formulate a well defined initial value problem (IVP).

  • Holographic thermalization

    Thermalization = Black hole formation

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 5/17

  • Entanglement entropy

    Divide the system into two parts A,B.The total Hilbert space factorizes:

    quantum field theory

    lattice model

    BA

    A B

    The reduced density matrix of A is obtained by the trace over

    Entanglement entropy is defined as the von Neumann entropy of A:

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 6/17

  • Entanglement entropy in a two quantum bit system

    Entanglement entropy is a measure for entanglement in a quantum system.

    A (maximally) entangled state in a two spin 1/2 system:

    Alice Bob

    ?

    A product state (not entangled) in a two spin 1/2 system:

    Alice Bob

    ? ?

    Consider a quantum system of two spin 1/2 dof's.Observer Alice has only access to one spin and Bob to the other spin.

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 7/17

  • Holographic entanglement entropy

    A B

    zextremal surface

    w, y, ...

    x

    t=const.

    [Ryu-Takayanagi 06,Hubeny-Rangamani-Takayanagi 07]

    D dim. CFT

    UV cut off in CFT

    D+1 dim. GR

    Within AdS/CFT entanglement entropy can be computed form the area of minimal (extremal) surfaces in the gravity theory.

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 8/17

  • Holographic entanglement entropy In practice computing extremal co-dim. 2 hyper-surfaces is

    numerically involved. Can we somehow simplify our lives?

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 9/17

    [E. Tonni 14: minimal surface for a star shaped boundary region (red) in AdS4 computed with Surface Evolver]

    Yes we can!

    [work in progress: CE-Grumiller-Khavari]

  • Entanglement entropy from geodesics

    z

    w, y, ...A

    homogeneousdirections

    B

    x

    t=const.D dim. CFT

    D+1 dim. GR

    UV (IR) cut off in CFT (GR)

    L

    Consider a stripe region of infinite extend in homogeneous directions of the geometry.The entanglement entropy is prop. to the geodesics length in an auxiliary spacetime.

    geodesic

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 10/17

  • Numerics: relax, don't shoot!

    There are two standard numerical methods for solving two point boundary value problems. Shooting:Very sensitive to initialization on asymptotic AdS spacetimes.

    Relaxation:Converges very fast if good initial guess is provided.

    Geodesic equation as two point boundary value problem.

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 11/17

    [see Numerical Recipes]

  • Holographic shock wave collisions

    HIC is modeled by two colliding sheets of energy with infinite extend in transverse direction and Gaussian profile in beam direction. [Chesler-Yaffe 10]

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 12/17

  • Wide vs. narrow shocks

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 13/17

    Two qualitatively different dynamical regimes Wide shocks (~RHIC): full stopping

    [Solana-Heller-Mateos-van der Schee 12]

    Narrow shocks (~LHC): transparency

    y

  • Geodesics and apparent horizon

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 14/17

  • Entanglement entropy

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 15/17

    t=const.

    L

    A

    z

    [CE-Grumiller-Van der Schee-Stanzer-Stricker 16XX.XXXXX]

    PreliminaryPreliminary

  • Is the numerics right?

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 16/17

  • Summary AdS/CFT allows to study the real time dynamics of strongly coupled

    QFT's by solving the IVP of (classical) supergravity theories. Entanglement entropy may serve as an order parameter for the full

    stoppingtransparency transition in holographic shock wave collisions.

    Christian Ecker (TU-Wien) Trento, Jun. 16, 2016 17/17

    Work in progress Going beyond supergravity: string corrections, semi-holography,

    On the field theory side the null energy condition (NEC) is violated in narrow shock wave collisions. The quantum null energy condition (QNEC) is conjectured to give an upper bound for this violation.

    [CE-Mukhopadhyay-Preiss-Rebhan-Stricker]

    [CE-Grumiller-Van der Schee-Stanzer-Stricker 16XX.XXXXX]

    [CE-Grumiller-Van der Schee-Stanzer]

    Slide 1Slide 2Slide 3Slide 4Slide 5Slide 6Slide 7Slide 8Slide 9Slide 10Slide 11Slide 12Slide 13Slide 14Slide 15Slide 16Slide 17Slide 18