Black holes as Information Scramblers

Computing Dynamical AdS spacetimesBlack hole formation as a model for thermalisationWilke van der Schee

Holography near and far-from equilibrium,University of Alabama, Tuscaloosa, 26 Oct 2015

1OutlineDynamical black hole formationEinstein equations in characteristic formulationCFT dual description

AdS/CFT already covered in other lectures, two points though:Always work in large coupling expansion, not necessarily close to QCDMay learn qualitative lessons, and can attempt quantitative comparison

Numerical problems, preferably in MathematicaSpectral methods to solve ordinary ODEs (non-linear on website)Some useful Mathematica

Flash of recent researchA simple complete model for off-central heavy ion collisions2/17Wilke van der Schee, MIThttp://sites.google.com/site/wilkevanderschee/Many states + linearized2Einstein equationsGeneralized Eddington-FinkelsteinDramatically simplifies equations for time evolutionevents near boundary propagate instantaneously into bulkevents near horizon (IR) require time to propagate to boundary (UV)

Fefferman-Graham coordinatesPreferred choice to do holographic renormalization (in literature)Sometimes easier analytically (i.e. single gravitational wave)

Quite often necessary to change coordinates (near boundary)

3/17Wilke van der Schee, MIT

Many states + linearized3Characteristic Einstein equationsThe schemeFirst solve for S (better for stability, as opposed to evolving)Non-homogeneous: same for fluxesThen dotted derivative of spatial metric (start with S)Then A

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H. Bondi, Gravitational Waves in General Relativity (1960)P.M. Chesler and L.G. Yaffe, Horizon formation and far-from-equilibrium isotropization in supersymmetric Yang-Mills plasma (2008)

Many states + linearized4Characteristic Einstein equationsThe characteristic formulationUse generalized Eddington-Finkelstein coordinatesExercise: show that wave equation in null coordinates becomes a first order system, at the price of the boundary condition propagating `instanteneouslySeparate out spatial determinant metric (S)Time derivatives along null geodesics:Tip: be careful with non-commuting operations!

Two constraintsOn equal time slice: consistent initial conditionsCan also be solved (i.e. B S), much easier than in ADM!!On constant r slice: usually at boundary, constraint on NB coefficientsEquivalent to SE-conservation in field theoryEssential check to see if numerics converge5/17Wilke van der Schee, MIT

Many states + linearized5Characteristic Einstein equationsA nested scheme, also called Bondi-Sachs formulationNon-linear Einstein equations decouple in nested set of ODEsNo approximations neededGeneralizes relatively straightforwardly to Maxwell and/or Scalar fields

Why isnt this always used?Null slice has non-trivial boundary behaviourExactly wanted in AdS/CFT, but awkward in i.e. binary mergersTwo null rays can meet, causing a caustic and coordinate singularityIn AdS this usually does not happen (balance horizon position and energy on boundary)Instantaneous propagation can make codes unstableSolved by using highly accurate spectral codes (perhaps mainly suitable for `simple problems)6/17Wilke van der Schee, MITMany states + linearized6Near-boundary expansionNear-boundary expansion till some orderSeries expand Einstein equationsSeries expand functions in metric (A, B, etc)First series expand till highest order, then solveSeries expand till order needed at every order

Find problems fixing 4th order coefficients, and possibly at 8th orderFind gauge freedom forCurved boundary: find logarithmic terms, or even log2

Requires a bit of playing, game is to go till very high order

7/17Wilke van der Schee, MIT

Many states + linearized7Holographic renormalisationVery useful formula from Skenderis et al

Refers to near-boundary expansion in Fefferman-Graham coordinatesg(2) generically zero for flat boundaryEasy solution: solve coordinate transform near boundary

In cases with conformal anomaly there is a scheme dependence (see also recent Kiritsis et al, 1503.07766)

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S. de Haro, S. N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence (2001)Many states + linearized8Frequently used mathematicaUse of pure functionsFunction[x, x^3][5] 125Shorthand: #^3&[5] 125

Replacement with pure function:

Making a packageInitialization cells are evaluated when loading packageMake equations and write to file, then load file with initialization cellEdit Preferences Advanced Option Inspector Notebook options File Options Autogenerated package Manual

Last trick: not too much Mathematica, to keep understanding what Mathematica does, and hence be able to optimise speed (Compile?)9/17Wilke van der Schee, MIT

Many states + linearized9Spectral methodsTwo examples:

Spectral methods:

Non-linear problem: use NewtonLazy? NDSolve`FiniteDifferenceDerivative[1,grid, order]

10/17Wilke van der Schee, MITBoyd: Chebyshev and Fourier spectral methods

Many states + linearized10Collisions at infinitely strong coupling11/15

Match longitudinal profile of energy density to nucleiApproximately homogeneous in transverse plane

J. Casalderrey-Solana, M.P. Heller, D. Mateos and WS, From full stopping to transparency in a holographic model of heavy ion collisions (2013)

Wilke van der Schee, MIT

Benchmark:11Colliding two nuclei:12/15Locally in transverse plane: use shock waves (i.e. ~Gaussian rapidity) Go and run hydro (MUSIC) and get particle spectra Wilke van der Schee, MIT

MUSIC Results, LHCDirected flow: right ball-park values

Note: somewhat subtle to measure; event-plane etcCould be very sensitive to viscosity13/15ALICE, Directed flow of charged particles at mid-rapidity relative to the spectator plane in PbPb collisions at sNN = 2.76 TeV (2013)Wilke van der Schee, MITWith Bjrn Schenke

MUSIC Results, LHCParticle spectra in longitudinal direction:

Rescaled initial energy density by factor 20 Profile is significantly too narrow

14/15ALICE, Bulk Properties of Pb-Pb collisions at sNN = 2.76 TeV measured by ALICE (2011)Wilke van der Schee, MIT

With Bjrn Schenke

Shock waves without symmetries15/17Challenging computation (grid 40x145x39x39, ~1 month on desktop)P. Chesler and L. Yaffe, Holography and off-center collisions of localized shock waves (2015)Wilke van der Schee, MITNumerics is a bit more involved, hard to keep AH fixed, use domain decomposition

15References for learning numericsCharacteristic formulation:Chesler-Yaffe: 1309.1439Casalderrey, Heller, Mateos, WS, Triana: 1407.1849, 1304.5172Balasubramanian, Herzog: 1312.4953

ADM formulationHeller, Janik, Witaszczyk : 1203.0755Bantilan, Gubser, Pretorius: 1201.2132

Elliptic Einstein-DeTurck: Donos, Gauntlett: 1409.6875

Reviews/books:Grandclment, Novak: livingreviews.org (see also Winicour)Casalderrey, Liu, Matoes, Rajagopal, Wiedemann: 1101.0618 (and book!)Boyd: Chebyshev and Fourier spectral methods16/17Wilke van der Schee, MIT

sites.google.com/site/wilkevanderschee/Many states + linearized16solve linearised Einstein Equations for arbitrary initial conditions, find QNMExpand the metric (slide 3) around the thermal metric (take T=1/p):

Show that the Einstein equations reduce to a simple equation for Optional: solve near-boundary expansion, and obtain

Rewrite equation in terms ofUse spectral methods to solve the ODE in z to obtainUse i.e. or something similar Use NDSolve for time-stepping, and obtainPlot the CFT pressure anisotropy and identify the oscillation and damping rate of the lowest AdS quasi-normal mode17/17Wilke van der Schee, MIT

Many states + linearized17