1. COMPUTING DYNAMICAL ADS SPACETIMES BLACK HOLE FORMATION AS A MODEL FOR THERMALISATION Wilke van der Schee Holography near and far-from equilibrium, University of Alabama, Tuscaloosa, 26 Oct 2015

2. OUTLINE Dynamical black hole formation Einstein equations in characteristic formulation CFT dual description AdS/CFT already covered in other lectures, two points though: Always work in large coupling expansion, not necessarily close to QCD May learn qualitative lessons, and can attempt quantitative comparison Numerical problems, preferably in Mathematica Spectral methods to solve ordinary ODEs (non-linear on website) Some useful Mathematica Flash of recent research A simple complete model for off-central heavy ion collisions 2/17 Wilke van der Schee, MIT http://sites.google.com/site/wilkevanderschee/

3. EINSTEIN EQUATIONS Generalized Eddington-Finkelstein Dramatically simplifies equations for time evolution ‘events’ near boundary propagate ‘instantaneously’ into bulk ‘events’ near horizon (IR) require time to propagate to boundary (UV) Fefferman-Graham coordinates Preferred choice to do holographic renormalization (in literature) Sometimes easier analytically (i.e. single gravitational wave) Quite often necessary to change coordinates (near boundary) 3/17 Wilke van der Schee, MIT

4. CHARACTERISTIC EINSTEIN EQUATIONS The scheme First solve for S (better for stability, as opposed to evolving) Non-homogeneous: same for fluxes Then dotted derivative of spatial metric (start with S) Then A 4/17 Wilke van der Schee, MIT 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)

5. CHARACTERISTIC EINSTEIN EQUATIONS The characteristic formulation Use generalized Eddington-Finkelstein coordinates Exercise: show that wave equation in null coordinates becomes a first order system, at the price of the boundary condition propagating `instanteneously’ Separate out spatial determinant metric (S) Time derivatives along null geodesics: Tip: be careful with non-commuting operations! Two constraints On equal time slice: consistent initial conditions Can also be solved (i.e. B  S), much easier than in ADM!! On constant r slice: usually at boundary, constraint on NB coefficients Equivalent to SE-conservation in field theory Essential check to see if numerics converge 5/17 Wilke van der Schee, MIT

6. CHARACTERISTIC EINSTEIN EQUATIONS A nested scheme, also called Bondi-Sachs formulation Non-linear Einstein equations decouple in nested set of ODEs No approximations needed Generalizes relatively straightforwardly to Maxwell and/or Scalar fields Why isn’t this always used? Null slice has non-trivial boundary behaviour Exactly wanted in AdS/CFT, but awkward in i.e. binary mergers Two null rays can meet, causing a caustic and coordinate singularity In AdS this usually does not happen (balance horizon position and energy on boundary) ‘Instantaneous’ propagation can make codes unstable Solved by using highly accurate spectral codes (perhaps mainly suitable for `simple’ problems) 6/17 Wilke van der Schee, MIT

7. NEAR-BOUNDARY EXPANSION Near-boundary expansion till some order Series expand Einstein equations Series expand functions in metric (A, B, etc) First series expand till highest order, then solve Series expand till order needed at every order Find problems fixing 4 th order coefficients, and possibly at 8 th order Find gauge freedom for Curved boundary: find logarithmic terms, or even log 2 Requires a bit of playing, game is to go till very high order 7/17 Wilke van der Schee, MIT

8. HOLOGRAPHIC RENORMALISATION Very useful formula from Skenderis et al Refers to near-boundary expansion in Fefferman-Graham coordinates g (2) generically zero for flat boundary Easy solution: solve coordinate transform near boundary In cases with conformal anomaly there is a scheme dependence (see also recent Kiritsis et al, 1503.07766) 8/17 Wilke van der Schee, MIT S. de Haro, S. N. Solodukhin and K. Skenderis, Holographic reconstruction of space-time and renormalization in the AdS/CFT correspondence (2001)

9. FREQUENTLY USED MATHEMATICA Use of pure functions Function[x, x^3][5]  125 Shorthand: #^3&[5]  125 Replacement with pure function: Making a package Initialization cells are evaluated when loading package Make equations and write to file, then load file with initialization cell Edit  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?) 9/17 Wilke van der Schee, MIT

10. SPECTRAL METHODS Two examples: Spectral methods: Non-linear problem: use Newton Lazy? NDSolve`FiniteDifferenceDerivative[1,grid, order…] 10/17 Wilke van der Schee, MIT Boyd: Chebyshev and Fourier spectral methods

11. COLLISIONS AT INFINITELY STRONG COUPLING 11/15 Match longitudinal profile of energy density to nuclei Approximately 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:

12. COLLIDING TWO NUCLEI: 12/15 Locally in transverse plane: use shock waves (i.e. ~Gaussian rapidity)  Go and run hydro (MUSIC) and get particle spectra Wilke van der Schee, MIT

13. MUSIC RESULTS, LHC Directed flow: right ball-park values Note: somewhat subtle to measure; event-plane etc Could be very sensitive to viscosity 13/15 ALICE, Directed flow of charged particles at mid-rapidity relative to the spectator plane in Pb–Pb collisions at √s NN = 2.76 TeV (2013) Wilke van der Schee, MIT With Björn Schenke

14. MUSIC RESULTS, LHC Particle spectra in longitudinal direction: Rescaled initial energy density by factor 20 Profile is significantly too narrow 14/15 ALICE, Bulk Properties of Pb-Pb collisions at √s NN = 2.76 TeV measured by ALICE (2011) Wilke van der Schee, MIT With Björn Schenke

15. SHOCK WAVES WITHOUT SYMMETRIES 15/17 Challenging 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, MIT Numerics is a bit more involved, hard to keep AH fixed, use domain decomposition

16. REFERENCES FOR LEARNING NUMERICS Characteristic formulation: Chesler-Yaffe: 1309.14391309.1439 Casalderrey, Heller, Mateos, WS, Triana: 1407.1849, 1304.51721407.18491304.5172 Balasubramanian, Herzog: 1312.49531312.4953 ADM formulation Heller, Janik, Witaszczyk : 1203.07551203.0755 Bantilan, Gubser, Pretorius: 1201.21321201.2132 Elliptic Einstein-DeTurck: Donos, Gauntlett: 1409.68751409.6875 Reviews/books: Grandclément, Novak: livingreviews.org (see also Winicour)livingreviews.orgWinicour Casalderrey, Liu, Matoes, Rajagopal, Wiedemann: 1101.0618 (and book!)1101.0618 Boyd: Chebyshev and Fourier spectral methods 16/17 Wilke van der Schee, MIT sites.google.com/site/wilkevanderschee/

17. SOLVE LINEARISED EINSTEIN EQUATIONS FOR ARBITRARY INITIAL CONDITIONS, FIND QNM Expand the metric (slide 3) around the thermal metric (take T=1/  ): Show that the Einstein equations reduce to a simple equation for Optional: solve near-boundary expansion, and obtain Rewrite equation in terms of Use spectral methods to solve the ODE in z to obtain Use i.e. or something similar Use NDSolve for time-stepping, and obtain Plot the CFT pressure anisotropy and identify the oscillation and damping rate of the lowest AdS quasi-normal mode 17/17 Wilke van der Schee, MIT