Numerical techniques

Numerical Techniques for Holography based on KB, Christopher P Herzog arXiv:1312.4953 Koushik Balasubramanian YITP, Stony Brook University New Frontiers in Dynamical Gravity, 2014 Saturday, March 29, 14 Numerical Techniques for Holography based on KB, Christopher P Herzog arXiv:1312.4953 Koushik Balasubramanian YITP, Stony Brook University New Frontiers in Dynamical Gravity, 2014 Saturday, March 29, 14 Motivation Saturday, March 29, 14 Motivation • What can we learn about hydrodynamics using gauge/gravity duality? Saturday, March 29, 14 Motivation • What can we learn about hydrodynamics using gauge/gravity duality? • What can we learn about gravity? Saturday, March 29, 14 Motivation • What can we learn about hydrodynamics using gauge/gravity duality? • What can we learn about gravity? • What happens far from equilibrium? Saturday, March 29, 14 Motivation • What can we learn about hydrodynamics using gauge/gravity duality? • What can we learn about gravity? • What happens far from equilibrium? • When is hydro not a good description? (Breakdown of gradient expansion) Saturday, March 29, 14 Thanks to Computers Saturday, March 29, 14 Why numerics? Saturday, March 29, 14 Why numerics? • I can’t think of any other way Saturday, March 29, 14 • Relaxation Time For large k, we can use gauge/gravity correspondence to obtain relaxation time scale • Solve Linearized Einstein’s equations for small Text log D Im(G(!)) 2! E • Dotted line is the large wavenumber behavior (simple WKB approximation is not good enough) • All dimensionful quantities are measured in units where 2.3k 10 T = 4⇡ The markers show values obtained by solving the full nonlinear equations Saturday, March 29, 14 Numerical Scheme • Pseudospectral methods for discretizing spatial derivatives • • Runge-Kutta and Adams-Bashforth for time stepping • In gravity, we need to solve boundary evolution equations, bulk graviton evolution equations and one evolution equation at the apparent horizon • Number of propagating degrees of freedom is the same as hydro Saturday, March 29, 14 We have used the null characteristic formulation for solving Einstein’s equations Numerical Scheme • Bondi-Sachs coordinates ds2 = ✓ e2 V z A hAB U U z2 B ◆ dt2 2e2 2hAB U B A hAB A B dt dz dt dx + dx dx 2 z z z • Einstein’s equations have a nested structure • Gauge Choice: The location of apparent horizon is fixed DA U A 2dt z=1 =0 • Error Monitoring: Check Bianchi constraint Saturday, March 29, 14 Numerical Scheme • Boundary Expansion = U A = V = (V0 e2 z z3 z@A (e + z )+z + z Vs ), s, ↵ = z ↵s , UsA , A ⇡ = Dz + I z s (Dz ) ⇡sA ◆ ✓ Dz + I U A s z (Dz ) dt Saturday, March 29, 14 s @A z z3 V ↵0 , dt ↵ = ↵˙ Equations • Einstein’s ✓ ◆ ✓ = z ✓s A + ⇡ s , dt ✓ = ✓˙ s) z3 V✓ = S (z, ↵s , ✓s , (1) = S⇡A (z, ↵s , ✓s , s, s) (2) = SU A z, ↵s , ✓s , s, A , ⇡ s s (3) = Sd t z, ↵s , ✓s , s, A s , Us (4) Numerical Scheme • Einstein’s equations ✓ ◆ Dz + I dt ↵s + C↵↵ dt ↵s + C↵✓ dt ✓s z ✓ ◆ Dz + I dt ✓s + C✓↵ dt ↵s + C✓✓ dt ✓s z = Sdt ↵ ( , dt s) = Sdt ✓ ( , dt s) • Elliptic Equation (at Apparent horizon) H Cxx Dx(2) VH + CxH Dx(1) VH + C0H VH = SVH ↵H , ✓H , Saturday, March 29, 14 A , U H H, H , d t H , d t ↵ H , dt ✓ H (1 Boundary Data • Boundary/Horizon Evolution Equations @t ↵s @ t ✓s @t • 1 = (dt ↵)s + (z↵s0 + 2↵s )z V z 1 = (dt ✓)s + (z✓s0 + 2✓s )z V z A = S VH , U H , H @t U3A = SU3A ↵3 , ✓3 , @ t V3 = S V ↵ , ✓3 , A , V3 , U , 0 Boundary Conditions @z s ⇡sA A @ z Us dt ↵s Saturday, March 29, 14 A , V , U 3 , = 0, = 3e = A U3 , = 0, U3A , dt ✓s = (1) (2) (3) (4) (5) Numerical Simulations 0 4⇡ 50 k= 5⇡ 50 k= 6⇡ 50 Ttx (t) Ttx (0) Ref D log k= ⇡ 50 k= 4⇡ 50 k= 5⇡ 50 k= 6⇡ 50 Ref log D Ttx (t) Ttx (0) E k= E k= ⇡ 50 2 k t 8⇡Ti 2 k t 8⇡Ti = 0.2, v = 0.2 • • Gravity and hydro agree initially Gradient corrections become important at late times Reference line shows the linear response theory result Saturday, March 29, 14 Difference in Stress Tensor Ttx 0.002 0.000 0.002 0.004 0.006 0.001 Ttt 0.000 0.001 0.002 0.003 Txx 0.000 0.001 0.002 0.003 0.0 0.5 1.0 1.5 2.0 2 2.5 3.0 k t 8⇡Ti k = 4⇡/50 = 0.2, v = 0.2 Saturday, March 29, 14 3.5 4.0 Gravity vs Hydro log D Ttx (t) Ttx (0) E Gravity Hydro Ref GR Ref Hydro 5 2 k t 8⇡Ti 20⇡ k= , 50 Saturday, March 29, 14 = 0.2, v = 0.2 Large Behavior(Hydro) = 0.2 = 0.3 = 0.4 = 0.5 Ref k = ⇡/50 0.5 log D Ttx (t) Ttx (t⇤ ) E 0.0 f ( ) = 2(1 1.0 1.5 0.0 0.2 0.4 0.6 0.8 1.0 1.2 1.4 k f ( )t 8⇡T0 2(1 = ⌧ Saturday, March 29, 14 p )⌘k 3✏0 T0 p 2) Large Behavior(Gravity) 0.2 0.0 k = ⇡/50 0.4 log D Ttx (t) Ttx (t⇤ ) E 0.2 0.6 0.8 1.0 0.0 f ( ) = 2(1 = 0.2 = 0.3 = 0.4 Ref 0.2 0.4 0.6 0.8 1.0 k f ( )t 8⇡T0 2(1 = ⌧ Saturday, March 29, 14 p )⌘k 3✏0 T0 p 2) T T T Late time solution 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 0.8 0.6 0.4 0.2 0.0 0.2 0.4 0.6 ⇥10 ⇥10 ⇥10 t = 2000 3 40 20 20 40 20 20 40 20 20 40 t = 4000 40 t = 10000 40 x k = 4⇡/50 Saturday, March 29, 14 T0 T =p , gtt u=0 Late time solution agrees w i th the exact analytical solution ` k = 0.2 = 0.3 = 0.4 Ref Ttx (t) Ttx (0) E result!!! • Relaxation seems slower D log • No known analytical for large lattice strength at large , k 10 20 40 60 k f ( )t 8⇡T0 Saturday, March 29, 14 80 100 Summary • Linear response theory seems to work for small values of lattice strength • For large lattice strengths, we can obtain analytical results for small lattice wave numbers • We need to use Numerical GR for all other cases Saturday, March 29, 14 Thank You Saturday, March 29, 14 ... • I can’t think of any other way • Numerical techniques are well-developed Saturday, March 29, 14 Why numerics? • I can’t think of any other way • Numerical techniques are well-developed • We.. .Numerical Techniques for Holography based on KB, Christopher P Herzog arXiv:1312.4953 Koushik Balasubramanian... PDEs with courage Saturday, March 29, 14 Why numerics? • I can’t think of any other way • Numerical techniques are well-developed • We can face nonlinear PDEs with courage • Computers can stay