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Recent Developments in Numerical Relativity

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Recent Developments in Numerical Relativity Recent Developments in Numerical Relativity Miguel Zilhão Departament de Física Fonamental & Institut de Ciències del Cosmos, Universitat de Barcelona August 17 2015, Equilibration Mechanisms in Weakly and Strongly Coupled Quantum Field Theory, Seattle WA Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 1 / 57 Outline Contents 1 Introduction Milestones 2 Formalism Cauchy-based approach 3+1 GHG Characteristic-based approach 3 Tools Einstein Toolkit 4 Recent developments 5 Final remarks Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 2 / 57 Introduction Outline 1 Introduction Milestones 2 Formalism Cauchy-based approach 3+1 GHG Characteristic-based approach 3 Tools Einstein Toolkit 4 Recent developments 5 Final remarks Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 3 / 57 Introduction What is numerical relativity Numerical Relativity: solving numerically the full GR equations, typically for dynamical spacetimes in the strong field regime, where no approximations hold. Goals: understanding gravity in its full non-linear glory. Challenges: very difficult problem. Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 4 / 57 Introduction Why numerical relativity Study of systems with strong and dynamical gravitational fields Gravitational radiation Astrophysics, gravitational wave astronomy Mathematical and theoretical Physics Cosmic censorship, Instabilities (Black hole interior, Myers-Perry) High-energy particle systems AdS/CFT correspondence; Black hole production at the LHC; Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 5 / 57 Introduction Gravitational waves Accelerated bodies emit gravitational radiation Detected indirectly by measurements of the Hulse-Taylor binary system (1993 Nobel Prize) Interact weakly with matter carry unique information about astronomical) phenomena New window to the universe ) Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 6 / 57 Introduction Gravitational waves Difficult to detect Need theoretical models for) the structure of the waveform Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 7 / 57 Introduction Mathematical and theoretical Physics Cosmic censorship hypothesis: does it hold under extreme conditions? Sperhake, Cardoso, Pretorius, Berti, Gonzales, 2008 No no-hair theorem for D > 4 black hole solutions with non-spherical topology.) (Non-)Linear stability of higher-dimensional black objects: Black string Choptuik, Lehner, Olabarrieta, Petryk, Pretorius, Villegas, 2003 Emparan & Reall, 2008 Lehner, Pretorius 2010 Myers-Perry black hole Shibata & Yoshino, 2010 Black ring ... Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 8 / 57 Introduction AdS/CFT duality Properties of strongly coupled thermal gauge theories are related to the physics of higher-dimensional black holes Formation of quark-gluon plasma at the RHIC black hole collisions in AdS5 , Issues with numerical simulations in AdS: AdS is not globally hyperbolic The boundary plays an active role Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 9 / 57 Introduction Milestones Outline 1 Introduction Milestones 2 Formalism Cauchy-based approach 3+1 GHG Characteristic-based approach 3 Tools Einstein Toolkit 4 Recent developments 5 Final remarks Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 10 / 57 Introduction Milestones History and milestones 1915 Einstein’s equations are published Einstein 1964 First documented attempts at numerical simulations: evolving Hahn & Lindquist two wormholes 1976 Head-on collision of two black holes (in axisymmetry) Smarr & Eppley 1990’s "Binary Black Hole Grand Challenge Project" Matzner et al 1993 Critical phenomena in gravitational collapse Choptuik 1997 Release of Cactus 1.0 Seidel et al 1998 Generic (3D) single BH simulation (using a characteristic ap- Gomez et al proach) 1999 BSSN evolution system Baumgarte & Shapiro; Shibata & Nakamura 2005 First simulations of BH binaries through inspiral, merger and Pretorius ringdown (Two-body problem in GR) (GHG code) 2006 "Moving puncture" simulations (BSSN code) UTB/RIT; NASA Goddard 2008 High-energy collision of two BHs Berti, Cardoso, Gonzalez, Sperhake, Pretorius 2010 Collision of gravitational shock waves in AAdS5 spacetimes Chesler & Yaffe (2+1 code) 2010 Black hole collisions in higher dimensions Witek, M.Z. et al; Yoshino & Shibata 2012 Simulations of AAdS5 spacetimes (GHG code) Bantilan, Pretorius, Gubser 2015 Off-center collisions of shock waves in AAdS5 spacetimes (4+1 Chesler &Yaffe code) Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 11 / 57 Formalism Outline 1 Introduction Milestones 2 Formalism Cauchy-based approach 3+1 GHG Characteristic-based approach 3 Tools Einstein Toolkit 4 Recent developments 5 Final remarks Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 12 / 57 Formalism Einstein’s equations 1 Rµν R gµν = 8π Tµν − 2 Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 13 / 57 Formalism Einstein’s equations α 1 X αδ Γ = g (@γgδβ + @βgδγ @δgβγ) βγ 2 − δ=t;x1;:::;xD−1 " # X δ δ X δ γ δ γ 8πTαβ = @δΓ @αΓ + (Γ Γ Γ Γ ) αβ − δβ αβ δγ − γβ δα δ γ ( ) 1 X δγ X h µ µ X µ ν µ ν i gαβ g @µΓ @δΓ + (Γ Γ Γ Γ ) − 2 δγ − µδ δγ µν − νγ µδ δ,γ µ ν Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 14 / 57 Formalism Before numerical evolution. Write Einstein’s equations as a well-posed Initial Boundary Value Problem (IBVP): solution’s behaviour depends continuously with the initial data; numerically suitable gauge conditions. Discretize resulting PDEs Specify constraint preserving, and physically cor- rect, boundary conditions Find a way to deal with singularities Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 15 / 57 Formalism During numerical evolution. Compute constraint-satisfying initial data repre- senting snapshot of physical system Implement mesh refinement, or similar, to effi- ciently handle different length scales (and paral- lelize resulting algorithms) Extract physical results in gauge-invariant fash- ion from numerical data Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 16 / 57 Formalism Cauchy-based approach Outline 1 Introduction Milestones 2 Formalism Cauchy-based approach 3+1 GHG Characteristic-based approach 3 Tools Einstein Toolkit 4 Recent developments 5 Final remarks Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 17 / 57 Formalism Cauchy-based approach 3+1 decomposition We write the metric as ds2 = α2dt2 + γ dxi + βi dt dxj + βj dt ; − ij γij is the metric on surfaces of t = const Kij is the extrinsic curvature Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 18 / 57 Formalism Cauchy-based approach ADM-York evolution equations Evolution equations (@t β) γ = 2αK ; − L ij − ij h k (@t β) K = α + α R + KK 2K K − L ij −∇i rj ij ij − ik j 8π i + (S E)γ 2S ; D 2 − ij − ij − Constraints 2 ij R + K Kij K = 16πE; − K ij γij K = 8πpi : rj − Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 19 / 57 Formalism Cauchy-based approach Electromagnetic analogy Evolution equations @t E~ + H~ = 4π~j − r × @t H~ + E~ = 0 − r × Constraints E~ = 4πρ r · H~ = 0 r · Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 20 / 57 Formalism Cauchy-based approach Well-posedness Consider evolution equations of the form i @t φ + M @i φ = S(φ) The system is well-posed iff φ(t; xi ) keαt φ(0; xi ) k k ≤ k k ie, the norm of the solution can be bounded by the same exponential for all initial data. Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 21 / 57 Formalism Cauchy-based approach Hyperbolicity i i Construct matrix P(ni ) = ni M , for arbitrary unit vector n (principal symbol). Strongly hyperbolic system: if P has real eigenvalues and complete set of eigenvectors for all ni . Weakly hyperbolic system: if P has real eigenvalues but does not have a complete set of eigenvectors. If a system is strongly hyperbolic, it is well-posed. Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 22 / 57 Formalism Cauchy-based approach BSSN equations . the ADM equations are only weakly hyperbolic. Alternative formulations to ADM system started being explored in the late 1980s, before full impact of the hyperbolicity properties of the different formulations had been realized. Baumgarte-Shapiro-Shibata-Nakamura (BSSN) formulation is de- rived from the ADM equations but works with conformally rescaled variables, a trace split of the extrinsic curvature and promotes the contracted Christoffel symbols to the status of independent vari- ables. Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 23 / 57 Formalism Cauchy-based approach BSSN equations k k 2 k ~ @t γ~ = β @ γ~ + 2γ~ @ β γ~ @ β 2αA ; ij k ij k(i j) − 3 ij k − ij k 2 k @t χ = β @ χ + χ(αK @ β ); k 3 − k ~ k ~ ~ k 2 ~ k TF @t A = β @ A + 2A @ β A @ β + χ αR @ α ij k ij k(i j) − 3 ij k ij − ri j S + α K A~ 2A~ k A~ 8πα χS γ~ ; ij − i kj − ij − 3 ij k k ~ij ~ 1 2 @t K = β @ K @ α + α A A + K + 4πα(E + S); k − r k ij 3 i k i k i 2 i k i ~jk 1 ij k @t Γ~ = β @ Γ~ Γ~ @ β + Γ~ @ β + 2αΓ~ A + γ~ @ @ β k − k 3 k jk 3 j k 4 +γ ~jk @ @ βi αγ~ij @ K A~ij 3αχ−1@ χ + 2@ α 16παχ−1ji j k − 3 j − j j − Miguel Zilhão (UB) Recent Developments in Numerical Relativity INT-15-2c, 17 Aug 24 / 57 Formalism Cauchy-based approach Moving Punctures This system, together with the moving puncture gauge conditions, al- lowed for the 2005-06 breakthrough simulations of the Brownsville/RIT and NASA-Goddard groups.
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