Introduction to Numerical Relativity I Intro & Geometry

Transcription

1 Ë ¼ Å ¼ Â ¼ Introduction to Numerical Relativity I Intro & Geometry S. Husa University of the Balearic Islands, sascha.husa@uib.es August 5, 2010 Ë ¾ Å ¾ Â ¾ Ë ½ Å ½ Â S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

2 Outline 1 Initial value problem S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

3 Outline 1 Initial value problem Decomposition S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

4 Outline 1 Initial value problem Decomposition 3 Conformal approach to solve the constraints S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

5 Outline 1 Initial value problem Decomposition 3 Conformal approach to solve the constraints 4 Black Holes S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

6 Outline 1 Initial value problem Decomposition 3 Conformal approach to solve the constraints 4 Black Holes 5 Asymptotics S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

7 General Relativity I Initial value problem General relativity is the modern (classical) theory of gravitation, formulated in terms of spacetime geometry, described by the Einstein equations: G ab [g cd ] = R ab 1 2 R c c g ab = 8πκT ab [g cd, φ A ] S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

8 General Relativity I Initial value problem General relativity is the modern (classical) theory of gravitation, formulated in terms of spacetime geometry, described by the Einstein equations: G ab [g cd ] = R ab 1 2 R c c g ab = 8πκT ab [g cd, φ A ], R bd = R a bad. R a bcd = Γ a bd,c Γ a bc,d + Γ m bdγ a mc Γ m bcγ a md, [ a, b ]v c = R c dabv d, Γ i kl = 1 2 g im (g mk,l + g ml,k g kl,m ). Written in terms of coordinate components and partial derivatives the EE correspond to a very complex system of coupled nonlinear PDEs. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

9 General Relativity I Initial value problem General relativity is the modern (classical) theory of gravitation, formulated in terms of spacetime geometry, described by the Einstein equations: G ab [g cd ] = R ab 1 2 R c c g ab = 8πκT ab [g cd, φ A ], R bd = R a bad. R a bcd = Γ a bd,c Γ a bc,d + Γ m bdγ a mc Γ m bcγ a md, [ a, b ]v c = R c dabv d, Γ i kl = 1 2 g im (g mk,l + g ml,k g kl,m ). Written in terms of coordinate components and partial derivatives the EE correspond to a very complex system of coupled nonlinear PDEs. Add equations for the dynamics of the matter fields φ A and solve consistently, do not prescribe g ab and then compute T ab! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

10 General Relativity I Initial value problem General relativity is the modern (classical) theory of gravitation, formulated in terms of spacetime geometry, described by the Einstein equations: G ab [g cd ] = R ab 1 2 R c c g ab = 8πκT ab [g cd, φ A ], R bd = R a bad. R a bcd = Γ a bd,c Γ a bc,d + Γ m bdγ a mc Γ m bcγ a md, [ a, b ]v c = R c dabv d, Γ i kl = 1 2 g im (g mk,l + g ml,k g kl,m ). Written in terms of coordinate components and partial derivatives the EE correspond to a very complex system of coupled nonlinear PDEs. Add equations for the dynamics of the matter fields φ A and solve consistently, do not prescribe g ab and then compute T ab! Case T ab = 0 is highly nontrivial: black holes & gravitational waves! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

11 General Relativity II Initial value problem While GR can be formulated in a very compact and elegant way, fully understanding the physical content of the theory remains a major challenge: What is the solution space and what is its physical interpretation? S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

12 General Relativity II Initial value problem While GR can be formulated in a very compact and elegant way, fully understanding the physical content of the theory remains a major challenge: What is the solution space and what is its physical interpretation? The complexity/nonlinearity of GR severly limits the physical relevance of most exact solutions, exceptions in 3+1 e.g. Kerr, FRW, (A)dS (+ BH). S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

13 General Relativity II Initial value problem While GR can be formulated in a very compact and elegant way, fully understanding the physical content of the theory remains a major challenge: What is the solution space and what is its physical interpretation? The complexity/nonlinearity of GR severly limits the physical relevance of most exact solutions, exceptions in 3+1 e.g. Kerr, FRW, (A)dS (+ BH). What does GR tell us about our universe? What effects can we observe? S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

14 General Relativity II Initial value problem While GR can be formulated in a very compact and elegant way, fully understanding the physical content of the theory remains a major challenge: What is the solution space and what is its physical interpretation? The complexity/nonlinearity of GR severly limits the physical relevance of most exact solutions, exceptions in 3+1 e.g. Kerr, FRW, (A)dS (+ BH). What does GR tell us about our universe? What effects can we observe? Quantum theory, singularity theorems how should we modify GR? S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

15 General Relativity II Initial value problem While GR can be formulated in a very compact and elegant way, fully understanding the physical content of the theory remains a major challenge: What is the solution space and what is its physical interpretation? The complexity/nonlinearity of GR severly limits the physical relevance of most exact solutions, exceptions in 3+1 e.g. Kerr, FRW, (A)dS (+ BH). What does GR tell us about our universe? What effects can we observe? Quantum theory, singularity theorems how should we modify GR? Can observations help to find a better (quantum) theory of gravity? S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

16 General Relativity II Initial value problem While GR can be formulated in a very compact and elegant way, fully understanding the physical content of the theory remains a major challenge: What is the solution space and what is its physical interpretation? The complexity/nonlinearity of GR severly limits the physical relevance of most exact solutions, exceptions in 3+1 e.g. Kerr, FRW, (A)dS (+ BH). What does GR tell us about our universe? What effects can we observe? Quantum theory, singularity theorems how should we modify GR? Can observations help to find a better (quantum) theory of gravity? S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

17 Initial value problem General Relativity II While GR can be formulated in a very compact and elegant way, fully understanding the physical content of the theory remains a major challenge: What is the solution space and what is its physical interpretation? The complexity/nonlinearity of GR severly limits the physical relevance of most exact solutions, exceptions in 3+1 e.g. Kerr, FRW, (A)dS (+ BH). What does GR tell us about our universe? What effects can we observe? Quantum theory, singularity theorems how should we modify GR? Can observations help to find a better (quantum) theory of gravity? Mathematical problems & exact solutions dominated GR for much of its history: Deep insights gained: positive mass theorem, singularity theorems, nonlinear stability of Minkowski,... GR research has often been decoupled from other areas of physics. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

18 Initial value problem Why Numerical Relativity? Astrophysics, cosmology, general understanding of the solution space of the EE require approximate solutions analytical and numerical! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

19 Initial value problem Why Numerical Relativity? Astrophysics, cosmology, general understanding of the solution space of the EE require approximate solutions analytical and numerical! Numerical solutions allow to study the equations with a minimum of simplifying physical assumptions, and allow mathematical control over the convergence of the approximation! Typically, numerical analysis provides theorems that a certain algorithm will converge to the correct solution for some finite time at a certain order for sufficient resolution. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

20 Initial value problem Why Numerical Relativity? Astrophysics, cosmology, general understanding of the solution space of the EE require approximate solutions analytical and numerical! Numerical solutions allow to study the equations with a minimum of simplifying physical assumptions, and allow mathematical control over the convergence of the approximation! Typically, numerical analysis provides theorems that a certain algorithm will converge to the correct solution for some finite time at a certain order for sufficient resolution. Nonlinear PDEs need to be studied with non-perturbative quantitative methods and in particular also by numerical methods. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

21 Initial value problem Why Numerical Relativity? Astrophysics, cosmology, general understanding of the solution space of the EE require approximate solutions analytical and numerical! Numerical solutions allow to study the equations with a minimum of simplifying physical assumptions, and allow mathematical control over the convergence of the approximation! Typically, numerical analysis provides theorems that a certain algorithm will converge to the correct solution for some finite time at a certain order for sufficient resolution. Nonlinear PDEs need to be studied with non-perturbative quantitative methods and in particular also by numerical methods. The Einstein equations pose a number of fundamental problems/challenges for treatment with numerical methods, but standard finite difference methods are often an excellent choice! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

22 Initial value problem NR comes in many flavours... physical model: black holes: formation, ringdown, binary dynamics & GWs,... Astrophysical matter: supernovae, neutron stars, BH NS,... Fundamental matter: hairy BH s, critical collapse, string dynamics... Cosmology! technical: slicing: asymptotics: aim: computational: spacelike or null cutoff or compactified toy or tool mathematical understanding or astrophysics problems? requires parallelism? mesh-refinement? multiple patches? dynamical regime: test fields on fixed background matter dynamics dominated rest mass dominated kinetic energy dominated waves dominated S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

23 Initial value problem Is NR difficult? The art: make continuum (physical!) features manifest in discrete system. The ideal numerical relativist is an expert (at least) in the geometric arts of GR, PDE theory, numerics, scientific programming, high performance computing, black hole physics and astrophysics/cosmology! Fear not! Don t get confused! But note first hints of trouble: GR can be viewed as gauge theory of diffeomorphism group constraints! Computational d.o.f. physical d.o.f. Construct spacetimes physics is in the geometry, not particular metric components! Numerical solution procedures require gauge fixing. Nonlinear PDEs typically require case-by-case study which PDE system do the EEs actually correspond to? Complicated nonlinear structure of GR makes it hard to carry over techniques addressing related issues in Electrodynamics or Yang-Mills. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

24 Methods options for NR Initial value problem Discretizing GR can be approached in very different ways: direct discretization of the geometry (e.g. Regge calculus: triangulation into simplices, spacetime curvature expressed in terms of deficit angles). discrete differential forms a standard method in computational electrodynamics. PDE problem, constrained or free evolution, at least 2 hyperbolic eqs.!) mimetic discretization (more directly model the underlying physics than with PDE system), but some no-go theorems for manifest solution of constraints. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

25 Methods options for NR Initial value problem Discretizing GR can be approached in very different ways: direct discretization of the geometry (e.g. Regge calculus: triangulation into simplices, spacetime curvature expressed in terms of deficit angles). discrete differential forms a standard method in computational electrodynamics. PDE problem, constrained or free evolution, at least 2 hyperbolic eqs.!) mimetic discretization (more directly model the underlying physics than with PDE system), but some no-go theorems for manifest solution of constraints. Seems fair to say that so far only the standard PDE approach has led to the solution of real problems. In the absence of matter fields, the EE do not lead to the formation of shocks (but happen for coordinates) or turbulence standard high order finite difference or spectral methods work very well! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

26 How? Initial value problem G ab [g cd ] = 8πκT ab [g cd, φ A ]. EEs are instrinsically 4-dimensional how/where should we specify boundary conditions = select the physical solution? Written in any coordinate system, EEs become underdetermined set of coupled nonlinear PDEs (and overdetermined because we have to solve constraints). First sort out what can be chosen and what is then determined by equations algorithmic method of solution. Q: Does the theory have an initial value formulation? Yes! Many! Unique (modulo diffeomorphisms) solution of EEs equation can be specified by initial data on a spacelike hypersurface! Predictability! Important source of physical intuition! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

27 Initial value problem Start simple: Maxwell 4-dimensional: df = 0, d F = 4π J. space+time split: introduce electric and magnetic field E a = F ab n b, B c = 1 2 F ab 3 ɛ abc. a E a = 4πρ, a B a = 0, t E a = ɛ abc b B c 4πj a, t B a = ɛ abc b E c. Initial value problem makes sense! Working with A a additional gauge issues appear! Numerical ED is difficult, but well understood (analytical formulation, numerical algorithms, comparison with experiment)! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

28 Initial value problem Start simple: Maxwell 4-dimensional: df = 0, d F = 4π J. space+time split: introduce electric and magnetic field E a = F ab n b, B c = 1 2 F ab 3 ɛ abc. a E a = 4πρ, a B a = 0, t E a = ɛ abc b B c 4πj a, t B a = ɛ abc b E c. Initial value problem makes sense! Working with A a additional gauge issues appear! Numerical ED is difficult, but well understood (analytical formulation, numerical algorithms, comparison with experiment)! Curved background: L n D i E i = KD i E i, L n D i B i = KD i B i In collapsing case (K < 0) instability of constraints! Solution for Maxwell: use ge a, gb a. GR?? S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

29 Fast track Decomposition Simplest way to get PDEs from EEs: chose coordinates {x i, t} (i = 1, 2, 3) read off metric in the form ds 2 = α 2 dt 2 + h ab (dx a + β a dt)(dx b + β b dt), (1) h ab is a positive definite matrix (Riemannian 3-metric), and α 0 (problems foliating BH spacetimes). 4 functions α(x i, t) and β j (x i, t) freely specifiable, steer coordinate system through spacetime as time evolution proceeds physical result is independent of this choice diffeo invariance. PDEs resulting from this ansatz are of 2nd order for h, split into 2 parts: 4 constraints (no second time derivatives) & 6 evolution equations. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

30 3+1 Decomposition Projections and the Induced Metric Given foliation: write all tensors in terms of horizontal and vertical parts! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

31 3+1 Decomposition Projections and the Induced Metric Given foliation: write all tensors in terms of horizontal and vertical parts! Let n a = α a t denote the future timelike unit normal to Σ & define N a b := n a n b, h a b := δ a b N a b, S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

32 3+1 Decomposition Projections and the Induced Metric Given foliation: write all tensors in terms of horizontal and vertical parts! Let n a = α a t denote the future timelike unit normal to Σ & define N a b := n a n b, h a b := δ a b N a b, check they are in fact the desired projection operators (exercise!): h a bh b c = h a c, N a bn b c = N a c, h a bn b c = 0, S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

33 3+1 Decomposition Projections and the Induced Metric Given foliation: write all tensors in terms of horizontal and vertical parts! Let n a = α a t denote the future timelike unit normal to Σ & define N a b := n a n b, h a b := δ a b N a b, check they are in fact the desired projection operators (exercise!): h a bh b c = h a c, N a bn b c = N a c, h a bn b c = 0, h a b projects onto the tangential, and N a b onto the normal directions, h a bn b = 0, N a bn b = n a. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

34 3+1 Decomposition Projections and the Induced Metric Given foliation: write all tensors in terms of horizontal and vertical parts! Let n a = α a t denote the future timelike unit normal to Σ & define N a b := n a n b, h a b := δ a b N a b, check they are in fact the desired projection operators (exercise!): h a bh b c = h a c, N a bn b c = N a c, h a bn b c = 0, h a b projects onto the tangential, and N a b onto the normal directions, h a bn b = 0, N a bn b = n a. Apply first to metric induces a tensor field h ab by h ab = g ab + n a n b = g cd h c ah d b. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

35 3+1 Decomposition Projections and the Induced Metric Given foliation: write all tensors in terms of horizontal and vertical parts! Let n a = α a t denote the future timelike unit normal to Σ & define N a b := n a n b, h a b := δ a b N a b, check they are in fact the desired projection operators (exercise!): h a bh b c = h a c, N a bn b c = N a c, h a bn b c = 0, h a b projects onto the tangential, and N a b onto the normal directions, h a bn b = 0, N a bn b = n a. Apply first to metric induces a tensor field h ab by h ab = g ab + n a n b = g cd h c ah d b. h ab is purely horizontal, positive definite and nondegenerate for horizontal vector fields (exercise!) natural Riemannian metric on Σ. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

36 3+1 Decomposition Induced Curvatures 2 curvatures associated with embedding of Σ in M: intrinsic: Riemann tensor extrinsic: describes how Σ bends in M. Natural derivative operator D a associated with h ab : D c T a1...ar b 1...b s := hc c ha1 a... h ar 1 a r hb 1 b 1... h b s b s c T a 1...a r b 1...b s (2) define Riemann tensor of 3 R abcd [h ef ]. Extrinsic curvature: K ab := h c ah d b c n d = 1 2 L nh ab = velocity, Relation of the intrinsic and extrinsic curvatures of Σ to the curvature of M is given by two crucial geometric identities, the Gauss-Codazzi Eqs.: 3 R abc d = ha a h b b h c c h d d R a b c d K ac K b d + K bc K a d, (3) D a K a b D b K a a = R cd n d h c b. (4) S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

37 3+1 Decomposition Threadings of spacetime Consider changes of tensors T along the integral curves of a time flow vector t a, given by Lie derivative: Ṫ := L t T = t T (x α, t) in adapted coordinates{x α, t}. Spacetime engineering: threading t a dynamically steers spacetime evolution. Decompose t a into a normal and a tangential component: t a = αn a + β a, β a n a = 0, t a timelike if β a β a α 2 < 0. (5) α n µ β µ Σ t Lapse α determines how fast time elapses. n µ tµ Σ 0 Shift vector β a shifts spatial coordinate points with time evolution. Lapse and shift are not determined by EEs plenty of rope to shoot yourself in the foot! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

38 Projecting G ab = Decomposition Now use EEs compute all projections of G ab = κt ab using Gauss-Codazzi (3,4)! Projections with n a yield: 0 = G bc n c h b a = h b ar bc = D a K a b D b K a a (6) 0 = G ab n a n b = 1 ( 3 R + (K a a) 2 K ab K ab) (7) 2 No time derivatives of K ab they are relations between the initial data h and K, which cannot be freely specified the constraint equations of GR! G ab h c a h d b yields another evolution equation (have L n h ab = 2K ab ): K ab = D a D b α + β c D c K ab + K cb D a β c K bc D a β c + α (3 R ab + K c c K ab ) (8) Bianchi-Id. ( a G ab = 0) Constraints propagate! (What happens for small initial violations?) Mid 70 s: let s just code them up and collide black holes! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

39 Conformal approach to solve the constraints Conformal approach to constraints: time symmetry Standard formalism to solve the constraints: conformal approach developed by Lichnerowicz, York, Ó Murchadha and others: geometrically intuitive, elegant and the most practical framework known (actually: thin sandwich!). Mathematically natural or physically meaningful? Consider the case of time symmetry : K ab = 0: R[h ab ] = 0, 1 equation for 6 variables! Idea: express h ab from base metric h ab and a conformal factor ψ as h ab = ψ 4 hab, ψ > 0. (9) Scalar curvature transforms as R h = ψ 5 (R h ψ 8 h ψ) h ψ + 1 R h ψ = 0 elliptic! 8 S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

40 Conformal approach to solve the constraints Conformal approach to constraints: general case A ab := K ab 1 3 hab K trace-free part of extrinsic curvature. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

41 Conformal approach to solve the constraints Conformal approach to constraints: general case Rescale A ab := K ab 1 3 hab K trace-free part of extrinsic curvature. A ab = ψ 10 Ā ab [ Aab = ψ 2 Ā ab ] Da A ab = ψ 10 D a Ā ab. (10) S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

42 Conformal approach to solve the constraints Conformal approach to constraints: general case Rescale A ab := K ab 1 3 hab K trace-free part of extrinsic curvature. A ab = ψ 10 Ā ab [ Aab = ψ 2 Ā ab ] Da A ab = ψ 10 D a Ā ab. (10) Decompose the traceless symmetric Āab as Ā ab = Āab TT + (LW ) ab, divergence-free (transverse) traceless part Ā ab TT and vector potential W a, (LW ) ab := D a W b + D b W a 2 3 h ab Dc W c. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

43 Conformal approach to solve the constraints Conformal approach to constraints: general case Rescale A ab := K ab 1 3 hab K trace-free part of extrinsic curvature. A ab = ψ 10 Ā ab [ Aab = ψ 2 Ā ab ] Da A ab = ψ 10 D a Ā ab. (10) Decompose the traceless symmetric Āab as Ā ab = Āab TT + (LW ) ab, divergence-free (transverse) traceless part Ā ab TT and vector potential W a, (LW ) ab := D a W b + D b W a 2 3 h ab Dc W c. Insertion Āab TT = Āab (LW ) ab into constraints yields coupled elliptic PDEs D a (LW ) ab = D a Ā ab ψ6 D b K + 8πψ 10 j b, (11) hψ R h ψ 1 8 ψ 7 (Āab (LW ) ab ) ψ5 K 2 = 2πψ 5 ρ. (12) S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

44 Conformal approach to solve the constraints Conformal approach to constraints: general case Rescale A ab := K ab 1 3 hab K trace-free part of extrinsic curvature. A ab = ψ 10 Ā ab [ Aab = ψ 2 Ā ab ] Da A ab = ψ 10 D a Ā ab. (10) Decompose the traceless symmetric Āab as Ā ab = Āab TT + (LW ) ab, divergence-free (transverse) traceless part Ā ab TT and vector potential W a, (LW ) ab := D a W b + D b W a 2 3 h ab Dc W c. Insertion Āab TT = Āab (LW ) ab into constraints yields coupled elliptic PDEs D a (LW ) ab = D a Ā ab ψ6 D b K + 8πψ 10 j b, (11) hψ R h ψ 1 8 ψ 7 (Āab (LW ) ab ) ψ5 K 2 = 2πψ 5 ρ. (12) Freely specifiable: h ab, K, symmetric tracefree Āab. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

45 Conformal approach to solve the constraints Conformal flatness, Bowen-York, and puncture data - I Typical simplifying assumptions for constraints: constant mean curvature (CMC) slices: K = const., Asymptotically Euclidean (K = 0) or hyperboloidal (K 0). Boundary condition for asymptotically Euclidean data: lim r ψ = 1. Spatial conformal flatness: seed metric h ab is flat. CMC appears not to be a strong restriction, but conformal flatness is (at least for strong data): no conformally flat slices in Kerr. NR simulations typically use CMC data decouple Hamiltonian and momentum constraints. Conformally flat data are very commonly used in NR, e.g. for neutron stars & even BHs. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

46 Black Holes Conformal flatness, Bowen-York, and puncture data - II An application of conformal compactification! Ë ¼ Å ¼ Â ¼ Ë Ë ¾ Å ¾ Â ¾ Ë ½ Å ½ Â ½ Ë puncture ID : wormhole topology & each asymptotic end is compactified to a single point at the price of a coordinate singularity. K ab = 0 ψ = 0 ψ = 1 + i m i 2 r r i. ID for N BHs: add extra s enforce minimal surfaces trapped surfaces. Good enough until recently: conformal flatness: h ab = δ ab analytic solution of momentum constraint for spinning and boosted BHs ( Bowen-York K ab ). Straightforward specification of momenta and spins great flexibility! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

47 Black Holes Conformal flatness, Bowen-York, and puncture data - III Exact solutions exist to the momentum constraint for conformally flat initial data, Bowen-York family describes BH with angular momentum S i & momentum P i : A ij = 3 2r 2 [ Pi n j + P j n i (δ ij n i n j )P k n k ] + 3 r 3 [ ɛkil S l n k n j + ɛ kjl S l n k n i ]. By linearity of the momentum constraint, we can superpose solutions to describe n BHs with total angular momentum J i = a S i a and momentum a Pi a. Hamiltonian constraint then takes the simple form ψ = 1 8 ψ 7 A ab A ab 1 12 ψ5 K 2. Variations of the conformal formalism to solve constraints are known as (conformal) thin sandwich formalism, there lapse α and shift β i appear explicitly in the construction. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

48 What is a black hole? Black Holes Trapped surface (TS): spacelike 2-surface where ingoing and outoing wavefronts decrease in area. Light cones tip over outflow boundaries BH excision. Under very general conditions: TS spacetime singularity. Marginally TS: expansion Θ ± = q ab a l ±b = 0 (q: metric on 2-surface, l ±al a ± = 0) Apparent horizon: the outermost marginally trapped surface. Event horizon: the boundary in the spacetime between points that can send light rays to infinity, and those that can not. Global concept! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

49 Black Holes Physical properties of BHs in a given spacetime The EH is a teleologically defined null surface local objects more natural in NR. apparent horizon outermost future marginally trapped surface, Θ + = 0. dynamical horizon spacelike surface foliated by marginally trapped surfaces. isolated horizon null surface foliated by marginally trapped surfaces dynamical horizon with no matter or radiation falling in. Laws of BH mechanics can be generalized for isolated and dynamical horizons, horizon properties can be computed locally on a spacelike cross-section S, e.g. angular momentum J (ϕ) = 1 ϕ a r a K ab d 2 V. 8πG S ϕ a is a Killing vector on S, r a is the unit radial normal to S in M. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

50 Black Holes Locating BHs in a given spacetime Locating EH requires evolution at least to a stationary state then track null surfaces backward in time, reasonable initial guesses will converge onto the EH. Describe EH by level surfaces: F (t, x i ) = 0, g ab a b = 0. t f = g ti g tt if (gti i f ) + 2 g tt g ij i f j f g t t For AH/IH/DH look for surfaces of constant null expansion: Θ ± = K r a r b K ab ± D a r a convertible to elliptic equation for level set surface. Solve directly as an elliptic equation or find them with a parabolic flow. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

51 Black Holes Find apparent horizons in 2D Represent axially symmetric 2-surface as level set surface F (ρ, z) = 0, spacelike unit vector to the 2-surface, r a, becomes r a = a F g ab a F b F, Brill wave ansatz for axisymmetric 3-metric: ds 2 = ψ 4 [ e 2q(r,ϑ) ( dr 2 + r 2 dϑ 2) + r 2 sin 2 ϑdϕ 2]. Simplest case: assume r(ϑ), choose F (r, ϑ) = r f (ϑ), and setting Θ = 0 in the case of time symmetry then yields a single nonlinear ODE boundary value problem of second order with singular boundaries (r := dr/dϑ): ( ( ) ) r r 2 [ ( = 1 + r 2 q r + 4 ψ ) ( r r cot ϑ + q ϑ + 4 ψ ) ] ϑ + 2r + r 2 r ψ ψ r. Solve ODE directly, or minimize Θ[F ] for some an ansatz for F (ρ, z). Minimal surfcaces (K ab = 0) and more general AHs are very smooth! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

52 Black Holes Find and analyse apparent and event horizons in 3D Overview: J. Thornburg. Living Rev. Rel. 10:3, 2007 AH: Solve for the surface as an elliptic equation or a parabolic flow method: λ x i = Θ i n i. Generalizations achieve faster convergence, e.g. Gundlach s fast flow. Performance: use spectral methods, but off-surface evaluations expensive. Parabolic eq. may require small time steps in numerical evolution! Flow methods are very robust w.r.t. choice of initial surface! What is efficient in higher dimensions? Try minimization or flows first? S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

53 Black Holes Find and analyse apparent and event horizons in 3D Overview: J. Thornburg. Living Rev. Rel. 10:3, 2007 AH: Solve for the surface as an elliptic equation or a parabolic flow method: λ x i = Θ i n i. Generalizations achieve faster convergence, e.g. Gundlach s fast flow. Performance: use spectral methods, but off-surface evaluations expensive. Parabolic eq. may require small time steps in numerical evolution! Flow methods are very robust w.r.t. choice of initial surface! What is efficient in higher dimensions? Try minimization or flows first? Location (existence) of AHs depends on the choice of foliation! Pathological foliations of a BH may not have AHs! EH is gauge-independent, the topology of spatial cross-sections is not. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

54 Black Holes Modelling BHs numerically The interior structure of a black hole is supposed to be very complicated may only be interested in the exterior, actually! Problems: How to avoid simulating the interior structure of a BH? Need hyperbolic analysis to check for pure outflow, unphysical characteristics can create problems (what happens for constrained evolutions?). Numerical dispersion may create outgoing modes with high frequency. Ping-pong feedback is possible due to reflections at outer boundary or refinement boundaries. Waves that travel toward BH get blue-shifted. When BHs travel across the grid, information is coming out of the BH! S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

55 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

56 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. 2 BCs should yield well-posed initial boundary value problem. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

57 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. 2 BCs should yield well-posed initial boundary value problem. 3 Implement the correct physics, e.g. outgoing waves. Can typically not be achieved without modeling a global spacetime, e.g. no local notion of GWs, nonlinear backscattering of GWs. Conditions (1+2+3) are easily achieved with symmetry BCs. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

58 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. 2 BCs should yield well-posed initial boundary value problem. 3 Implement the correct physics, e.g. outgoing waves. Can typically not be achieved without modeling a global spacetime, e.g. no local notion of GWs, nonlinear backscattering of GWs. Conditions (1+2+3) are easily achieved with symmetry BCs. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

59 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. 2 BCs should yield well-posed initial boundary value problem. 3 Implement the correct physics, e.g. outgoing waves. Can typically not be achieved without modeling a global spacetime, e.g. no local notion of GWs, nonlinear backscattering of GWs. Conditions (1+2+3) are easily achieved with symmetry BCs. Conditions (1) and (2) have only first been met by Friedrich and Nagy, S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

60 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. 2 BCs should yield well-posed initial boundary value problem. 3 Implement the correct physics, e.g. outgoing waves. Can typically not be achieved without modeling a global spacetime, e.g. no local notion of GWs, nonlinear backscattering of GWs. Conditions (1+2+3) are easily achieved with symmetry BCs. Conditions (1) and (2) have only first been met by Friedrich and Nagy, Practical result for (1)+(2) Kreiss & Winicour 2006, more recent work. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

61 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. 2 BCs should yield well-posed initial boundary value problem. 3 Implement the correct physics, e.g. outgoing waves. Can typically not be achieved without modeling a global spacetime, e.g. no local notion of GWs, nonlinear backscattering of GWs. Conditions (1+2+3) are easily achieved with symmetry BCs. Conditions (1) and (2) have only first been met by Friedrich and Nagy, Practical result for (1)+(2) Kreiss & Winicour 2006, more recent work. In practice: often use causal isolation of the boundaries from region of interest, easy in 1+1 and 2+1 dimensions, harder in 3+1. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

62 Modeling of global spacetime by compactification methods open problem e.g. for BBH spacetimes. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30 Asymptotics Boundary conditions for evolving EEs Required on a noncompact manifold or one with boundaries: 1 BCs should be consistent with the EEs preserve constraints. 2 BCs should yield well-posed initial boundary value problem. 3 Implement the correct physics, e.g. outgoing waves. Can typically not be achieved without modeling a global spacetime, e.g. no local notion of GWs, nonlinear backscattering of GWs. Conditions (1+2+3) are easily achieved with symmetry BCs. Conditions (1) and (2) have only first been met by Friedrich and Nagy, Practical result for (1)+(2) Kreiss & Winicour 2006, more recent work. In practice: often use causal isolation of the boundaries from region of interest, easy in 1+1 and 2+1 dimensions, harder in 3+1.

63 Asymptotics Isolated systems as models of sources of GW Key concepts to describe astrophysical processes in GR: essential independence of the large-scale structure of the universe, radiation leaves system. physical idealization: isolated system geometry flattens at large distances or approaches some cosmological background geometry. GR: mass, momentum, emitted gravitational radiation can not be defined unambiguously in local/quasilocal way only make sense in asymptotic limits. Formalize as asymptotically flat or asymptotically de Sitter/AdS spacetimes AF spacetimes usually used to model sources of GWs. Beware: there are 3 directions toward infinity: timelike / spacelike / null! Compactification in these 3 directions behaves very differently. Key idea: use conformal rescalings (Penrose). S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

64 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

65 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

66 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

67 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. asymptotically anti-de Sitter: null infinity is time-like boundary data required. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

68 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. asymptotically anti-de Sitter: null infinity is time-like boundary data required. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

69 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. asymptotically anti-de Sitter: null infinity is time-like boundary data required. Different approaches to compactified equations: Evolution along null surfaces works well, but not general due to caustics. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

70 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. asymptotically anti-de Sitter: null infinity is time-like boundary data required. Different approaches to compactified equations: Evolution along null surfaces works well, but not general due to caustics. Compactification along spacelike directions: standard method to construct initial data, underresolved blue-shifted waves in evolution. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

71 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. asymptotically anti-de Sitter: null infinity is time-like boundary data required. Different approaches to compactified equations: Evolution along null surfaces works well, but not general due to caustics. Compactification along spacelike directions: standard method to construct initial data, underresolved blue-shifted waves in evolution. K = const. spacelike hyperboloidal surfaces reaching null infinity (t 2 x 2 = const. in Minkowski). S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

72 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. asymptotically anti-de Sitter: null infinity is time-like boundary data required. Different approaches to compactified equations: Evolution along null surfaces works well, but not general due to caustics. Compactification along spacelike directions: standard method to construct initial data, underresolved blue-shifted waves in evolution. K = const. spacelike hyperboloidal surfaces reaching null infinity (t 2 x 2 = const. in Minkowski). S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

73 Asymptotics Conformal Compactification Using conformal rescaling to an unphysical spacetime, we can discuss asymptotics in terms of local differential geometry. g ab = Ω 2 g ab, M = {p M Ω(p) > 0}, = M = {p M Ω(p) = 0}. Einstein s vacuum equations in terms of Ω & g ab : G ab [Ω 2 g] = G ab [g] 2 Ω ( a b Ω g ab c c Ω) 3 Ω 2 g ab ( c Ω) c Ω. singular for Ω = 0, multiplication by Ω 2 degenerate principal Ω = 0. asymptotically flat: null infinity is null. asymptotically de Sitter: null infinity is space-like. asymptotically anti-de Sitter: null infinity is time-like boundary data required. Different approaches to compactified equations: Evolution along null surfaces works well, but not general due to caustics. Compactification along spacelike directions: standard method to construct initial data, underresolved blue-shifted waves in evolution. K = const. spacelike hyperboloidal surfaces reaching null infinity (t 2 x 2 = const. in Minkowski). S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30

74 Asymptotics Radiation lives at null infinity Taking appropriate limit in M, worldlines of increasingly distant geodesic observers converge to null geodesic generators of J + (proper time Bondi time)! Compactification at i 0 leads to piling up of waves, at J + this effect does not appear waves leave the physical spacetime through the boundary J +. Observers situated at astronomical distances are J +. E.g. computing the signal at a GW detector, J more realistically corresponds to an observer sufficiently far way from the source to treat the radiation linearly, but not so far away that cosmological effects have to be taken into account. S. Husa (UIB) Numerical Relativity ESI Vienna, August / 30