Sergio Daín (1969-2016)
Understanding isolated system dynamics in General Relativity A perspective on Sergio Dain’s contribution to General Relativity Jos´ e Luis Jaramillo Institut de Math´ ematiques de Bourgogne (IMB) Universit´ e de Bourgogne Franche-Comt´ e Jose-Juis.Jaramillo@u-bourgogne.fr VIII International Meeting on Lorentzian Geometry, GeLoMa 2016 M´ alaga, 21 September 2016 Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 1 / 35
Gravitational collapse in General Relativity: the general framework 1 Aspects of the Cauchy problem in General Relativity 2 Elliptic problems in General Relativity Einstein equations: physical content Geometric inequalities: the role of angular momentum 3 Global inequalities: | J | ≤ m 2 Local inequalities: A ≥ 8 π | J | Perspective 4 Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 2 / 35
Gravitational collapse in General Relativity: the general framework Scheme Gravitational collapse in General Relativity: the general framework 1 Aspects of the Cauchy problem in General Relativity 2 Elliptic problems in General Relativity Einstein equations: physical content Geometric inequalities: the role of angular momentum 3 Global inequalities: | J | ≤ m 2 Local inequalities: A ≥ 8 π | J | Perspective 4 Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 3 / 35
Gravitational collapse in General Relativity: the general framework A general framework of research Understanding Einstein equations: interplay of Geometry, Analysis and Physics R µν − 1 2 R g µν = 8 πG c 4 T µν Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 4 / 35
Gravitational collapse in General Relativity: the general framework A general framework of research Understanding Einstein equations: interplay of Geometry, Analysis and Physics R µν − 1 2 R g µν = 8 π T µν Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 4 / 35
Gravitational collapse in General Relativity: the general framework A general framework of research Understanding Einstein equations: interplay of Geometry, Analysis and Physics R µν − 1 2 R g µν = 8 π T µν Classical gravitational collapse picture Singularity Theorems : incomplete inextendible causal 1 geodesic, given “strong gravitational field” data on ˜ S Trapped surfaces [Penrose, Hawking, 65, 67, 70, 73...] . Non-simply connected data [Gannon, Lee 75, 76...] . (Weak) Cosmic Censorship conjecture [Penrose 69] : 2 Complete I + and Black Hole region and Horizon. Spacetime settles down to a stationary final state : 3 Positivity mass theorems [Schoen & Yau 79, 80, Witten 81] . BH uniqueness “theorems” [e.g. Chru´ sciel et al. 12] : 4 Final state given by Kerr spacetime, ( m, J ) . Initial value problem : 3+1 approach. Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 4 / 35
Aspects of the Cauchy problem in General Relativity Scheme Gravitational collapse in General Relativity: the general framework 1 Aspects of the Cauchy problem in General Relativity 2 Elliptic problems in General Relativity Einstein equations: physical content Geometric inequalities: the role of angular momentum 3 Global inequalities: | J | ≤ m 2 Local inequalities: A ≥ 8 π | J | Perspective 4 Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 5 / 35
Aspects of the Cauchy problem in General Relativity Elliptic problems in General Relativity Plan Gravitational collapse in General Relativity: the general framework 1 Aspects of the Cauchy problem in General Relativity 2 Elliptic problems in General Relativity Einstein equations: physical content Geometric inequalities: the role of angular momentum 3 Global inequalities: | J | ≤ m 2 Local inequalities: A ≥ 8 π | J | Perspective 4 Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 6 / 35
Aspects of the Cauchy problem in General Relativity Elliptic problems in General Relativity The problem of the gravitational field dynamics A physical formulation “Suppose we want to describe an isolated self-gravitating system . For example a star, a binary system, a black hole or colliding black holes. Typically these astrophysical systems are located far away from the Earth, so that we can receive from them only electromagnetic and gravitational radiation. How is this radiation? For example one can ask how much energy is radiated, or which are the typical frequencies for some systems. This is the general problem we want to study .” [S. Dain, Lecture Notes in Physics 604, 161-182 (2002)] A first step in the mathematical study: elliptic systems “ Elliptic problems appear naturally in physics mainly in two situations: as equations which describe equilibrium (for example, stationary solutions in General Relativity) and as constraints for the evolution equations (for example, constraint equations in Electromagnetism and General Relativity). In addition, in General Relativity they appear often as gauge conditions for the evolution equations.” [S. Dain, Lecture Notes in Physics 692, 117-139 (2006)] Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 7 / 35
Aspects of the Cauchy problem in General Relativity Elliptic problems in General Relativity Initial data problem in General Relativity: Cauchy problem Initial data set for Einstein equations ( ˜ S, ˜ h ij , ˜ K ij , µ, j i ) ˜ S : connected three-dimensional manifold. ˜ h ij : Riemannian metric. ˜ K ij : symmetric tensor field. µ : scalar field. j i : vector field on S . Constraint equations K ij + ˜ K 2 = 16 πµ R − ˜ ˜ K ij ˜ (Hamiltonian constraint) K ij − ˜ D i ˜ D j ˜ ˜ K = − 8 πj i (Momentum constraint) Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 8 / 35
Aspects of the Cauchy problem in General Relativity Elliptic problems in General Relativity Asymptotic flatness Asymptotically Euclidean initial data Data on ˜ S are asymptotically flat with N asymptotic ends, if for some compact set Ω we have ˜ S \ Ω = � N k =1 ˜ S ( k ) where ˜ S ( k ) are open sets that can be mapped by a coordinate x j diffeomorphically onto the complement of a closed ball in R 3 such that system ˜ � 1 + 2 m � ˜ r − 2 ) , ∂ k ˜ r − 2 ) , ∂ l ∂ k ˜ r − 3 ) h ij = δ ij + O (˜ h ij = O (˜ h ij = O (˜ r ˜ ˜ ∂ k ˜ r − 2 ) , r − 3 ) K ij = O (˜ K ij = O (˜ x ) 2 � 1 2 → ∞ in each “asymptotic end” ˜ �� 3 as ˜ r = j =1 (˜ S ( k ) . Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 9 / 35
Aspects of the Cauchy problem in General Relativity Elliptic problems in General Relativity Mass m and angular momentum J Conserved quantities at spatial infinity ADM mass m : 1 � � � ∂ j ˜ h ij − ∂ i ˜ ν i dA m = 16 π lim h jj ˜ ˜ r →∞ S r ADM momentum P i : P i = 1 � � � K jk − ˜ ˜ K ˜ ( ∂ i ) j ˜ ν k dA 8 π lim h ij ˜ r →∞ S r Angular momentum J i at spatial infinity: J i = 1 � � � K jk − ˜ ˜ K ˜ ( φ i ) j ˜ ν k dA , x j ∂ k 8 π lim with φ i = ǫ ijk ˜ h jk r →∞ ˜ S r Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 10 / 35
Aspects of the Cauchy problem in General Relativity Elliptic problems in General Relativity Conformal compactification: from Geometry to Analysis Conformal compactification of the data Focus on vacuum data: µ = 0 , j i = 0 . Conformally compactified data: ( S, h ij , K ij ) with ˜ S = S \ { i } ( i point at infinity) and K ij = ψ − 2 K ij + 1 ˜ ˜ h ij = ψ 4 h ij , ˜ h ij ˜ K 3 Constraint equations � D i D i − 1 � − 1 8 K ij K ij ψ − 7 + 1 K 2 ψ 5 (Lichnerowicz equation) ˜ 8 R ψ = 12 2 3 ψ 6 D i ˜ D j K ij = K Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 11 / 35
Aspects of the Cauchy problem in General Relativity Elliptic problems in General Relativity Conformal compactification: from Geometry to Analysis Conformal compactification of the data Focus on vacuum data: µ = 0 , j i = 0 . Conformally compactified data: ( S, h ij , K ij ) with ˜ S = S \ { i } ( i point at infinity) and K ij = ψ − 2 K ij + 1 ˜ ˜ h ij = ψ 4 h ij , ˜ h ij ˜ K 3 Constraint equations: maximal slicing ˜ K = 0 � � D i D i − 1 − 1 8 K ij K ij ψ − 7 (Lichnerowicz equation) = 8 R ψ D j K ij = 0 Jos´ e Luis Jaramillo Understanding isolated system dynamics in GR M´ alaga, 21 September 2016 11 / 35
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