THE NONLINEAR STABILITY OF MINKOWSKI SPACE FOR SELF-GRAVITATING - PowerPoint PPT Presentation

THE NONLINEAR STABILITY OF MINKOWSKI SPACE FOR SELF-GRAVITATING MASSIVE MATTER Philippe LeFloch Laboratoire Jacques-Louis Lions & CNRS Universit´ e Pierre et Marie Curie (Paris 6, Jussieu) contact@philippelefloch.org GENERAL RELATIVITY § Global geometry of spacetimes p M , g αβ q with signature p´ , ` , ` , `q § Einstein equations for self-gravitating matter G αβ “ 8 π T αβ § Einstein curvature G αβ “ R αβ ´ p R { 2 q g αβ § Riccci curvature R αβ “ B 2 g ` B ‹ B g § scalar curvature R : “ R α α “ g αβ R αβ CAUCHY PROBLEM § Global nonlinear stability of Minkowski spacetime § initial data prescribed on a spacelike hypersurface § small perturbation of an asymptotically flat slice in Minkowski space § Vacuum spacetimes T αβ “ 0 or massless matter fields § Christodoulou - Klainerman (1993), Lindblad - Rodnianski (2010) § Massive matter fields massive matter T αβ , open since 1993 § LeFloch - Yue Ma (2016)

CHALLENGES § Gravitational waves § Weyl curvature (vacuum), Ricci curvature (matter) § Nonlinear wave interactions § exclude dynamical instabilities, self-gravitating massive modes § avoid gravitational collapse (trapped surfaces, black holes) § Global dynamics § (small) perturbation disperses in timelike directions § asymptotic convergence to Minkowski spacetime § future timelike geodesically complete spacetime MAIN STRATEGY § Nonlinear wave systems § Einstein equations in wave gauge § PDE system which couples wave and Klein-Gordon equations § no longer scale-invariant, time-asymptotics drastically different § Hyperboloidal Foliation Method PLF-YM, Monograph, 2014 § foliation of the spacetime by asymptotically hyperbolidal slices § sharp time-decay estimates (metric, matter fields) for wave and Klein-Gordon equations § quasi-null structure of the Einstein equations

OUTLINE of the lecture § Einstein gravity and f(R)-gravity § Nonlinear global stability: geometric statements § Overview of the Hyperboloidal Foliation Method § Nonlinear global stability: statements in wave coordinates § Quasi-null hyperboloidal structure of the Einstein equations ELEMENTS of proof § Second-order formulation of the f(R)-gravity theory § Wave-Klein-Gordon systems § null interactions (simplest setup, better energy bounds) § weak metric interactions (simplest setup) § strong metric interactions (simpler setup) § Sharp pointwise bounds for wave-Klein-Gordon equations on curved space

EINSTEIN GRAVITY AND F(R)-GRAVITY Self-gravitating massive fields Massive scalar field with potential U p φ q , for instance U p φ q “ c 2 2 φ 2 , described by the energy-momentum tensor ´ 1 ¯ 2 g α 1 β 1 ∇ α 1 φ ∇ β 1 φ ` U p φ q T αβ : “ ∇ α φ ∇ β φ ´ g αβ Einstein-Klein-Gordon system for the unknown p M , g αβ , φ q ´ ¯ R αβ ´ 8 π ∇ α φ ∇ β φ ` U p φ q g αβ “ 0 l g φ ´ U 1 p φ q “ 0 Field equations of the f p R q -modified gravity Generalized Hilbert-Einstein functional § Gravitation mediated by additional fields § Functional ż ´ ¯ f p R g q ` 16 π L r φ, g s dV g M 2 R 2 ` κ 2 O p R 3 q and κ ą 0 § f p R q “ R ` κ § long history in physics: Weyl 1918, Pauli 1919, Eddington 1924, . . .

Critical point equation N αβ “ 8 π T αβ ´ ¯ ´ ¯` ˘ N αβ “ f 1 p R g q G αβ ´ 1 f p R g q ´ R g f 1 p R g q f 1 p R g q g αβ ` g αβ l g ´ ∇ α ∇ β 2 § If f linear, N αβ reduces to G αβ . § Vacuum Einstein solutions are vacuum f(R)-solutions § Fourth-order derivatives of g Gravity/matter coupling ∇ α R αβ “ 1 Bianchi identities (geometry) 2 ∇ β R § imply ∇ α G αβ “ 0, but also ∇ α N αβ “ 0 . § Euler equations ∇ α T αβ “ 0 Energy-momentum tensor of a massive field § Jordan’s coupling ´ 1 ¯ 2 ∇ γ φ ∇ γ φ ` U p φ q l g φ ´ U 1 p φ q “ 0 T αβ : “ ∇ α φ ∇ β φ ´ g αβ § Einstein’s coupling ´ ´ 1 ¯ ¯ T αβ : “ f 1 p R g q 2 ∇ γ φ ∇ γ φ ` U p φ q ∇ α φ ∇ β φ ´ g αβ ill-posed PDE for φ

WAVE-KLEIN-GORDON FORMULATION Field equations in coordinates G αβ “ 8 π T αβ Einstein equations § Second-order system with no specific PDE type § Wave coordinates l g x α “ 0 § Second-order system of 11 nonlinear wave-Klein-Gordon equations § Hamiltonian-momentum Einstein’s contraints N αβ “ 8 π T αβ Modified gravity equations § Fourth-order system with no specific PDE type § The augmented formulation § conformal transformation g : αβ : “ f 1 p R g q g αβ § evolution equation for the scalar curvature ρ : “ 1 κ ln f 1 p R g q (new degree of freedom) § Wave coordinates l g : x α “ 0 § Second-order system of 12 nonlinear wave-Klein-Gordon equations § More involved algebraic structure, and additional constraints

Ricci curvature in wave gauge 2 g : αβ B β g : αγ “ g : αβ B γ g : αβ (following Lindblad-Rodnianski, 2005) 2 R : l g : g : αβ “ ´ r αβ ` Q αβ ` P αβ “ ´ g : α 1 β 1 B α 1 B β 1 g : αβ ` Q αβ ` P αβ (i) terms satisfying Klainerman’s null condition (good decay in time) Q αβ : “ g : λλ 1 g : δδ 1 B δ g : αλ 1 B δ 1 g : βλ g : δδ 1 ` ˘ ´ g : λλ 1 B δ g : αλ 1 B λ g : βδ 1 ´ B δ g : βδ 1 B λ g : αλ 1 g : δδ 1 ` ˘ ` g : λλ 1 B α g : λ 1 δ 1 B δ g : λβ ´ B α g : λβ B δ g : ` . . . . . . . . . . . . λ 1 δ 1 (ii) “quasi-null terms” (need again the gauge conditions) 2 g : λλ 1 g : δδ 1 4 g : δδ 1 g : λλ 1 P αβ : “ ´ 1 B α g : δλ 1 B β g : λδ 1 ` 1 B β g : δδ 1 B α g : λλ 1 Notation § Functions V κ “ V κ p ρ q and W κ “ W κ p ρ q § defined from f p R q » R ` κ 2 R 2 § quadratic in ρ § Quadratic potential U p φ q “ c 2 2 φ 2

f(R)-gravity for a self-gravitating massive field l g : g : αβ “ F αβ p g : , B g : q ´ 3 κ 2 B α ρ B β ρ ` κ V κ p ρ q g : r αβ ` ˘ 2 e ´ κρ B α φ B β φ ` c 2 φ 2 e ´ 2 κρ g : ´ 8 π αβ ´ 2 e ´ κρ φ 2 ¯ g : αβ B α φ B β φ ` c 2 3 κ r l g : ρ ´ ρ “ κ W κ p ρ q ´ 8 π l g : φ ´ c 2 φ “ c 2 ` ˘ e ´ κρ ´ 1 φ ` κ g : αβ B α φ B β ρ r § wave gauge conditions g : αβ Γ : λ αβ “ 0 § curvature compatibility e κρ “ f 1 p R e ´ κρ g : q § Hamiltonian and momentum constraints (propagating from a Cauchy hypersurface) ` 2 φ 2 ˘ g αβ ∇ α φ ∇ β φ ` c 2 In the limit κ Ñ 0 one has g : Ñ g and ρ Ñ 8 π Einstein system for a self-gravitating massive field ` ˘ 2 B α φ B β φ ` c 2 φ 2 g αβ r l g g αβ “ F αβ p g , B g q ´ 8 π l g φ ´ c 2 φ “ 0 r Main issues § Time-asymptotic decay (energy, sup-norm), global existence theory § Dependence in f and singular limit f p R q Ñ R

NONLINEAR GLOBAL STABILITY: geometric statements Earlier works on vacuum spacetimes or massless matter § Christodoulou-Klainerman 1993 § fully geometric proof, Bianchi identities, geometry of null cones § null foliation, maximal foliation § Lindblad-Rodnianski 2010 § first global existence result in coordinates § wave coordinates (despite an “instability” result by Choquet-Bruhat) § asymptotically flat foliation § their proof relies strongly on the scaling field r B r ` t B t of Minkowski spacetime § Extensions to massless models § same time asymptotics, same Killing fields § Bieri (2009), Zipser (2009), Speck (2014) Initial value problem § geometry of the initial hypersurface p M 0 » R 3 , g 0 , k 0 q § matter fields φ 0 , φ 1 § initial data sets close to a spacelike, asympt. flat slice in Minkowski spacetime

Dynamics of self-gravitating massive matter § Spatially compact problem § compactly supported massive scalar field § Positive mass theorem § no solution can be exactly Minkowski “at infinity” § coincides with a slice of Schwarzschild near space infinity, with ADM mass m ăă 1 § Compact Schwarzschild perturbation Theorem 1. Nonlinear stability of Minkowski spacetime with self-gravitating massive fields Consider the Einstein-massive field system when the initial data set p M 0 » R 3 , g 0 , k 0 , φ 0 , φ 1 q is a compact Schwarzschild perturbation satisfying the Einstein constraint equations. Then, the initial value problem § admits a globally hyperbolic Cauchy development, § which is foliated by asymptotically hyperbolic hypersurfaces. § Moreover, this spacetime is future causally geodesically complete and asymptotically approaches Minkowski spacetime.

Theorem 2. Nonlinear stability of Minkowski spacetime in f(R)-gravity Consider the field equations of f p R q -modified gravity when the initial data set p M 0 » R 3 , g 0 , k 0 , R 0 , R 1 , φ 0 , φ 1 q is a compact Schwarzschild perurbation satisfy- ing the constraint equations of modified gravity. Then, the initial value problem § admits a globally hyperbolic Cauchy development, § which is foliated by asymptotically hyperbolic hypersurfaces. § Moreover, this spacetime is future causally geodesically complete and asymptotically approaches Minkowski spacetime. Limit problem κ Ñ 0 § relaxation phenomena for the spacetime scalar curvature § passing from a second-order wave equation to an algebraic equation Theorem 3. f(R)-spacetimes converge toward Einstein spacetimes The Cauchy developments of modified gravity in the limit κ Ñ 0 when the nonlinear function f “ f p R q (the integrand in the Hilbert-Einstein action) approaches the scalar curvature function R converge (in every bounded time interval, in a sense specified quantitatively in Sobolev norms) to Cauchy developments of Einstein’s gravity theory.