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Rigidity Results for Stationary Spacetimes

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Rigidity Results for Stationary Spacetimes Rigidity results for stationary spacetimes Marc Mars University of Salamanca April 2015 Andrejewski Days 2015 Schloss Gollwitz, Brandenburg an der Havel Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 1/121 Table of Contents 1 Lecture 1. Introduction. Properties of Killing vectors. Examples of time-independent spacetimes. 2 Lecture 2. Quotient formalism. Harmonic map formulation 3 Lecture 3. Global properties of stationary spacetimes. Rigidity of strictly stationary spacetimes. Rigidity of stationary black hole spacetimes. Stationary and axially symmetric black holes. 4 Lecture 4. Non-rotating black hole spacetimes. Rigidity of static, asymptotically flat Killing initial data. Black hole rigidity without analyticity Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 2/121 Introduction Equilibrium configurations play a fundamental role to understand any physical theory. This is particularly important in Gravity because of the geometric nature of the field. The aim of this course is to present some of the main results in this area of research. The topic is vast and cannot be covered in a few hours. I will restrict myself to the case of pure gravitation, i.e. vacuum spacetimes. Even with this restriction, the presentation will by no means be exhaustive = Many interesting results will be left out. ⇒ The hope is to give an idea of what the subject is about, what are the main techniques, the difficulties and main results. Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 3/121 Stationarity in Gravitation In many physical theories there is a canonical notion of time. Equilibrium there simply means: independent of time. In General Relativity, there is no canonical notion of time. The definition of “equilibrium state” needs to be formulated in geometric terms. Require the metric to remain invariant under a suitable one-parameter isometry group. Recall that an isometry of a (pseudo-)Riemannian manifold ( , g M) is a diffeomorphism M ⋆ Φ: satisfying Φ (g M) = g M. M −→M A one-parameter isometry group is a differentiable map Ψ: R ×M −→ M (s, p) Ψ(s, p) := Ψs (p) satisfying −→ s, t R : (i)Ψs is a diffeomorphism , (ii)Ψs Ψt = Ψs+t (iii)Ψs is an isometry . ∀ ∈ ◦ dΨs (p) (i) and (ii) define a one-parameter group of transformations: Generator ξ(p) = ds s=0 Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 4/121 If Ψs is a one parameter isometry group then { } ⋆ ⋆ dΨs (g M) Ψ (g M) = g M = =0 ξg M =0. s ⇒ ds ⇐⇒ L Definition (Killing vector) A Killing field is a vector field ξ satisfying ξg M = 0. L Given a complete vector field ξ the one-parameter group of transformations can be reconstructed. Recall: A vector field ξ X( ) is complete iff all their integral curves ∈ M γ : I =(a, b) R , γ˙ (s) = ξ(γ(s)) have maximal domain I = R ⊂ −→ M A non-complete ξ generates a local one-parameter group of transformations. Theorem (Local one-parameter group of transformations) Let ( , g M) be a (pseudo-)Riemannian manifold and ξ any vector field. For any point M p , there exists an open set Up and a map ∈M Ψ:(a, b) Up × −→ M dΨs (q) such that (i) Ψs := Ψ(s, ) is a diffeomorphism onto the image, (ii) ds = ξ(q) and · s=0 (iii) Ψs Ψt = Ψs+t for all points and values of s, t where this expression is defined. ◦ ⋆ If, moreover, ξ is a Killing vector then Ψs (g M)=0. (Local isometry group) . Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 5/121 To talk about “equilibrium states” one still needs some relation of ξ with “time”. Definition (Spacetime) A spacetime ( , g M) is a smooth n-dimensional connected manifold (n 4) with a M ≥ smooth metric g M of Lorentzian signature , +, , + and a time orientation. {− ··· } Time orientation: Existence of a timelike vector field u, declared to be future. Write g M(u, v) also by u, v . h i A causal vector v is future if v, u 0. v is past if v is future. h i ≤ − The weakest possible notion of “equilibrium state” in General Relativity is Definition (Time-independent) A spacetime is ( , g M) time-independent if it admits a Killing vector ξ which is timelike M somewhere. Aim: classify time-independent spacetimes satisfying suitable field equations: e.g.. vacuum, electrovacuum, Yang-Mills, scalar field, etc. For many results, stronger notions of “equilibrium state” will be necessary. Theory mostly developed in dimension n = 4. Less clear that any reasonable classification exists in higher dimensions. Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 6/121 Setup and notation Given a spacetime ( , g M) denote by the Levi-Civita covariant derivative. M M ∇ Curvature operator of : Riem (X , Y )Z = X Y Y X Z. ∇ ∇ ∇ −∇ ∇ −∇[X ,Y ] Riemann tensor: RiemM(X , Y , Z, W ) = X , RiemM(Z, W )Y h i Ricci tensor, curvature scalar and Einstein tensor denoted: RicM, ScalM, EinM. Einstein field equations: M Ein +Λg M = χT . 8πG Λ : cosmological constant, χ = c4 =8π, T : energy-momentum tensor, Vacuum: T =0, 1 2 Scalar field: T = dΦ dΦ dΦ M g M. sc ⊗ − 2 | |g Perfect fluid: T =(ρ + p) u u + pg M, u, u = 1 pf ⊗ h i − Spacetime indices: α,β,γ, =0,, n 1. ··· ··· − µ 1 µν Electromagnetic field: (TEM)αβ = FαµF Fµν F g M, Fαβ two-form. β − 4 αβ + Equations for matter fields. E.g. in electrovacuum: dF =0, divg M F = 0. We will be mostly concerned with vacuum spacetimes RicM =0. Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 7/121 Basic properties of Killing vectors = ξ X( ),ξ Killing vector field endowed with [ , ] is a Lie algebra. G { ∈ M } A Killing vector satisfies the Killing equations: ξg M =0 αξβ + β ξα =0. L ⇐⇒ ∇ ∇ (i) F ξ and F ( ) F = const. Imply for any ξ : ∈G ∈F M ⇐⇒ ∈G M α (ii) β µξν = Riem αβµν ξ . ∇ ∇ Consequences of (ii): Fix p and define ξ = g M(ξ, ). The linear map ∈M · 2 ϕ : Tq Λ (Tp ) G −→ M × M ξ (v = ξ p, ω = dξ p) −→ | | is an isomorphism onto its image p := ϕ( ). n(n+1) H G dim and equality if and only if ( , g M) is of constant curvature k, i.e. G ≤ 2 M RiemM(X , Y )Z = k ( Y , Z X X , Z Y ) , k R. h i −h i ∈ (i) Prove that ϕ is an isomorphism (onto the image) Exercises: n(n+1) (ii) Prove that a spacetime is maximally symetric (dim( ) = ) if G 2 and only if has constant curvature. Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 8/121 Covariantly constant vectors A particular case of Killing vector is a covariantly constant vector field. Definition (Covariantly constant vector field) A covariantly constant vector field is a vector field satisfying αξβ =0. ∇ Theorem The dimension of the vector space = ξ X( ) ξ covariantly constant in any C { ∈ M } (pseudo)-Riemannian space of dimension n is dim( ) n. C ≤ M Moreover dim( ) = n if and only if ( , g M) is locally flat (Riem =0). C M Exercises: (i) Prove the theorem. (ii) Prove: If dim( ) = n 1, then dim( ) = n. C − C Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 9/121 Hypersurface orthogonal Killing vectors Definition (Integrable Killing vector) A vector field ξ X( ) is integrable iff ξ dξ =0 (ξ := g M(ξ, )). ∈ M ∧ · By Fr¨obenius: the set of points ξ =0 is foliated by maximal, injectively immersed, M\{ 6 } codimension-one submanifolds Σa satisfying ξ T Σa =0 for all Σa and p Σa. { } | p ∈ Integrable (also called hypersurface orthogonal) Killing vectors play a very important role in the classification of stationary spacetimes. Definition (Time-invariant spacetime) ( , g M) is time-invariant if it is time-independent and the corresponding Killing vector M ξ is integrable. Name is motivated by the following result (exercise): Let ( , g M) be time-invariant. For any p with ξ p timelike a neighbourhood M ∈Mi | ∃ Up =( a, a) Σ of p with coordinates (t, x ) (i, j =1, , n) such that − × ··· 2 2 i j g M = N (x)dt + hij (x)dx dx , ξ = ∂t , t(p)=0. − Note: The transformation (t, x) ( t, x) is an isometry of Up leaving p invariant. −→ − Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 10/121 Examples of time-independent spacetimes Most fundamental example: Minkowski spacetime. n 2 2 2 ( = R , η), η = dt + dx1 + + dxn 1 M − ··· − Has maximal number of Killings vectors and of covariantly constant fields. ζ = ∂t : globally timelike, integrable, Killing vector Boost ξ := x1∂t + t∂x1 is also a Killing vector. Timelike in x1 > t , spacelike in x1 < t , null in x1 = t . | | | | | | | | | | | | In the region x1 > t , the coordinate change | | + t = X sinh T , x1 = X cosh T , (X , T ) R R ∈ × transforms the metric into static flat Kasner: = R R+ R2, M × × 2 2 2 2 2 g M = X dT + dX + dx + dx , ξ = ∂T − 2 ··· n This spacetime is extendible. Static flat Kasner belongs to a larger family of, generarically inextendible, vacuum, time-independent spacetimes: Static Kasner spacetimes: + n 2 2α1 2 2 2α2 2 2αn 2 = R R R − , g M = X dT + dX + X dx2 + + X dxn 1 M × × − ··· − n 1 n 1 − − 2 αi =1, αi =1 i=1 i=1 X X Marc Mars (University of Salamanca) Rigidity results for stationary spacetimes April2015 11/121 Groups of isometries One-parameter groups of isometries is a particular case of groups of isometries.
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