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MIT Lectures on the Geometry and Analysis of Black Hole Spacetimes in General Relativity

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MIT Lectures on the Geometry and Analysis of Black Hole Spacetimes in General Relativity MIT lectures on the geometry and analysis of black hole spacetimes in general relativity Mihalis Dafermos∗ November 8, 2007 Contents 1 Lecture I: What is a black hole? 3 1.1 BasicLorentziangeometry. 3 1.1.1 Lorentzianmetrics . 3 1.1.2 The Levi-Civita connection, geodesics, curvature . ... 4 1.1.3 Timeorientationandcausalstructure . 5 1.1.4 GlobalhyperbolicityandCauchysurfaces . 6 1.2 R1+1 .................................. 6 1.3 R3+1 .................................. 7 1.4 Exercises ............................... 10 1.5 Penrose(-Carter)diagrams. 10 1.6 Schwarzschild............................. 11 1.6.1 M > 0............................. 12 1.6.2 M < 0............................. 13 1.6.3 Historicaldigression . 13 1.6.4 Note on the definition of black holes . 14 1.7 Reissner-Nordstr¨om . 15 1.7.1 M > 0, 0 <e2 <M 2 ..................... 16 1.7.2 M > 0, e2 = M 2 ....................... 18 1.7.3 e2 >M 2 or M < 0...................... 18 1.8 Schwarzschild-deSitter. 18 1.8.1 0 < 3M√Λ < 1........................ 19 1.8.2 3M√Λ=1 .......................... 19 1.9 Exercises ............................... 20 ∗University of Cambridge, Department of Pure Mathematics and Mathematical Statistics, Wilberforce Road, Cambridge CB3 0WB United Kingdom 1 2 Lecture II: The analysis of the Einstein equations 21 2.1 Formulationoftheequations . 21 2.1.1 Thevacuumequations . 22 2.1.2 TheEinstein-Maxwellsystem . 22 2.1.3 TheEinstein-scalarfieldsystem. 23 2.1.4 TheEinstein-Vlasovsystem . 24 2.2 Theinitialvalueproblem . 25 2.2.1 Theconstraintequations . 25 2.3 Initialdata .............................. 26 2.3.1 Thevacuumcase....................... 26 2.3.2 Thecaseofmatter .. .. .. .. .. .. .. .. .. 26 2.3.3 Cosmology .......................... 26 2.3.4 Isolated self-gravitating systems: asymptotic flatness.. 27 2.4 Themaximaldevelopment. 28 2.5 HyperbolicityandtheproofofTheorem2.2 . 29 2.5.1 Wave coordinates and Einstein’s linearisation . .. 29 2.5.2 Local existence for quasilinear wave equations . .. 30 2.5.3 From quasilinear wave equations to the Einstein vacuum equations ........................... 31 2.6 Penrose’sincompletenesstheorem. 31 2.7 Thecosmiccensorshipconjectures . 32 2.7.1 Strongcosmiccensorship . 32 2.7.2 Weakcosmiccensorship . 33 2.8 Exercises ............................... 33 3 Lecture III: The linear wave equation on black hole backgrounds 34 3.1 DecayonMinkowskispace. 34 K 3.1.1 The Morawetz current Jµ .................. 34 3.1.2 Angular momentum operators Ωi .............. 34 3.1.3 Digression: Penrose’s conformal compactification . ... 34 3.2 Schwarzschild:Acloserlook. 34 3.2.1 Trapping ........................... 34 3.2.2 Theredshifteffect . 34 K Y X 3.3 The currents Jµ , Jµ , Jµ ...................... 34 3.4 The wave equation on Schwarzschild-de Sitter . .. 34 3.5 ThegeometryoftheKerrsolution . 34 3.6 Openproblems,furtherreading . 34 4 Lecture IV: Non-linear spherically symmetric problems 34 4.1 The spherically symmetric Einstein equations with matter.... 34 4.1.1 The Hawking mass, the trapped set and the regular region 34 4.1.2 Ageneralextensionprinciple . 34 4.1.3 Spherically symmetric Penrose inequalities . .. 34 4.2 Spherically symmetric “no hair” theorems . 34 4.3 Decay rates for a self-gravitating scalar field . .... 34 4.4 Openproblems,furtherreading . 34 2 1 Lecture I: What is a black hole? General relativity concerns Lorentzian manifolds satisfying the so-called Ein- stein equations. We will turn to a more detailed discussion of the analysis of the Einstein equations in the next lecture. The purpose of this lecture is to in- troduce one of the most central notions of general relativity, that of black hole. This notion is tied to global Lorentzian causality. In the present lecture, we shall give the basics of Lorentzian geometry and then take the shortest route to the notion of black holes, that provided by Penrose diagrams. 1.1 Basic Lorentzian geometry 1.1.1 Lorentzian metrics Definition 1.1. Let n+1 be a C1 oriented n +1-dimensional manifold. A M 0 Lorentzian metric gµν is a C non-degenerate covariant symmetric 2-tensor of signature ( , +,..., +). We call the pair ( ,g) a Lorentzian manifold. − M In general relativity, physical spacetime is represented by a 3+1-dimensional Lorentzian manifold ( 3+1,g) with a piece of extra structure to be defined in Section 1.1.3. AlthoughM the most important case is n = 3, note that one often considers submanifolds, quotient manifolds, etc. of lower dimension, which again inherit Lorentzian metrics. The case n > 3 is central for current (c. 2007) speculation amongst high energy physicists. 1 Definition 1.2. Let v Tp . We call v timelike if g(v, v) < 0, null if v =0, g(v, v)=0, and spacelike∈ M if2 g(v, v) > 0. If v =0, g(v, v) 0, we say that6 v is causal. 6 ≥ These appellations are inherited by vector fields and curves: Definition 1.3. Let S , and let V Γ(T S). We will call V timelike, etc., if V (p) is timelike⊂M for all p S. ∈ M| ∈ Definition 1.4. Let γ : I be a C1 curve where I R. We call γ timelike, etc., if γ′(t) is timelike,→ M etc., for all t. ⊂ For general C1 immersed submanifolds i : d , with d<n +1 we make the following definitions N → M Definition 1.5. We call d spacelike if i∗g is Riemannian, timelike if i∗g is Lorentzian, and null if Ni∗g is degenerate. The above definition agrees with the previous for curves. For d = n, note that any hyperplane P through the origin in the tangent bundle has a unique 1-dimensional orthogonal complement. We will call a gen- erator of this orthogonal complement a normal for P . n is spacelike iff its N 1in some conventions denoted lightlike 2in some conventions, the 0 vector is also counted as spacelike 3 normal is timelike, n is timelike iff its normal is spacelike, and n is null iff its normal is null. NoteN that in the latter case, the normal is containedN in the tangent space. 1.1.2 The Levi-Civita connection, geodesics, curvature First let us note that the inverse metric gµν can be defined as in Riemannian µν µ geometry by g gνλ = δλ where δ here denotes the Kronecker symbol. Here, following the Einstein summation convention repeated indices are to be summed. Indices will be raised with the inverse metric in the usual way. In addition, all notations ever invented to denote geometric objects will be used interchangably when convenient in what follows. Definition 1.6. Let be a C3 manifold with g a C2 Lorentzian metric. We define the Levi-CivitaM connection to be the connection in T defined by ( V )ν = ∂ V ν +Γν V λ where ∇ M ∇µ µ µλ . 1 Γν = gνα(∂ g + ∂ g ∂ g ). µλ 2 λ µα µ λα − α µλ Definition 1.7. Let be a C3 manifold with g a C2 Lorentzian metric. We M µ define the Riemann curvature tensor Rνλρ to be the (1, 3)-tensor defined by . Rµ = ∂ Γµ ∂ Γµ +Γα Γµ Γα Γµ , νλρ λ ρν − ρ λν ρν λα − λν ρα the Ricci curvature Rµν to be the covariant 2-tensor defined by . α Rµν = Rµαν and the scalar curvature R to be the scalar defined by . µν R = g Rµν . Note: Our physical spacetimes ( ,g) will satisfy the Einstein equations M 1 R g R =8πT Λg (1) µν − 2 µν µν − µν where here Λ is a fixed real constant, called the cosmological constant, and where Tµν is a symmetric tensor which is determined by matter fields which themselves must satisfy a system of equations to close the system (1). Equations (1) implies that Tµν is divergence free. In the special case where Tµν = 0, the system (1) is equivalent to Rµν = Λgµν . These are known as the vacuum Einstein equations. In the study of astrophysical isolated self-gravitating systems, it is appropriate to set Λ = 0. Current models for cosmology depend on a small Λ > 0. We will return to the system (1) later on. Definition 1.8. Let be a C3 manifold with g a C2 Lorentzian metric. A C2 parametrized curveMγ : I is a geodesic if →M ′ ′ γ =0, ∇γ 4 i.e., if, in local coordinates xα d2xµ(γ(t)) dxν (γ(t)) dxλ(γ(t)) +Γµ =0. dt2 νλ dt dt Proposition 1.1.1. If γ : I is a geodesic and γ′(0) is timelike, spacelike, null, causal, resp., then γ is→M a timelike, spacelike, null, causal, resp., curve. Proof. ′ ′ ′ ′ ′ γ (g(γ ,γ )) = 2( ′ γ ,γ )=0. ∇γ Thus we may define Definition 1.9. A geodesic γ is said to be a timelike, spacelike, null, causal, resp., geodesic if its tangent vector at a single point is timelike, spacelike, null, causal, resp. Of these geodesics, the timelike and null have special importance in the physical interpretation of general relativity. Timelike geodesics correspond to “freely falling observers” or “test particles”. In principle, the metric g is “mea- sured” by examining the trajectories of these. Null geodesics correspond to the trajectories of “light rays” in the geometric optics limit. They also correspond to the characteristics of the Einstein equations for the metric g. Definition 1.10. A parametrized C1 curve γ : I is inextendible if it is not the restriciton to I of a C1 curve γ : J →Mwhere J I properly. →M ⊃ Definition 1.11. Let ( be a C3 manifold with C2 Lorentzian metric g. We will say that ( ,g) isMgeodesically complete if all inextendible geodesics have domain RM. Otherwise we say that ( ,g) is geodesically incomplete. We say that ( ,g) is timelike, null, spacelikeM , causally, resp., geodesi- cally (in)completeM if the above holds where only timelike, etc., geodesics are considered. 1.1.3 Time orientation and causal structure Let Γ0(T ) denote the subset of continuous timelike vector fields.3 Define T ⊂ M an equivalence relation on by T1 T2 if g(T1(p),T2(p)) < 0, for all p.
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