Lecture On Black Holes - KEK
Lecture on Black HolesAt Osaka UniversityApril 25-27, 2011Hideo KodamaTheory Center, IPNS, KEK
To Index- 2-ContentsCh.1 Basics of Black Holes1.1 Concept of Infinity . . . . . . . . . . . . . . . . .1.1.1 Conformal Embedding: example . . . . . .1.1.2 Conformal Infinity . . . . . . . . . . . . .1.2 Definition . . . . . . . . . . . . . . . . . . . . . .1.2.1 Causal sets . . . . . . . . . . . . . . . . .1.2.2 Causality conditions . . . . . . . . . . . .1.2.3 Horizon . . . . . . . . . . . . . . . . . . .1.3 Raychaudhuri equation . . . . . . . . . . . . . . .1.3.1 Jacobi equation . . . . . . . . . . . . . . .1.3.2 Representation in terms of the Fermi basis1.3.3 Expansion, shear, rotation . . . . . . . . .1.3.4 Geodesic deviation equations . . . . . . .1.3.5 Conjugate point . . . . . . . . . . . . . . .1.4 Area Theorem . . . . . . . . . . . . . . . . . . . .1.5 Symmetry . . . . . . . . . . . . . . . . . . . . . .1.5.1 Killing vector . . . . . . . . . . . . . . . .1.5.2 Stationary spacetime . . . . . . . . . . . .1.5.3 Killing horizon . . . . . . . . . . . . . . .1.6 Examples . . . . . . . . . . . . . . . . . . . . . .1.6.1 Static spherically symmetric black holes .1.6.2 Ernst formalism . . . . . . . . . . . . . . .1.6.3 Kerr-Newman Solution . . . . . . . . . . .1.6.4 Degenerate solutions . . . . . . . . . . . .1.7 Smarr Formula . . . . . . . . . . . . . . . . . . .1.7.1 Komar integral . . . . . . . . . . . . . . .1.7.2 Integral at infinity . . . . . . . . . . . . .1.7.3 Integral at horizon . . . . . . . . . . . . .1.8 Wald Formulation for BH Thermodynamcis . . .1.8.1 Noether charge . . . . . . . . . . . . . . .1.8.2 Bekenstein formula . . . . . . . . . . . . 272829To Index
3 To IndexCh.2 Solutions with High Symmetries2.1 Black-Brane Type Spacetime . . . . . . . . . . . . . .2.1.1 Assumptions . . . . . . . . . . . . . . . . . . .2.1.2 General Robertson-Walker spacetime . . . . .2.1.3 Braneworld model . . . . . . . . . . . . . . .2.1.4 Higher-dimensional static Einstein black holes2.1.5 Black branes . . . . . . . . . . . . . . . . . . .2.2 Higher-dimensional rotating black hole . . . . . . . .2.2.1 GLPP solution . . . . . . . . . . . . . . . . .2.2.2 Simply rotating solution . . . . . . . . . . . .2.3 Special MP Solution . . . . . . . . . . . . . . . . . .2.3.1 Invariant bases of S 3 . . . . . . . . . . . . . .2.3.2 S 1 fibring of S 2N 1 . . . . . . . . . . . . . . .2.3.3 U(N) MP solution . . . . . . . . . . . . . . .2.3.4 Internal Structure . . . . . . . . . . . . . . . .2.4 Black Ring . . . . . . . . . . . . . . . . . . . . . . . .2.4.1 Generalised Weyl Formulation . . . . . . . . .2.4.2 Static black ring solution . . . . . . . . . . . .2.4.3 Rotating black ring . . . . . . . . . . . . . . .Ch.3 Rigidity and Uniqueness3.1 Initial value problem . . . . . . . . . . . .3.1.1 Spacetime Decomposition . . . . .3.1.2 (n 1)-decomposition . . . . . . .3.2 Positive Energy Theorem . . . . . . . . . .3.3 Rigidity Theorem for Static Black Holes .3.3.1 Birkhoff’s Theorem . . . . . . . . .3.3.2 Rigidity theorem: 4D . . . . . . . .3.3.3 Rigidity theorem: High D . . . . .3.4 Uniqueness Theorem: 4D Einstein-Maxwell3.4.1 Outline . . . . . . . . . . . . . . .3.4.2 Non-linear σ model . . . . . . . . .3.5 Rigidity Theorem in General Dimensions .3.6 Spacetime Topology . . . . . . . . . . . . .3.6.1 Topological censorship . . . . . . .3.6.2 Horizon topology . . . . . . . . . .Ch.4 Gauge-invariant Perturbation4.1 Background Solution . . . . . . .4.1.1 Examples . . . . . . . . .4.2 Perturbations . . . . . . . . . . .4.2.1 Perturbation equations . .4.2.2 Gauge problem . . . . . .Theory. . . . . . . . . . . . . . . . . . . . 5151525354545556575758.606061626263To Index
4 To Index4.2.3 Tensorial decomposition of4.3 Tensor Perturbation . . . . . . .4.3.1 Tensor Harmonics . . . . .4.3.2 Perturbation equations . .4.4 Vector Perturbation . . . . . . . .4.4.1 Vector harmonics . . . . .4.4.2 Perturbation equations . .4.5 Scalar Perturbation . . . . . . . .4.5.1 Scalar harmonics . . . . .4.5.2 Perturbation equations . .perturbations. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Ch.5 Instabilities of Black Holes5.1 Various Instabilities: summary . . . . . . . . .5.1.1 4D black holes . . . . . . . . . . . . . .5.1.2 Higher-dimensional BHs . . . . . . . .5.2 Superradiance . . . . . . . . . . . . . . . . . .5.2.1 Klein-Gordon field around a Kerr BH .5.2.2 Integral on a null surface . . . . . . . .5.2.3 Flux integral . . . . . . . . . . . . . .5.2.4 Superradiance condition . . . . . . . .5.2.5 Penrose process . . . . . . . . . . . . .5.3 Superradiant Instability . . . . . . . . . . . .5.3.1 Massive scalar around a (adS-)Kerr BH5.3.2 4D Kerr . . . . . . . . . . . . . . . . .5.4 Stability of Static Black Holes . . . . . . . . .5.4.1 Tensor Perturbations . . . . . . . . . .5.4.2 Vector Perturbations . . . . . . . . . .5.4.3 Scalar Perturbations . . . . . . . . . .5.4.4 Summary of the Stability Analysis . .5.5 Flat black brane . . . . . . . . . . . . . . . . .5.5.1 Strategy . . . . . . . . . . . . . . . . .5.5.2 Tensor perturbations . . . . . . . . . .5.5.3 Vector perturbations . . . . . . . . . .5.5.4 Scalar perturbations . . . . . . . . . .Ch.6 Lovelock Black Holes6.1 Lovelock theory . . . . . . . . . . . . . . . . .6.2 Static black hole solution . . . . . . . . . . . .6.2.1 Constant curvature spacetimes . . . . .6.2.2 Black hole solution . . . . . . . . . . .6.3 Perturbation equations for the static solution .6.3.1 Tensor perturbations . . . . . . . . . .6.3.2 Vector perturbations . . . . . . . . . 8687919595969797100.108. 109. 110. 110. 111. 112. 113. 113To Index
5 To Index6.3.3 Scalar perturbations . . . . . . . . . . . . . . . . . . . . . . 1156.4 Instability of Lovelock BHs . . . . . . . . . . . . . . . . . . . . . . . 116To Index
To Index- 6-Basics of Black Holes1§1.1Concept of Infinity1.1.1Conformal Embedding: exampleThe Minkowski Spacetime with metricg(E n,1) dt2 dr 2 r 2 g(S n 1)(1.1.1)can be embedded into the static Einstein universeΩ2 g(E n,1) dη 2 dχ2 sin2 χ g(S n 1) dη 2 g(S n ),η χη χcosΩ cos22(1.1.2)(1.1.3)by the transformationη χ,2η χ.t r 2 tan2t r 2 tan(1.1.4a)(1.1.4b) r 0 t- SnTo Index
Ch.1Basics of Black Holes7 To IndexThe image is η π χ,0 χ π.(1.1.5)The boundary of this image in SEU is given by M I I ,1.1.2I R S n 1(1.1.6)Conformal InfinityBy generalising the previous example, Penrose proposed the following definitionof the infinity boundary I of a spacetime M in terms of a conformal mappingf : M Mˆ [Penrose R 1963[Pen63]]:1. I f (M ) Mˆ: smooth2. ĝ Ω2 f g: Ω : Mˆ R3. Ω I 0, dΩ I 0For the Weyl transformationĝμν Ω2 gμν ;Ω eΦ ,(1.1.7)the Christofell symbol and the curvature tensor of an (n 1)-dimensional Riemannian space(time) transform asˆ ν Φδ μ ˆ λ Φδνμ ˆ μ Φĝνλ ,Γμνλ Γ̂μνλ λˆ λ] ˆ ν Ω 2Ωĝν[σ ˆ λ] ˆ μΩΩ2 Rμ νλσ Ω2 R̂μ νλσ 2Ωδ μ (1.1.8)[σμ2ˆδ[λ ĝσ]ν , 2( Ω)Ω2 Rμν Ω2 R̂μν(1.1.9) ˆ ν Ω ĝμν Ω ˆ 2 Ω n( Ω)ˆ μ ˆ 2 , (1.1.10) (n 1)Ω ˆ 2 Ω n(n 1)( Ω)ˆ 2.R Ω2 R̂ 2nΩ From this, it follows that for vacuum spacetime satisfying 2ΛT2gμν κ Tμν gμν ,Rμν n 1n 1(1.1.11)(1.1.12)r aar 0r r 0r 0 -To Index
Ch.1Basics of Black Holes8 To Indexˆ has toor with the energy-momentun tensor decreasing as O(Ω) at infinity, Ωsatisfy the conditionˆ 2 ( Ω)2Λ.n(n 1)(1.1.13)This implies thatΛ 0 Λ 0 Λ 0 I : NullI : SpacelikeI : Timelike【Definition 1.1.1 (WAS spacetime)】In general, when a spacetime M hasa neighborhood of infinity that is isomorphic to a neighborhood of infinityof either E n,1 , dSn 1 or adSn 1 , M is called to be weakly asymptoticallysimple. §1.2Definition1.2.1Causal sets【Definition 1.2.1 (Chronological past/future)】For a set S in a spacetime M , its chronological past I (S , M ) (future I (S , M )) is defined as the setof all points that can be connected to a point in S by a future (past)-directedtimelike curve of non-zero length in M . 【Definition 1.2.2 (Causal past/future)】For a set S in a spacetime M , its causal past J (S , M ) (future J (S , M )) is defined as the set of all pointsthat can be connected to a point in S by a future (past)-directed causalcurve in M . 【Proposition 1.2.3】The boundary of J (S , M ) is a union of a nullhypersurface N and an acausal subset S0 J (S , M ) J (S , M ).Each null geodesic generator of N (N ) has no past (future) end pointexcept in S . Proof. Let us consider the case of J (S , M ). Let p be a point in B J (S , M ).Then, if p is not a point in S0 , there exists a sequence of points qj converging top and future-directed causal curves γj connecting qj to a point in S0 , A subset ofγj converges to a null geodesic passing through p in B.To Index
Ch.1Basics of Black Holes9 To Index【Definition 1.2.4 (Cauchy development)】in M ,For an acausal hypersurface S Future (past) Cauchy development (or domain of dependence) D (S ) (D (S )) the set of all points p such that any past-directed (future-directed) nonextendible curve passing through p intersects §. Cauchy development (or domain of dependence) D ( S ) D (S ) D (S ). Cauchy horizon H (S ) D (S ) I (D (S )). 1.2.2Causality conditions【Definition 1.2.5 (Various causality conditions)】Chornonology conditionCausality condition There exists no closed timelike curve.There exists no closed causal curve.Strong causality condition at p Every neighborhood of p containa neighborhood of p which no causal curve intersects more than once.Stable causality condition Chronogy condition holds for anymetric in an open neighborhood of the metric in the space of metric There is a global time function t whose gradient is everywhere timelike. 【Definition 1.2.6 (global hyperbolicity (Leray 1952))】A set N in M issaid to be globally hyperbolic if the strong causality condition holds on Nand if for any two points p, q N , J (p) J (q) is compact and containedin N . 【Theorem 1.2.7 (HE1973)】If S is a closed achronal set, then int(D(S )) : D(S ) Ḋ(S ) is globally hyperbolic. 【Theorem 1.2.8 (Geroch 1970)】If an open set N is globally hyperbolic,then N R S with S a spacelike manifold, and for each t R, {t} Sis a Cauchy surface for N . In particular, N is stably causal. To Index
Ch.1Basics of Black HolesI10 To IndexB HI ΣDOCI1.2.3 I HorizonLet M be an AS spacetime and I be its conformal infinity． Asymoptotically predictableI D(Σ) in Mˆ(1.2.1) HorizonH (J (I )) J (I )(1.2.2) Black hole regionB M J (I )(1.2.3) DOC（Domain of outer communiation）DOC J (I , M ) J (I , M )(1.2.4)To Index
Ch.1Basics of Black Holes11 To Index§1.3Raychaudhuri equation1.3.1Jacobi equationFor a congruence of curves Γ : xμ xμ (τ, z), let uμ ẋμ be its tangent vectorfield and Z μ δz i xμ / z i be the relative dispacement of neighboring curves of areference curve. Then, from u Z Z u(1.3.1)it follows that 2u Z u Z u R(u, Z)u Z u u.(1.3.2)Hence, we have 2u Z R(u, Z)u Z A(1.3.3)where u u A.(1.3.4)In particular, when Γ is a geodesic congruence, i.e., A 0, this equation is calledthe Jacobi equation.1.3.2Representation in terms of the Fermi basisWhen uμ is a timelike vector field (a velocity field of particles),u τ ; u · u 1.(1.3.5)let us take an orthonormal basis Ea satisfying the conditionE0 u, u · EI 0, ĖI u EI AI u.(1.3.6)Then, because AI can be expressed asAI u · ĖI u̇ · EI A · EI ,(1.3.7)Ea is uniquely determined if one specifies its value at a point on the flow line.Let us express the relative position vector Z between two fluid line in terms ofa Fermi basis asZ Z 0 u Z I EI .(1.3.8)To Index
Ch.1Basics of Black Holes12 To IndexThen, we haveŻ I ĖI · Z EI · u Z AI Z 0 EI · Z u AI Z 0 Z 0 EI · u̇ EI · EJ uZ J ,(1.3.9)MIJ EI · EJ u.(1.3.10)hence,Ż I MIJ Z J ;Further,Ż 0 u̇ · Z u · u Z u̇ · EI Z I u · Z u AI Z I .(1.3.11)These can be combined into the single expressionŻ Z 0 u̇ MIJ Z J E I .1.3.3(1.3.12)Expansion, shear, rotationLet us define a symmetric tensor θIJ and an anti-symmetric tensor ωIJ byθIJωIJ1 σIJ δIJ θ : M(IJ) ; σII 0,d: M[IJ] .(1.3.13)(1.3.14)Then, fromMIJ EμI EνJ (EμI EIλ )(EνJ EJσ ) σ uλ (δμλ uμ uλ )(δνσ uν uσ ) σ uλ ν uμ uν u̇μ(1.3.15)it follows that ν uμ θμν ωμν u̇μ uν ; θμν : EμI EνJ θIJ , ωμν : EμI EνJ ωIJ .1.3.4(1.3.16)Geodesic deviation equationsFrom [u, Z] 0, it follows thatZ̈ u Z u R(u, Z)u Z u̇.(1.3.17)Here, by differentiating the expression for Ż by τ , we obtainZ̈ Z u̇ AI Z I u̇ Z I EI u̇ AI MIJ Z J u (Ṁ M 2 )IJ Z J EI .(1.3.18)Hence, we have(Ṁ M 2 )IJ RIμJν uμ uν AI AJ EI · EJ u̇.(1.3.19)To Index
Ch.1Basics of Black Holes13 To IndexThis matrix equation is equivalent to1θ̇ θ2 2σ 2 2ω 2 Rμν uμ uν μ u̇μ ,d222σ̇IJ θσIJ σIK σJK σ 2 δIJ ωIK ωJ K ω 2δIJddd11 RIuJu Ruu δIJ AI AJ E(I · EJ ) u̇ · u̇δIJ ,dd2ω̇IJ θωIJ σIK ωKJ σJK ωKI E[I · EJ ] u̇,d(1.3.20a)(1.3.20b)(1.3.20c)where2σ 2 : σIJ σIJ ,2ω 2 : ωIJ ωIJ .(1.3.21)In particular, when u̇ 0, the equation for θ̇ is called the Raychaudhuri equation.Similarly, in the case of a null geodesic congruence, the same set of equationswith d replaced by d 1 and the range of I restricted to 1, · · · , d 1 hold byconsidering a parallel null basis instead of the orthonormal Fermi basis.1.3.5Conjugate point【Definition 1.3.1】Two points on a geodesic is called conjugate if thereexists a Jacobi field that vanishes at these points. Similarly, a point p issaid conjugate to a surface S along a geodesic γ that passes through p andcrosses S orthogonally at q, if there is a Jacobi field that vanishes at p andis tangential to S at q. 【Proposition 1.3.2】If p and q (S ) are conjugate along a null geodesicγ, any point on the extension of γ past p can be connected to q (S ) bu atimelike curve. §1.4Area Theorem【Definition 1.4.1 (Strong energy condition)】The Ricci curvature satisfya bthe condition Rab V V for any timeline (null) vector V , it is said that thespacetime satisfies the timelike (null) strong energy condition or convergencecondition. 【Proposition 1.4.2 (θ̇ 0)】If the horizon H is future complete andthe null strong energy condition is satisfied, the expansion rate θ of the nullgeodesic generators of H is non-negative. To Index
Ch.1Basics of Black Holes14 To IndexProof. Let u be a non-degenerate function that is defined around H and constanton H . Then, the tangent vector k of each null geodesic generator of H isorthogonal to the normal vector of H that is propotional to u: k f u.From this we obtain dk df du df k , hence ω 0. Therefore, theRaychaudhuri equation can be writtenθ̇ 1 2θ 2σ 2 Rμν k μ k ν 0.d 1(1.4.1)If θ is negative at some v v0 , it behaves asθ(v) θ0 .01 θ0 v vd 1(1.4.2)Hence, θ diverges at v v1 , a finite affine distance from v v0 . The point v v1is conjugate to a spacetlike 2-surface on the horizon. Hence, the null geodesicgenerator passing through this point has to go outside J (I ) beyond this point.This contradict the fact that each null geodesic generator of H has no futureend point.【Theorem 1.4.3 (Area Increase Theorem)】If the horizon is null geodesically future complete and if the null strong energy condition is satisfied, thehorizon area does not decrease. §1.5Symmetry1.5.1Killing vectorLet Φt : M M be a one-parameter family of isometries:Φ t g g,Φt Φs Φt s ,(1.5.1a)Φ0 idM .(1.5.1b)Then, its infinitesimal transformation X is defined as the vector field that is tangent to the curve Φt (p) at each point p: dΦt (p) .(1.5.2)Xp dt t 0This infinitesimal transformation X is called a Killing vector and satisfiesΦ t g g 0.t 0t X g limL(1.5.3)To Index
Ch.1Basics of Black Holes15 To IndexFrom X (g(Y, Z)) g([X, Y ], Z) g(Y, [X, Z])(L X g)(Y, Z) L X (g(Y, z)) g( X Y Y X, Z) g(Y, X Z Z X) g( Y X, Z) g(Y, Z X),(1.5.4)this equation can be written μ Xν ν Xμ 0.(1.5.5)Conversely, any vector field satisfying this Killing equation generate a local oneparameter family of isometries uniquely.【Formula 1.5.1】 · X 0,(1.5.6a) Xa Rab Xb 0,(1.5.6b) a b Xc Rbca Xd .(1.5.6c)d Proof. Summing a b Xc b a Xc Rabc d Xd , b c Xa c b Xa Rbca d Xd , c a Xb a c Xb Rcab d Xd ,we obtain2 a b Xc 2Rbca d Xd .1.5.2Stationary spacetime【Definition 1.5.2 (Stationary spacetime)】A spacetime M is said to bestationary if there is a Killing vector ξ that is timelike in some region. The metric of a stationary spacetime can be writtends2 e2U (x) (dt A(x))2 gij (x)dxi dxj ,(1.5.7)where x (xi ) is the spatial coordinates. The Killing vector ξ can be writtenξ t in this coordinate system, henceξ e2U (dt A(x)).(1.5.8)The rotation of the Killing vector is defined as (ξ dξ ) e3U n dA.(1.5.9)To Index
Ch.1Basics of Black Holes16 To Index【Definition 1.5.3 (Static spacetime)】A stationary spacetime M withthe time translation Killing vector ξ is called static when the rotation of ξvanishes. When a spacetime (M , g) is static, from the rotation free condition, we can finda coordinate system locally in which the metric can be writtends2 e2U (x) dt2 gij (x)dxi dxj .(1.5.10)【Definition 1.5.4 (Axisymmetric spacetime)】A spacetime M is said tobe axisymmetric if there is a Killing vector field η whose orbits are all closed. 1.5.3Killing horizon【Definition 1.5.5 (Killing horizon)】A null hypersurface H in a stationaryspacetime is called a Killing horizon when there is a Killing vector that isparallel to the null geodesic generators on H . 【Proposition 1.5.6 (Killing horizon of static BHs)】A horizon of anasymptotically simple and static spacetime with respect to infinity I is aKilling horizon if the spacetime is asymptotically predictable and the timetranslation Killing vector ξ is timelike in a neighborhood of I . Proof. The Killing vector ξ is always tangent to the horizon H . If ξ is spacelike,there must exist a hypersurface S outside H on which ξ is null and outside ofwhich ξ is timelike. In this region where ξ is timelike, ξ can be written ξ e2U twith g(ξ, ξ) e2U , where t const hypersurfaces Σt are spacelike. Hence,S limt Σt . Because ξ is orthgonal to Σt , it is also orthogonal to S . However,because ξ is tangent to S , this means that S is a null hypersurface, and ξ isparallel to its null geodesic generators. If S intersects with I , the assumptionon the timelikeness of ξ near I is violated. This implies that S H
Ch.1 Basics of Black Holes 8 To Index or with the energy-momentun tensor decreasing as O(Ω) at infinity, ˆΩhasto satisfy the condition ( ˆΩ)2 2Λ n(n 1) (1.1.13) This implies that
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Introduction of Chemical Reaction Engineering Introduction about Chemical Engineering 0:31:15 0:31:09. Lecture 14 Lecture 15 Lecture 16 Lecture 17 Lecture 18 Lecture 19 Lecture 20 Lecture 21 Lecture 22 Lecture 23 Lecture 24 Lecture 25 Lecture 26 Lecture 27 Lecture 28 Lecture
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Lecture 1: A Beginner's Guide Lecture 2: Introduction to Programming Lecture 3: Introduction to C, structure of C programming Lecture 4: Elements of C Lecture 5: Variables, Statements, Expressions Lecture 6: Input-Output in C Lecture 7: Formatted Input-Output Lecture 8: Operators Lecture 9: Operators continued
By Rumki Basu 4 administration dichotomy has to do with its normative implications. In other words, the proposed principle is that elected officials have the legal right to make policy decisions, and it is the duty of career civil servants to carry out those policies in good faith. The politics – administration dichotomy has since its inception, been a contested area of public administration .
