Part 3 Black Holes - University Of Cambridge
Part 3 Black HolesHarvey Reall
Part 3 Black Holes March 3, 2020iiH.S. Reall
ContentsPrefacevii1 Spherical stars1.1 Cold stars . . . . . . . . . . . . . . . . . . .1.2 Spherical symmetry . . . . . . . . . . . . . .1.3 Time-independence . . . . . . . . . . . . . .1.4 Static, spherically symmetric, spacetimes . .1.5 Tolman-Oppenheimer-Volkoff equations . . .1.6 Outside the star: the Schwarzschild solution1.7 The interior solution . . . . . . . . . . . . .1.8 Maximum mass of a cold star . . . . . . . .1123456782 arzschild black holeBirkhoff’s theorem . . . . . . . . . . .Gravitational redshift . . . . . . . . . .Geodesics of the Schwarzschild solutionEddington-Finkelstein coordinates . . .Finkelstein diagram . . . . . . . . . . .Gravitational collapse . . . . . . . . . .Black hole region . . . . . . . . . . . .Detecting black holes . . . . . . . . . .White holes . . . . . . . . . . . . . . .The Kruskal extension . . . . . . . . .Einstein-Rosen bridge . . . . . . . . . .Extendibility . . . . . . . . . . . . . .Singularities . . . . . . . . . . . . . . .11111213141718192125262930303 The3.13.23.3initial value problemPredictability . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .Extrinsic curvature . . . . . . . . . . . . . . . . . . . . . . . . . . .The Gauss-Codacci equations . . . . . . . . . . . . . . . . . . . . .33333538iii.
CONTENTS3.43.53.63.74 The4.14.24.34.44.54.64.74.84.94.104.114.12The constraint equations . . . .The initial value problem in GRAsymptotically flat initial dataStrong cosmic censorship . . . .40414344singularity theoremNull hypersurfaces . . . . . . . . . . . . .Geodesic deviation . . . . . . . . . . . . .Geodesic congruences . . . . . . . . . . . .Null geodesic congruences . . . . . . . . .Expansion, rotation and shear . . . . . . .Expansion and shear of a null hypersurfaceTrapped surfaces . . . . . . . . . . . . . .Raychaudhuri’s equation . . . . . . . . . .Energy conditions . . . . . . . . . . . . . .Conjugate points . . . . . . . . . . . . . .Causal structure . . . . . . . . . . . . . . .Penrose singularity theorem . . . . . . . 09294.9797991001011035 Asymptotic flatness5.1 Conformal compactification5.2 Asymptotic flatness . . . . .5.3 Definition of a black hole . .5.4 Weak cosmic censorship . .5.5 Apparent horizon . . . . . .6 Charged black holes6.1 The Reissner-Nordstrom solution6.2 Eddington-Finkelstein coordinates6.3 Kruskal-like coordinates . . . . .6.4 Cauchy horizons . . . . . . . . . .6.5 Extreme RN . . . . . . . . . . . .6.6 Majumdar-Papapetrou solutions .7 Rotating black holes7.1 Uniqueness theorems . . . . . . . .7.2 The Kerr-Newman solution . . . .7.3 The Kerr solution . . . . . . . . . .7.4 Maximal analytic extension . . . .7.5 The ergosphere and Penrose processPart 3 Black Holes March 3, 2020iv.H.S. Reall
CONTENTS8 Mass, charge and angular momentum8.1 Charges in curved spacetime . . . . .8.2 Komar integrals . . . . . . . . . . . .8.3 Hamiltonian formulation of GR . . .8.4 ADM energy . . . . . . . . . . . . . .9 Black hole mechanics9.1 Killling horizons and surface gravity .9.2 Interpretation of surface gravity . . .9.3 Zeroth law of black holes mechanics .9.4 First law of black hole mechanics . .9.5 Second law of black hole mechanics .10 Quantum field theory in curved spacetime10.1 Introduction . . . . . . . . . . . . . . . . . . . . .10.2 Quantization of the free scalar field . . . . . . . .10.3 Bogoliubov transformations . . . . . . . . . . . .10.4 Particle production in a non-stationary spacetime10.5 Rindler spacetime . . . . . . . . . . . . . . . . . .10.6 Wave equation in Schwarzschild spacetime . . . .10.7 Hawking radiation . . . . . . . . . . . . . . . . .10.8 Black hole thermodynamics . . . . . . . . . . . .10.9 Black hole evaporation . . . . . . . . . . . . . . .Part 3 Black Holes March 3, 2020v.107. 107. 109. 111. 115.117. 117. 119. 120. 121. 125.129. 129. 130. 133. 134. 135. 141. 143. 152. 153H.S. Reall
CONTENTSPart 3 Black Holes March 3, 2020viH.S. Reall
PrefaceThese are lecture notes for the course on Black Holes in Part III of the CambridgeMathematical Tripos.AcknowledgmentI am grateful to Andrius Štikonas and Josh Kirklin for producing most of thefigures.ConventionsWe will use units such that the speed of light is c 1 and Newton’s constant isG 1. This implies that length, time and mass have the same units.The metric signature is ( )The cosmological constant is so small that is is important only on the largestlength scales, i.e., in cosmology. We will assume Λ 0 in this course.We will use abstract index notation. Greek indices µ, ν, . . . refer to tensorcomponents with respect to some basis. Such indices take values from 0 to 3. Anequation written with such indices is valid only in a particular basis. Spacetimecoordinates are denoted xµ . Abstract indices are Latin indices a, b, c . . . Theseare used to denote tensor equations, i.e., equations valid in any basis. Any objectcarrying abstract indices must be a tensor of the type indicates by its indices e.g.X a b is a tensor of type (1, 1). Any equation written with abstract indices can bewritten out in a basis by replacing Latin indices with Greek ones (a µ, b νetc). Conversely, if an equation written with Greek indices is valid in any basisthen Greek indices can be replaced with Latin ones.For example: Γµνρ 12 g µσ (gσν,ρ gσρ,ν gνρ,σ ) is valid only in a coordinatebasis. Hence we cannot write it using abstract indices. But R g ab Rab is a tensorequation so we can use abstract indices.Riemann tensor: R(X, Y )Z X Y Z Y X Z [X,Y ] Z.vii
CHAPTER 0. PREFACEBibliography1. N.D. Birrell and P.C.W. Davies, Quantum fields in curved space, CambridgeUniversity Press, 1982.2. Spacetime and Geometry, S.M. Carroll, Addison Wesley, 2004.3. V.P. Frolov and I.D. Novikov, Black holes physics, Kluwer, 1998.4. S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time,Cambridge University Press, 1973.5. R.M. Wald, General relativity, University of Chicago Press, 1984.6. R.M. Wald, Quantum field theory in curved spacetime and black hole thermodynamics, University of Chicago Press, 1994.Most of this course concerns classical aspects of black hole physics. The booksthat I found most useful in preparing this part of the course are Wald’s GR book,and Hawking and Ellis. The final chapter of this course concerns quantum fieldtheory in curved spacetime. Here I mainly used Birrell and Davies, and Wald’ssecond book. The latter also contains a nice discussion of the laws of black holemechanics.Part 3 Black Holes March 3, 2020viiiH.S. Reall
Chapter 1Spherical stars1.1Cold starsTo understand the astrophysical significance of black holes we must discuss stars.In particular, how do stars end their lives?A normal star like our Sun is supported against contracting under its owngravity by pressure generated by nuclear reactions in its core. However, eventuallythe star will use up its nuclear “fuel”. If the gravitational self-attraction is to bebalanced then some new source of pressure is required. If this balance is to lastforever then this new source of pressure must be non-thermal because the star willeventually cool.A non-thermal source of pressure arises quantum mechanically from the Pauliprinciple, which makes a gas of cold fermions resist compression (this is calleddegeneracy pressure). A white dwarf is a star in which gravity is balanced byelectron degeneracy pressure. The Sun will end its life as a white dwarf. Whitedwarfs are very dense compared to normal stars e.g. a white dwarf with the samemass as the Sun would have a radius around a hundredth of that of the Sun. UsingNewtonian gravity one can show that a white dwarf cannot have a mass greaterthan the Chandrasekhar limit 1.4M where M is the mass of the Sun. A starmore massive than this cannot end its life as a white dwarf (unless it somehowsheds some of its mass).Once the density of matter approaches nuclear density, the degeneracy pressureof neutrons becomes important (at such high density, inverse beta decay convertsprotons into neutrons). A neutron star is supported by the degeneracy pressure ofneutrons. These stars are tiny: a solar mass neutron star would have a radius ofaround 10km (the radius of the Sun is 7 105 km). Recall that validity of Newtoniangravity requires Φ 1 where Φ is the Newtonian gravitational potential. At thesurface of a such a neutron star one has Φ 0.1 and so a Newtonian description1
CHAPTER 1. SPHERICAL STARSis inadequate: one has to use GR.In this chapter we will see that GR predicts that there is a maximum massfor neutron stars. Remarkably, this is independent of the (unknown) propertiesof matter at extremely high density and so it holds for any cold star. As wewill explain, detailed calculations reveal the maximum mass to be around 3M .Hence a hot star more massive than this cannot end its life as a cold star (unlessit sheds some mass e.g. in a supernova). Instead the star will undergo completegravitational collapse to form a black hole.In the next few sections we will show that GR predicts a maximum mass fora cold star. We will make the simplifying assumption that the star is sphericallysymmetric. As we will see, the Schwarzschild solution is the unique sphericallysymmetric vacuum solution and hence describes the gravitational field outsideany spherically symmetric star. The interior of the star can be modelled using aperfect fluid and so spacetime inside the star is determined by solving the Einsteinequation with a perfect fluid source and matching onto the Schwarzschild solutionoutside the star.1.2Spherical symmetryWe need to define what we mean by a spacetime being spherically symmetric. Youare familiar with the idea that a round sphere is invariant under rotations, whichform the group SO(3). In more mathematical language, this can be phrased asfollows. The set of all isometries of a manifold with metric forms a group. Considerthe unit round metric on S 2 :dΩ2 dθ2 sin2 θ dφ2 .(1.1)The isometry group of this metric is SO(3) (actually O(3) if we include reflections).Any 1-dimensional subgroup of SO(3) gives a 1-parameter group of isometries, andhence a Killing vector field. A spacetime is spherically symmetric if it possessesthe same symmetries as a round S 2 :Definition. A spacetime is spherically symmetric if its isometry group containsan SO(3) subgroup whose orbits are 2-spheres. (The orbit of a point p under agroup of diffeomorphisms is the set of points that one obtains by acting on p withall of the diffeomorphisms.)The statement about the orbits is important: there are examples of spacetimeswith SO(3) isometry group in which the orbits of SO(3) are 3-dimensional (e.g.Taub-NUT space: see Hawking and Ellis).Definition. In a spherically symmetricspacetime, the area-radius function r :pM R is defined by r(p) A(p)/4π where A(p) is the area of the S 2 orbitPart 3 Black Holes March 3, 20202H.S. Reall
1.3. TIME-INDEPENDENCEthrough p. (In other words, the S 2 passing through p has induced metric r(p)2 dΩ2 .)1.3Time-independenceDefinition. A spacetime is stationary if it admits a Killing vector field k a whichis everywhere timelike: gab k a k b 0.We can choose coordinates as follows. Pick a hypersurface Σ nowhere tangentto k a and introduce coordinates xi on Σ. Assign coordinates (t, xi ) to the pointparameter distance t along the integral curve through the point on Σ with coordinates xi . This gives a coordinates chart such that k a ( / t)a . Since k a is aKilling vector field, the metric is independent of t and hence takes the formds2 g00 (xk )dt2 2g0i (xk )dtdxi gij (xk )dxi dxj(1.2)where g00 0. Conversely, given a metric of this form, / t is obviously a timelikeKilling vector field and so the metric is stationary.Next we need to introduce the notion of hypersurface-orthogonality. Let Σ be ahypersurface in M specified by f (x) 0 where f : M R is smooth with df 6 0on Σ. Then the 1-form df is normal to Σ. (Proof: let ta be any vector tangent toΣ then df (t) t(f ) tµ µ f 0 because f is constant on Σ.) Any other 1-formn normal to Σ can be written as n gdf f n0 where g is a smooth function withg 6 0 on Σ and n0 is a smooth 1-form. Hence we have dn dg df df n0 f dn0so (dn) Σ (dg n0 ) df . So if n is normal to Σ then(n dn) Σ 0(1.3)Conversely:Theorem (Frobenius). If n is a non-zero 1-form such that n dn 0 everywherethen there exist functions f, g such that n gdf so n is normal to surfaces ofconstant f i.e. n is hypersurface-orthogonal.Definition. A spacetime is static if it admits a hypersurface-orthogonal timelikeKilling vector field. (So static implies stationary.)For a static spacetime, we know that k a is hypersurface-orthogonal so when defining coordinates (t, xi ) we can choose Σ to be orthogonal to k a . But Σ is thesurface t 0, with normal dt. It follows that, at t 0, kµ (1, 0, 0, 0) in ourchart, i.e., ki 0. However ki g0i (xk ) so we must have g0i (xk ) 0. So inadapted coordinates a static metric takes the formds2 g00 (xk )dt2 gij (xk )dxi dxjPart 3 Black Holes March 3, 20203(1.4)H.S. Reall
CHAPTER 1. SPHERICAL STARSwhere g00 0. Note that this metric has a discrete time-reversal isometry:(t, xi ) ( t, xi ). So static means ”time-independent and invariant under timereversal”. For example, the metric of a rotating star can be stationary but notstatic because time-reversal changes the sense of rotation.1.4Static, spherically symmetric, spacetimesWe’re interested in determining the gravitational field of a time-independent spherical object so we assume our spacetime to be stationary and spherically symmetric.By this we mean that the isometry group is R SO(3) where the the R factorcorresponds to “time translations” (i.e., the associated Killing vector field is timelike) and the orbits of SO(3) are 2-spheres as above. It can be shown that anysuch spacetime must actually be static. (The gravitational field of a rotating starcan be stationary but the rotation defines a preferred axis and so the spacetimewould not be spherically symmetric.) So let’s consider a spacetime that is bothstatic and spherically symmetric.Staticity means that we have a timelike Killing vector field k a and we can foliateour spacetime with surfaces Σt orthogonal to k a . One can argue that the orbit ofSO(3) through p Σt lies within Σt . We can define spherical polar coordinates onΣ0 as follows. Pick a S 2 symmetry orbit in Σ0 and define spherical polars (θ, φ) onit. Extend the definition of (θ, φ) to the rest of Σ0 by defining them to be constantalong (spacelike) geodesics normal to this S 2 within Σ0 . Now we use (r, θ, φ) ascoordinates on Σ0 where r is the area-radius function defined above. The metricon Σ0 must take the formds2 e2Ψ(r) dr2 r2 dΩ2(1.5)drdθ and drdφ terms cannot appear because they would break spherical symmetry.Note that r is not “the distance from the origin”. Finally, we define coordinates(t, r, θ, φ) with t the parameter distance from Σ0 along the integral curves of k a .The metric must take the formds2 e2Φ(r) dt2 e2Ψ(r) dr2 r2 dΩ2(1.6)The matter inside a star can be described by a perfect fluid, with energy momentumtensorTab (ρ p)ua ub pgab(1.7)where ua is the 4-velocity of the fluid (a unit timelike vector: gab ua ub 1), andρ,p are the energy density and pressure measured in the fluid’s local rest frame(i.e. by an observer with 4-velocity ua ).Part 3 Black Holes March 3, 20204H.S. Reall
1.5. TOLMAN-OPPENHEIMER-VOLKOFF EQUATIONSSince we’re interested in a time-independent and spherically symmetric situation we assume that the fluid is at rest, so ua is in the time direction: a a Φ(1.8)u e tOur assumptions of staticity and spherical symmetry implies that ρ and p dependonly on r. Let R denote the (area-)radius of the star. Then ρ and p vanish forr R.1.5Tolman-Oppenheimer-Volkoff equationsRecall that the fluid’s equations of motion are determined by energy-momentumtensor conservation. But the latter follows from the Einstein equation and thecontracted Bianchi identity. Hence we can obtain the equations of motion fromjust the Einstein equation. Now the Einstein tensor inherits the symmetries of themetric and so there are only three non-trivial components of the Einstein equation.These are the tt, rr and θθ components (spherical symmetry implies that the φφcomponent is proportional to the θθ component). You are asked to calculate theseon examples sheet 1.Define m(r) by 1 2m(r)2Ψ(r)(1.9)e 1 rand note that the LHS is positive so m(r) r/2. The tt component of the Einsteinequation givesdm 4πr2 ρ(1.10)drThe rr component of the Einstein equation givesm 4πr3 pdΦ drr(r 2m)(1.11)The final non-trivial component of the Einstein equation is the θθ componentThis gives a third equation of motion. But this is more easily derived from ther-component of energy-momentum conservation µ T µν 0, i.e., from the fluidequations of motion. This givesdp(m 4πr3 p) (p ρ)drr(r 2m)(1.12)We have 3 equations but 4 unknowns (m, Φ, ρ, p) so we need one more equation.We are interested in a cold star, i.e., one with vanishing temperature T . Thermodynamics tells us that T , p and ρ are not independent: they are related by thePart 3 Black Holes March 3, 20205H.S. Reall
CHAPTER 1. SPHERICAL STARSfluid’s equation of state e.g. T T (ρ, p). Hence the condition T 0 implies arelation between p and ρ, i.e, a barotropic equation of state p p(ρ). So, for acold star, p is not an independent variable so we have 3 equations for 3 unknowns.These are called the Tolman-Oppenheimer-Volkoff equations.We assume that ρ 0 and p 0, i.e., the energy density and pressure ofmatter are positive. We also assume that p is an increasing function of ρ. If thiswere not the case then the fluid would be unstable: a fluctuation in some regionthat led to an increase in ρ would decrease p, causing the fluid to move into thisregion and hence further increase in ρ, i.e., the fluctuation would grow.1.6Outside the star: the Schwarzschild solutionConsider first the spacetime outside the star: r R. We then have ρ p 0.For r R (1.10) gives m(r) M , constant. Integrating (1.11) givesΦ 1log (1 2M/r) Φ02(1.13)for some constant Φ0 . We then have gtt e2Φ0 as r . The constant Φ0can be eliminated by defining a new time coordinate t0 eΦ0 t. So without loss ofgenerality we can set Φ0 0 and we have arrived at the Schwarzschild solution 12M2M2ds 1 dr2 r2 dΩ2dt 1 rr2(1.14)The constant M is the mass of the star. One way to see this is to note thatfor large r, the Schwarzschild solution reduces to the solution of linearized theorydescribing the gravitational field far from a body of mass M (a c
3.V.P. Frolov and I.D. Novikov, Black holes physics, Kluwer, 1998. 4.S.W. Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge University Press, 1973. 5.R.M. Wald, General relativity, University of Chicago Press, 1984. 6.R.M. Wald, Quantum eld theory in curved spacetime and black hole ther- modynamics, University of Chicago Press, 1994. Most of this course concerns .
BLACK HOLES IN 4D SPACE-TIME Schwarzschild Metric in General Relativity where Extensions: Kerr Metric for rotating black hole . 2 BH s GM c M BH 32 BH 1 s BH M rM Uvv Jane H MacGibbon "The Search for Primordial Black Holes" Cosmic Frontier Workshop SLAC March 6 - 8 2013 . BLACK HOLES IN THE UNIVERSE SUPERMASSIVE BLACK HOLES Galactic .
mass black holes, no credible formation process is known, and indeed no indications have been found that black holes much lighter than this \Chandrasekhar limit" exist anywhere in the Universe. Does this mean that much lighter black holes cannot exist? It is here that one could wonder about all those fundamental assumptions that underly the theory of quantum mechanics, which is the basic .
However, in addition to black holes formed by stellar collapse, there might also be much smaller black holes which were formed by density fluctua-202 S. W. Hawking tions in the early universe [9, 10]. These small black holes, being at a higher temperature, would radiate more than they absorbed. They would therefore pre- sumably decrease in mass. As they got smaller, they would get hotter and .
Black holes have entropy S. Black holes have Hawking temperature T H, consistent with thermodynamic relation between energy, entropy and temperature. Thermodynamics S A 4 where Ais the area of the event horizon. T H 2 ˇ where in the surface gravity of the black hole. Luke Barclay Durham, CPT luke.barclay@durham.ac.uk Supervisor: Ruth GregoryBlack Holes, Vortices and Thermodynamics. Path .
Sep 27, 2015 · Beams - Straight Beams are measured by counting the number of holes. Beams come in odd numbers when counting the holes, with one exception. Beams start with15 holes and go down in size by two holes to the 3 hole beam and include one even-numbered beam with 2 holes. The number of holes corresponds to the length
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Session 10 – Black Holes. Brief Description. Students learn about black holes, the densest objects in the Universe. They learn that the collapsing . core of a star forms a black hole and do an activity that shows how the density of a stellar core increases as the core collapses even though the mass remains the same. They then engage in a kinesthetic activity to model how a black hole affects .
Gene example Black and Liver B Locus is the gene responsible for the Black / liver coat colours: The B Locus has two alleles : B Black b Liver The black parent alleles are B / B (Black / Black) The liver parent alleles are b / b (liver / liver) The offspring is black and its alleles are B / b (Black / liver) The offspring inherited the black allele from the black
