May 2007 Black Hole Laws And The Attractor Mechanism
May 2007Black Hole Laws and theAttractor MechanismDaniel PeavoySupervisor: Dr Thomas MohauptAbstractWe discuss black hole physics, beginning with an explanation of generalrelativity from a variational principle. The Schwarzschild and ReissnerNordström metrics are derived with discussion of singularities and coordinate systems. The computer program ‘Maple’ is employed in this work.An analysis of the Kerr metric follows before discussion of black holesand thermodynamics. The zeroth and first laws are derived with a sketchof the proof of the second law. Finally we review more recent work byFerrera and Kallosh (1996) on the ‘Attractor Mechanism’ by deriving thefield equations for this model of a black hole and verifying that they aresatisfied by the given solution.
Contents1 Introduction22 General Relativity from a variationalprinciple2.1 Differential forms and integration . . .2.2 The Einstein-Hilbert Lagrangian . . .2.3 The full field equations . . . . . . . . .2.4 Geodesics . . . . . . . . . . . . . . . .2.5 Killing’s equation . . . . . . . . . . . .3 Static, spherically symmetric black holes3.1 Static, spherically symmetric spacetimes .3.2 The Schwarzschild solution . . . . . . . .3.3 Singularities . . . . . . . . . . . . . . . . .3.4 The event horizon . . . . . . . . . . . . .3.5 Eddington-Finkelsteincoordinates. . . . . . . . . . . . . . . . . .3.6 Gravitational red shift . . . . . . . . . . .3368910.1111141516. . . . . . . . . . . . . . . . . . . . . . . . .18204 Charged black holes214.1 The Reissner-Nordström metric . . . . . . . . . . . . . . . . . . . 235 Stationary, axisymmetric black holes6 Geodesics and Causality6.1 The Equation of Geodesic Deviation.6.2 Geodesic congruences . . . . . . . .6.3 Conjugate points . . . . . . . . . . .6.4 Causality . . . . . . . . . . . . . . .26.28282932347 The Surface Gravity of a Black Hole367.1 Constancy of the surface gravity . . . . . . . . . . . . . . . . . . 388 Black holes and thermodynamics8.1 The mass of a rotating black hole .8.2 Differential formula for the mass of8.3 Particle creation by black holes . .8.4 The laws of black hole mechanics .a. . . . .rotating. . . . . . . . . . . . . .black hole. . . . . . . . . . .40414344459 The Attractor Mechanism4610 Astrophysical black holes50A Appendix521
1IntroductionThe term ‘black hole’ was first used by Wheeler in 1967. However, such objects,characterised by the complete collapse of matter, have been studied theoreticallylong before the advent of relativity theory. In the 18th century Michell andLaplace studied Newtonian theory and showed that it was possible for matterto completely collapse on itself.In Newtonian theory the collapse of matter to a point is completely analogousto the singularity in the Coulomb potential. However, as is popularly known,relativity theory has far deeper implications for the physics of collapsed matter.It implies that there exists a singularity in spacetime itself.General relativity is based on extending special relativity (to include noninertial observers) and on the priciple of equivalence: A frame linearly accelerated relative to an inertial frame in special relativity is locally identical to aframe at rest in a gravitational field. Einstein made use of the fact that the motion of a particle in a gravitational field is independent of it’s mass. Therefore,there exists special paths in spacetime, determined by the gravitational field ofmassive objects, on which all inertial particles move. In general the spacetimewill be curved and these special paths, called geodesics, are the curved spaceanalogues of straight lines.Of course the equations describing the motion of particles will be highlynon-linear since in principle each particle moving on a geodesic also has it’sown gravitational field. In this project we consider the gravitational field dueto a single massive body only and denote it’s mass M the geometrical mass, i.ethe mass that affects the geometry. We neglect the mass of particles/observersmoving in the field of M .In the spirit of relativity all observers are equivalent in the sense that eachcan ‘know’ the laws of physics. However, in curved spacetime there is no preffered coordinate system, as different to Minkowski space. We must considerthe most general physical manifolds (see [3] for an introduction to manifolds),which in general are covered by any number of coordinate patches.The principle of general covariance states that the laws of physics shouldbe invariant under a coordinate transformation. However, there often existcoordinate systems in which the symmetries of the system are most apparentand can be used to simplify the mathematics.After formulating the physical principles, Einstein realised that a great dealof the mathematics of Riemannian geometry and tensor calculus must be usedto describe a theory of curved spacetime. In the appendix of this project we haveincluded some of the basic mathematical relations used in black hole physics.For an introduction to this material see [2] and [1].General relativity has had much success in it’s verification by experiment.For example, it successfully explains the advance of the perihelion of Mercury,see [2], and gravitational lensing effects. However, most of the phenomenapresent in the relatively weak gravitational fields of the solar system can beexplained by Newtonian theory.The curvature of spacetime in the region of a completely collapsed bodyof infinite density is so large that Newtonian theory is completely inadaquateto describe the physics of this region. So, for the description of black holes,relativity is employed to the full. General relativity relies upon the language oftensors. The appendix of the project has a collection of tensorial relations for2
the reader to refer to.A black hole is viewed as a point of infinite curvature surrounded by an eventhorizon, from which nothing can escape. So called ‘naked singularities’, with noevent horizon, are thought to be unphysical.During the 1970s remarkable properties of black holes were studied theoretically resulting in the black hole laws that are analogous to the thermodynamicallaws. In 1975 Hawking studied the quantum field theory in the region of a blackhole event horizon. He found that black holes emit radiation and that this radiation had a thermal spectrum. This is a thermodynamical phenomenon resultingfrom a combination of classical general relativity and quantum theory. Thus,black hole physics remains an important topic in theories that unite generalrelativity and quantum theory up to the present day. Issues such as how oneexplains the non-zero entropy of a black hole in terms of Boltzmanns law microstate counting are currently being addressed in theories of quantum gravitysuch as superstring theory.The theoretical work, up to the 1980s, was undertaken without physicalconfirmation that black holes existed. However, today there is numerous astrophysical data, which indicates that there is indeed a so called supermassive blackhole at the centre of our galaxy and many other possible black hole candidates.2General Relativity from a variationalprincipleFor many purposes it is useful to express General Relativity in a Lagrangianformulation. As well as providing an easy method for computing the field equations, this approach is fundamental in any theory of quantum mechanics incurved spacetime. The idea is the same as nonrelativistic mechanics; that wehave a collection of tensor fields ψi , and we define a funtional I(ψi ), which mapsthese field configurations, and their derivatives, to a number. I(ψi ) is called theaction and, as usual, we extremise the action to obtain the field equations. Asin flat space we express the action in terms of a Lagrangian density L(ψi , ψi ):ZI(ψi ) LΩ,(2.0.1)Mwhere Ω is a volume element of our spacetime manifold M . However, in generalrelativity the field variable is the metric gµν and we realise that the volumeelement in the above is not constant over the spacetime. We must be moreprecise when defining integration over a more general manifold and we nowintroduce the mathematical tools to do this.2.1Differential forms and integrationHere we give an outline of the mathematics of differential forms relevant to ourdiscussion. For a more in depth treatment refer to [1] and [3]. A differentialp-form is a totally antisymmetric tensor of type (0,p) (see Appendix). If ωa1 ···apare the components of a p-form thenωa1 ···ap ω[a1 ···ap ] .3(2.1.1)
If the dimension of our manifold is n then any p-form, where p n, vanishes.Note that a one-form is simply a dual vector. The space of dual vectors is isomorphic to the space of vectors at any point on a pseudo-Riemannian manifold.The isomorphism is given by the metricg : v µ gνµ v µ vν .(2.1.2)On a manifold we define the basis vectors eµ of the tangent space aseµ , xµ(2.1.3)where xµ are the coordinates of a coordinate neighbourhood of the vector space.Similarly the basis e µ of dual vectors is given bye µ dxµ .(2.1.4)One can verify that these basis obey the correct coordinate transformation lawsfor covariant/contravariant vectors. We can then define the inner product between a vector V V µ x µ and a dual vector ω ων dxν by hω, V i ων V µ dxν , µ ων V µ δµν ων V ν .(2.1.5) xWe can construct a basis for arbitrary rank p-forms by taking the wedgeproduct between one-forms. We define the wedge product asXdxµ1 dxµ2 . . . dxµr sgn(P)dxµP (1) dxµP (2) . . . dxµP (r) , (2.1.6)P Srwhere P (r) is a permutation belonging to the permutation group Sr of r elements. Then sgn(P) is 1 (-1) depending on whether P (r) is an even (odd)permutation. Then a general p-form can be writtenωp 1ωµ µ .µ dxµ1 dxµ2 . . . dxµp ,p! 1 2 p(2.1.7)where ωµ1 µ2 .µp is totally antisymmetric. We define the exterior derivative d,which maps a p-form to a (p 1)-form. It can be easily shown (see [1]) that d isa derivative operator independent of the metric. We can therefore express it interms of the ordinary derivative a . Then 1 dωp ωdxµ1 dxµ2 . . . dxµp .(2.1.8)µµ.µp! xν 1 2 pω0 is just an ordinary function, f say, ω1 is a dual-vector and in threedimensional space we have ωx ωz ωy ωx ωz ωy dx dy dy dz dz dx.dω1 x y y z z x(2.1.9)Hence, d acting on ω1 is the curl of a (dual) vector. It is easy to verify thatdω0 is the gradient of a function, dω2 the divergence and dω3 0 because theresulting (3 1)-form has dimension greater than that of the manifold, d 3.4
We have seen how forms are used to reproduce the usual differential calculusin Euclidean space. Their generality means that they can be applied to anymanifolds and hence are important in studies of general relativity. We now lookat how they are used to provide a well-defined volume element on a manifoldand hence a means of integration.Firstly we must define the orientation of a manifold as the integration of adifferential form is only defined when the manifold is orientable. If p is a pointof a manifold such that two coordinate patches xµ , y ν overlap at this point thenwe transform the basis vectors as µ xeµ .(2.1.10)ẽν y νIf J det( xµ / y ν ) 0 then two coordinate patches are said to define the sameorientation at p. If J 0 they define opposite orientations. We then define anorientable manifold to be one covered by coordinate patches that everywheredefine the same orientation.If we have an n-dimensional orientable manifold M then there exists an nform ω which vanishes nowhere. This n-form can be used to find the invariantvolume element, which we use to integrate a function on M .Two orientations, ω and ω̃, are said to be equivalent if ω̃ hω where his a strictly positive function. We now have the volume element, up to a positive function. We are able to define the invariant volume element, on an ndimensional manifold M , byp(2.1.11)ΩM g dx1 dx2 . . . dxn ,where g det gµν . We now show that ΩM is indeed invariant as we move fromone coordinate patch xµ to another y λ on the manifold. The invariant volumeelement is thens µ ν x xdetgdy 1 dy 2 . . . dy nµν y α y λ µ λ p x y det g detdx1 dx2 . . . dxnα y xνp g dx1 dx2 . . . dxn .(2.1.12)If the manifold is orientable then det( y λ / xν ) 0 and we see that ΩM isinvariantthe coordinate change. The signature of our metric is 2 so wep under write g g. We now define the integration of a function f on a manifoldto beZZ f ΩM f g dx1 dx2 . . . dxn .(2.1.13)MMIf an object can be written in the form abc.abc.Tµνρ. g T̃µνρ.,(2.1.14)abc.where T̃µνρ.is a tensor, then we call it a tensor density. We see that a Lagrangian for gravity must be a scalar density.Finally, for later use, we include here Stokes’ theorem for an arbitrarymanifold, for a proof see [3]. For oriented manifold M , dimension d, with5
d-dimensional submanifold N with boundary N , and (d-1)-form ω, Stokes’theorom statesZZdω ω.(2.1.15)N NNotice that the integration over a volume, on the left hand side, is over a d-formas discussed above.2.2The Einstein-Hilbert LagrangianIn the last section we have seen that our Lagrangian must contain the factor g. Then the simplest scalar density we can build out of the metric and it’sderivatives is LG g R,(2.2.1)where R is the Ricci scalar (see Appendix). We call this the Einstein-HilbertLagrangian and we now show that varying this Lagrangian with respect to gableads to the Einstein field equations. Firstly, we show some preliminary results.Let gab be the metric with inverse g ab . Then we have1 baA .gg ab (2.2.2)where Aba is the transpose of the matrix of cofactors of gab and g is the determinant. Summing only over the column index, the determinant is given byg nXgab Aab .(2.2.3)b 1This gives g Aab gg ba . gab(2.2.4)We can then deduce that1 ( g) 2 gab1 g ( g) 2 gab g11 gg ab ( g) 2211 ab g ( g) 2 ,2 (2.2.5)recalling that the metric is symmetric. Notice that g ab gab is a constant tensorso thatδ(g ab gab ) gab δg ab g ab δgab 0 gab δg ab g ab δgab ,(2.2.6)and (2.2.5) givesδ( g)1/21( g)1/2 g ab δgab21 ( g)1/2 gab δg ab .2 6(2.2.7)
It follows that11 ( g) 21 gab ( g) 2 . g ab2(2.2.8)Note the different sign to (2.2.5).We now prove the Palatini equation, which gives an expression for the variation of the Riemann tensor (see Appendix). We use geodesic coordinates, wherethe connection vanishes at some point p, so that Γabc 0 and, from (A.0.46),the Riemann tensor reduces toRa bcd c Γabd d Γabc .(2.2.9)The connection is not a tensor so this is only valid at p. However, we can forma tensor by varying the connection,aΓabc Γbc Γabc δΓabc .(2.2.10)δΓabc is the difference of two connections, and therefore a tensor (see [2] chapter6). The above variation induces a variation in the Riemann tensor:a Ra bcd δRa bcd .(2.2.11) c (δΓabc ) d (δΓabc ) c (δΓabc ) d (δΓabc ) ,(2.2.12)Ra bcd RbcdIn geodesic coordinates we haveδRa bcdwhere a is the covariant derivative (see Appendix). Notice that δRa bcd is thedifference of two tensors so is itself, a tensor and (2.2.12), the Palatini equation,holds in any coordinates. Contracting a and c gives the useful resultδRbd a (δΓabd ) d (δΓaba ) ,(2.2.13)where Rbd is the Ricci tensor (see Appendix).We now compute the partial and covariant derivatives of the metric determinant g. Using (2.2.4) we have g xc g gab gab xc gab gg ba c . x (2.2.14)The metric is symmetric so we can write c g gg ab c gabgg ab (Γdac gdb Γdbc gad )gδda Γdac gδdb Γdbc2gΓaac ,(2.2.15)where we have used c gab 0 in the second line (see section 2.5).It is easy to verify that the covariant derivative of g vanishes. It can bewritteng det(gab ) µνρσ gµ1 gν2 gρ3 gσ4 ,(2.2.16)7
where is the alternating symbol. Each gµ1 is covariantly constant so c g 0.This implies (2.2.17) c g 0 . It is interesting to note that c g 6 c g so we see that g is not ascalar but a scalar density.Now, writing LG from (2.2.1) in terms of the Ricci tensor the action isZ gg ab Rab dΩ .(2.2.18)I ΩWe now perform the variation, givingZ δI δ( g g ab )Rab g g ab δRab dΩ .(2.2.19)ΩUsing (2.2.13) and (2.2.17) the second term on the right hand side becomesZZ ab g g δRab dΩ g g ab ( c δΓcab b δΓcac ) dΩΩΩZ c ( g g ab δΓcab ) b ( g g ab δΓcac ) .Ω(2.2.20)The last line is the integral of a divergence so is equal to a surface integral byStokes’ theorem. Therefore, if all variations are zero on the boundary of thespacetime, (2.2.20) vanishes and we are left withZ δI δ( g g ab )Rab dΩZΩ Rab g ab δ g Rab gδg ab dΩ ZΩ 1 Rab g ab ( ggab δg ab ) Rab gδg ab dΩ2Ω Z 1 g Rab Rgab δg ab dΩ2ZΩ gGab δg ab dΩ ,(2.2.21) Ωwhere we have used (2.2.7) and the definition of the Ricci scalar (A.0.53). Gabis the Einstein tensor. δI must vanish and δg ab is arbitrary, so we see thatGab 0 and we have the vacuum field equations.2.3The full field equationsTo obtain the full field equations we include the matter Lagrangian LM in theexpression for the action:ZI (LG κLM ) dΩ ,(2.3.1)Ω8
where κ is the coupling constant. We then setand δLG gGababδg(2.3.2) δLM gTab ,abδg(2.3.3)where the last equation defines the energy-momentum tensor Tab of the matterfields. The variation of the action must vanish so we have reproduced the fullfield equationsGab κTab .(2.3.4)For a discussion of the numerical constant κ see [2], section 12.10, where itis argued that κ 8π. Here, and henceforth, we use geometric units withGN c 1, where GN is Newtons constant and c is the velocity of light invacuum.2.4GeodesicsCurves in spacetime can be timelike, traversed by massive particles; lightlike,for photons; or spacelike which connect points not in physical contact (communication). Further discussion can be found in books on special (general)relativity.The infinitesimal length ds along a timelike curve can be definedds2 gab dxa dxb ,(2.4.1)where dxa is the change in coordinate along the curve and gab is the metric. Letu be a parameter along the curve and write dsdu 2 gabdxa dxb.du du(2.4.2)If u is linearly related to s then it is an affine parameter. For example, ifs αu β(2.4.3)α2 gab α2 ẋa ẋb(2.4.4)gab ẋa ẋb 1 ,(2.4.5)then (2.4.2) becomesandwhere a dot is differentiation with respect to affine parameter s. Similarly for aspacelike curvegab ẋa ẋb 1 .(2.4.6)Null curves are possible on a manifold with metric of indeterminate form. Thesecurves connect points such that ds 0. Summarising if timelike 10if nullgab ẋa ẋb (2.4.7) 1 if spacelike9
Returning to (2.4.2) we haveZs Zds dsdu duZ gabdxa dxbdu du 1/2du .(2.4.8)A geodesic is an extremal curve, so that δs 0, giving the Euler-Lagrangeequation d L L 0,(2.4.9) xadu ẋawhere L (gab ẋa ẋb )1/2 . We can evaluate the Euler-Lagrange equation explicitlyto give the geodesic equation (see [2] chapter 6). However, we will not need thisfor our purposes. We will be mostly interested in null geodesics, which are usefulto elucidate the structure of a spacetime and it will be enough for us to statethat they must obey equations (2.4.7) and (2.4.9).2.5Killing’s equationWe include here a discussion of Killing vectors. A spacetime, in general, willpossess continuous and discrete symmetries, collectively called isometries. Thetangents to continuous isometries are Killing vectors. Killing vectors are usefulfor understanding the structure of a spacetime. For example a Killing vectorcould be timelike in which case the spacetime would be stationary, i.e ‘looks’the same for all times.We can derive a useful equation satisfied by Killing vectors. The reader isreferred to the Appendix for definitions in the following. Firstly note thatX b b Y a Y
During the 1970s remarkable properties of black holes were studied theoreti-cally resulting in the black hole laws that are analogous to the thermodynamical laws. In 1975 Hawking studied the quantum field theory in the region of a black hole event horizon. He found that black holes emit radiation and that this radia- tion had a thermal spectrum. This is a thermodynamical phenomenon resulting .
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Susannah G Tringe*‡, Andreas Wagner† and Stephanie W Ruby* Addresses: *Department of Molecular Genetics and Microbiology, University of New Mexico Health Sciences Center, Albuquerque, NM 87131, USA. †Department of Biology, University of New Mexico, Albuquerque, NM 87131, USA. ‡Current address: DOE Joint Genome Institute, 2800
