Derivation of the Metric of Reissner-Nordström and Kerr-Newman Black Holes

UNIVERSIDAD DE SEVILLA DOBLE GRADO EN F´ ISICA Y MATEM ´ ATICAS1 TRABAJO DE FIN DE GRADO INSTITUT F ¨ UR THEORETISCHE PHYSIK WWU M ¨ UNSTER Derivation of the Metric of Reissner-Nordstr¨ om and Kerr-Newman Black Holes Author Carlos Francisco ROMERO MADRID2 Supervisor Pr. Dr. Jochen HEITGER3 July 25, 2018 1Facultad de F´ ısica, Facultad de Matem´ aticas. 2[email protected] [email protected] ”El Universo es enorme, se calcula que es varias veces la Manga del Mar Menor.” Pep´ ın Tre4 4https://youtu.be/hcoiez3PenU 1 Abstract In this thesis, both the geometrical and action principle approach to Einstein’s field equations are developed, providing an intuitive fundamental path also with a more formal development given by an extremal principle. After this solid introduction, derivations of the metric of two distinct theoretical models of black holes are presented. Firstly, the Reissner-Nordstr¨ om model is studied and its metric is obtained by solving the differential field equations. Secondly, the Kerr-Newman model is approached by the Newman-Jannis algorithm that provides an easy and straightaway procedure to obtain its metric and energymomentum tensor just by identifying a seed metric and applying a change of variables. Finally, the solutions are studied, horizons and regions of interest of both black holes are commented. Keywords: general relativity, gravitation, Reissner-Nordstr¨ om, Kerr-Newman, Schwarzschild, Minskowski, inertial frame, curvilinear coordinates, covariant derivative, general covariance, Einstein’s field equations, metric, affine connection, energy-momentum, action principle, Newman- Jannis algorithm, tetrad formalism, horizon. 2 Contents 1 Introduction 4 2 Geometric Approach 4 2.1 PrincipleofEquivalence ............................. 4 2.2 GravitationalForces................................ 5 2.3 Relation between gµν and Γλ µν ........................... 7 2.4 TheNewtonianLimit............................... 8 2.5 Principle of General Covariance . . . . . . . . . . . . . . . . . . . . . . . . . 9 2.6 Energy-MomentumTensor............................ 9 2.7 CurvatureTensor ................................. 10 2.8 Gravitation and Curvilinear Coordinates . . . . . . . . . . . . . . . . . . . . . 11 2.9 Derivations of the Field Equations . . . . . . . . . . . . . . . . . . . . . . . . 12 3 Action Principle Approach 15 3.1 The Gravitational Action Principle . . . . . . . . . . . . . . . . . . . . . . . . 15 3.2 The Comprehensive Action Principle . . . . . . . . . . . . . . . . . . . . . . . 17 3.3 FurtherReading.................................. 18 4 Black Holes 19 4.1 Introduction.................................... 19 4.2 Reissner-Nordstr¨ omBlackHole ......................... 19 4.2.1 Reissner-Nordstr¨ omMetric ....................... 20 4.2.2 Derivation of the Metric and the Energy-Momentum Tensor . . . . . . 20 4.2.3 Horizons ................................. 23 4.3 Kerr-NewmanBlackHole ............................ 24 4.3.1 Kerr-Newman Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 24 4.3.2 TetradFormalism............................. 25 4.3.3 Newman-Janis Algorithm . . . . . . . . . . . . . . . . . . . . . . . . 25 4.3.4 Derivation of the Metric and the Energy-Momentum Tensor . . . . . . 27 4.3.5 Horizons ................................. 28 5 Conclusion and Discussion of the Results 31 3 1 Introduction General relativity is one of the most shocking and best experimentally-proved theories of the last century with enormous changes on the development of Physics due to its profound mathematical structure and geometrical results. Nowadays, general relativity is still a matter of difficulties for many students and scientists due to a deep need of interpretation and mathematical background that intuition is not always willing provide our minds. The geometrical approach of general relativity is, therefore, a matter of need for most of us to relate curvature and gravitation. This path is of course much more technical mathematically and will take us some time but, once we have developed our own intuition of this subject, the following derivations will be easily, substantially, derived. As done in Weinberg (1972) [1], Schutz (2009) [2], Dirac (1975) [3], Hamilton (2015) [4] or Chandrasekhar (1983) [5], geometry is developed and linked to gravitation through identification of geodesics and curvature. A much more elegant way of deriving Einstein’s field equations is, of course, an extremal principle, which in Physics gets the name of action principle as done firstly by David Hilbert in gravitation and reproduced in Weinberg (1972) [1] and Hamilton (2015) [4]. This action principle will provide us with a highly sophisticated mathematical tool developed properly in Gelfand & Fomin (1963) [6], the Calculus of Variations, that will help us, once we have defined and obtained all proper quantities and functionals, to obtain the very same equations as we did using only geometrical procedures. Finally, we will be studying black holes as done by most of the authors, but mainly focusing on the works of Chandrasekhar (1983) [5], Hamilton (2015) [4] and Weinberg (1972) [1], in that order of relevance, with certain parameters that make them have physically special properties. We will be studying firstly the Reissner-Nordstr¨ om black hole, which consists of a static charged mass, no more no less. This charge will lead us to different solutions like the uncharged case, known as Schwarzschild’s solution. From this watchtower, a way simpler derivation of the Kerr-Newman black hole will be possible just by ”turning on” the rotation and giving significant electric charge to the mass. A discussion of physically interesting properties of these two solutions will be exposed, which makes the really important part of a physicist investigation. 2 Geometric Approach Weinberg (1972) [1] will be mostly followed in this section because of its straightaway development, Schutz (2009) [2], which goes backwards from the field equations to its foundations in Newtonian gravity and Dirac (1975) [3], whose compact notation is more than recommendable, eventhough we will be using that of [1] to avoid mistakes through the derivations, are the books that can be consulted to verify and follow, more deeply, this section. 2.1 Principle of Equivalence The principle of equivalence of gravitation and inertia gives a subtle but important property of Newtonian transformations of reference systems matching gravitational forces in an arbitrary 4 reference frame with inertial forces in a locally inertial frame. From the preceding sentence we will have to clarify some terms because, what is a locally inertial frame? In order to develop a theory of gravitation, a mathematical structure must be assumed and, in the case of general relativity, its foundations are based on Riemannian geometry. For this interpretation, spacetime will be formalized by a four-dimensional differentiable manifold that will locally resemble an Euclidean space. Locally will therefore mean an open subset of the manifold small enough such that it will behave, in our case, as a Minkowskian spacetime. This will translate to the vanishing of the first derivatives of the metric in an arbitrary inertial frame that acts as the subject of study. More visually, if a system of Nparticles where forces act upon has as equations of motion the following expression mNÜ xN xN xN=mNg g g+Õ M F F F(xM xM xM−xN xN xN)(1) where the primes have been neglected in the right side of the equation because an affine transformation of the coordinates doesn’t change the distance. By a simple Galilean transformation x x x0=x x x−1 2g g gt2and t0=t, we get a new system in which the gravitational force is suppressed by an inertial force of the same value. This can be done due to the equivalence of inertial mass and gravitational mass, which has been studied up to a high level of accuracy leading to the equality of both values. After the transformation, the equation is invariant in the right side, where the non-gravitational forces are mNÜ xN xN xN0=Õ M F F F(xM xM xM−xN xN xN)(2) Of course, in this case g g gwas a constant, but we can do this transformations also when the gravitational field depends on time. This is done by a differentiable change of coordinates that will lead to derivatives of the transformation x x x→ξ ξ ξas factors ∂ξµ/∂xνof the transformation. So, the question here must be, are the laws of nature the same in an gravitationally-accelerating system than in an inertial reference frame absent of gravitation? The answer is yes, and when we extrapolate this to every point in space, due to homogeneity of spacetime, we get the principle of equivalence that we can state as: ”at every spacetime point in an arbitrary gravitational field it is possible to choose a locally inertial coordinate system such that [...] the laws of nature take the same form as in unaccelerated Cartesian coordinate systems in the absence of gravitation.” (Weinberg, 1972, p.68) [1]. 2.2 Gravitational Forces Take a test particle in reference frame where non-zero gravitational forces are the only forces acting on the particle. This particle could be observed in another reference frame with such a coordinate system, called freely falling, that makes gravitational forces vanish, and we know this is true due to the principle of equivalence that we just stated. So, therefore, in this new system, the particle moves freely, and we know the equations of motion for this case, leading 5 to ∂2ξλ ∂τ2=0(3) with the proper time as dτ2=−ηαβdξαdξβ. Remember that proper time along a timelike world line is defined as the time as measured by a clock following that line; the particle’s clock. If we now take any other arbitrarily chosen coordinate system xµ, we could obtain the same information of the previous coordinate system ξνby asking that the coordinate transformation between the proper system and the new one are regularly differentiable. This means nothing but that we could locally ”see” the old coordinates as a differentiable function of the new, x x x=f(ξ). With this, applying the chain rule, we will obtain the equation of motion in the new system ∂2xλ ∂τ2+Γλ µν dxµ dτ dxν dτ=0(4) where we define the following quantity as Christoffel symbols or affine connection Γλ µν ≡∂xλ ∂ξα ∂2ξα ∂xµ∂xν(5) The proper time, as long as it is also a function of the coordinates, will also experience a change. Its expression in the arbitrarily chosen coordinates is dτ2=−gµν dxµdxν(6) where gµν is the metric tensor, given by gµν ≡ηαβ ∂ξα ∂xµ ∂ξβ ∂xν(7) In the case of a massless particle, as a photon, dτ2=0as the particle is travelling at the speed of light. If the proper interval is null, then derivatives using proper time as a variable make no sense, see equation (3). As a solution, we can take another parameter σ≡ξ0that we do not need to know in order to find the path of the particle, as its only purpose is parametrization, and applying again the chain rule we get the very same equations ∂2xλ ∂σ2+Γλ µν dxµ dσ dxν dσ=0(8) −gµν dxµ dσ dxν dσ=0(9) Analyzing the results we can see that equation (4), as well as (8), gives the differential equation of motion and equation (6) tells us how to compute the proper time, while in (9) sets initial conditions to massless particles. To argue that both gµν and Γλ µν are relevant quantities that comprise the effects of gravitation, we would expect that coordinate transformations from one coordinate system xµto another ξα 6 will only depend on the metric, the affine connection and the coordinates, xµ. As we are working with frames that are only locally inertial, this means that the coordinates’ ξαTaylor’s series up to second order around a spacetime point can be expressed not uniquely, but up to an inhomogeneous Lorentz transformation right hand side of equation (12), determined. In order to do this, take the affine connection and multiply by ∂ξβ/∂xλto get Γλ µν ∂ξα ∂xλ=∂2ξα ∂xµ∂xν(10) with solution, expanded as a Taylor’s series ξα(x x x)=aα+bα µ(xµ−Xµ)+ 1 2bα λΓλ µν(xµ−Xµ)(xν−Xν)+... (11) Thus, by identifying coefficients we clearly see aα=ξα(X X X)bα=∂ξα(x x x) ∂xλx x x=X X X (12) 2.3 Relation between gµν and Γλ µν Just as a remark before deriving the relation between metric and affine connection, we need to be completely clear about the mathematical procedure of derivation and fixing coordinate systems. When we fix a locally inertial coordinate system ξα(x x x), we need a specific point in spacetime to do so, call it X, and the coordinates should reflect this information by a proper labelling ξα X X X(x x x). Of course, if X X Xis the origin of our locally inertial coordinate system, we should be able to move in a neighborhood small enough with smooth properties, by means of differentiable functions. Derivatives with arguments the components X X Xα, which are not related at all with the metric or affine connection, appear as new terms. The solution is to use the principle of equivalence by means of another interpretation, that ”the locally inertial coordinates ξα X X X(x x x) that we construct at a given point X X Xcan be chosen so that the first derivatives of the metric tensor vanish at X X X”as developed in Weinberg (1972), p.73-77 [1]. These ”derivatives” are respect to X X X, giving us the expression we will be giving in the following lines, as there will be no such terms independent of gµν and Γλ µν. By differentiating the gµν and identifying terms with Γλ µν we get ∂gµν ∂xλ=Γρ µλgρν +Γρ λνgµρ (13) and it can be easily seen that, with proper summation Γσ µλ = 1 2gσν ∂gµν ∂xλ+∂gλν ∂xµ−∂gµλ ∂xν(14) This shows that gµν acts as a gravitational potential of the gravitational force, which we find by clearing the term gµν in equation (4) to be proportional to the affine connection Γλ µν. 7 2.4 The Newtonian Limit A new theory is considered valid only when it can predict results that previous theories already predicted. In the case of gravity, the previous theory was Newtonian theory of gravity and we will be not only confirming but also using its statements to complete our results for a better comprehension. Before beginning the approximation, it is interesting to point out the interpretation of the metric tensor done in Dirac (1975), p.26 [3]: ”At first sight Einstein’s law of gravitation does not look anything like Newton’s. To see a similarity, we must look on the gµν as potentials describing the gravitational field. There are ten of them, instead of just the one potential of the Newtonian theory. They describe not only the gravitational field, but also the system of coordinates. The gravitational field and the system of coordinates are inextricably mixed up in the Einstein’s theory, and one cannot describe the one without the other.” Let’s start with a particle whose velocity is Newtonian, non-relativistic, in a weak stationary gravitational field and therefore we consider that |dx x x/dτ|  dt/dτ, with τbeing the proper time as in equation (6). With this assumption, the equation of motion given by (4) is ∂2xλ ∂τ2+Γλ 00 dt dτ2 =0(15) Stationary tells us that all time derivatives of gµν are null, giving Γλ 00 =−1 2gλρ ∂g00 ∂xρ(16) and weak gives us the prerequisite to be able to have a coordinate system such that, in an open set Garound a point X X X, we have gµν(x x x)=ηµν(x x x)+hµν(x x x) |hµν(x x x)|  1∀x x x∈G(X X X)(17) and to first order in hµν, the rest of the terms can be neglected in this approximation, Γλ 00 =−1 2ηλρ ∂h00 ∂xρ(18) ∂2x x x ∂τ2= 1 2dt dτ2 ∇h00 (19) and the Newtonian analogue, for φ=−GM/r,is ∂2x x x ∂τ2=−∇φ(20) 8 3 Action Principle Approach Dirac (1975) [3] reproduces the approach done originally by David Hilbert of an action principle applied to relativity that we consider in this section. As well as Gelfand (1963) [6], a book on the foundations of Calculus of Variations that might be consulted in case of mathematical need. This formulation of the dynamic equations has the great advantage of allowing us to establish a connection between symmetries and conservation laws [1]. 3.1 The Gravitational Action Principle Let us define the action scalar Ias I=∫Ω Rpg(x x x)d4x x x(40) where the integration volume can be compactly finite or even infinity, if the Ricci scalar R has proper vanishing behaviour. The fact that we have defined this action quantity as the integral of Rmakes sense because the integral must have a scalar argument, and the only possible scalar formed from the metric tensor and the affine connection, as argued in (2.7), a contraction of the curvature tensor, the Ricci scalar. It is also worth to mention that the element pg(x x x)can be regarded as the Jacobian of the coordinate transformation. Take the mathematical theory from Gelfand (1975) [6], and make small variations δgµν, while keeping the gµν and their derivatives of first order constant at the boundaries δgµν(∂Ω)=0(41) δ∂gµν ∂xα∂Ω =0(42) We will assume that, for a properly defined action, making δI=0will provide us with a functional equation leading to Einstein’s field equations in vacuum. We have R=gµν Rµν =R∗−L(43) with R∗=gµν(∂Γσ µσ ∂xν−∂Γσ µν ∂xσ) ≡ gµν(Γσ µσ,ν −Γσ µν,σ)(44) L=gµν(Γσ µν Γρ σρ −Γρ µσΓσ νρ)(45) where second derivatives of gµν are only occurring in R∗and only linearly and can be removed by partial integration. Also, we find perfect differentials, or the product rule, that will contribute nothing to the action Iin the first to terms of R∗√g=(gµν Γσ µσ√g),ν −(gµνΓσ µν√g),σ −(gµν√g),νΓσ µσ +(gµν√g),σΓσ µν (46) 15 Expanding and using gνβΓµ νβ√g=(gµν√g),ν, we get that the remaining two terms are 2L√g, so the action becomes I=∫Ω L√gd4x x x(47) which only contains the gµν and its first derivatives. If we write L=L√g, which may be considered a Lagrangian density in three dimensions or as an action density in spacetime, as I=∫ΩLd4x x x=∫Ω0 dx0∫Ω0Ldx1dx2dx3(48) being L, the argument correspondent to the time variable of integration dx0, the Lagrangian. Now, applying the rules of variational calculus, we vary Lto obtain δL=Γα µνδ(gµν√g),α −Γβ αβδ(gαν√g),ν +(Γβ µαΓα βν −Γβ αβΓα µν)δ(gµν√g)(49) we get, after adding and subtracting terms to make the first two terms of equation (49) a perfect differential, δI=δ∫ΩLd4x x x=∫Ω Rµνδ(gµν√g)d4x x x(50) where Rµν =Γα µα,ν −Γα µν,α −Γα µν Γβ αβ +Γα µβΓβ αν. And we can also deduce, from the usual partial derivation of the factor, that the variation must take the following form (due to the behaviour of Leibniz’s rule) δgµν =−gµαgβνδgαβ (51) δ√g= 1 2√ggαβδgαβ (52) Providing us with δ(gµν√g)=−(gµαgβν −1 2gµνgαβ)√gδgαβ (53) turning equation (50) into δI=−∫Ω Rµν(gµαgβν −1 2gµνgαβ)√gδgαβd4x x x(54) which gives the vanishing functional condition that of Einstein’s field equations for vacuum as it must vanish ∀δgαβ arbitrarily chosen: Rαβ −1 2gαβ R=0(55) 16 3.2 The Comprehensive Action Principle In this section we will rename our action from I→Igin order to denote that this is the gravitational part of the action and, I0will be the action of all the other fields, consisting of a sum of terms corresponding each one to one field in particular. A really interesting feature of the action principle is linearity in the variational derivative. We only need to obtain the action for each of the fields and add them all together. Let’s start with a renormalized equation (48), where L → (16π)−1L, so (we will omit the region of integration Ωfrom now on) Ig=∫Ld4x x x(56) and applying a variation and partial integration in the second term δIg=∫∂L ∂gµν δgµν +∂L ∂gµν,α δgµν,αd4x x x=∫"∂L ∂gµν −∂L ∂gµν,α ,α#δgµν d4x x x(57) where the work on section (3.1) leads to pµν√g≡ −(16π)−1Rµν −1 2gµν R√g=∂L ∂gµν −∂L ∂gµν,α ,α (58) where we have defined the quantity pµν so as to reduce notation. Now, let’s calculate the part of the action corresponding to the rest of the fields φn(n∈N), each of them assumed to be independent but unspecified. The action looks I0=∫L0d4x x x(59) where L0=f(φn, φn,ν), in the form of an integral of a scalar density. The total variation of the action leads to 0=δ(Ig+I0)=∫ pµνδgµν +Nµνδgµν +Õ n χnδφn!√gd4x x x(60) where pµν√gis the argument of integration coming from equation (58) of the Igderivation (note that in [3] these quantities pµν are renamed while we keep their first definition and introduce the Nµν as other new terms outside pµν), the second tensor in the argument Nµν√g≡1 2 ∂f(φn, φn,ν) ∂√g√ggµν =∂L0 ∂gµν (61) is formed from L0,√g, as what we are really varying is the metric; and, the last terms in the 17 argument of the variation of the action χn≡1 √g"∂f(φn, φn,ν) ∂φn +Õ ν∂f(φn, φn,ν) ∂φn,ν ,ν#=0(62) are the coefficients due to the partial derivation of the scalar fields, which are assumed to be independent and that is the reason why they should vanish to fulfill the variational principle. To make the argument of equation (60) completely vanish, equation (62) must give us Rµν −1 2gµν R−16πNµν =0(63) where both pµν and Nµν are symmetrical. These are no less than the Einstein’s field equations, when a proper normalization of Nµν is taken, derived from an action principle we can define the energy-momentum tensor to be the tensor that Rµν −1 2gµν R+8πTµν =0(64) where any tensor produced by matter must fulfill, in order to be consistent, Tµν ;ν=0(65) This same procedure is derived, for example, in Weinberg (1972) p. 387-365, [1] by defining the energy-momentum tensor as the functional derivative (in our case, variational derivative) of the gravitational action itself, obtaining therefore the same results as with the normalized Nµν. 3.3 Further Reading For a deeper understanding of the subject, several generalizations have been proposed over the decades after General Relativity came to light. Among them, both Brans-Dicke theory and Einstein-Cartan theory are of high interest for those who would like to continue into more general gravitation theories. Brans-Dicke theory [1] provides a different approach by defending that the gravitational constant Gis not so. Instead, gravitational interaction is mediated by a scalar field and that gives Gas an average of the interaction of the field with all the matter in the Universe. In fact, general relativity theory can be obtained from Brans-Dicke’s as a limit. Einstein-Cartan theory [7] relaxes the assumption of General Relativity that the affine connection has vanishing antisymmetric part (torsion tensor), so that the torsion can be coupled to the intrinsic angular momentum (spin) of matter, much in the same way in which the curvature is coupled to the energy and momentum of matter. This theory provides a possible different origin of the Universe, the Big Bounce, that will make the Big Bang an obsolete theory. It also provides a mathematical basis for the theory of wormholes that may be interesting for further reading. 18 All three Einstein’s, Brans-Dicke’s and Einstein-Cartan’s theories are relativistic classical field theories of gravitation, called metric theories. For a further insight, also see quantum theories of gravity [1]. 4 Black Holes For this interesting part of the paper, we will be following the both deep and conceptual books by Hamilton (2015) [4] and Chandrasekhar (1983) [5], as well as Nordebo (2016) [8] and Newman & Adamo (2004) [11]. This will provide us with an intuition of the particular stellar configurations we are going to work with by learning key concepts of the subject. 4.1 Introduction Ablack hole is defined [4] to be a region where space itself falls faster than the speed of light, whose main result is that, of course, no light can escape from a black hole from inside. A way to see it is by applying the cosmological expansion analogue: imagine that you are a photon inside a black hole, you know you are the fastest entity ever known. You follow a path that hypothetically carries you out of the black hole, but this path will be longer and longer and, somehow, space is being created just in front of you at a rate that you cannot surpass in any possible way. So, theoretical physics had demonstrated the possibility of existence of black holes in our Universe, but we know that theoretical physics has not always been correct. We need evidence, and there is non-direct observational evidence of their ubiquitous presence by means of the study of their surroundings. To the day, there are two kinds of black holes that can be observed: stellar-sized black holes in X-ray binary systems, mostly in our own Milky Way galaxy, and supermassive black holes in Active Galactic Nuclei (AGN) found at the centres of our own and other galaxies [1]. Secondary evidences of the presence of a black hole can also be inferred by facts like high luminosity, non-stellar spectrum, rapid variability and relativistic jets. To the day, there is also evidence of gravitational waves measured at Earth by the experiment LIGO, setting the general theory of relativity as a more-than-proved gravitational theory. The types of black holes that we will be studying are a generalization of the so-called ”ideal black holes” which are stationary, static and electrovac (null-energy-momentum tensor except for the contribution of a stationary electromagnetic field). The Schwarzschild solution of a black hole may be the most known of all of them, we assume, because of their simplicity and importance. 4.2 Reissner-Nordstr¨ om Black Hole The first attempt we are dealing with, in the matter of black holes, is the Reissner-Nordstr¨ om metric surrounding a stationary, non-rotating, charged spherically-symmetric mass discovered independently by Hans Reissner, Hermann Weyl and Gunnar Nordstr¨ om. This metric describes the unique solution in asymptotically flat spacetime. 19 This types of black holes are of little interest, regarding their charge, as astrophysical objects like stars or clouds tend to neutralize electrically, providing a little net charge for the body. The interest comes from its solution, the physical interpretation and its resemblance to the Kerr solution. 4.2.1 Reissner-Nordstr¨ om Metric The Reissner-Nordstr¨ om metric for a black hole of mass Mand charge Q, in geometric units c=G=k=1, is given by ds2=−∆dt2+∆−1dr2+r2(dθ2+sin2(θ)dφ2)(66) where the so-called horizon function is given by ∆(r) ≡ 1−2M r +Q2 r2(67) If you have seen or studied Schwarzschild metric before, this particular metric will sound familiar to you or, at least, resemble that other metric. Just take Q→0to obtain the most simple case of a black hole. Another possible interpretation of the mass is to take Schwarzschild’s MS and identify its analogue in Reissner-Nordstr¨ om’s MRN : MS→MRN (r)=M−Q2 2r(68) Where the quantity MRN (r)has the coordinate-independent interpretation [4] as the mass interior to radius r, being the mass seen by an observer at infinity Mminus the mass (or energy, thanks to mass-energy equality in relativity theory) in the electric field E=Q/r2outside r 4π∫∞ r E2 8πr02dr0=Q 2r(69) This may look like a Newtonian calculation, but it turns out to be true as the electromagnetic field is unchanged under Lorentz boosts in the radial direction of movement. 4.2.2 Derivation of the Metric and the Energy-Momentum Tensor If we consider spacetime to be spherically symmetric, we can write a general metric, as concisely developed in Nordebo (2016), p.25-30 [8], ds2=−A(t,r)dt2+B(t,r)dr2+r2(dθ2+sin2(θ)dφ2)(70) where we have assumed dependence on time t, eventhough we will later find out that the dependence is not so, and the unknowns functions of the metric will only depend on r. Since we are working in vacuum, the energy-momentum tensor we need is given by the 20 electromagnetic energy-momentum tensor Tµν = 1 µ0Fµγ Fναgαγ −1 4gµν FγδFγδ(71) and as we know, the trace of the electromagnetic Tµν vanishes, turning Einstein’s field equations to be Rµν =−8πTµν (72) Finally we also need the source-free Maxwell’s equations Fµν ;ν=0(73) Fµν;γ+Fγµ;ν+Fνγ;µ=0(74) So, let’s compute the affine connection applying equation (14) to the functions A(t,r)and B(t,r); and, thereafter, we will compute the Ricci tensor using contracted equation (25) by λ=ν. Firstly, all non-vanishing Christoffel symbols are given Γ0 00 =Û A 2A Γ0 01 =Γ0 10 =A0 2A Γ0 11 =Û B 2A Γ1 00 =A0 2B Γ1 01 =Γ0 10 =Û B 2B Γ1 11 =B0 2B Γ1 22 =−r B Γ1 33 =−rsin2(θ) B Γ2 12 =Γ2 21 = 1 r Γ2 33 =−sin(θ)cos(θ) Γ3 13 =Γ3 31 = 1 r Γ3 23 =Γ3 32 =cot(θ) (75) where a dot represents differentiation with respect to t, and the prime, with respect to r. Analogously, all the non-vanishing component of the Ricci tensor are R00 =A0 4BA0 A +B0 B−A00 2B−A0 rB +Ü B 2B−Û B 4BÛ A A−Û B B(76) R11 =−A0 4AA0 A +B0 B+A00 2A−B0 rB +Ü B 2A +Û B 4AÛ A A−Û B B(77) R22 =r 2BA0 A−B0 B+ 1 B−1(78) R33 =R22 sin2(θ)(79) R01 =R10 =−Û B rB (80) By generalizing the spherically symmetric gravitational field, we can get no further. Now, 21 if we want to determine the functions Aand B, what we need is to plug these relations into the Einstein’s field equations with a proper energy-momentum tensor, computed with the electromagnetic tensor and the metric tensor. So, as we have only a radial component of the electric field, we can write Er(t,r)=E1(t,r)=F01(t,r)=F10(t,r)(81) and using equation (71) we get T00 = 1 2µ0 AF01F01 (82) T11 =−1 2µ0 BF01F01 (83) T22 = 1 2µ0 r2F01F01 (84) T33 = 1 2µ0 g33F01F01 =T22sin2(θ)(85) All non-diagonal components of the energy-momentum tensor are zero, which means T01 =0=R01 =Û B rB →Û B=0→B=B(r)(86) the dependence on time of Bhas been eliminated. Also note that T00 A +T11 B =0→0=R00 A +R11 B = 1 rB A0 A +B0 B(87) which implies ∂ln(AB) ∂r =A0 A +B0 B=0→AB =f(t)(88) as it must be constant with respect to r. Solving Maxwell’s equations, given in a compact form by equation (74), and applying Gauss’ law to determine the arbitrary constant of integration, we get Er=Q 4π0r2(89) And now, using explicitly Einstein’s field equations we can write by identifying B=A/f and B0=−f A0/A2, −8πT22 =R22 =r 2BA0 A−B0 B+ 1 B−1= 1 f(t) ∂(r A(r)) ∂r−1(90) 22 and by plugging in the expression of T22 1 f(t) ∂(r A(r)) ∂r−1=−1 f(t)8π1 2µ0 r2E2 r(91) which gives ∂(r A(r)) ∂r =f−Q2 4π0r2(92) and integrating and dividing by r, we obtain the result A(t,r)=f(t)+C(t) r +Q2 4π0r2(93) Finally, to get the values of the unknown functions of time fand C, we must take the limit to the Schwarzschild metric Q→0, and therefore g00 must approach g00 =1−2M r(94) giving f(t)=1,C(t)=−2M≡rS,r2 Q≡Q2 4π0 (95) g00(r)=A(r)=1−rS r + r2 Q r2(96) So, a nice visualization of the metric in this particular case would be to show the tensor gµν(r, θ)= −1−rS r+ r2 Q r20 0 0 01−rS r+ r2 Q r2−1 0 0 0 0 r20 0 0 0 r2sin2(θ)  and by setting k=1/4π0=1, we get the result from section 4.2.1. 4.2.3 Horizons In the Reissner-Nordstr¨ om metric we find two possible horizons or singularities in the term g11, given by the inverse of the horizon function in equation (67), that make this term tend to infinity. The horizons occur where an object at rest in the geometry, dr =dθ=dφ=0, follows a null geodesic, ds2=0, which occurs where the horizon function vanishes. This gives us a quadratic equation that we can easily solve R±=M±pM2−Q2(97) 23 We can check that the time coordinate has different behaviours depending on the value of r:tis a timelike coordinate when r>r+and r<r−, so outside the outer horizon and inside the inner horizon; and tis a spacelike coordinate between the inner and outer horizons r−<r<r+. We can interpret this differing behaviour from the Schwarzschild solution: outside the outer horizon, spacetime is falling at the speed of light, as it does inside the inner horizon. This is caused by electromagnetic repulsion (negative pressure of the electric field). In between the horizons, space behaves as in the Schwarzschild black hole, spacetime is falling at velocities higher than the speed of light. If we made M2=Q2, with their correspondent units, we would see that both the inner and outer horizon are the same, providing just a 2-dimensional spherical surface where spacetime is falling at exactly the speed of light. Eventhough there is only a ”shell” of spacetime with velocity c, light could not get out of the inside of the black hole. And if we take a vanishing amount of charge we get, as expected from last section, only one horizon that corresponds to that of Schwarzschild’s solution, r=2M. 4.3 Kerr-Newman Black Hole The Kerr-Newman metric is the metric corresponding to a stationary, rotating, charged black hole in an asymptotically flat empty space. The original solution was given by Kerr [12] for the uncharged case and completed by Newman for the case with non-vanishing charge. We can also state that is a generalization of the Reissner-Nordstr¨ om metric developed in the previous section, where the angular momentum was assumed to vanish. 4.3.1 Kerr-Newman Geometry The metric for the Kerr-Newman geometry is better given by the Boyer-Lindquist metric [4], which states ds2=−R2∆ ρ2(dt −asin2(θ)dφ)2+ρ2 R2∆dr2+ρ2dθ2+R4sin2(θ) ρ2dφ−a R2dt2 (98) where R≡pr2+a2ρ≡pr2+a2cos2(θ)(99) and ∆is the horizon function given in equation (67), and ais the term related to the angular momentum of the system. It is a matter of checking that if we set the constants M=Q=0→∆=1, the Boyer- Lindquist metric gives the Minkowski metric. Also, if we set a=0, we get the Reissner- Nordstr¨ om metric. The Boyer-Lindquist line-element of equation (98) defines not only a metric but also a tetrad carefully chosen to exhibit the symmetries of the geometry. In the locally inertial frame defined by the Boyer-Lindquist tetrad, the energy-momentum tensor (which is non-vanishing for charged Kerr-Newman) is diagonal. These assertions only become apparent in the tetrad frame, and are obscure in the coordinate frame. 24 5 Conclusion and Discussion of the Results Throughout this thesis we have been able to fully understand what theory general relativity is and why it was considered to be an enlightening theory of gravity since its beginning up to the day of today. Successfully, Einstein’s field equations have been obtained both by geometrical and by means of an action principle, having provided the reader with a wide range of mathematical and physical theory and interpretation of the subject that could also be a basis for further approaches and deeper books on general relativity subjects, as gravitational waves, light deflection and another phenomena worth to study. I personally recommend both Berman (2007) [13] and Fern´ andez Barb´ on (2005) [14], apart from the literature that has been used and followed through the development of the thesis. The first gives an overview of all the so-called metric theories and, most concisely, black holes classification. The second provides a deeper common link between mathematics, quantum physics and gravity theory with key concepts and interpretations that are more than worth to read. We also obtained, in a successful way, not only the metrics but also the energy-momentum tensors in both cases. We encourage the reader to think on the importance of the huge massive objects spread through the Universe and their cosmic relevance. It is an interesting exercise to do to imagine yourself around one of these bodies. Of course, there is no need to say that the rest of the books and articles are more than recommended in order to understand and be able to provide other students and reader with a bibliography that can cover all subjects and doubts that may arise. I personally find really interesting the last section were we spoke about regions around the black holes. I think it provides a imaginative reader with the visual help that is needed in order to consider these astrophysical and theoretical objects a completely astonishing physical entity. I would also like to mention to the reader to be careful with the notation when following different books and articles. Do not focus in the notation, which is, of course, of value, but I would recommend to get the underlying idea of what is being studied with that particular language. Also, I would like to encourage universities, in particular my own, University of Sevilla, to provide students with the opportunity of studying this beautiful subject, that nowadays is not present in the curricular programme of any of the degrees done in the Faculty of Physics. It is also really important to do calculations with these new objects here defined: differentiate, make your own exercises on the subject, do not trust other people’s calculations until you have got the same results. We, scientist, know that this is the only way of doing real science. Study, read, think, do, compare and repeat. 31 References [1] WEINBERG, S. (1972). Gravitation and Cosmology, Principles and Applications of the General Theory of Relativity. NEW YORK, USA: WILEY. [2] SCHUTZ, B. (2009). A First Course in General Relativity (2ND ED.). NEW YORK, USA: CAMBRIDGE UNIVERSITY PRESS. [3] DIRAC, P.A.M. (1975). General Theory of Relativity. NEW YORK, USA: JOHN WILEY AND SONS. [4] HAMILTON, A. J. S. (2015). General Relativity, Black Holes and Cosmology, http://jila.colorado.edu/ ajsh/home.html. Last visited 16 May 2018. [5] CHANDRASEKHAR, S. (1983). The Mathematical Theory of Black Holes. NEW YORK, USA: OXFORD UNIVERSITY PRESS. [6] GELFAND, I. M.; FOMIN, S. V. (1963). CALCULUS OF VARIATIONS.(RICHARD A. SILVERMAN, TRANS.) NEW JERSEY, USA: PRENTICE-HALL, INC. [7] CARTAN,´ E. (1922). Sur une g´ en´ eralisation de la notion de courbure de Riemann et les espaces ` a torsion. C. R. ACAD. SCI. (PARIS) 174, 593–595. [8] NORDEBO, J. (2016). The Reissner-Nordstr¨ om metric, http://www.diva-portal.org/smash/get/diva2:912393/FULLTEXT01.pdf. Last visited 01 June 2018. [9] DRAKE S. P., SZEKERES P. (1998). An explanation of the Newman-Janis Algorithm, https://arxiv.org/abs/gr-qc/9807001. Last visited 04 June 2018. [10] DEFELICE, F.; CLARKE, C.J.S. (1990). Relativity on Curved Manifolds. NEW YORK, USA: CAMBRIDGE UNIVERSITZ PRESS. [11] NEWMAN, T.; ADAMO, T. (2004). Kerr-Newman metric, http://www.scholarpedia.org/article/Kerr-Newman metric. Last visited 04 June 2018. [12] KERR, ROY. (1963). Gravitational Field of a Spinning Mass as an Example of Algebraically Special Metrics. PHYSICAL REVIEW LETTERS - PHYS REV LETT. 11. 237- 238. 10.1103/PhysRevLett.11.237 [13] BERMAN, M. S. (FEB 2007). Simple Derivation of Schwarzschild, Lense-Thirring, Reissner-Nordtr¨ om, Kerr and Kerr-Newman Metrics,arXiv:physics/0702014v4. [14] FERN ´ ANDEZ BARB ´ ON, J. L. (MARCH 2005). Geometr´ ıa no conmutativa y espaciotiempo cu´ antico [Non-conmutative geometry and quantum spacetime]. Investigaci´ on y Ciencia (Madrid) 342, 60-61. 32