DOCSLIB.ORG

General Relativity and Cosmology

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
General Relativity and Cosmology General Relativity and Cosmology Badis Ydri Departement de Physique, Faculté des Sciences Université d’Annaba Annaba, Algerie 2 YDRI GR Contents 1 Introduction and References 1 2 Differential Geometry Primer and Einsetin’s Equation 3 2.1 SpecialRelativity............................... ..... 3 2.1.1 PostulatesofSpecialRelativity . ....... 3 2.1.2 LorentzTransformations. .... 3 2.1.3 Interval ..................................... 5 2.1.4 Metric...................................... 7 2.2 Manifolds ....................................... 9 2.2.1 Maps,OpenSetandCharts. 9 2.2.2 Manifold: Definition and Examples . ..... 10 2.2.3 Vectors ..................................... 12 2.2.4 Tensors ..................................... 14 2.2.5 MetricTensor.................................. 17 2.3 Curvature....................................... 18 2.3.1 CovariantDerivative . ... 18 2.3.2 ParallelTransport . .. .. .. .. .. .. .. .. .. .. .. .. .. 20 2.3.3 TheRiemannCurvature .. .. .. .. .. .. .. .. .. .. .. .. 22 2.3.4 Geodesics .................................... 25 2.4 Einsetin’sEquation. .. .. .. .. .. .. .. .. .. .. .. .. ..... 27 2.4.1 RelativisticMechanics . .... 27 2.4.2 PerfectFluid .................................. 29 2.4.3 GeodesicDeviation. .. .. .. .. .. .. .. .. .. .. .. .. .. 31 2.4.4 Einsetin’s Equation and Minimal Coupling . ....... 32 2.4.5 Killing Vectors and Symmetric Spaces . ...... 36 3 Gravitational Physics 41 3.1 Cosmology....................................... 41 3.1.1 ExpandingUniverse .. .. .. .. .. .. .. .. .. .. .. .. 41 3.1.2 HomogeneityandIsotropy. ... 44 3.1.3 The Friedmann-Lemaître-Robertson-Walker Universe ............ 45 3.1.4 TheScaleFactor ................................ 50 3.1.5 CosmologicalRedshift andDistances . ....... 54 3.2 SphericalStar................................... ... 59 3.2.1 TheSchwarzschildMetric . ... 59 3.2.2 Particle Motion in Schwarzschild Spacetime . ......... 62 3.2.3 Precession of Perihelia and Gravitational Redshift . ............ 66 3.2.4 FreeFall..................................... 69 4 YDRI GR 4 Black Holes 71 4.1 SchwarzschildBlackHole . ..... 71 4.2 The Kruskal-Szekres Diagram: Maximally Extended Schwarzschild Solution . 73 4.3 VariousTheoremsandResults . ..... 76 4.4 Reissner-Nordström(Charged)Black Hole . .......... 79 4.4.1 Maxwell’sEquationsandChargesinGR. ..... 79 4.4.2 Reissner-NordströmSolution . ..... 80 4.4.3 Extremal Reissner-NordströmBlack Hole . ....... 83 4.5 KerrSpacetime................................... .. 84 4.5.1 Kerr (Rotating) and Kerr-Newman (Rotating and Charged) Black Holes . 84 4.5.2 KillingHorizons ............................... 86 4.5.3 SurfaceGravity................................ 86 4.5.4 Event Horizons, Ergosphere and Singularity . ......... 87 4.5.5 PenroseProcess................................ 89 4.6 BlackHolesThermodynamics . .... 90 1 Introduction and References • R.M.Wald,General Relativity. • S.M.Carroll, An Introduction to General Relativity: Spacetime and Geometry. • G.’t Hooft, Introduction to General Relativity. • G.’t Hooft, Introduction to the Theory of Black Holes. • T.Eguchi, P.B.Gilkey and A.J.Hanson, Gravitation, Gauge Theories and Differential Ge- ometry. 2 YDRI GR 2 Differential Geometry Primer and Einsetin’s Equation 2.1 Special Relativity 2.1.1 Postulates of Special Relativity The two fundamental postulates of special relativity are: • The principle of relativity: The laws of physics take the same form in all inertial reference frames. • The constancy of the speed of light: The speed of light in vacuum is the same in all inertial reference frames. These postulates lead to a host of relativistic effects such as relativity of simultaneity, time dilation and Lorentz (length) contraction. The precise content of the principle of relativity is the invariance/covariance of the laws of physics under Lorentz transformations. In the non- relativistic limit the Lorentz group becomes the Galilean group. The postulate of the constancy of the speed of light can be derived from the principle of relativity (in other words from the in- variance/covariance under Lorentz transformations) but we conformed here to the usual practice of stating it independently. 2.1.2 Lorentz Transformations Any physical process consists of a collection of events. Any event takes place at a given point (x,y,z) of space at an instant of time t. The four numbers (x,y,z) and t define a point in spacetime (also called Minkowski spacetime). The trajectory of a particle in spacetime is called a worldline. Even a particle at rest will trace a wolrdline since time flows always. Lorentz transformations relate the coordinates (x,y,z,t) of a given event in an inertial ref- ′ ′ ′ ′ erence frame O to the coordinates (x ,y ,z ,t ) of the same event in another inertial reference ′ frame O . 4 YDRI GR Let (x,y,z,t) be the coordinates in O of an event E. The projection of E onto the x axis is given by the point P which has the coordinates (x, 0, 0,t). For simplicity we will assume that ′ the observer O moves with respect to the observer O at a constant speed v along the x axis. At ′ ′ time t =0 the two observers O and O coincides. After time t the observer O moves a distance ′ vt on the x axis. Let d be the distance between O and P as measured by O. Then clearly x = d + vt. (2.1) ′ ′ Before the theory of special relativity the coordinate x of the event E in the reference frame O is taken to be equal to the distance d. We get therefore the transformation laws ′ x = x vt ′ − y = y ′ z = z ′ t = t. (2.2) This is a Galilean transformation. It is a well established result that the postulates of special relativity lead to Lorentz con- ′ ′ traction. In other words the distance between O and P measured by the observer O which is ′ precisely the coordinate x is larger than d. More precisely ′ x = γd. (2.3) Hence ′ x = γ(x vt). (2.4) − Einstein’s postulates lead also to time dilation and relativity of simultaneity. Thus the time of ′ ′ the event E measured by O is different from t. Since the observer O moves with respect to O at a speed v in the negative x direction we must have ′ ′ x = γ(x + vt ). (2.5) Thus we get ′ v t = γ(t x). (2.6) − c2 In summary we get the transformation laws ′ x = γ(x vt) ′ − y = y ′ z = z ′ v t = γ(t x). (2.7) − c2 This is a Lorentz transformation. ′ ′ Let us look at the clock found at the origin of the reference frame O . We set then x =0 in the above equations. We get immediately the time dilation effect, viz ′ t t = . (2.8) γ YDRI GR 5 ′ At time t =0 the clocks in O read different times depending on their location since ′ v t = γ x. (2.9) − c2 Hence moving clocks can not be synchronized. We consider now two events A and B with coordinates (x ,t ) and (x ,t ) in O and ′ ′ ′ ′ ′ A A B B coordinates (xA,tA) and (xB ,tB) in O . We can immediately compute ′ v ∆t = γ(∆t ∆x). (2.10) − c2 Thus if the two events are simultaneous with respect to O, i.e. ∆t =0 they are not simultaneous ′ with respect to O since ′ v ∆t = γ ∆x. (2.11) − c2 Let us now derive Einstein’s velocity addition rule. We consider a particle in the reference frame O moving a distance dx in the x direction during a time interval dt. The velocity with respect to O is dx u = . (2.12) dt ′ ′ ′ In the reference frame O the particle moves a distance dx in a time interval dt given by ′ dx = γ(dx vdt). (2.13) − ′ v dt = γ(dt dx). (2.14) − c2 ′ The velocity with respect to O is therefore ′ ′ dx u v u = ′ = −vu . (2.15) dt 1 2 − c ′ ′ In general if V~ and V~ are the velocities of the particle with respect to O and O respectively ′ and ~v is the velocity of O with respect to O. Then ′ V~ ~v V~ = − . (2.16) V~ ~v 1 2 − c 2.1.3 Interval The above Lorentz transformations can also be rewritten as ′ x0 = γ(x0 βx1) ′ − x1 = γ(x1 βx0) ′ − x2 = x2 ′ x3 = x3. (2.17) 6 YDRI GR In the above equation x0 = ct, x1 = x , x2 = y , x3 = z. (2.18) v β = , γ = 1 β2. (2.19) c − p Lorentz transformation can also be rewritten as 4 ′ µ µ ν x = Λν x . (2.20) ν=0 X γ γβ 0 0 γβ− γ 0 0 Λ= . (2.21) −0 0 10 0 0 01 The matrix Λ is the Lorentz transformation matrix. A generalized Lorentz transformation can be ′ obtained if the relative motion of the two inertial reference frames O and O is along an arbitrary direction in space. The transformation law of the coordinates xµ will still be given by (2.20) with a more complicated matrix Λ. The set of coordinates (x0, x1, x2, x3) which transforms ′ µ µ ν under Lorentz transformations as x = Λν x will be called a 4 vector in analogy with the set of coordinates (x1, x2, x3) which is called a vector because it transforms− under rotations as ′ a a b x = Rb x . In fact the set of all Lorentz transformations contains rotations as a subset. Thus in general a 4 vector a is any set of numbers (a0,a1,a2,a3) which transforms as (x0, x1, x2, x3) under Lorentz− transformations, viz 4 ′ µ µ ν a = Λν a . (2.22) ν=0 X For the particular Lorentz transformation (2.21) we have ′ a0 = γ(a0 βa1) ′ − a1 = γ(a1 βa0) ′ − a2 = a2 ′ a3 = a3. (2.23) The numbers aµ are called the contravariant components of the 4 vector a. We define the − covariant components aµ by a = a0 , a = a1 , a = a2 , a = a3. (2.24) 0 − 1 2 3 By using the Lorentz transformation (2.23) we verify given any two 4 vectors a and b the identity − ′ ′ ′ ′ ′ ′ ′ ′ a0 b0 + a1 b1 + a2 b2 + a3 b3 = a0b0 + a1b1 + a2b2 + a3b3. (2.25) − − In fact we can show that this identity holds for all Lorentz transformations. We recall that under rotations the scalar product ~a~b of any two vectors ~a and ~b is invariant,i.e.
Load more

Recommended publications
  • Aspects of Black Hole Physics
    Aspects of Black Hole Physics Andreas Vigand Pedersen The Niels Bohr Institute Academic Advisor: Niels Obers e-mail: [email protected] Abstract: This project examines some of the exact solutions to Einstein’s theory, the theory of linearized gravity, the Komar definition of mass and angular momentum in general relativity and some aspects of (four dimen- sional) black hole physics. The project assumes familiarity with the basics of general relativity and differential geometry, but is otherwise intended to be self contained. The project was written as a ”self-study project” under the supervision of Niels Obers in the summer of 2008. Contents Contents ..................................... 1 Contents ..................................... 1 Preface and acknowledgement ......................... 2 Units, conventions and notation ........................ 3 1 Stationary solutions to Einstein’s equation ............ 4 1.1 Introduction .............................. 4 1.2 The Schwarzschild solution ...................... 6 1.3 The Reissner-Nordstr¨om solution .................. 18 1.4 The Kerr solution ........................... 24 1.5 The Kerr-Newman solution ..................... 28 2 Mass, charge and angular momentum (stationary spacetimes) 30 2.1 Introduction .............................. 30 2.2 Linearized Gravity .......................... 30 2.3 The weak field approximation .................... 35 2.3.1 The effect of a mass distribution on spacetime ....... 37 2.3.2 The effect of a charged mass distribution on spacetime .. 39 2.3.3 The effect of a rotating mass distribution on spacetime .. 40 2.4 Conserved currents in general relativity ............... 43 2.4.1 Komar integrals ........................ 49 2.5 Energy conditions ........................... 53 3 Black holes ................................ 57 3.1 Introduction .............................. 57 3.2 Event horizons ............................ 57 3.2.1 The no-hair theorem and Hawking’s area theorem ....
    [Show full text]
  • The Schwarzschild Metric and Applications 1
    The Schwarzschild Metric and Applications 1 Analytic solutions of Einstein©s equations are hard to come by. It©s easier in situations that exhibit symmetries. 1916: Karl Schwarzschild sought the metric describing the static, spherically symmetric spacetime surrounding a spherically symmetric mass distribution. A static spacetime is one for which there exists a time coordinate t such that i) all the components of g are independent of t ii) the line element ds2 is invariant under the transformation t -t A spacetime that satisfies (i) but not (ii) is called stationary. An example is a rotating azimuthally symmetric mass distribution. The metric for a static spacetime has the form where xi are the spatial coordinates and dl2 is a time-independent spatial metric. Cross-terms dt dxi are missing because their presence would violate condition (ii). [Note: The Kerr metric, which describes the spacetime outside a rotating 2 axisymmetric mass distribution, contains a term ∝ dt d.] To preserve spherical symmetry, dl2 can be distorted from the flat-space metric only in the radial direction. In flat space, (1) r is the distance from the origin and (2) 4r2 is the area of a sphere. Let©s define r such that (2) remains true but (1) can be violated. Then, A(xi) A(r) in cases of spherical symmetry. The Ricci tensor for this metric is diagonal, with components SP 10.1 Primes denote differentiation with respect to r. The region outside the spherically symmetric mass distribution is empty. 3 The vacuum Einstein equations are R = 0. To find A(r) and B(r): 2.
    [Show full text]
  • The Schwarzschild Metric and Applications 1
    The Schwarzschild Metric and Applications 1 Analytic solutions of Einstein's equations are hard to come by. It's easier in situations that e hibit symmetries. 1916: Karl Schwarzschild sought the metric describing the static, spherically symmetric spacetime surrounding a spherically symmetric mass distribution. A static spacetime is one for which there exists a time coordinate t such that i' all the components of g are independent of t ii' the line element ds( is invariant under the transformation t -t A spacetime that satis+es (i) but not (ii' is called stationary. An example is a rotating azimuthally symmetric mass distribution. The metric for a static spacetime has the form where xi are the spatial coordinates and dl( is a time*independent spatial metric. -ross-terms dt dxi are missing because their presence would violate condition (ii'. 23ote: The Kerr metric, which describes the spacetime outside a rotating ( axisymmetric mass distribution, contains a term ∝ dt d.] To preser)e spherical symmetry& dl( can be distorted from the flat-space metric only in the radial direction. In 5at space, (1) r is the distance from the origin and (2) 6r( is the area of a sphere. Let's de+ne r such that (2) remains true but (1) can be violated. Then, A,xi' A,r) in cases of spherical symmetry. The Ricci tensor for this metric is diagonal, with components S/ 10.1 /rimes denote differentiation with respect to r. The region outside the spherically symmetric mass distribution is empty. 9 The vacuum Einstein equations are R = 0. To find A,r' and B,r'# (.
    [Show full text]
  • Part 3 Black Holes
    Part 3 Black Holes Harvey Reall Part 3 Black Holes March 13, 2015 ii H.S. Reall Contents Preface vii 1 Spherical stars 1 1.1 Cold stars . .1 1.2 Spherical symmetry . .2 1.3 Time-independence . .3 1.4 Static, spherically symmetric, spacetimes . .4 1.5 Tolman-Oppenheimer-Volkoff equations . .5 1.6 Outside the star: the Schwarzschild solution . .6 1.7 The interior solution . .7 1.8 Maximum mass of a cold star . .8 2 The Schwarzschild black hole 11 2.1 Birkhoff's theorem . 11 2.2 Gravitational redshift . 12 2.3 Geodesics of the Schwarzschild solution . 13 2.4 Eddington-Finkelstein coordinates . 14 2.5 Finkelstein diagram . 17 2.6 Gravitational collapse . 18 2.7 Black hole region . 19 2.8 Detecting black holes . 21 2.9 Orbits around a black hole . 22 2.10 White holes . 24 2.11 The Kruskal extension . 25 2.12 Einstein-Rosen bridge . 28 2.13 Extendibility . 29 2.14 Singularities . 29 3 The initial value problem 33 3.1 Predictability . 33 3.2 The initial value problem in GR . 35 iii CONTENTS 3.3 Asymptotically flat initial data . 38 3.4 Strong cosmic censorship . 38 4 The singularity theorem 41 4.1 Null hypersurfaces . 41 4.2 Geodesic deviation . 43 4.3 Geodesic congruences . 44 4.4 Null geodesic congruences . 45 4.5 Expansion, rotation and shear . 46 4.6 Expansion and shear of a null hypersurface . 47 4.7 Trapped surfaces . 48 4.8 Raychaudhuri's equation . 50 4.9 Energy conditions . 51 4.10 Conjugate points .
    [Show full text]
  • Sufficient Conditions for Periodicity of a Killing Vector Field Walter C
    PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY Volume 38, Number 3, May 1973 SUFFICIENT CONDITIONS FOR PERIODICITY OF A KILLING VECTOR FIELD WALTER C. LYNGE Abstract. Let X be a complete Killing vector field on an n- dimensional connected Riemannian manifold. Our main purpose is to show that if X has as few as n closed orbits which are located properly with respect to each other, then X must have periodic flow. Together with a known result, this implies that periodicity of the flow characterizes those complete vector fields having all orbits closed which can be Killing with respect to some Riemannian metric on a connected manifold M. We give a generalization of this characterization which applies to arbitrary complete vector fields on M. Theorem. Let X be a complete Killing vector field on a connected, n- dimensional Riemannian manifold M. Assume there are n distinct points p, Pit' " >Pn-i m M such that the respective orbits y, ylt • • • , yn_x of X through them are closed and y is nontrivial. Suppose further that each pt is joined to p by a unique minimizing geodesic and d(p,p¡) = r¡i<.D¡2, where d denotes distance on M and D is the diameter of y as a subset of M. Let_wx, • ■ ■ , wn_j be the unit vectors in TVM such that exp r¡iwi=pi, i=l, • ■ • , n— 1. Assume that the vectors Xv, wx, ■• • , wn_x span T^M. Then the flow cptof X is periodic. Proof. Fix /' in {1, 2, •••,«— 1}. We first show that the orbit yt does not lie entirely in the sphere Sj,(r¡¿) of radius r¡i about p.
    [Show full text]
  • Constructing $ P, N $-Forms from $ P $-Forms Via the Hodge Star Operator and the Exterior Derivative
    Constructing p, n-forms from p-forms via the Hodge star operator and the exterior derivative Jun-Jin Peng1,2∗ 1School of Physics and Electronic Science, Guizhou Normal University, Guiyang, Guizhou 550001, People’s Republic of China; 2Guizhou Provincial Key Laboratory of Radio Astronomy and Data Processing, Guizhou Normal University, Guiyang, Guizhou 550001, People’s Republic of China Abstract In this paper, we aim to explore the properties and applications on the operators consisting of the Hodge star operator together with the exterior derivative, whose action on an arbi- trary p-form field in n-dimensional spacetimes makes its form degree remain invariant. Such operations are able to generate a variety of p-forms with the even-order derivatives of the p- form. To do this, we first investigate the properties of the operators, such as the Laplace-de Rham operator, the codifferential and their combinations, as well as the applications of the arXiv:1909.09921v2 [gr-qc] 23 Mar 2020 operators in the construction of conserved currents. On basis of two general p-forms, then we construct a general n-form with higher-order derivatives. Finally, we propose that such an n-form could be applied to define a generalized Lagrangian with respect to a p-form field according to the fact that it incudes the ordinary Lagrangians for the p-form and scalar fields as special cases. Keywords: p-form; Hodge star; Laplace-de Rham operator; Lagrangian for p-form. ∗ [email protected] 1 1 Introduction Differential forms are a powerful tool developed to deal with the calculus in differential geom- etry and tensor analysis.
    [Show full text]
  • A Review on Metric Symmetries Used in Geometry and Physics K
    A Review on Metric Symmetries used in Geometry and Physics K. L. Duggala aUniversity of Windsor, Windsor, Ontario N9B3P4, Canada, E-mail address: [email protected] This is a review paper of the essential research on metric (Killing, homothetic and conformal) symmetries of Riemannian, semi-Riemannian and lightlike manifolds. We focus on the main characterization theorems and exhibit the state of art as it now stands. A sketch of the proofs of the most important results is presented together with sufficient references for related results. 1. Introduction The measurement of distances in a Euclidean space R3 is represented by the distance element ds2 = dx2 + dy2 + dz2 with respect to a rectangular coordinate system (x, y, z). Back in 1854, Riemann generalized this idea for n-dimensional spaces and he defined element of length by means of a quadratic 2 i j differential form ds = gijdx dx on a differentiable manifold M, where the coefficients gij are functions of the coordinates system (x1, . , xn), which represent a symmetric tensor field g of type (0, 2). Since then much of the subsequent differential geometry was developed on a real smooth manifold (M, g), called a Riemannian manifold, where g is a positive definite metric tensor field. Marcel Berger’s book [1] includes the major developments of Riemannian geome- try since 1950, citing the works of differential geometers of that time. On the other hand, we refer standard book of O’Neill [2] on the study of semi-Riemannian geometry where the metric g is indefinite and, in particular, Beem-Ehrlich [3] on the global Lorentzian geometry used in relativity.
    [Show full text]
  • Lecture A2 Conformal Field Theory Killing Vector Fields the Sphere Sn
    Lecture A2 conformal field theory Killing vector fields The sphere Sn is invariant under the group SO(n + 1). The Minkowski space is invariant under the Poincar´egroup, which includes translations, rotations, and Lorentz boosts. For µ a general Riemannian manifold M, take a tangent vector field ξ = ξ ∂µ and consider the infinitesimal coordinate transformation, xµ xµ + ǫξµ, ( ǫ 1). → | |≪ Question 1: Show that the metric components gµν transforms as ρ 2 2 gµν gµν + ǫ (∂µξν + ∂νξµ + ξ ∂ρ) + 0(ǫ )= gµν + ǫ ( µξν + νξµ) + 0(ǫ ). → ∇ ∇ The metric is invariant under the infinitesimal transformation by ξ iff µξν + νξµ = 0. A ∇ ∇ tangent vector field satisfying this equation is called a Killing vector field. µ Question 2: Suppose we have two tangent vector fields, ξa = ξa ∂µ (a = 1, 2). Show that their commutator ν µ ν µ [ξ ,ξ ] = (ξ ∂νξ ξ ∂ν ξ )∂µ 1 2 1 2 − 2 1 ν µ is also a tangent vector field. Explain why ξ1 ∂νξ2 is not a tangent vector field in general. Question 3: Suppose there are two Killing vector fields, ξa (a = 1, 2). One can consider an infinitesimal transformation, ga, corresponding to each of them. Namely, µ µ µ ga : x x + ǫξ , (a = 1, 2). → a −1 −1 Show that g1g2g1 g2 is generated by the commutator, [ξ1,ξ2]. If the metric is invariant under infinitesimal transformations generated by ξ1,ξ2, it should also be invariant under their commutator, [ξ1,ξ2]. Thus, the space of Killing vector fields is closed under the commutator – it makes a Lie algebra.
    [Show full text]
  • The Reissner-Nordström Metric
    The Reissner-Nordström metric Jonatan Nordebo March 16, 2016 Abstract A brief review of special and general relativity including some classi- cal electrodynamics is given. We then present a detailed derivation of the Reissner-Nordström metric. The derivation is done by solving the Einstein-Maxwell equations for a spherically symmetric electrically charged body. The physics of this spacetime is then studied. This includes gravitational time dilation and redshift, equations of motion for both massive and massless non-charged particles derived from the geodesic equation and equations of motion for a massive charged par- ticle derived with lagrangian formalism. Finally, a quick discussion of the properties of a Reissner-Nordström black hole is given. 1 Contents 1 Introduction 3 2 Review of Special Relativity 3 2.1 4-vectors . 6 2.2 Electrodynamics in Special Relativity . 8 3 Tensor Fields and Manifolds 11 3.1 Covariant Differentiation and Christoffel Symbols . 13 3.2 Riemann Tensor . 15 3.3 Parallel Transport and Geodesics . 18 4 Basics of General Relativity 19 4.1 The Equivalence Principle . 19 4.2 The Principle of General Covariance . 20 4.3 Electrodynamics in General Relativity . 21 4.4 Newtonian Limit of the Geodesic Equation . 22 4.5 Einstein’s Field Equations . 24 5 The Reissner-Nordström Metric 25 5.1 Gravitational Time Dilation and Redshift . 32 5.2 The Geodesic Equation . 34 5.2.1 Comparison to Newtonian Mechanics . 37 5.2.2 Circular Orbits of Photons . 38 5.3 Motion of a Charged Particle . 38 5.4 Event Horizons and Black Holes . 40 6 Summary and Conclusion 44 2 1 Introduction In 1915 Einstein completed his general theory of relativity.
    [Show full text]
  • Chapter 5 Differential Geometry
    Chapter 5 Differential Geometry III 5.1 The Lie derivative 5.1.1 The Pull-Back Following (2.28), we considered one-parameter groups of diffeomor- phisms γt : R × M → M (5.1) such that points p ∈ M can be considered as being transported along curves γ : R → M (5.2) with γ(0) = p. Similarly, the diffeomorphism γt can be taken at fixed t ∈ R, defining a diffeomorphism γt : M → M (5.3) which maps the manifold onto itself and satisfies γt ◦ γs = γs+t. We have seen the relationship between vector fields and one-parameter groups of diffeomorphisms before. Let now v be a vector field on M and γ from (5.2) be chosen such that the tangent vectorγ ˙(t) defined by d (˙γ(t))( f ) = ( f ◦ γ)(t) (5.4) dt is identical with v,˙γ = v. Then γ is called an integral curve of v. If this is true for all curves γ obtained from γt by specifying initial points γ(0), the result is called the flow of v. The domain of definition D of γt can be a subset of R× M.IfD = R× M, the vector field is said to be complete and γt is called the global flow of v. If D is restricted to open intervals I ⊂ R and open neighbourhoods U ⊂ M, thus D = I × U ⊂ R × M, the flow is called local. 63 64 5 Differential Geometry III Pull-back Let now M and N be two manifolds and φ : M → N a map from M onto N.
    [Show full text]
  • The Case for Astrophysical Black Holes
    33rd SLAC Summer Institute on Particle Physics (SSI 2005), 25 July - 5 August 2005 Trust but Verify: The Case for Astrophysical Black Holes Scott A. Hughes Department of Physics and MIT Kavli Institute, 77 Massachusetts Avenue, Cambridge, MA 02139 This article is based on a pair of lectures given at the 2005 SLAC Summer Institute. Our goal is to motivate why most physicists and astrophysicists accept the hypothesis that the most massive, compact objects seen in many astrophysical systems are described by the black hole solutions of general relativity. We describe the nature of the most important black hole solutions, the Schwarzschild and the Kerr solutions. We discuss gravitational collapse and stability in order to motivate why such objects are the most likely outcome of realistic astrophysical collapse processes. Finally, we discuss some of the observations which — so far at least — are totally consistent with this viewpoint, and describe planned tests and observations which have the potential to falsify the black hole hypothesis, or sharpen still further the consistency of data with theory. 1. BACKGROUND Black holes are among the most fascinating and counterintuitive objects predicted by modern physical theory. Their counterintuitive nature comes not from obtuse features of gravitation physics, but rather from the extreme limit of well-understood and well-observed features — the bending of light and the redshifting of clocks by gravity. This limit is inarguably strange, and drives us to predictions that may reasonably be considered bothersome. Given the lack of direct observational or experimental data about gravity in the relevant regime, a certain skepticism about the black hole hypothesis is perhaps reasonable.
    [Show full text]
  • Conformal Killing Forms on Riemannian Manifolds
    Conformal Killing forms on Riemannian manifolds Uwe Semmelmann October 22, 2018 Abstract Conformal Killing forms are a natural generalization of conformal vector fields on Riemannian manifolds. They are defined as sections in the kernel of a conformally invariant first order differential operator. We show the existence of conformal Killing forms on nearly K¨ahler and weak G2-manifolds. Moreover, we give a complete de- scription of special conformal Killing forms. A further result is a sharp upper bound on the dimension of the space of conformal Killing forms. 1 Introduction A classical object of differential geometry are Killing vector fields. These are by definition infinitesimal isometries, i.e. the flow of such a vector field preserves a given metric. The space of all Killing vector fields forms the Lie algebra of the isometry group of a Riemannian manifold and the number of linearly independent Killing vector fields measures the degree of symmetry of the manifold. It is known that this number is bounded from above by the dimension of the isometry group of the standard sphere and, on compact manifolds, equality is attained if and only if the manifold is isometric to the standard sphere or the real projective space. Slightly more generally one can consider conformal vector fields, i.e. vector fields with a flow preserving a given conformal class of metrics. There are several geometric conditions which force a conformal vector field to be Killing. Much less is known about a rather natural generalization of conformal vector fields, the so-called conformal Killing forms. These are p-forms ψ satisfying for any vector field X the differential equation 1 y 1 ∗ ∗ ∇X ψ − p+1 X dψ + n−p+1 X ∧ d ψ = 0 , (1.1) arXiv:math/0206117v1 [math.DG] 11 Jun 2002 where n is the dimension of the manifold, ∇ denotes the covariant derivative of the Levi- Civita connection, X∗ is 1-form dual to X and y is the operation dual to the wedge product.
    [Show full text]