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.