1 To my mother Lectures on General Theory of Relativity 6 Emil T. Akhmedov 1 0 2 These lectures are based on the following books: v o N Textbook On Theoretical Physics. Vol. 2: Classical Field Theory, by L.D. Landau and • 9 E.M. Lifshitz 1 ] Relativist’s Toolkit: The mathematics of black-hole mechanics, by E.Poisson, Cambridge c • q University Press, 2004 - r g General Relativity, by R. Wald, The University of Chicago Press, 2010 [ • 5 General Relativity, by I.Khriplovich, Springer, 2005 v • 6 9 An Introduction to General Relativity, by L.P. Hughston and K.P. Tod, Cambridge Uni- • 9 versity Press, 1994 4 0 . Black Holes (An Introduction), by D. Raine and E.Thomas, Imperial College Press, 2010 1 • 0 6 They were given to students of the Mathematical Faculty of the Higher School of Economics 1 : in Moscow. At the end of each lecture I list some of those subjects which are not covered in the v i lectures. If not otherwise stated, these subjects can be found in the above listed books. I have X r assumed that students that have been attending these lectures were familiar with the classical a electrodynamics and Special Theory of Relativity, e.g. with the first nine chapters of the second volume of Landau and Lifshitz course. I would like to thank Mahdi Godazgar and Fedor Popov for useful comments, careful reading and corrections to the text. 2 Contents LECTURE I General covariance. Transition to non–inertial reference frames in Minkowski space–time. Geodesic equation. Christoffel symbols. 4 LECTURE II Tensors. Covariant differentiation. Parallel transport. Locally Minkowskian reference system. Curvature or Riemann tensor and its properties. 13 LECTURE III Einstein–Hilbert action. Einstein equations. Matter energy–momentum tensor. 23 LECTURE IV Schwarzschild solution. Schwarzschild coordinates. Eddington–Finkelstein coordinates. 30 LECTURE V Penrose–Carter diagrams. Kruskal–Szekeres coordinates. Penrose–Carter diagram for the Schwarzschild black hole. 37 LECTURE VI Killing vectors and conservation laws. Test particle motion on Schwarzschild black hole background. Mercury perihelion rotation. Light ray deviation in the vicinity of the Sun. 46 LECTURE VII Energy–momentum tensor for a perfect relativistic fluid. Interior solution. Kerr’s rotating black hole. Proper time. Gravitational redshift. Concise comments on Cosmic Censorship Hypothesis and on the No Hair Theorem. 54 LECTURE VIII Oppenheimer–Snyder collapse. Concise comments on the origin of the thermal nature of the Hawking’s radiation and on black hole creation. 63 LECTURE IX Energy–momentum pseudo–tensor for gravity. Weak field approximation. Energy–momentum pseudo–tensor in the weak field limit on the Minkowskian background. Free gravitational waves.73 LECTURE X Gravitational radiation by moving massive bodies. Shock gravitational wave or Penrose parallel plane wave. 81 LECTURE XI Homogeneous three–dimensional spaces. Friedmann–Robertson–Walker metric. Homogeneous isotropic cosmological solutions. Anisotropic Kasner cosmological solution. 89 3 LECTURE XII Geometry of the de Sitter space–time. Geometry of the anti–de Sitter space–time. Penrose–Carter diagrams for de Sitter and anti de Sitter. Wick rotation. Global coordinates. Poincare coordinates. Hyperbolic distance. 99 4 LECTURE I General covariance. Transition to non–inertial reference frames in Minkowski space–time. Geodesic equation. Christoffel symbols. 1. Minkowski space-time metric is as follows: ds2 = η dxµdxν = dt2 d~x2. (1) µν − Throughout these lectures we set the speed of light to one c = 1, unless otherwise stated. Here µ,ν = 0,...,3 and Minkowskian metric tensor is η = Diag(1, 1, 1, 1). (2) µν || || − − − The bilinear form defining the metric tensor is invariant under the hyperbolic rotations: t′ = tcoshα+xsinhα, x′ = tsinhα+xcoshα, α =const, y′ = y, z′ =z, (3) i.e. η dxµdxν = dt2 d~x2 = (dt′)2 (d~x′)2 = η dx′µdx′ν. µν µν − − This is the so called Lorentz boost, where coshα = γ = 1/√1 v2, sinhα = vγ. Its physical − meaningisthetransformationfromaninertialreferencesystemtoanotherinertialreferencesystem. The latter one moves along the x axis with the constant velocity v with respect to the initial reference system. Under an arbitrary coordinate transformation (not necessarily linear), xµ = xµ(x¯ν), the metric can change in an unrecognizable way, if it is transformed as the second rank tensor (see the next lecture): ∂xµ ∂xν g (x¯) = η . (4) αβ µν ∂x¯α ∂x¯β But it is important to note that, as the consequence of this transformation of the metric, the interval does not change under such a coordinate transformation: ds2 = η dxµdxν = g (x¯)dx¯αdx¯β. (5) µν αβ In fact, it is natural to expect that if one has a space–time, then the distance between any its two– points does not depend on the way one draws the coordinate lattice on it. (The lattice is obtained by drawing three–dimensional hypersurfaces of constant coordinates x¯µ for each µ = 0,...,3 with 5 fixed lattice spacing in every direction.) Also it is natural to expect that the laws of physics should not depend on the choice of the coordinates in the space–time. This axiom is referred to as general covariance and is the basis of the General Theory of Relativity. 2. Lorentz transformations in Minkowski space–time have the meaning of transitions between inertial reference systems. Then what is a meaning of an arbitrary coordinate transformation? To answer this question let us start with the transition into a non–inertial reference system in Minkowski space–time. The simplest non–inertial motion is the one with the constant linear acceleration. Three– acceleration cannot be constant in a relativistic situation. Hence, we have to consider a motion of a particle with a constant four–acceleration, wµw = a2 = const, where wµ = d2zµ(s)/ds2 and µ − zµ(s) = z0(s),~z(s) is the world–line of the particle parametrized by the proper time1 s. Let us choose the spatial reference system such that the acceleration will be directed along the first axis. (cid:2) (cid:3) Then we have that: d2z0 2 d2z1 2 = a2. (6) ds2 − ds2 − (cid:18) (cid:19) (cid:18) (cid:19) Thus, the components of the four–acceleration compose a hyperbola. Hence, the standard solution of this equation is as follows: 1 1 z0(s)= sinh(as), z1(s) = [cosh(as) 1]. (7) a a − The integration constant in z1(s) is chosen for the future convenience. Thus, one has the following relation between z1 and z0 themselves: 2 1 1 z1+ z0 2 = . (8) a − a2 (cid:18) (cid:19) (cid:0) (cid:1) I.e. the world–line of a particle which moves with constant eternal acceleration is just a hyperbola (see fig. (1)). Note that the three–dimensional part of the acceleration is always along the positive direction of the x axis: d2z1/(dz0)2 = a 1 tanh2(as) > 0. Hence, for the negative s the cosh(as) − particleisactually decelerating, whileforthepositivesitaccelerates. (Notethats = 0corresponds (cid:2) (cid:3) to t = 0, as is shown on the fig. (1).) The asymptotes of the hyperbola are the light–like lines, z1 = z0 1/a. Hence, even if one moves with eternal constant acceleration, he cannot exceed ± − the speed of light, because the motion with the speed of light would be along one of the above asymptotes of the hyperbola. Moreover, for small az0 we find from (8) that: z1 a z0 2/2. In fact, for small proper times, ≈ as 1, we have that z0 s, v dz1/ds az0 1 and obtain the standard nonrelativistic (cid:0) (cid:1) ≪ ≈ ≈ ≈ ≪ 1 Note that four–velocity, u = dz (s)/ds, obeys the relation u uµ = 1, i.e. it is time–like vector. Differentiating µ µ µ both sides of this equality we obtain that wµu = 0. Hence, w should be space–like. As the result wµw = µ µ µ −a2<0. 6 Figure 1: In this picture and also in the other pictures of this lecture we show only slices of fixed y and z. acceleration, which, however, gets modified according to (8) once the particle reaches high enough velocities. It is important to stress at this point that eternal constant acceleration is physically im- possibleduetotheinfiniteenergyconsumption. I.e. herewearejustdiscussingsomemathematical abstraction, which, however, is helpful to clarify some important issues. These observations will allow us to find the appropriate coordinate system for accelerated ob- servers. The motion with a constant eternal acceleration is homogeneous, i.e. accelerated observer cannot distinguish any moment of its proper time from any other. Hence, it is natural to expect that there should be static (invariant under both time–translations and time–reversal transforma- tions) reference frame seen by accelerated observers. Inspired by (7), we propose the following coordinate change: t = ρ sinhτ, x = ρ coshτ, ρ 0, ≥ y′ = y, and z′ = z. (9) Please note that these coordinates cover only quarter of the entire Minkowski space–time. Namely — the right quadrant. In fact, since coshτ sinhτ , we have that x t . It is not hard to guess ≥ | | ≥ | | the coordinates which will cover the left quadrant. For that one has to choose ρ 0 in (9). ≤ Under such a coordinate change we have: dt = dρ sinhτ +ρdτ coshτ, dx = dρ coshτ +ρdτ sinhτ. (10) Then dt2 dx2 = ρ2dτ2 dρ2 and: − − 7 t=const t ρ=const x=const x O τ=const Figure 2: ds2 = dt2 dx2 dy2 dz2 = ρ2dτ2 dρ2 dy2 dz2 (11) − − − − − − is the so called Rindler metric. It is not constant, g = Diag(ρ2, 1, 1, 1), but is time– µν || || − − − independent and diagonal (i.e. static), as we have expected. In this metric the levels of the constant coordinate time τ are straight lines t/x = tanhτ in the x t plane (or three–dimensional flat planes in the entire Minkowski space). The levels of the − constant ρ are the hyperbolas x2 t2 = ρ2 in the x t plane. The latter ones correspond to world– − − lines of observers which are moving with constant four–accelerations equal to 1/ρ on a slice of fixed y and z. The hyperbolas degenerate to light-like lines x = t as ρ 0. These are asymptotes of ± → the hyperbolas for all ρ. As one takes ρ closer and closer to zero the corresponding hyperbolas are closer and closer to their asymptotes. Note also that τ = corresponds to x = t and τ = + −∞ − ∞ — to x = t. As a result we get a change of the coordinate lattice, which is depicted on the fig. (2). 3. Theimportantfeatureof theRindler’smetric (11)isthat itdegenerates atρ= 0. Thisis the so called coordinate singularity. It is similar to the singularity of the polar coordinates dr2+r2dϕ2 at r = 0. The space–time itself is regular at ρ = 0. It is just flat Minkowski space–time at the light–like lines x = t. Another important feature of the Rindler’s metric is that the speed of light ± is coordinate dependent: dρ If ds2 = 0, then = ρ, when dy = dz = 0. (12) dτ (cid:12) (cid:12) (cid:12) (cid:12) Atthesametime,inthepropercoordinates(cid:12)the(cid:12)speedoflightisjustequaltoonedρ/ds = dρ/ρdτ = (cid:12) (cid:12) 1. Furthermore, as ρ 0 the speed of light, dρ/dτ, becomes zero. This phenomenon is related → to the fact that if an observer starts an eternal acceleration with a = 1/ρ, say at the moment of time t = 0 = τ, then there is a region in Minkowski space–time from which light rays cannot reach him. In fact, as shown on the fig. (2) if a light ray was emitted from a point like O it is parallel to the asymptote x = t of the world–line of the observer in question. As the result, the light ray never intersects with hyperbolas, i.e. never catches up with eternally accelerating observer. These are the reasons why one cannot extend the Rindler metric beyond the light–like 8 Figure 3: lines x = t. The three–dimensional surface x = t of the entire Minkowski space–time is referred ± to as the future event horizon of the Rindler’s observers (those which are staying at the constant ρ positions throughout their entire life time). At the same time x = t is the past event horizon − of the Rindler’s observers. Note that if an observer accelerates duringa finite period of time, then, after the moment when the acceleration is switched off his world–line will be a straight line, which is tangential to the corresponding hyperbola. (I.e. the observer will continue moving with the gained velocity.) The angle this tangential line has with the Minkowskian time axis is less that 45o. Hence, sooner or later the light ray emitted from a point like O will actually reach such an observer. I.e. this observer does not have an event horizon. Another interesting phenomenon which is seen by the Rindler’s observers is shown on the fig. (3). A stationary object, x = const, in Minkowski space–time cannot cross the event horizon of the Rindler’s observers during any finite period of the coordinate time τ, which according to (7) is linearly related to the proper times of the eternally accelerating observers. This object just slows down and only asymptotically approaches the horizon. Note that, as ρ 0 a fixed finite portion → of the proper time, ds = ρdτ, corresponds to increasing portions of the coordinate time, dτ. Recall also that τ = corresponds to x = t and τ = + — to x = t. −∞ − ∞ All these peculiarities of the Rindler metric is the price one has to pay for the consideration of the physically impossible eternal acceleration. However, if one were transferring to a reference system of observers which are moving with accelerations only during finite proper times, then he would obtain a non–stationary metric due to the inhomogeneity of such a motion. 9 Themainlessontodrawfromtheseobservationsisasfollows. Thephysicalmeaningofageneral coordinate transformation that mixes spatial and time coordinates is a transition to another, not necessary inertial, reference system. In this case curves corresponding to fixed spatial coordinates (e.g. dρ = dy = dz = 0) are world–lines of (non–)inertial observers. As the result, the essence of the general covariance is that physical laws, in a suitable form/formualtion, should not depend on the choice of observers. 4. If even in flat space–time one can choose curvilinear coordinates and obtain a non–trivial metric tensor g (x), then how can one distinguish flat space–time from the curved one? Further- µν more, since we understood the physics behind the curvilinear coordinates in flat space–time, then it is also natural to ask: What is the physics behind curved space–times? To start answering these questions in the following lectures let us solve here a simple problem. Namely, let us consider a free particle moving in a space–time with a metric ds2 = g (x)dxµdxν. Let us find its world–line via the minimal action principle. If one considers a µν world–line zµ(τ) parametrized by a parameter τ (that, e.g., could be either a coordinate time or the proper one), then the simplest invariant characteristic that one can associate to the world–line is its length. Hence, the natural action for the free particle should be proportional to the length of its world–line. The reason why we are looking for an invariant action is that we expect the corresponding equations of motion to be covariant (i.e. to have the same form in all coordinate systems), according to the above formulated principle of general covariance. If one approximates the world–line by a broken line consisting of a chain of small intervals, then its length can be approximated by the expression as follows: N L = g [z ] [z z ]µ [z z ]ν, (13) µν i i+1 i i+1 i − − Xi=1q which follows from the definition of the metric. In the limit N and z z 0 we obtain i+1 i → ∞ | − | → an integral instead of the sum. As a result, the action should be as follows: 2 τ2 S = m ds = m dτ g [z(τ)] z˙µz˙ν. (14) µν − − Z1 Zτ1 q Here z˙ = dz/dτ. The coefficient of the proportionality between the action, S, and the length, L, is minus the mass, m, of the particle. This coefficient follows from the complementarity — from the fact that when g (x) = η we have to obtain the standard action for the relativistic particle µν µν in the Special Theory of Relativity. Note that the action (14) is invariant under the coordinate transformations, zµ z¯µ(z), and → also under the reparametrizations, τ f(τ), if one respects the ordering of points along the → world–line, df/dτ 0. In fact, then: ≥ dzµ dzν dzµ dzν dτ g = df g . µν µν r dτ dτ s df df 10 Let us find equations of motion that follow from the minimal action principle applied to (14). The first variation of the action is: τ2 δ[g (z)z˙µz˙ν] µν δS = m dτ = − Zτ1 2√z˙2 τ2 dτ √z˙2 = m δg (z) z˙µz˙ν +g (z)δz˙µz˙ν +g (z)z˙µδz˙ν . (15) µν µν µν − Zτ1 2√z˙2√z˙2 h i Here we denote z˙2 g (z)z˙αz˙β. Using the fact that √z˙2dτ = g dzµdzν = ds we can change αβ µν ≡ in(15)theparametrization fromτ tothepropertimes. After thatweintegrate by partsinthelast p two terms in the last line of (15). This way we get rid from the differential operator acting on δz: δz˙ = d δz. Then, using the Dirichlet boundary conditions, i.e. assuming that δz(s ) = δz(s ) = 0, ds 1 2 we arrive at the following expression for the first variation of S: s2 ds d d δS = m ∂ g (z)δzαz˙µz˙ν g (z) z˙ν δzµ g (z) z˙µ δzν = α µν µν µν − 2 − ds − ds Zs1 (cid:26) h i h i (cid:27) s2 ds = m ∂ g δzαz˙µz˙ν ∂ g z˙αz˙νδzµ ∂ g z˙αz˙µδzν 2g z¨µδzν = α µν α µν α µν µν − 2 − − − Zs1 (cid:26) (cid:27) s2 1 = m ds ∂ g ∂ g ∂ g z˙µz˙ν g z¨µ δzα. (16) α µν µ αν ν µα µα − 2 − − − Zs1 (cid:20) (cid:18) (cid:19) (cid:21) In these expressions z˙ = dz/ds and also we have used that g z¨µδzν = g δzµz¨ν because g = µν µν µν g . Taking into account that according to the minimal action principle δS should beequal to zero νµ for any δzα, we arrive at the following relation: z¨µ+Γµ (z)z˙ν z˙α = 0, (17) να which is referred to as the geodesic equation. Here 1 Γµ = gµβ ∂ g +∂ g ∂ g (18) να 2 ν αβ α βν − β να (cid:18) (cid:19) are the so called Christoffel symbols and gµβ is the inverse metric tensor, gµβg = δµ. βν ν Problems: Show that the metric ds2 = (1+ah)2 dτ2 dh2 dy2 dz2 (homogeneous gravitational • − − − field) also covers the Rindler space–time. Find the coordinate change from this metric to the one used in the lecture. Find the coordinates which cover the lower and upper quadrants (complementary to those • which are covered by Rindler’s coordinates) of the Minkowski space–time.