Physik-Institut der Universität Zürich. in conjunction with ETH Zürich. General Relativity. Autumn semester Prof.

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1 Physik-Institut der Universität Zürich in conjunction with ETH Zürich General Relativity Autumn semester 2016 Prof. Philippe Jetzer Original version by Arnaud Borde Revision: Antoine Klein, Raymond Angélil, Cédric Huwyler, Simone Balmelli, Yannick Boetzel Last revision of this version: January 4, 2017

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3 Sources of inspiration for this course include S. Carroll, Spacetime and Geometry, Pearson, 2003 S. Weinberg, Gravitation and Cosmology, Wiley, 1972 N. Straumann, General Relativity with applications to Astrophysics, Springer Verlag, 2004 C. Misner, K. Thorne and J. Wheeler, Gravitation, Freeman, 1973 R. Wald, General Relativity, Chicago University Press, 1984 T. Fliessbach, Allgemeine Relativitätstheorie, Spektrum Verlag, 1995 B. Schutz, A first course in General Relativity, Cambridge, 1985 R. Sachs and H. Wu, General Relativity for mathematicians, Springer Verlag, 1977 J. Hartle, Gravity, An introduction to Einstein s General Relativity, Addison Wesley, 2002 H. Stephani, General Relativity, Cambridge University Press, 1990, and M. Maggiore, Gravitational Waves: Volume 1: Theory and Experiments, Oxford University Press, A. Zee, Einstein Gravity in a Nutshell, Princeton University Press, 2013 As well as the lecture notes of Sean Carroll ( Matthias Blau ( Gian Michele Graf 2

4 CONTENTS Contents I Introduction 6 1 Newton s theory of gravitation 6 2 Goals of general relativity 7 II Special Relativity 9 3 Lorentz transformations Galilean invariance Lorentz transformations Proper time Relativistic mechanics Equations of motion Energy and momentum Equivalence between mass and energy Tensors in Minkowski space 14 6 Electrodynamics 17 7 Accelerated reference systems in special relativity 18 III Towards General Relativity 20 8 The equivalence principle About the masses About the forces Riemann space Physics in a gravitational field Equations of motion Christoffel symbols Newtonian limit Time dilation Proper time Redshift Photon in a gravitational field

5 CONTENTS 11 Geometrical considerations Curvature of space IV Differential Geometry Differentiable manifolds Tangent vectors and tangent spaces The tangent map Vector and tensor fields Flows and generating vector fields Lie derivative Differential forms Exterior derivative of a differential form Stokes theorem The inner product of a p-form Affine connections: Covariant derivative of a vector field Parallel transport along a curve Round trips by parallel transport Covariant derivatives of tensor fields Local coordinate expressions for covariant derivative Curvature and torsion of an affine connection, Bianchi identities Bianchi identities for the special case of zero torsion Riemannian connections 60 V General Relativity Physical laws with gravitation Mechanics Electrodynamics Energy-momentum tensor Einstein s field equations The cosmological constant The Einstein-Hilbert action 68 4

6 CONTENTS 22 Static isotropic metric Form of the metric Robertson expansion Christoffel symbols and Ricci tensor for the standard form Schwarzschild metric General equations of motion Trajectory VI Applications of General Relativity Light deflection Perihelion precession Quadrupole moment of the Sun Lie derivative of the metric and Killing vectors Maximally symmetric spaces Friedmann equations 91 5

7 1 NEWTON S THEORY OF GRAVITATION Part I Introduction 1 Newton s theory of gravitation In his book Principia in 1687, Isaac Newton laid the foundations of classical mechanics and made a first step in unifying the laws of physics. The trajectories of N point masses, attracted to each other via gravity, are the solutions to the equation of motion m i d 2 r i dt 2 = G N j=1 j i m i m j ( r i r j ) r i r j 3 i = 1... N, (1.1) with r i (t) being the position of point mass m i at time t. Newton s constant of gravitation is determined experimentally to be G = ± m 3 kg 1 s 2 (1.2) The scalar gravitational potential φ( r) is given by φ( r) = G N j=1 m j r r j = G d 3 r ρ( r ) r r, (1.3) where it has been assumed that the mass is smeared out in a small volume d 3 r. The mass is given by dm = ρ( r )d 3 r, ρ( r ) being the mass density. For point-like particles we have ρ( r ) m j δ (3) ( r r j ). The gradient of the gravitational potential can then be used to produce the equation of motion: m d2 r = m φ( r). (1.4) dt2 According to (1.3), the field φ( r) is determined through the mass of the other particles. The corresponding field equation derived from (1.3) is given by 1 φ( r) = 4πGρ( r) (1.5) The so called Poisson equation (1.5) is a linear partial differential equation of 2 nd order. The source of the field is the mass density. Equations (1.4) and (1.5) show the same structure as the field equation of electrostatics: and the non-relativistic equation of motion for charged particles φ e ( r) = 4πρ e ( r), (1.6) m d2 r dt 2 = q φ e( r). (1.7) Here, ρ e is the charge density, φ e is the electrostatic potential and q represents the charge which acts as coupling constant in (1.7). m and q are independent characteristics of the considered body. In 1 1 r r = 4πδ(3) ( r r ) 6

8 2 GOALS OF GENERAL RELATIVITY analogy one could consider the gravitational mass (right side) as a charge, not to be confused with the inertial mass (left side). Experimentally, one finds to very high accuracy ( ) that they are the same. As a consequence, all bodies fall at a rate independent of their mass (Galileo Galilei). This appears to be just a chance in Newton s theory, whereas in GR it will be an important starting point. For many applications, (1.4) and (1.5) are good enough. It must however be clear that these equations cannot be always valid. In particular (1.5) implies an instantaneous action at a distance, what is in contradiction with the predictions of special relativity. We therefore have to suspect that Newton s theory of gravitation is only a special case of a more general theory. 2 Goals of general relativity In order to get rid of instantaneous interactions, we can try to perform a relativistic generalization of Newton s theory (eqs. (1.4) and (1.5)), similar to the transition from electrostatics (eqs. (1.6) and (1.7)) to electrodynamics. The Laplace operator is completed such as to get the D Alembert operator (wave equation) = 1 c 2 2 t 2 (2.1) Changes in ρ e travel with the speed of light to another point in space. If we consider inertial coordinate frames in relative motion to each other it is clear that the charge density has to be related to a current density. In other words, charge density and current density transform into each other. In electrodynamics we use the current density j α (α = 0, 1, 2, 3): ρ e (ρ e c, ρ e v i ) = j α, (2.2) where the v i are the cartesian components of the velocity v (i = 1, 2, 3). An analogous generalization can be performed for the potential: The relativistic field equation is then φ e (φ e, A i ) = A α. (2.3) φ e = 4πρ e A α = 4π c jα. (2.4) In the static case, the 0-component reduces to the equation on the left. Equation (2.4) is equivalent to Maxwell s equations (in addition one has to choose a suitable gauge condition). Since electrostatics and Newton s theory have the same mathematical structure, one may want to generalize it the same way. So in (1.5) one could introduce the change. Similarly one generalizes the mass density. But there are differences with electrodynamics. The first difference is that the charge q of a particle is independent on how the particle moves; this is not the case for the mass: m = 1 m0. v2 c 2 As an example, consider a hydrogen atom with a proton (rest mass m p, charge +e) and an electron (rest mass m e, charge e). Both have a finite velocity within the atom. The total charge of the atom 7

9 2 GOALS OF GENERAL RELATIVITY is q = q e + q p = 0, but for the total mass we get m H m p + m e (binding energy). Formally this means that charge is a Lorentz scalar (does not depend on the frame in which the measurement is performed). Therefore we can assign a charge to an elementary particle, and not only a charge at rest, whereas for the mass we must specify the rest mass. Since charge is a Lorentz scalar, the charge density (ρ e = δq δv ) transforms like the 0-component of a Lorentz vector ( 1 δv gets a factor γ = 1 due to length contraction). The mass density 1 v 2 /c 2 (ρ = δm δv ) transforms instead like the 00-component of a Lorentz tensor, which we denote as the energy-momentum tensor T αβ. This follows from the fact that the energy (mass is energy E = mc 2 ) is the 0-component of a 4-vector (energy-momentum vector p α ) and transforms as such. Thus, instead of (2.2), we shall have ρ ( ρc 2 ρcv i ) ρcv i T αβ i, j = 1, 2, 3 (2.5) ρv i v j This implies that we have to generalize the gravitational potential φ to a quantity depending on 2 indices which we shall call the metric tensor g αβ. Hence we get φ = 4πGρ g αβ GT αβ. (2.6) In GR one finds (2.6) for a weak gravitational field (linearized case), e.g. used for the description of gravitational waves. Due to the equivalence between mass and energy, the energy carried by the gravitational field is also mass and thus also a source of the gravitational field itself. This leads to non-linearities. One can note that photons do not have a charge and thus Maxwell s equations can be linear. To summarize: 1. GR is the relativistic generalization of Newton s theory. Several similarities between GR and electrodynamics exist. 2. GR requires tensorial equations (rather than vectorial as in electrodynamics). 3. There are non-linearities which will lead to non-linear field equations. 8

10 3 LORENTZ TRANSFORMATIONS Part II Special Relativity 3 Lorentz transformations A reference system with a well defined choice of coordinates is called a coordinate system. Inertial reference systems (IS) are (from a practical point of view) systems which move with constant speed with respect to distant (thus fixed) stars in the sky. Newton s equations of motion are valid in IS. Non- IS are reference systems which are accelerated with respect to an IS. In this chapter we will establish how to transform coordinates between different inertial systems. 3.1 Galilean invariance Galilei stated that all IS are equivalent, i.e. all physical laws are valid in any IS: the physical laws are covariant under transformations from an IS to another IS. Covariant means here form invariant. The equations should have the same form in all IS. With the coordinates x i (i = 1, 2, 3) and t, an event in an IS can be defined. In another IS, the same event has different coordinates x i and t. A general Galilean transformation can then be written as: x i = α i kx k + v i t + a i, (3.1) t = t + τ, (3.2) where x i, v i and a i are cartesian components of vectors v = v i e i where e i is a unit vector we use the summation rule over repeated indices: α i kx k = k α i kx k latin indices run on 1,2,3 greek indices run on 0,1,2,3 v is the relative velocity between IS and IS a is a constant vector (translation) α i k is the relative rotation of coordinates systems, α = (α i k) is defined by α i n(α T ) n k = δ i k or αα T = I i.e. α 1 = α T (3.3) 9

11 3 LORENTZ TRANSFORMATIONS The condition αα T = I ensures that the line element ds 2 = dx 2 + dy 2 + dz 2 (3.4) remains invariant. α can be defined by giving 3 Euler angles. Eqs. (3.1) and (3.2) define a 10 (a = 3, v = 3, τ = 1 and α = 3) parametric group of transformations, the so-called Galilean group. The laws of mechanics are left invariant under transformations (3.1) and (3.2). But Maxwell s equations are not invariant under Galilean transformations, since they contain the speed of light c. This led Einstein to formulate a new relativity principle (special relativity, SR): All physical laws, including Maxwell s equations, are valid in any inertial system. This leads us to Lorentz transformations (instead of Galilean), thus the law of mechanics have to be modified. 3.2 Lorentz transformations We start by introducing 4-dimensional vectors, glueing time and space together to a spacetime. The Minkowski coordinates are defined by x 0 = ct, x 1 = x, x 2 = y, x 3 = z. (3.5) x α is a vector in a 4-dimension space (or 4-vector). An event is given by x α in an IS and by x α in an IS. Homogeneity of space and time imply that the transformation from x α to x α has to be linear: x α = Λ α βx β + a α, (3.6) where a α is a space and time translation. The relative rotations and boosts are described by the 4 4 matrix Λ. Linear means in this context that the coefficients Λ α β and a α do not depend on x α. In order to preserve the speed of light appearing in Maxwell s equations as a constant, the Λ α β have to be such that the square of the line element ds 2 = η αβ dx α dx β = c 2 dt 2 d r 2 (3.7) remains unchanged, with the Minkowski metric η αβ = (3.8) Because of ds 2 = ds 2 c 2 dτ 2 = c 2 dτ 2, the proper time is an invariant under Lorentz transformations. Indeed for light dτ 2 = dt 2 dx2 +dy 2 +dz 2 c = 0. Thus, c 2 = d x 2 2 dt and c = d x dt. Applying a Lorentz. transformation results in c = d x dt This has the important consequence that the speed of light c is the same in all coordinate systems (what we intended by the definition of (3.7)). 10

12 3 LORENTZ TRANSFORMATIONS A 4-dimensionial space together with this metric is called a Minkowski space. Inserting (3.6) into the invariant condition ds 2 = ds 2 gives ds 2 = η αβ dx α dx β = η αβ Λ α γdx γ Λ β δdx δ = η γδ dx γ dx δ = ds 2. (3.9) Then we get Λ α γλ β δη αβ = η γδ or Λ T ηλ = η. (3.10) Rotations are special subcases incorporated in Λ: x α = Λ α βx β with Λ i k = α i k, and Λ 0 0 = 1, Λ i 0 = Λ 0 i = 0. The entire group of Lorentz transformations (LT) is the so called Poincaré group (and has 10 parameters). The case a α 0 corresponds to the Poincaré group or inhomogeneous Lorentz group, while the subcase a α = 0 can be described by the homogeneous Lorentz group. Translations and rotations are subgroups of Galilean and Lorentz groups. Consider now a Lorentz boost in the direction of the x-axis: x 2 = x 2, x 3 = x 3. v denotes the relative velocity difference between IS and the boosted IS. Then Λ 0 0 Λ Λ α β = Λ 1 0 Λ (3.11) Evaluating eq. (3.10): (γ, δ) = (0, 0) (Λ 0 0) 2 (Λ 1 0) 2 = 1 (3.12a) = (1, 1) (Λ 0 1) 2 (Λ 1 1) 2 = 1 (3.12b) = (0, 1) or (1, 0) Λ 0 0Λ 0 1 Λ 1 0Λ 1 1 = 0 (3.12c) The solution to this system is ( ) ( Λ 0 0 Λ 0 1 cosh Ψ = Λ 1 0 Λ 1 1 sinh Ψ ) sinh Ψ cosh Ψ. (3.13) For the origin of IS we have x 1 = 0 = Λ 1 0ct + Λ 1 1vt. This way we find and as a function of velocity: tanh Ψ = Λ1 0 Λ 0 = v 0 c, (3.14) Λ 0 0 = Λ = γ =, 1 v2 c 2 (3.15a) Λ 0 1 = Λ 1 0 = v/c. 1 v2 c 2 (3.15b) 11

13 3 LORENTZ TRANSFORMATIONS A Lorentz transformation (called a boost) along the x-axis can then be written explicitly as x = y = y, z = z, x vt, 1 v2 c 2 (3.16a) (3.16b) (3.16c) ct = ct x v c, 1 v2 c 2 (3.16d) which is valid only for v < c. For v c, (3.16) recovers the special (no rotation) Galilean transformation x = x vt, y = y, z = z and t = t. The parameter Ψ = arctanh v c (3.17) is called the rapidity. From this we find for the addition of parallel velocities: 3.3 Proper time Ψ = Ψ 1 + Ψ 2 v = v 1 + v v1v2 c 2 (3.18) The time coordinate t in IS is the time shown by clocks at rest in IS. We next determine the proper time τ shown by a clock which moves with velocity v(t). Consider a given moment t 0 an IS, which moves with respect to IS with a constant velocity v 0 (t 0 ). During an infinitesimal time interval dt the clock can be considered at rest in IS, thus: dτ = dt = 1 v2 0 dt. c2 (3.19) Indeed (3.16) with x = v 0 t gives t = t(1 v2 0 /c2 ) = t 1 v2 0 c 2 and thus (3.19). 1 v2 0 c 2 At the next time t 0 + dt, we consider an IS with velocity v 0 = v(t 0 + dt) and so on. Summing up all infinitesimal proper times dτ gives the proper time interval: τ = t 2 t 1 dt 1 v2 (t) c 2 (3.20) This is the time interval measured by an observer moving at a speed v (t) between t 1 and t 2 (as given by a clock at rest in IS). This effect is called time dilation. 12

14 4 RELATIVISTIC MECHANICS 4 Relativistic mechanics Let us now perform the relativistic generalization of Newton s equation of motion for a point particle. 4.1 Equations of motion The velocity v can be generalized to a 4-velocity vector u α : v i = dxi dt uα = dxα dτ Since dτ = ds c, dτ is invariant. With dx α = Λ α β dxβ it follows that u α transforms like dx α : (4.1) u α = Λ α βu β (4.2) All quantities which transforms this way are Lorentz vectors or form-vectors. The generalized equation of motion is then Both duα dτ m duα dτ = f α. (4.3) and f α are Lorentz vectors, therefore, (4.3) is a Lorentz vector equation: if we perform a Lorentz transformation, we get m du α dτ and for v c it reduces to Newton s equations. (left hand side becomes m ( 0, d v side ( f 0, f ) = ( 0, K LT: f α = Λ α βf β. For example v = v e 1 with γ = = f α. Eq. (4.3) is covariant under Lorentz transformations ) ) dt and the right hand ). The Minkowski force f α is determined in any IS through a corresponding ( ) 1/2, 1 v2 c leads to f 0 = γvk1 2 c, f 1 = γk 1, f 2 = K 2 and f 3 = K 3. For a general direction of velocity ( v) we get: f 0 = γ 4.2 Energy and momentum The 4-momentum p α = mu α = m dxα dτ This yields the relativistic v K, f = K c v K + (γ 1) v v 2. (4.4) is a Lorentz vector. With (3.19) we get p α = mc mv i, = 1 v2 c 1 v2 2 c 2 ( ) E c, p. (4.5) energy : E = mc2 = γmc 2 (4.6a) 1 v2 c 2 momentum : p = m v 1 v2 c 2 = γm v v c p = m v. (4.6b) With (4.4), the 0-component of (4.3) becomes (in the case v c) de dt = v. K }{{} power given. (4.7) to the particle 13

15 5 TENSORS IN MINKOWSKI SPACE This justifies to call the quantity E = γmc 2 an energy. From ds 2 = c 2 dτ 2 = η αβ dx α dx β it follows η αβ p α p β = m 2 c 2 and thus E 2 = m 2 c 4 + c 2 p 2, (4.8) the relativistic energy-momentum relation. The limit cases are E = mc 2 + p2 m 2 c 4 + c 2 p 2 2m v c or p mc2 (4.9) cp v c or p mc 2 with p = p. For particles with no rest mass (photons): E = cp (exact relation). 4.3 Equivalence between mass and energy One can divide the energy into the rest energy E 0 = mc 2 (4.10) and the kinetic energy E kin = E E 0 = E mc 2. The quantities defined in (4.6) are conserved when more particles are involved. Due to the equivalence between energy and mass, the mass or the mass density becomes a source of the gravitational field. 5 Tensors in Minkowski space Let us discuss the transformation properties of physical quantities under a Lorentz transformation. We have already seen how a 4-vector is transformed: V α V α = Λ α βv β. (5.1) This is a so-called contravariant 4-vector (indices are up). The coordinate system transforms according to X α X α = Λ α βx β. A covariant 4-vector is defined through V α = η αβ V β. (5.2) Let us now define the matrix η αβ as the inverse matrix to η αβ : η αβ η βγ = δγ α. (5.3) In our case η αβ = η αβ = (5.4) With (5.3) we can express (5.2) equally as V α = η αβ V β. (5.5) 14

16 5 TENSORS IN MINKOWSKI SPACE The transformation of a covariant vector is then given by V α = η αβ V β = η αβ Λ β γv γ = η αβ Λ β γη γδ V δ = Λ δ αv δ, (5.6) with Λ δ α = η αβ Λ β γη γδ (5.7) (one can use Λ α β instead of Λ β α but one should be very careful in writings since Λ αβ Λ β α). Thanks to (3.10), we find Λ γ αλ α β = η αδ η γɛ Λ δ ɛλ α β = η γɛ η ɛβ = δ γ β (5.8) And similarly, we get Λ β α Λ α γ = δ β γ. In matrix notation, we have Λ Λ = ΛΛ = I and thus Λ = Λ 1. To summarize the transformations of 4-vectors: A contravariant vector transforms with Λ A covariant vector transforms with Λ 1 = Λ The scalar product of two vectors V α and U β is defined by V α U α = V α U α = η αβ V α U β = η αβ V α U β (5.9) and is invariant under Lorentz transformations: V α U α = Λ α Λ β δ α V β U δ = V β U β. }{{} δ δ β The operator Λ β α x transforms like a covariant vector: α x α = xβ x α x β. Since xβ x α = Λ β α x α =. We will now use the notations x β α x (covariant vector) and α α x α (contravariant vector). The D Alembert operator can be written as = α α = η αβ α β = 1 2 c and is a 2 Lorentz scalar. t 2 A quantity is a rank r contravariant tensor if its components transform like the coordinates x α : T α1...αr = Λ α1 β 1... Λ αr β r T β1...βr (5.10) Tensors of rank 0 are scalars, tensors of rank 1 are vectors. For mixed tensors we have for example: T α βγ = Λ α Λ δ ɛ Λ β µ γt δ ɛµ The following operations can be used to form new tensors: 1. Linear combinations of tensors with the same upper and lower indices: T α β = ar α β + bs α β 2. Direct products of tensors: T αβ V γ (works with mixed indices) 3. Contractions of tensors: T αβ β or T αβ V β (lowers a tensor by 2 in rank) 4. Differentiation of a tensor field: α T βα (the derivative α of any tensor is a tensor with one additional lower index α: α T βγ R βγ α ) 5. Going from a covariant to a contravariant component of a tensor is defined like in (5.2) and (5.5) (lowering and raising indices with η αβ, η αβ ). 15

17 5 TENSORS IN MINKOWSKI SPACE One must be aware that the order of the upper and lower indices is important, Λ α β is not a tensor. η can be considered as a tensor: η = η αβ = η αβ is the Minkowski tensor. η αβ = Λ γ α Λ δ βη γδ (3.10) = Λ γ α Λ δ βλ µ γλ ν δη µν (5.8) = η αβ η appears in the line element (ds 2 = η αβ dx α dx β ) and is thus the metric tensor in Minkowski space. We also have η α β = η αγ η γβ = δ α β = η β α, and thus the Kronecker symbol is also a tensor. We define the totally antisymmetric tensor or (Levi-Civita tensor) as +1 (α, β, γ, δ) is an even permutation of (0, 1, 2, 3) ɛ αβγδ = 1 (α, β, γ, δ) is an odd permutation of (0, 1, 2, 3) 0 otherwise (5.11) Without proof we have: (det (Λ) = 1) ɛ αβγδ = ɛ αβγδ, ɛ αβγδ = η αα η ββ η γγ η δδ ɛ α β γ δ = ɛ αβγδ. The functions S(x), V α (x), T αβ... with x = (x 0, x 1, x 2, x 3 ) are a scalar field, a vector field, or a tensor field... respectively if: S (x ) = S(x) V α (x ) = Λ α βv β (x) T αβ (x ) = Λ α δ Λ β γt δγ (x). Also the argument has to be transformed, thus x has to be understood as x α = Λ α βx β. 16

18 6 ELECTRODYNAMICS 6 Electrodynamics Maxwell s equations relate the fields E( r, t), B( r, t), the charge density ρ e ( r, t) and the current density j( r, t): The continuity equation div E = 4πρ c inhomogeneous rot B = 4π c j + 1 c E t div B = 0 homogeneous rot E = 1 c B t (6.1) div j + ρ c = 0 α j α = 0 (6.2) with j α = (cρ e, j) follows from the conservation of charge, which for an isolated system implies t j 0 d 3 r = 0. α j α is a Lorentz scalar. We can define the field strength tensor which is given by the antisymmetric matrix 0 E x E y E z F αβ = E x 0 B z B y E y B z 0 B x. (6.3) E z B y B x 0 Using this tensor we can rewrite the inhomogeneous Maxwell equations α F }{{ αβ = 4π } c 4 vector j β }{{}, (6.4) 4 vector and also the homogeneous ones: ɛ αβγδ β F γδ = 0. (6.5) Both equations are covariant under a Lorentz transformation. Eq. (6.5) allows to represent F αβ as a curl of a 4-vector A α : F αβ = α A β β A α. (6.6) We can then reformulate Maxwell s equations for A α = (φ, A i ). From (6.6) it follows that the gauge transformation A α A α + α Θ (6.7) of the 4-vector A α leaves F αβ unchanged, where Θ(x) is an arbitrary scalar field. The Lorenz gauge α A α = 0 leads to the decoupling of the inhomogeneous Maxwell s equation (6.4) to The generalized equation of motion for a particle with charge q is A α = 4π c jα. (6.8) m duα dτ = q c F αβ u β (6.9) 17

19 7 ACCELERATED REFERENCE SYSTEMS IN SPECIAL RELATIVITY The spatial components give the expression of the Lorentz force d p ( dt = q E + v ) c B with p = γm v. The energy-momentum tensor for the electromagnetic field is T αβ em = 1 4π (F αγf γβ + 14 ηαβ F γδ F γδ ) (6.10) ( The 00-component represents the energy density of the field Tem 00 = u em = 1 E 2 8π + B ) 2 and the 0i-components the Poynting vector S ( ) i. i = ctem 0i = E B In terms of these tensors, Maxwell s equations are α T αβ em = 1 c F βγ j γ. T αβ em c 4π is symmetric and conserved: α T αβ em = 0. Setting β = 0 leads to energy conservation whereas α T αk em = 0 leads to conservation of the k th component of the momentum. One should note that α T αβ em = 0 is valid only if there is no external force, otherwise we can write α T αβ em = f β, where f β is the external force density. Such an external force can often be included in the energy-momentum tensor. 7 Accelerated reference systems in special relativity Non inertial systems can be considered in the context of special relativity. However, then the physical laws no longer have their simple covariant form. In e.g. a rotating coordinate system, additional terms will appear in the equations of motion (centrifugal terms, Coriolis force, etc.). Let us look at a coordinate system KS (with coordinates x µ ) which rotates with constant angular speed with respect to an inertial system IS (x α ): x = x cos(ωt ) y sin(ωt ), y = x sin(ωt ) + y cos(ωt ), z = z, (7.1) t = t, and assume that ω 2 (x 2 + y 2 ) c 2. Then we insert (7.1) into the line element ds (in the known IS form): ds 2 = η αβ dx α dx β = c 2 dt 2 dx 2 dy 2 dz 2 = [ c 2 ω 2 (x 2 + y 2 ) ] dt 2 + 2ωy dx dt 2ωx dy dt dx 2 dy 2 dz 2 = g µν dx µ dx ν. (7.2) The resulting line element is more complicated. For arbitrary coordinates x µ, ds 2 is a quadratic form of the coordinate differentials dx µ. Consider a general coordinate transformation from x µ (in IS) to x µ (in KS ): x α x α (x ) = x α (x 0, x 1, x 2, x 3 ), (7.3) 18

20 7 ACCELERATED REFERENCE SYSTEMS IN SPECIAL RELATIVITY then we get for the line element with ds 2 = η αβ dx α dx β = η αβ x α x µ x β x ν dx µ dx ν = g µν (x )dx µ dx ν, (7.4) g µν (x x α x β ) = η αβ x µ. (7.5) x ν The quantity g µν is the metric tensor of the KS system. It is symmetric (g µν = g νµ ) and depends on x. It is called metric because it defines distances between points in coordinate systems. In an accelerated reference system we get inertial forces. In the rotating frame we expect to experience the centrifugal force Z, which can be written in terms of a centrifugal potential φ: This enables us to see that g 00 from (7.2) is φ = ω2 2 (x 2 + y 2 ) and Z = m φ. (7.6) g 00 = 1 + 2φ c 2. (7.7) The centrifugal potential appears in the metric tensor. We will see later that the first derivatives of the metric tensor are related to the forces in the relativistic equations of motion. To get the meaning of t in KS we evaluate (7.2) at a point with dx = dy = dz = 0: dτ = ds clock = g 00 dt = 1 + 2φ c c 2 dt = 1 v2 }{{ c 2 } correspond to clocks time computed in an inertial system dt (7.8) τ represents the time of a clock at rest in KS. In an inertial system we have g µν = η µν and the clock moves with speed v = ωρ (ρ = x 2 + y 2 ). With (7.6) we see that both expressions in (7.8) are the same. The coefficients of the metric tensor g µν (x ) are functions of the coordinates. Such a dependence will also arise when one uses curved coordinates. Consider for example cylindrical coordinates: This results in the line element Here g µν is diagonal: x 0 = ct = x 0, x 1 = ρ, x 2 = θ, x 3 = z. ds 2 = c 2 dt 2 dx 2 dy 2 dz 2 = c 2 dt 2 dρ 2 ρ 2 dθ 2 dz 2 = g µν (x )dx µ dx ν. (7.9) 1 g µν = 1 ρ 2. (7.10) 1 The fact that the metric tensor depends on the coordinates can be either due to the fact that the considered coordinate system is accelerated or that we are using non-cartesian coordinates. 19

21 8 THE EQUIVALENCE PRINCIPLE Part III Towards General Relativity 8 The equivalence principle The principle of equivalence of gravitation and inertia tells us how an arbitrary physical system responds to an external gravitational field (with the help of tensor analysis). The physical basis of general relativity is the equivalence principle as formulated by Einstein: 1. Inertial and gravitational mass are equal 2. Gravitational forces are equivalent to inertial forces 3. In a local inertial frame, we experience the known laws of special relativity without gravitation 8.1 About the masses The inertial mass m t is the quantity appearing in Newton s law F = m t a which acts against acceleration by external forces. In contrast, the gravitational mass m s is the proportionality constant relating the gravitational force between mass points to each other. For a particle moving in a homogeneous gravitational field, we have the equation m t z = m s g, whose solution is z(t) = 1 m s gt 2 (+v 0 t + z 0 ). (8.1) 2 m t Galilei stated that all bodies fall at the same rate in a gravitational field, i.e. m s m t is the same for all bodies. Another experiment is to consider the period T of a pendulum (in the small amplitude approximation): ( ) T 2 2π = m s l m t g, where l is the length of the pendulum. Newton verified that this period is independent on the material of the pendulum to a precision of about Eötvös ( 1890), using torsion balance, got a precision of about Today s precision is about , this is way we can make the assumption m s = m t on safe grounds. Due to the equivalence between energy and mass (E = mc 2 ), all forms of energy contribute to mass, and due to the first point of the equivalence principle, to the inertial and to the gravitational masses. 8.2 About the forces As long as gravitational and inertial masses are equal, then gravitational forces are equivalent to inertial forces: going to a well-chosen accelerated reference frame, one can get rid of the gravitational field. As an example take the equation of motion in the homogeneous gravitational field at Earth s surface: m t d 2 r dt 2 = m s g (8.2) 20

22 8 THE EQUIVALENCE PRINCIPLE This expression is valid for a reference system which is at rest on Earth s surface ( to a good approximation an IS). Then we perform the following transformation to an accelerated KS system: r = r gt 2, t = t, (8.3) and we assume gt c. The origin of KS r = 0 moves in IS with r(t) = 1 2 gt2. Then, inserting (8.3) into (8.2) results in d 2 m t ( r dt ) gt 2 = m s g m t d 2 r dt 2 = (m s m t ) g. (8.4) If m s = m t, the resulting equation in KS is the one of a free moving particle d2 r = 0; the gravitational force vanishes. As another example in a free falling elevator the observer does not feel any gravity. Einstein generalized this finding postulating that (this is the Einstein equivalence principle) in a free falling accelerated reference system all physical processes run as if there is no gravitational field. Notice that the mechanical finding is now expanded to all types of physical processes (at all times and places). Moreover also non-homogeneous gravitational fields are allowed. The equality of inertial and gravitational mass is also called the weak equivalence principle (or universality of the free fall). As an example of a freely falling system, consider a satellite in orbit around Earth (assuming that the laboratory on the satellite is not rotating). Thus the equivalence principle states that in such a system all physical processes run as if there would be no gravitational field. The processes run as in an inertial system: the local IS. However, the local IS is not an inertial system, indeed the laboratory on the satellite is accelerated compared to the reference system of the fixed distant stars. The equivalence principle implies that in a local IS the rules of special relativity apply. For an observer on the satellite laboratory all physical processes follow special relativity and there are neither gravitational nor inertial forces. For an observer on Earth, the laboratory moves in a gravitational field and moreover inertial forces are present, since it is accelerated. The motion of the satellite laboratory, i.e. its free falling trajectory, is such that the gravitational forces and inertial forces just compensate each other (cf (8.4)). The compensation of the forces is exactly valid only for the center of mass of the satellite laboratory. Thus the equivalence principle applies only to a very small or local satellite laboratory ( how small depends on the situation). dt 2 The equivalence principle can also be formulated as follows: At every space-time point in an arbitrary gravitational field, it is possible to choose a locally inertial coordinate system such that, within a sufficiently small region around the point in question, the laws of nature take the same form as in non-accelerated Cartesian coordinate systems in the absence of gravitation. 2 2 Notice the analogy with the axiom Gauss took as a basis of non-euclidean geometry: he assumed that at any point on a curved surface we may erect a locally Cartesian coordinate system in which distances obey the law of Pythagoras. 21

23 8 THE EQUIVALENCE PRINCIPLE The equivalence principle allows us to set up the relativistic laws including gravitation; indeed one can just perform a coordinate transformation to another KS: } special relativity laws without gravitation coordinate transformation { relativistic laws with gravitation The coordinate transformation includes the relative acceleration between the laboratory system and KS which corresponds to the gravitational field. Thus from the equivalence principle we can derive the relativistic laws in a gravitational field. However, it does not fix the field equations for g µν (x) since those equations have no analogue in special relativity. From a geometrical point of view the coordinate dependence of the metric tensor g µν (x) means that space is curved. In this sense the field equations describe the connection between curvature of space and the sources of the gravitational field in a quantitative way. 8.3 Riemann space We denote with ξ α the Minkowski coordinates in the local IS (e.g. the satellite laboratory). From the equivalence principle, the special relativity laws apply. In particular, we have for the line element ds 2 = η αβ dξ α dξ β. (8.5) Going from the local IS to a KS with coordinates x µ is given by a coordinate transformation ξ α = ξ α (x 0, x 1, x 2, x 3 ). Inserting this into (8.5) results in ds 2 = η αβ ξ α x µ ξ β x ν dxµ dx ν = g µν (x)dx µ dx ν, (8.6) ξ α ξ β and thus g µν (x) = η αβ x µ. A space with such a path element of the form (8.6) is called a Riemann xν space. The coordinate transformation (expressed via g µν ) also describes the relative acceleration between KS and the local IS. Since at two different points of the local IS the accelerations are (in general) different, there is no global transformation in the form (8.6) that can be brought to the Minkowski form (8.5). We shall see that g µν are the relativistic gravitational potentials, whereas their derivatives determine the gravitational forces. 22

24 8 THE EQUIVALENCE PRINCIPLE Figure 1: An experimenter and his two stones freely floating somewhere in outer space, i.e. in the absence of forces. Figure 2: Constant acceleration upwards mimics the effect of a gravitational field: experimenter and stones drop to the bottom of the box. 23

25 8 THE EQUIVALENCE PRINCIPLE Figure 3: The effect of a constant gravitational field: indistinguishable for our experimenter from that of a constant acceleration as in figure 2. Figure 5: The experimenter and his stones in a non-uniform gravitational field: the stones will approach each other slightly as they fall to the bottom of the elevator. Figure 4: Free fall in a gravitational field has the same effect as no gravitational field (figure 1): experimenter and stones float. Figure 6: The experimenter and stones freely falling in a non-uniform gravitational field: the experimenter floats, so do the stones, but they move closer together, indicating the presence of some external forces. 24

26 9 PHYSICS IN A GRAVITATIONAL FIELD 9 Physics in a gravitational field 9.1 Equations of motion According to the equivalence principle, in a local IS the laws of special relativity hold. For a mass point on which no forces act we have d 2 ξ α = 0, (9.1) dτ 2 where the proper time τ is defined through ds 2 = η αβ dξ α dξ β = c 2 dτ 2. We can also define the 4-velocity as u α = dξα dτ. Solutions of (9.1) are straight lines ξ α = a α τ + b α. (9.2) Light (or a photon) moves in the local IS on straight lines. However, for photons τ cannot be identified with the proper time since on the light cone ds = cdτ = 0. Thus we denote by λ a parameter of the trajectory of photons: d 2 ξ α = 0. (9.3) dλ2 Let us now consider a global coordinate system KS with x µ and metric g µν (x). At all points x µ, one can locally bring ds 2 into the form ds 2 = η αβ dξ α dξ β. Thus at all points P there exists a transformation ξ α (x) = ξ α (x 0, x 1, x 2, x 3 ) between ξ α and x µ. The transformation varies from point to point. Consider a small region around point P. Inserting the coordinate transformation into the line element, we get We write (9.1) in the form ds 2 = η αβ dξ α dξ β ξ α ξ β = η αβ } x {{ µ x ν } dx µ dx ν. (9.4) g µν(x) metric tensor 0 = d dτ ( ξ α dx µ ) x µ = ξα d 2 x µ dτ x µ dτ ξ α dx µ dx ν x µ x ν dτ dτ, multiply it by xκ ξ and make use of ξα x κ α x µ ξ = δ κ µ. This way we can solve for d2 x µ α dτ and get the following 2 equation of motion in a gravitational field with d 2 x κ dτ 2 = dx µ dx ν Γκ µν dτ dτ, (9.5) Γ κ µν = xκ 2 ξ α ξ α x µ x ν. (9.6) The Γ κ µν are called the Christoffel symbols and represent a pseudo force or fictive gravitational field (like centrifugal or Coriolis forces) that arises whenever one describes inertial motion in non-inertial coordinates. Eq. (9.5) is a second order differential equation for the functions x µ (τ) which describe the trajectory of a particle in KS with g µν (x). Eq. (9.5) can also be written as m duα dτ = f α, u α = dxα dτ. Comparing with (4.3) one infers that the right hand side of (9.5) describes the gravitational forces. Due to (9.4), the velocity dxµ dτ has to satisfy the condition c 2 = g µν dx µ dτ dx ν dτ 25 (for m 0) (9.7)

27 9 PHYSICS IN A GRAVITATIONAL FIELD (assume dτ 0 and m 0). Due to (9.7) only 3 of the 4 components of dxµ dτ are independent (this is also the case for the 4-velocity in special relativity). For photons (m = 0) one finds instead, using (9.3), a completely analogous equation for the trajectory: and since dτ = ds = 0, one has instead of (9.7): 9.2 Christoffel symbols d 2 x κ dλ 2 = dx µ dx ν Γκ µν dλ dλ, (9.8) 0 = g µν dx µ dλ dx ν dλ (for m = 0). The Christoffel symbols can be expressed in terms of the first derivatives of g µν. Consider with (9.4): g µν x λ Using η αβ = η βα this becomes + g λν x µ g µλ x ν =η 2 ξ α ξ β αβ x µ x λ x ν + ξ α 2 ξ β x } µ x {{ ν x λ } 1 + η αβ 2 ξ α ξ β x λ x µ x ν + ξ α 2 ξ β x } λ x {{ ν x µ } 2 η αβ 2 ξ α ξ β ξ α 2 ξ β } x µ x {{ ν x λ + } x } µ x {{ λ x ν. } 2 1 On the other hand = 2η αβ 2 ξ α x µ x λ ξ β x ν. (9.9) g νσ Γ σ µλ {}}{{}}{ g νσ Γ σ ξ α ξ β x σ 2 ξ ρ µλ = η αβ x ν x } σ ξ {{ ρ x } µ x λ δ β ρ ξ α 2 ξ β = η αβ x ν x µ x λ = 1 [ gµν 2 x λ + g λν x µ g ] µλ x ν. (9.10) We introduce the inverse matrix g µν such that g µν g νσ = δ µ σ. Therefore we can solve with respect to the Christoffel symbols: Γ κ µλ = 1 2 gκν [ gµν x λ 26 + g λν x µ g ] µλ x ν. (9.11)

28 9 PHYSICS IN A GRAVITATIONAL FIELD Note that the Γ s are symmetric in the lower indices Γ κ µν = Γ κ νµ. The gravitational forces on the right hand side of (9.6) are given by derivatives of g µν. Comparing with the equation of motion of a particle in a electromagnetic field shows that the Γ λ µν correspond to the field F αβ, whereas the g µν correspond to the potentials A α. 9.3 Newtonian limit Let us assume that v i c and the fields are weak and static (i.e. not time dependent). Thus dx i dτ dx0 dτ. Inserting this into (9.5) leads to For static fields we get from (9.11): small d 2 x κ dτ 2 = dx µ dx ν velocity ( ) {}}{ dx Γκ µν Γ κ (9.12) dτ dτ dτ Γ κ 00 staticity {}}{ = gκi g 00 2 x i (i = 1, 2, 3) (9.13) (the other terms contain partial derivative with respect to x 0 which are zero by staticity). We write the metric tensor as g µν = η µν + h µν. For weak fields we have h µν = g µν η µν 1. In this case the coordinates (ct, x i ) are almost Minkowski coordinates. Inserting the expansion for g µν into (9.13) (taking only linear terms in h) gives Γ κ 00 = Then, let us compute (9.12) for κ = 0, κ = j: Taking (x j ) = r, we can write ( 0, 1 ) h 00 2 x i δk i. (9.14) d 2 t dτ 2 = 0 dt choice dτ = constant {}}{ = 1, (9.15a) d 2 x j ( ) 2 dτ 2 = h 00 dt c2 2 x j. dτ }{{} (9.15b) 1 2 d 2 r dt 2 = c2 2 h 00( r), (9.16) which is to be compared with the Newtonian case d2 r = φ( r). Therefore: dt2 g 00 ( r) = 1 + h 00 ( r) = 1 + 2φ( r) c 2. (9.17) Notice that the Newtonian limit (9.16) gives no clue on the other components of h µν. The quantity 2φ c 2 is a measure of the strength of the gravitational field. Consider a spherically symmetric mass 27

29 10 TIME DILATION distribution. Then 2φ(R) c at Earth surface, on the Sun (and similar stars), on a white dwarf, on a neutron star GR required. 10 Time dilation We study a clock in a static gravitational field and the phenomenon of gravitational redshift Proper time The proper time τ of the clock is defined through the 4-dimensional line element as dτ = ds clock = 1 ) ( g µν (x)dx c c µ dx ν, (10.1) clock x = (x µ ) are the coordinates of the clock. The time shown by the clock depends on both the gravitational field and of its motion (the gravitational field being described by g µν ). Special cases: 1. Moving clock in an IS without gravity : (g µν = η µν, dx i = v i dt, dx 0 = cdt). dτ = 2. Clock at rest in a gravitational field (dx i = 0) For a weak static field, one has with (9.17): dτ = 1 v2 c 2 dt dτ = g 00 dt φ(r) c 2 dt ( φ c 2 ). (10.2) The fact that φ is negative implies that a clock in a gravitational field goes more slowly than a clock outside the gravitational field Redshift Let us now consider objects which emit or absorb light with a given frequency. Consider only a static gravitational field (g µν does not depend on time). A source in r A (at rest) emits a monochromatic 28

30 10 TIME DILATION electromagnetic wave at a frequency ν A. An observer at r B, also at rest, measures a frequency ν B. At source: dτ A = g 00 ( r A )dt A At observer: dτ B = g 00 ( r B )dt B (10.3) As a time interval we consider the time between two following peaks departing from A or arriving at B. In this case dτ A and dτ B correspond to the period of the electromagnetic waves at A and B, respectively, and therefore dτ A = 1 ν A, dτ B = 1 ν B. (10.4) Going from A to B needs the same time t for the first and the second peak of the electromagnetic wave. Consequently, they will arrive with a time delay which is equal to the one with which they were emitted, thus dt A = dt B. With (10.3) and (10.4) we get: ν A ν B = The quantity z is the gravitational redshift: For weak fields with g 00 = 1 + 2φ c 2 g 00 ( r B ) g 00 ( r A ), with z = ν A ν B 1 = λ B λ A 1. (10.5) we have z = g 00 ( r B ) 1. (10.6) g 00 ( r A ) z = φ( r B) φ( r A ) c 2 ( φ c 2 ), (10.7) such a redshift is observed by measuring spectral lines from stars. As an example take solar light with (10.7) z = φ(r B) φ(r A ) c 2 φ(r A) c 2 = GM c , R with M kg and R m. For a white dwarf we find z 10 4 and for a neutron star z In general there are 3 effects which can lead to a modification in the frequency of spectral lines: 1. Doppler shift due to the motion of the source (or of the observer) 2. Gravitational redshift due to the gravitational field at the source (or at the observer) 3. Cosmological redshift due to the expansion of the Universe (metric tensor is time dependent) 10.3 Photon in a gravitational field Consider a photon with energy E γ = ω = 2π ν, travelling upwards in the homogeneous gravity field of the Earth, covering a distance of h = h B h A (h small). The corresponding redshift is z = ν A 1 = φ(r B) φ(r A ) ν B c 2 = g(h B h A ) c 2 = gh c 2, (10.8) 29

31 11 GEOMETRICAL CONSIDERATIONS resulting in a frequency change ν = ν B ν A (ν A > ν B, ν B = ν) and thus ν ν = gh c 2. (10.9) The photon changes its energy by E γ = Eγ c gh (like a particle with mass Eγ 2 c 2 = m). This effect has been measured in 1965 (through the Mössbauer effect) as ν exp ν th = 1.00 ± 0.01 (1% accuracy) Geometrical considerations In general, the coordinate dependence of g µν (x) means that spacetime, defined through the line element ds 2, is curved. The trajectories in a gravitational field are the geodesic lines in the corresponding spacetime Curvature of space The line element in an N-dimensional Riemann space with coordinates x = (x 1,..., x N ) is given as ds 2 = g µν dx µ dx ν (µ, ν = 1,..., N). Let us just consider a two dimensional space x = (x 1, x 2 ) with ds 2 = g 11 dx 1 dx 1 + 2g 12 dx 1 dx 2 + g 22 dx 2 dx 2. (11.1) Examples: Plane with Cartesian coordinates (x 1, x 2 ) = (x, y): ds 2 = dx 2 + dy 2, (11.2) or with polar coordinates (x 1, x 2 ) = (ρ, φ): ds 2 = dρ 2 + ρ 2 dφ 2 (11.3) Surface of a sphere with angular coordinates (x 1, x 2 ) = (θ, φ): ds 2 = a 2 ( dθ 2 + sin 2 θdφ 2) (11.4) The line element (11.2) can, via a coordinate transformation, be brought into the form (11.3). However, there is no coordinate transformation which brings (11.4) into (11.2). Thus: The metric tensor determines the properties of the space, among which is also the curvature. The form of the metric tensor is not uniquely determined by the space, in other words it depends on the choice of coordinates. 3 Pound, R. V. and Snider, J. L., Effect of Gravity on Gamma Radiation, Physical Review,

32 11 GEOMETRICAL CONSIDERATIONS The curvature of the space is determined via the metric tensor (and it does not depend on the coordinate choice) 4. If g ik = const then the space is not curved. In an Euclidian space, one can introduce Cartesian coordinates g ik = δ ik. For a curved space g ik const (does not always imply that space is curved). For instance by measuring the angles of a triangle and checking if their sum amounts to 180 degrees or differs, one can infer if the space is curved or not (for instance by being on the surface of a sphere). 4 Beside the curvature discussed here, there is also an exterior curvature. We only consider intrinsic curvatures here. 31

33 12 DIFFERENTIABLE MANIFOLDS Part IV Differential Geometry 12 Differentiable manifolds A manifold is a topological space that locally looks like the Euclidean R n space with its usual topology. A simple example of a curved space is the S 2 sphere: one can setup local coordinates (θ, ϕ) which map S 2 onto a plane R 2 (a chart). Collections of charts are called atlases. There is no one-to-one map of S 2 onto R 2 ; we need several charts to cover S 2. Definition: Given a (topological) space M, a chart on M is a one-to-one map φ from an open subset U M to an open subset φ(u) R n, i.e. a map φ : M R n. A chart is often called a coordinate system. A set of charts with domain U α is called an atlas of M, if U α = M, {φ α α I}. α Definition: dim M = n Definition: Two charts φ 1, φ 2 are C -related if both the maps φ 2 φ 1 1 : φ 1 (U 1 U 2 ) φ 2 (U 1 U 2 ) and its inverse are C. φ 2 φ 1 1 is the so-called transition function between the two coordinate charts. A collection of C related charts such that every point of M lies in the domain of at least one chart forms an atlas (C : derivatives of all orders exist and are continuous). The collection of all such C -related charts forms a maximal atlas. If M is a space and A its maximal atlas, the set (M, A) is a (C )-differentiable manifold. (If for each φ in the atlas the map φ : U R n has the same n, then the manifold has dimension n.) Important notions: A differentiable function f : M R belongs to the algebra F = C (M), sum and product of such functions are again in F = C (M). F p is the algebra of C -functions defined in any neighbourhood of p M (f = g means f(q) = g(q) in some neighbourhood of p). A differentiable curve is a differentiable map γ : R M. Differentiable maps F : M M are differentiable if φ 2 F φ 1 1 is a differentiable map for all suitable charts φ 1 of M and φ 2 of M. The notions have to be understood by means of a chart, e.g. f : M R is differentiable if x f(p(x) ) f(x) is differentiable. This is independent of the chart representing a neighbourhood of p. }{{} M 32

34 12 DIFFERENTIABLE MANIFOLDS M U 1 U 2 p φ 1 φ 2 R n x x R n X φ 2 φ 1 1 X (Chart φ 1 ) R n (Chart φ 2 ) R n Figure 7: Manifold, charts and transition function Tangent vectors and tangent spaces At every point p of a differentiable manifold M one can introduce a linear space, called tangent space T p (M). A tensor field is a (smooth) map which assigns to each point p M a tensor of a given type on T p (M). Definition: a C -curve in a manifold M is a map γ of the open interval I = (a, b) R M such that for any chart φ, φ γ : I R n is a C map. 33

35 12 DIFFERENTIABLE MANIFOLDS Let f : M R be a smooth function on M. Consider the map f γ : I R, t f(γ(t)). This has a well-defined derivative: the rate of change of f along the curve. Consider f φ 1 φ γ and }{{}}{{} R n R I R n x i f(x i ) t x i (γ(t)) =f(φ 1 (x i )) use the chain rule: d n (f γ) = dt i=1 ( ) dx i (γ(t)) x i f(xi ). (12.1) dt [ γ)] d dt (f Thus, given a curve γ(t) and a function f, we can obtain a qualitatively new object the rate of change of f along the curve γ(t) at t = t 0. t=t 0, Definition: The tangent vector γ p to a curve γ(t) at a point p is a map from the set of real functions f defined in a neighbourhood of p to R defined by γ p : f [ ] d (f γ) = (f γ) p dt = γ p(f). (12.2) p Given a chart φ with coordinates x i, the components of γ p with respect to the chart are [ ] d (x i γ) p = dt xi (γ(t)). (12.3) p The set of tangent vectors at p is the tangent space T p (M) at p. Theorem: If the dimension of M is n, then T p (M) is a vector space of dimension n (without proof). We set γ(0) = p (t = 0), X p = γ p, and X p f = γ p (f). Eq. (12.3) determines X p (x i ), the components of X p with respect to a given basis: X p f = [f γ] (0) = [ f φ 1 φ γ ] (0) = n i=1 x i (f φ 1 ) d dt (xi γ)(0) (12.4) = i ( ) (Xp x i f(x1,..., x n ) (x i ) ). This way we see that X p = ( Xp (x i ) ) ( ) x i, (12.5) i p ( ) and so the x i span T p (M). From (12.5) we see that X p (x i ) are the components of X p with p respect to the given basis (X p (x i ) = Xp i or X i ). 34

36 12 DIFFERENTIABLE MANIFOLDS Suppose that f, g are real functions on M and fg : M R is defined as fg(p) = f(p)g(p). X p T p (M), then (Leibniz rule) If Notation: (Xf)(p) = X p f, p M. X p (fg) = (X p f)g(p) + f(p)(x p g). (12.6) Basis of T p (M): T p = T P (M) has dimension n. In any basis (e 1,..., e n ) we have X = X i e i. Changes of basis are given by ē i = φ i k e k, Xi = φ i kx k. (12.7) The transformations φ i k and φ i k are inverse transposed of each other. In particular, e i = coordinate basis (with respect to a chart). Upon change of chart x x, x i is called φ i k = xk x i, φi k = xi x k. (12.8) Definition: The cotangent space T p (or dual space T p of T p ) consists of covectors ω T p, which are linear one-forms ω : X ω(x) < ω, X > R (ω : T p R). In particular for functions f, df : X Xf is an element of T p. The elements df = f,i dx i = form a linear space of dimension n, therefore all of T p. ( ) f x i We can define a dual basis (e 1,..., e n ) of T p : ω = ω i e i. In particular the dual basis of a basis (e 1,..., e n ) of T p is given by < e i, X >= X i or < e i, X j e j >= X j < e i, e j > = X i. Thus ω i =< ω, e i >. }{{} δ i j Upon changing the basis, the ω i transform like the e i and the e i like the X i (see (12.7)). In particular we have for the coordinate basis e i = basis is: x, e i = dx i (< e i, e i j >=< dx i, x j dx i >= δ i j). The change of (Similar to co- and contravariant vectors.) x i = xk x i x k = φ i k x k d x i = xi x k dxk = φ i kdx k Tensors on T p are multilinear forms on T p and T p, i.e. a tensor T of type ( 1 2) (for short T 1 2 T p ): T (ω, X, Y ) is a trilinear form on T p T p T p. The tensor product is defined between tensors of any type, i.e. T (ω, X, Y ) = R(ω, X)S(Y ) : T = R S. In components: T (ω, X, Y ) = T (e i, e j, e k ) }{{} T i jk ω i X j Y k }{{} e i(ω)e j (X)e k (Y ), (12.9) hence T = T i jk e i e j e k. Any tensor of any type can therefore be obtained as a linear combination of tensor products X ω ω with X T p, ω, ω T p. A change of basis can be performed similarly 35

37 12 DIFFERENTIABLE MANIFOLDS to the ones for vectors and covectors: T i jk = T α βγφ i αφ j β φ k γ (12.10) Trace: any bilinear form b T p T p determines a linear form l (T p T p ) such that l(x ω) = b(x, ω). In particular tr T is a linear form on tensors T of type ( 1 1), defined by tr(x ω) =< ω, X >. In components with respect to a dual pair of bases we have: tr T = T α α. Similarly T i jk S k = T i ik defines for instance a map from tensors of type ( 1 2) to tensors of type ( 0 1) The tangent map Definition: Let ϕ be a differentiable map: M map ( push-forward ): which we can describe in two ways: M and let p M, p = ϕ(p). Then ϕ induces a linear ϕ : T p (M) T p ( M), (a) For any f F p ( M) (F: space of all smooth functions on M (or M), that is C map f : M R): (ϕ X) f = X( f ϕ) (b) Let γ be a representative of X (X = γ p, see (12.2) and (12.3)). Then let γ = ϕ γ be a representative of ϕ X. This agrees with (a) since d dt f( γ(t)) t=0 = d dt ( f ϕ)(γ(t)) t=0. With respect to bases (e 1,..., e n ) of T p and (ē 1,..., ē n ) of T p ( M), this reads X = ϕ X: Xi = (ϕ ) i k Xk with (ϕ ) i k =< ēi, ϕ e k > or in case of coordinate bases: (ϕ ) i k = xi x k. Definition: The adjoint map (or pull-back ) ϕ of ϕ is defined as ϕ : T p T p, ω ϕ ω (= ω in T p ) with < ϕ ω, X >=< ω, ϕ X >. The same result is obtained from the definition In components, ω = ϕ ω reads ω k = ω i (ϕ ) i k. ϕ : d f d( f ϕ), f F( M). (12.10a) Consider (local) diffeomorphisms, i.e. maps ϕ such that ϕ 1 exists in a neighbourhood of p. Note that dim M = dim M ( ) x and det i x 0. Then ϕ j and ϕ, as defined above, are invertible and may be extended to tensors of arbitrary types. Example: tensor of type ( ) 1 1 (ϕ T )( ω, X) = T (ϕ ω, ϕ 1 X ), }{{}}{{} ω X (ϕ T )(ω, X) = T ((ϕ ) 1 ω, ϕ X). }{{}}{{} ω X 36

38 13 VECTOR AND TENSOR FIELDS Here, ϕ and ϕ are the inverse of each other and we have ϕ (T S) = (ϕ T ) (ϕ S), tr(ϕ T ) = ϕ (tr T ), (12.11) and similarly for ϕ. In components T = ϕ T reads T k i = T α x i x β β x α x k (in a coordinate basis). (12.12) This is formally the same as for transformation (12.10) when changing basis. 13 Vector and tensor fields Definition: If to every point p of a differentiable manifold M a tangent vector X p T p (M) is assigned, then we call the map X: p X p a vector field on M. Given a coordinate system x i and associated basis ( x i )p for each T p(m), X p has components X i p with X p = X i p ( x i )p and Xi p = X p (x i ) (see (12.5)). Eq. (12.8) shows how X i p transform under coordinate transformations. The quantity Xf is called the derivative of f with respect to the vector field X. The following rules apply: X(f + g) = Xf + Xg, X(fg) = (Xf)g + f(xg) (Leibnitz rule). (13.1) The vector fields on M form a linear space on which the following operations are defined as well: X fx (multiplication by f F), X, Y [X, Y ] = XY Y X (commutator). [X, Y ], unlike XY, satisfies the Leibniz rule (13.1). The components of the commutator of two vector fields X, Y relative to a local coordinate basis can be obtained by its action on x i. X = X i x and Y = Y k i x k we get [X, Y ] j = (XY Y X)x j Y x j = Y k xj x k = Y k δ j k = Y j XY j = X k x k (Y j ) = X k Y j,k }{{} Y j x k Thus using XY j Y X j = X k Y j,k Y k X j,k In a local coordinate basis, the bracket [ k, j ] clearly vanishes (X k = 1 and Y k = 1, and thus Y k,j = 0). The Jacobi identity holds: [X, [Y, Z]] + [Y, [Z, X]] + [Z, [X, Y ]] = 0. (13.2) 37

39 13 VECTOR AND TENSOR FIELDS Definition: Let T p (M) r s be the set of all tensors of rank (r, s) defined on T p (M) (contravariant of rank r, covariant of rank s). If we assign to every p M a tensor t p T p (M) r s, then the map t : p t p defines a tensor field of type ( r s). Algebraic operations on tensor fields are defined point-wise; for instance the sum of two tensor fields is defined by (t + s) p = t p + s p where t, s T p (M) r s. Tensor products and contractions of tensor fields are defined analogously. Multiplication by a function f F(M) is given by (ft) p = f(p)t p. In a neighbourhood U of p, having coordinates (x 1,..., x n ) a tensor field can be expanded in the form t = ( t i1...ir j }{{ 1...j } s x... ) ( dx j1... dx js). i1 x ir (13.3) components of t relative to the coordinate system (x 1,..., x n ) If the coordinates are transformed to ( x 1,..., x n ) the components of t transform according to t i1...ir j 1...j s t k1...kr l 1...l s x i1 x k1... xir x l1 x kr x j1... xls. (13.4) js x (We shall consider C tensor fields). Covariant tensors of order 1 are also called one-forms. The set of tensor fields of type ( ) r s is denoted by T r s (M). Definition: A pseudo-riemannian metric on a differentiable manifold M is a tensor field g T 0 2 (M) having the properties: (i) g(x, Y ) = g(y, X) for all X, Y (ii) For every p M, g p is a non-degenerate ( 0) bilinear form on T p (M). This means that g p (X, Y ) = 0 for all X T p (M) if and only if Y = 0. The tensor field g T 0 2 (M) is a (proper) Riemannian metric if g p is positive definite at every point p. Definition: A (pseudo-)riemannian manifold is a differentiable manifold M, together with a (pseudo-) Riemannian metric g Flows and generating vector fields A flow is a 1-parametric group of diffeomorphisms: ϕ t : M M, s, t R with ϕ t ϕ s = ϕ t+s. In particular ϕ 0 = id. Moreover, the orbits (or integral curves) of any point p M, t ϕ t (p) γ(t) shall be differentiable. A flow determines a vector field X by means of Xf = d dt (f ϕ t) (13.5) t=0 i.e. X p = d dt γ(t) t=0 = γ(0) (see (12.2) and (12.3)). γ(0) is the tangent vector to γ at the point p = γ(0). At the point γ(t) we have then γ(t) = d dt ϕ t(p) = d ds (ϕ s ϕ t ) (p) = X ϕt(p) s=0 38

40 13 VECTOR AND TENSOR FIELDS i.e. γ(t) solves the ordinary differential equation: γ(t) = X γ(t), γ(0) = p. (13.6) The generating vector field determines the flow uniquely. Not always does (13.6) admit global solutions (i.e. for all t R), however for most purposes, local flows are good enough. 39

41 14 LIE DERIVATIVE 14 Lie derivative The derivative of a vector field V rests on the comparison of V p and V p at nearby points p, p. Since V p T p and V p T p belong to different spaces their difference can be taken only after V p has been transported to V p. This can be achieved by means of the tangent map ϕ (Lie transport). The Lie derivative L X R of a tensor field R in direction of a vector field X is defined by L X R = d dt ϕ t R, (14.1) t=0 or more explicitly (L X R) p = d dt ϕ t R ϕt(p). Here ϕ t is the (local) flow generated by X, where t=0 ϕ t R ϕt(p) is a tensor on T p depending on t. R ϕ t (p) R p ϕ t (R ϕ t (p)) = ϕ t (R ϕ t (p)) p ϕ t (p) L X R = d dt ϕ t R 1 = lim t 0 t=0 t (ϕ t R R) t ϕ t (p) = γ(t); X p = d dt γ(t) t=0 = γ(0) (ϕ is the inverse of ϕ ) Figure 8: Illustration of the Lie derivative In order to express L X in components we write ϕ t in a chart: ϕ t : x x(t), and linearize it for small t: x i = x i + tx i (x) + O(t 2 ), x i = x i tx i ( x) + O(t 2 ), thus 2 x i x k t = 2 x i x k t = Xi,k at t = 0. As an example, let R be of type ( ) 1 1. By (12.12) we have (ϕ t R) i j (x) = Rα β( x) xi x β x α x. Taking j (according to (14.1)) a derivative with respect to t at t = 0 yields (first term: x k Rα β( x) }{{} R α β,k( x) x k t }{{} X k (L X R) i j = Ri j,kx k R α jx i,α + R i βx β,j (14.2) x i x β x α x j = R i j,kx k ). t=0 40

42 14 LIE DERIVATIVE Properties of L X : (a) L X is a linear map from tensor fields to tensor fields of the same type. (b) L X (tr T ) = tr(l X T ) (c) L X (T S) = (L X T ) S + T (L X S) (d) L X f = Xf (f F(M)) (e) L X Y = [X, Y ] (Y vector field) (proof: (a) follows from (14.1), (b) and (c) from (12.11), (d) from (13.5), whereas (e) is more involved). Further properties of L X : if X,Y are vector fields and λ R, then (i) L X+Y = L X + L Y, L λx = λl X (ii) L [X,Y ] = [L X, L Y ] = L X L Y L Y L X Proof of (ii): Apply it to f F(M), [L X, L Y ]f = L X L Y f L Y L X f = L X (Y f) L Y (Xf) = XY f Y Xf = [X, Y ]f = L [X,Y ] f. Next apply it on a vector field Z: [L X, L Y ]Z = [X, [Y, Z]] [Y, [X, Z]] = [[X, Y ], Z] = L [X,Y ] Z. (e) Jacobi identity For higher rank tensors the derivation follows from the use of (c). If [X, Y ] = 0 then L X L Y = L Y L X and for φ and ψ, which are the flows generated by X and Y, one finds: φ s ψ t = ψ t φ s. 41

43 15 DIFFERENTIAL FORMS 15 Differential forms Definition: A p-form Ω is a totally antisymmetric tensor field of type ( ) 0 p Ω(X π(1),..., X π(p) ) = (sign π)ω(x 1,..., X p ) for any permutation π of {1,..., p} (π S p (group of permutations)) with sign π being its parity. For p > dim M, Ω 0. Any tensor field of type ( 0 p) can be antisymmetrized by means of the operation A: (AT )(X 1,..., X p ) = 1 (sign π)t (X π(1),..., X π(p) ) (15.1) p! π S p with A 2 = A. The exterior product of a p 1 -form Ω 1 with a p 2 -form Ω 2 is the (p 1 + p 2 )-form: Properties: Ω 1 Ω 2 = ( 1) p1p2 Ω 2 Ω 1 Ω 1 Ω 2 = (p 1 + p 2 )! A(Ω 1 Ω 2 ) (15.2) p 1! p 2! Ω 1 (Ω 2 Ω 3 ) = (Ω 1 Ω 2 ) Ω 3 = (p 1 + p 2 + p 3 )! A(Ω 1 Ω 2 Ω 3 ) p 1! p 2! p 3! The components in a local basis (e 1,..., e n ) of 1-forms are Ω = Ω i1...i p e i1... e ip = AΩ = Ω i1...i p A(e i1... e ip ) = = n i 1=1 n i p=1 1 i 1<...<i p n Ω i1...i p 1 p! ei1... e ip Ω i1...i p e i1... e ip (15.3) A covariant tensor of rank p, which is antisymmetric under exchange of any pair of indices (i.e. is a p-form), in n dimensions has ( ) n p = n! (n p)!p! independent components. Examples: For 1-forms A, B (vector fields) we have (A B) ik = A i B k A k B i = ( 1)(B A) ik. For a 2-form A and a 1-form B (A B) ikl = A ik B l + A kl B i + A li B k, (15.4) 42

44 15 DIFFERENTIAL FORMS since A B = (1 + 2)! A(A B) 1! 2! = 3! 1! 2! (A ikb l ) 1 3! ei e k e l = 1 2 (A ikb l )e i e k e l Thus by comparing with (15.3) we get (15.4). = (A ikb l + cyclic permutations)e i e k e l = (A ik B l + cyclic permutations) 1 3! ei e k e l Exterior derivative of a differential form The derivative df of a 0-form f F is the 1-form df(x) = Xf: the argument X (vector) acts as a derivation. In a local coordinate basis: df = f x i dxi. The exterior derivative is performed by an operator d applied to forms, converting p-forms to (p + 1)-forms. The derivative dω of a 1-form Ω is given by This expression is verified as follows: X 1 Ω(X 2 ) = X 1 Ω, X 2 }{{} 1-form dω(x 1, X 2 ) = X 1 Ω(X 2 ) X 2 Ω(X 1 ) Ω([X 1, X 2 ]). (15.5) = X i 1 x i }{{},i ( Ωk X2 k ) = X i 1 Ω k,i X2 k + X1Ω i k X2,i, k X 2 Ω(X 1 ) = X k 2 Ω i,k X i 1 + X k 2 Ω i X i 1,k, Ω([X 1, X 2 ]) = Ω, X 1 X 2 X 2 X 1 = Ω i (X 1 X 2 X 2 X 1 ) i = Ω i ( X k 1 X i 2,k X k 2 X i 1,k), then dω(x 1, X 2 ) = (Ω k,i Ω i,k )X i 1X k 2. 1 (1+1)! This is manifestly a 2-form (the coefficient also fits the expectations: 2! 1!1! = 1). One can easily verify that For Ω f = fω (as f is a 0-form), the product rule dω(fx 1, X 2 ) = fdω(x 1, X 2 ). (15.6) d(ω f) = dω f Ω df 43

45 15 DIFFERENTIAL FORMS applies, as one can verify and So d(ω f)(x 1, X 2 ) = (15.5) X 1 (fω)(x 2 ) X 2 (fω)(x 1 ) (fω)([x 1, X 2 ]), X 1 (fω)(x 2 ) = X1 i x i (fω kx2 k ) = fx1 i ( Ωk x i X2 k ) + X i f 1 }{{} x i Ω kx2 k. }{{} fx 1Ω(X 2) df(x 1)Ω(X 2) d(f Ω)(X 1, X 2 ) = fdω(x 1, X 2 ) + Ω(X 2 )df(x 1 ) Ω(X 1 )df(x 2 ). (15.7) }{{}}{{} dω f Ω df Moreover we have d 2 f = 0, since d 2 f(x 1, X 2 ) = (15.5) X 1 df(x 2 ) X 2 df(x 1 ) df([x 1, X 2 ]) The generalization of the definition to a p-form Ω gives dω(x 1,.., X p+1 ) = = X 1 X 2 f X 2 X 1 f [X 1, X 2 ]f = 0. p+1 ( 1) i 1 X i Ω(X 1,.., ˆX i,.., X p+1 ) i=1 (15.8) + p+1 ( 1) i+j Ω([X i, X j ], X 1,.., ˆX i,.., ˆX j,.., X p+1 ), (15.9) i<j whereˆmeans omission, e.g. (X 1, ˆX 2, X 3 ) = (X 1, X 3 ). One can show that the following properties hold: (a) d is a linear map from p-forms to p + 1-forms, (b) d(ω 1 Ω 2 ) = dω 1 Ω 2 + ( 1) p1 Ω 1 dω 2, (c) d 2 = 0, i.e. d(dω) = 0, (d) df(x) = Xf (f F), By means of (a)-(d) we have an alternative definition of d. By eq. (15.3) we have with respect to a coordinate basis Ω = 1 p! Ω i 1...i p dx i1... dx ip, and hence (15.10a) dω }{{} = 1 p! dω i 1...i p dx i1... dx ip (15.10b) ddx ip =0 44

46 15 DIFFERENTIAL FORMS Components: p! dω = Ω i1...i p,i 0 dx i0 dx i1... dx ip = Ω i0i 2...i p,i 1 dx i0... dx ip = ( 1) k Ω i0...î k...i p,i k dx i0... dx ip (k = 0,..., p) dω = 1 1 p! p + 1 }{{} 1 (p+1)! p ( 1) k Ω i0...î k...i p,i k dx i0... dx ip (15.11) k=0 } {{ } (dω) i0...ip Examples: p = 1: p = 2: Consider a map ϕ : M (dω) ik = Ω k,i Ω i,k (15.12) (dω) ikl = Ω ik,l + Ω kl,i + Ω li,k (15.13) M and ϕ : T p ( M) T p (M); then ϕ d = d ϕ. (15.14) A proof is found by using (15.10), (12.11) and property (b). It suffices to verify (15.14) on 0-forms and 1-forms. For 0-forms f, (15.14) is identical to (12.10a). For 1-forms which are differentials d f, due to (c) we have (ϕ d)(d f) = 0 (d 2 f = 0), (d ϕ )(d f) = d(ϕ d f) = (12.10a) (ϕ d f) =d( f ϕ) d(d( f ϕ)) = d 2 ( f ϕ) = 0. Setting ϕ = ϕ t (the flow generated by X) and forming (14.1) (L X R = d dt ϕ t R t=0 ), one obtains the infinitesimal version of (15.14): L X d = d L X. (15.15) Definition: A p-form ω with ω = dη is exact dω = 0 is closed An exact p-form is always closed (d 2 η = 0), but the converse is not generally true (Poincaré lemma gives conditions under which the converse is valid) η is not unique since gauge transformations η η + dρ, with ρ any (p 1)-form, leave dη unchanged. 6 This is a generalization of the results of three-dimensional vector analysis: rot grad f = 0 and div rot k = 0. 45

47 15 DIFFERENTIAL FORMS The integral of an n-form: ( ) x i M is orientable within an atlas of positively oriented charts, if det x j > 0 for any change of coordinates. For an n-form ω (n = dim M): 1 ω = ω i1...i n n! dxi1... dx in = ω } 1...n dx {{} 1... dx n (15.16) ω(x) is determined by the single component ω(x); under a change of coordinates ω(x) transforms like ω( x) = ω 1...n = ( ) x i1 ω i1...i }{{ n } x 1 xin x i x n = ω(x) det x j. (15.17) totally antisymmetric The integral of a n-form is defined as follows: ω = dx 1... dx n ω(x 1,..., x n ) M U (if the support of ω is contained in a chart U). This integral is independent of the choice of coordinates, ( ) since in different coordinates dx 1... dx n ω(x) = d x 1... d x n ω(x) x i det x j and (15.17) applies Stokes theorem Let D be a region in a n dimensional differentiable manifold M. The boundary D consists of those p D whose image x in some chart satisfies e.g. x 1 = 0. One can show that D is a closed (n 1) dimension submanifold of M. If M is orientable then D is also orientable. D shall have a smooth boundary and be such that D is compact. Then for every (n 1)-form ω we have dω = ω (15.18) D D 15.3 The inner product of a p-form Definition: Let X be a vector field on M. For any p-form Ω we define the inner product as (i X Ω)(X 1,..., X p 1 ) Ω(X, X 1,..., X p 1 ) (15.19) (and zero if p = 0). Properties: 7 Actually, it is often impossible to cover the whole manifold with a single set of coordinates. In the general case it is necessary to introduce different sets of coordinates in different overlapping patches of the manifold, with the constraint that in the overlap between the patch with coordinate x i and another patch with coordinate x i, the x i can be expressed in a smooth one-to-one way as functions of x i and vice-versa (orientable manifold). 8 The integral over( a p-form ) over the overlap between two patches (x i and x i ) can be evaluated using either coordinate x system, provided det i x j > 0. 46

48 15 DIFFERENTIAL FORMS (a) i X is a linear map from p-forms to (p 1)-forms, (b) i X (Ω 1 Ω 2 ) = (i X Ω 1 ) Ω 2 + ( 1) p1 Ω 1 (i X Ω 2 ), (c) i X 2 = 0, (d) i X df = Xf = df, X with f F(M), (e) L X = i X d + d i X. Proof of (e): for 0-forms f we have and for 1-forms df Application: Gauss theorem L X f = Xf, i X df + d i X f = i X df = Xf, =0 L X df = (15.15) L X d=d L X d(l X f) = d(xf), i X ddf + d i X df = d(xf). =0 Let X be a vector field. Then d(i X η) is an n-form with dim M = n. η is an n-form, and if η p 0 p M, then η is a volume form. A function div η X F is defined through (div η X)η = d(i X η) = L X η. 9 (15.20) We can apply Stokes theorem since d(i X η) is an n-form and thus i X η an (n 1)-form: d(i X η) = (div η X)η = i X η. (15.21) The standard volume form η is given by η = g dx 1... dx n. D D D 9 dη = 0, thus L X = i X d + d i X applied on η gives L X η = i X dη + d(i X η). 47

49 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD Expression for div η X in local coordinates: Let η = a(x) dx 1... dx n, X = X i x. Then since (div i η X)η = L X η, we have (using property (c) of the Lie derivative): L X η = (Xa) dx 1... dx n + a n dx 1... d(xx i )... dx n. i=1 Since d(xx i ) = d(x k x k xi ) = dx i (x) = X i,jdx j, but dx 1... dx j... dx n 0 only if j = i }{{} δ i k (otherwise we have two identical dx i ) we find n L X η = }{{} Xa dx 1... dx n + a X i,idx 1... dx n X i a i=1 x i = (X i a,i + ax i,i) 1 a η div η X = 1 a (axi ),i = 1 g ( g X i ),i for the standard η. (15.22) 16 Affine connections: Covariant derivative of a vector field Definition: An affine (linear) connection or covariant differentiation on a manifold M is a mapping which assigns to every pair X, Y of C vector fields on M another C vector field X Y with the following properties: (i) X Y is bilinear in X and Y, (ii) if f F(M), then fx Y = f X Y, X (fy ) = f X Y + X(f)Y. (16.1) Lemma: Let X and Y be vector fields. If X vanishes at the point p on M, then X Y also vanishes at p. Proof: Let U be a coordinate neighbourhood of p. On U we have the representation X = ξ i ξ i F(U) with ξ i (p) = 0. Then ( X Y ) p = Y = ξ i ξi (p)[ x i }{{} Y ] p = 0. x i Since X Y produces again a vector field, the result of the covariant differentiation can only be a linear combination of again the basis in the current chart. This leads us to the following statement: =0 x i, 48

50 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD Definition: One sets, relative to a chart (X 1,..., X n ) for U M: x i ( ) x j = Γ k ij x k (16.2) The n 3 functions Γ k ij F(U) are called Christoffel symbols (or connection coefficients) of the connection in a given chart. 10 The Christoffel symbols are not tensors: If we use (16.1): x a ( ) x b = ( x i x a x a = xi x a x j ( x b ( ) x j x i x b x b Γk ij x k + ) = Γ c ab x c = Γ c x k ab x c ) [ x j = xi x j x a 2 x j x a x b x j. x b Γk ij x k. (16.3) x k + x i ( x j x b ) ] x j Comparison with 16.3: x k x Γ c c ab = xi x j x a x b Γk ij + 2 x k x a x b The second term is not compatible with being a tensor. Γ c ab = xi x j x c x a x b x k Γk ij + xc 2 x k x k x a x b (16.4) If for every chart there exist n 3 functions Γ k ij which transform according to (16.4) under a change of coordinates, then one can show that there exists a unique affine connection on M which satisfies (16.3). Definition: for every vector field X we can introduce the tensor X T 1 1 (M) defined by where ω is a one-form. X is called the covariant derivative of X. X(Y, ω) ω, Y X, (16.5) In a chart (x 1,..., x n ), let X = ξ i i and X = ξ i ;jdx j i (< dx i, i >= δ i k ): ξ i ;j = X( j, dx i ) = dx i, j X = dx i, ξ k,j k + ξ k Γ δ jk δ = ξ i,j + Γ i jkξ k 11 (16.6) 10 For a pseudo-riemannian manifold, the corresponding connection coefficients are given by (9.6) or (9.11). 11 semicolon shall denote the covariant derivative ( normal derivative + additional terms, that vanish in (cartesian) Euclidean or Minkowski space) 49

51 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD 16.1 Parallel transport along a curve Definition: let γ : I M be a curve in M with velocity field γ(t), and let X be a vector field on some open neighbourhood of γ(i). X is said to be autoparallel along γ if γ X = 0. (16.7) The vector γ X is sometimes denoted as DX dt or X dt (covariant derivative along γ). In terms of coordinates, we have X = ξ i i, γ = dxi dt i (see (12.3)). With (16.1) and (16.2) we get γ X = dx i dt i (ξ k k ) = dxi dt i (ξ k k ) = dxi dt [ξ k Γ jik j + i ξ k k ] = dxi [ ξ j Γ k dt ij k + i ξ k ] k [ dξ k = dt + dx i ] Γk ij dt ξj k, (16.8) where we used dxi ξ k dt x i = dξk dt. This shows that γx only depends on the values of X along γ. In terms of coordinates we get for (16.7) dξ k dt + dx i Γk ij dt ξj = 0. (16.9) For a curve γ and any two point γ(s) and γ(t) consider the mapping τ t,s : T γ(s) (M) T γ(t) (M), which transforms a vector v(s) at γ(s) into the parallel transported vector v(t) at γ(t). The mapping τ t,s is the parallel transport along γ from γ(s) to γ(t). We have τ s,s = 1 and τ r,s τ s,t = τ r,t. We can now give a geometrical interpretation of the covariant derivative that will be generalized to tensors. Let X be a vector field along γ, then γ X(γ(t)) = d ds τ t,s X(γ(s)), (16.10) s=t Proof: Let s work in a given chart. By construction, v(t) = τ t,s v(s) with v(s) T γ(s) (M) and due to (16.8) it satisifies: v i + Γ i kjẋk v j = 0. If we write (τ t,s v(s)) i = (τ t,s ) i j vj (s) = v i (t) (with τ t,s = (τ s,t ) 1 50

52 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD τ t,s v(t) v(s) M γ(t) γ γ(s) Figure 9: Illustration of parallel transport. and τ s,s = 1), we get v i (s) = d dt v i (t) t=s = d ] dt [(τ t,s ) i j vj (s) t=s ( ) d = dt (τ t,s ) i j v j (s) t=s = Γ i kjẋ k v j (s). d dt (τ t,s ) i j = Γi kjẋ k (16.11) t=s Since τ t,s = (τ s,t ) 1 d, ds s=t (τ t,s ) i j = d dt t=s (τ t,s ) i j = Γi kjẋk. Then ( ) d ds [τ t,s X(γ(s))] i d i = s=t ds τ t,s s=t jx j + d ds X i (γ(s)) s=t = Γ i kjẋ k X j + X i,j which is again (16.8) (X = ξ i i and the second term gives dξi dt ). 51 dx j (γ(s)) ds, s=t

53 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD Definition: If X Y = 0, then Y is said to be parallel transported with respect to X. Geometrical interpretation of parallel transport: Consider the differential da i = A i,jdx j = A i (x + dx) A i (x). In order that the difference of two vectors be a vector, we have to consider them at the same position. The transport has to be chosen such that for cartesian coordinates there is no change in transporting it. The covariant derivative exactly achieves this. Definition: Let X be a vector field such that X X = 0. Then the integral curves of X are called geodesics. In local coordinates x i the curve is given by (using (12.3) and (13.6)) the requirement d dt xi (t) = X i (x(t)). Inserting this into (16.8) and using d2 x i dt 2 = dxi dt, we get ẍ k + Γ k ijẋ i ẋ j = 0. (16.12) For a vector parallel transported along a geodesic, its length and angle with the geodesic does not change Round trips by parallel transport Consider (16.8) and denote ξ i = v i, thus v i = Γ i kjẋ k v j. (16.13) Let γ : [0, 1] M be a closed path, wih γ(0) = p = γ(1). Displace a vector v 0 T p (M) parallel along γ and obtain the field v(t) = τ t,0 v 0 T γ(t) (M). We assume that the closed path is sufficiently small (such that we can work in the image of some chart), thus we can expand Γ i kj (x) around the point x(0) = x 0 on the curve: Γ i kj(x) Γ i kj(x 0 ) + (x ρ x ρ 0 ) x ρ Γi kj(x) + (16.14) x=x0 Thus (16.13) is to first order in (x k x k 0): t 0 t v i dt = v i (t) v0 i = 0 Γ i kj }{{} ẋ k dt Γ i kj(x 0 )v j 0 v j v j 0 t 0 ẋ k dt, }{{} x k (t) x k 0 taking only the first term in the expansion of Γ. And hence, v i (t) = v i 0 Γ i kj(x 0 )(x k (t) x k 0)v j 0 + (16.15) 52

54 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD N 1 2 W C 4 α E 3 S Figure 10: Illustration of the path dependence of parallel transport on a curved space: vector 1 at N can be parallel transported along the geodesic N-S to C, giving rise to vector 2. Alternatively, it can be first transported along the geodesic N-S to E (vector 3) and then along E-W to C to give the vector 4. Clearly these two are different. The angle α between them reflects the curvature of the two-sphere. By plugging (16.14) and (16.15) into (16.13), we obtain an equation valid to second order: 1 1 v i dt = Γ i kjẋ k v j dt (16.16) 0 0 v i (1) v i ( Γ i kj(x 0 ) + (x ρ x ρ 0 ) ) x ρ Γi kj(x 0 ) + ( v j 0 Γj k j (x 0)(x k(t) x k 0 )v j 0 + )ẋ k dt. 53

55 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD Multiplying out and discarding terms of third order or higher in x k x k 0, we get: v i (1) v i 0 Γ i kj(x 0 )v j ẋ k dt }{{} x k (1) x k (0) =0 [ ] x ρ Γi kj(x 0 ) Γ i k j (x 0)Γ j ρj (x 0) v j 0 Since we are considering a closed path ( 1 0 ẋ dt = xk (1) x k (0) = 0), with [ ] v i = v i (1) v i (0) = x ρ Γi kj(x 0 ) Γ i kl(x 0 )Γ l ρj(x 0 ) v j x ρ ẋ k dt, 1 0 (x ρ x ρ 0 )ẋk dt x ρ ẋ k dt = d dt (xρ x k ) dt 0 } {{ } =0 1 0 ẋ ρ x k dt = 1 0 ẋ ρ x k dt, antisymmetric in (ρ, k). Then v i = 1 [ 2 x ρ Γi kj Γ i klγ l ρj ] x k Γi ρj + Γ i ρlγ l kj (x 0 ) }{{} R i jkρ v i = 1 2 Ri jkρ(x 0 )v j 0 We shall see that Rjkρ i is the curvature tensor. 1 0 R i jkρ = v j x ρ ẋ k dt, x ρ ẋ k dt. (16.17) x k Γi ρj x ρ Γi kj + Γ l ρjγ i kl Γ l kjγ i ρl (16.18) Thus an arbitrary vector v i will not change when parallel transported around an arbitrary small closed curve at x 0 if and only if R i jkρ vanishes at x Covariant derivatives of tensor fields The parallel transport is extended to tensors by means of the requirements: τ s,t (T S) = (τ s,t T ) (τ s,t S), τ s,t tr(t ) = tr(τ s,t T ), τ s,t c = c (c R). 54

56 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD For e.g. a covariant vector ω, τ s,t ω, τ s,t X γ(s) = ω, X γ(t) and for a tensor of type ( 1 1) : τs,t T (τ s,t ω, τ s,t X) = T (ω, X). In components: (τ s,t T ) i k = T α β(τ s,t ) i α (τ β s,t) k (16.19) (τ i k is inverse transpose of τ i k). The covariant derivative X (X vector field, T tensor field) associated to τ is ( X T ) p = d dt τ 0,tT γ(t) t=0, (16.20) with γ(t) any curve with γ(0) = p and γ(0) = X p (generalization of (16.10)). Properties of the covariant derivative: (a) X is a linear map from tensor fields to tensor fields of the same type ( r r), (b) X f = Xf, (c) X (tr T ) = tr( X T ), (d) X (T S) = ( X T ) S + T ( X S). This follows from the properties of τ s,t. For a 1-form ω we have: ( X ω)(y ) = tr( X ω Y ) = tr( X (ω Y )) tr(ω X Y ) = X tr(ω Y ) ω( X Y ) General differentiation rule for a tensor field of type ( 1 1) : = Xω(Y ) ω( X Y ). (16.21) ( X T )(ω, Y ) = XT (ω, Y ) T ( X ω, Y ) T (ω, X Y ) (16.22) Due to (a)-(d), the operation X is completely determined by its action on vector fields Y, which are the affine connections (see (16.1) and (16.2)) Local coordinate expressions for covariant derivative Let T Tp q (U) be a tensor of rank (p, q) with local coordinates (x 1,..., x n ) valid in a region U. We have T i1...ip j 1...j q i1... ip dx j1... dx jq and X = X k k. Let us use XT i1...ip j 1...j q = X k T i1...ip j 1...j q,k (16.23) and write (16.2): X ( i ) = X k k i = X k Γ l ki l. (16.24) 55

57 16 AFFINE CONNECTIONS: COVARIANT DERIVATIVE OF A VECTOR FIELD Moreover, ( X dx j )( i ) (16.21) = X dx j, i dx j, X i }{{} } δ j i {{ } 0 = X k Γ j ki, or X dx j = X k Γ j ki dxi. (16.25) Using (16.23), (16.24) and (16.25) for ω j = dx j, Y i = i we obtain the following expression for X T : T i1...ip j 1...j q;k = T i1...ip j 1...j q,k + Γ i1 kl T li2...ip j 1...j q Γ ip kl T i1...ip 1l j 1...j q Γ l kj 1 T i1...ip lj 2...j q... Γ l kj q T i1...ip j 1...j q 1l. (16.26) Examples: Contravariant and covariant vector fields: ξ i ;k = ξ i,k + Γ i klξ l, η i;k = η i,k Γ l kiη l, Kronecker tensor: Tensor ( 1 1) : δ i j;k = 0, T i k;r = T i k,r + Γ i rlt l k Γ l rkt i l. The covariant derivative of a tensor is again a tensor. Consider the covariant derivative of the metric g µν : g µν;λ = g µν x λ Γρ λµ g ρν Γ ρ λν g ρµ. (16.27) Inserting into this the expressions of Γ ρ λµ given by (9.11) leads us to g µν;λ = 0. (16.28) This is not surprising since g µν;λ vanishes in locally inertial coordinates and being a tensor it is then zero in all systems. Covariance principle: Write the appropriate special relativistic equations that hold in the absence of gravitation, replace η αβ by g αβ, and replace all derivatives with covariant derivatives (, ;). The resulting equations will be generally covariant and true in the presence of gravitational fields. 56

58 17 CURVATURE AND TORSION OF AN AFFINE CONNECTION, BIANCHI IDENTITIES 17 Curvature and torsion of an affine connection, Bianchi identities Let an affine connection be given on M, let X, Y, Z be vector fields. Definition: T (X, Y ) = X Y Y X [X, Y ] (17.1) R(X, Y ) = X Y Y X [X,Y ] (17.2) T (X, Y ) is antisymmetric and f-linear in X, Y and then defines a tensor of type ( 1 2) through: (ω, X, Y ) ω, T (X, Y ) is thus a ( 1 2) tensor field called the torsion tensor. f-linearity: T (fx, gy ) = fgt (X, Y ) f, g, F(M). In local coordinates, the components of the torsion tensor are given by: T k ij = dx k, T ( i, j ) = dx k, i j j i [ i, j ] }{{}}{{} =Γ l ij l =0 = Γ k ij Γ k ji (17.3) (using that dx k, l = δ k l ). In particular, we have T k ij = 0 Γ k ij = Γk ji. R(X, Y ) = R(Y, X) is antisymmetric in X,Y. The vector field R(X, Y )Z is f-linear in X, Y, Z: (R(fX, gy ) hz = fgh R(X, Y )Z; f, g, h F(M)). R determines a tensor of type ( 1 3) : the Riemann tensor or curvature tensor. In components with respect to local coordinates: (ω, Z, X, Y ) ω, R(X, Y )Z R i jklω i Z j X k Y l R i jkl = dx i, R( k, l ) j = dx i, ( k l l k ) j 12 = dx i, k (Γ s lj s ) l (Γ s kj s ) = Γ i lj,k Γ i kj,l + Γ s ljγ i ks Γ s kjγ i ls. (17.4) 12 Notice that [ k, l ] j = 0. }{{} =0 57

59 17 CURVATURE AND TORSION OF AN AFFINE CONNECTION, BIANCHI IDENTITIES Eq. (17.4) is exactly the the same as defined in (16.18). It is antisymmetric in the last two indices: R i jkl = R i jlk. Definition: The Ricci tensor is the following contraction of the curvature tensor: R jl R i jil = Γ i lj,i Γ i ij,l + Γ s ljγ i is Γ s ijγ i ls (17.5) The scalar curvature is the trace of the Ricci tensor: R g lj R jl = R l l (17.6) Example: For a pseudo-riemannian manifold the connection coefficients are given by (9.11). Consider a two-sphere (which is a pseudo Riemannian manifold) with the metric ds 2 = a 2 (dθ 2 + sin 2 θdφ 2 ), then: The non-zero Γ are: The Riemann tensor is given by ( ) ( ) g θφ = a 2 1 0, g θφ = sin 2 θ a sin 2 θ Γ θ φφ = sin θ cos θ, Γ φ θφ = Γφ φθ = cot θ. R θ φθφ = θ Γ θ φφ φ Γ θ θφ + Γ θ θλγ λ φθ Γ θ φλγ λ θφ = (sin 2 θ cos 2 θ) ( sin θ cos θ) cot θ = sin 2 θ. The Ricci tensor has the following components: The Ricci scalar is R = g θθ }{{} 1 a 2 R φφ = Rφθφ θ + R φ φφφ = sin 2 θ, }{{} =0 R θθ = 1, R θφ = R φθ = 0. R θθ }{{} 1 + g φφ }{{} 1 a 2 sin 2 θ R φφ }{{} sin 2 θ = 1 a a 2 sin 2 θ sin2 θ = 2 a 2. +g θφ R θφ }{{} 0 +g φθ R φθ }{{} 0 58

60 17 CURVATURE AND TORSION OF AN AFFINE CONNECTION, BIANCHI IDENTITIES The Ricci scalar is constant over this two-sphere and positive, thus the sphere is positively curved Bianchi identities for the special case of zero torsion X, Y and Z are vector fields, then R(X, Y )Z + cyclic = 0 (1st Bianchi identity), (17.7) ( X R)(Y, Z) + cyclic = 0 (2nd Bianchi identity). (17.8) Proof of the 1st identity: Torsion = 0 X Y Y X = [X, Y ]. Then ( X Y Y X )Z + ( Z X X Z )Y + ( Y Z Z Y )X [X,Y ] Z [Z,X] Y [Y,Z] X = X ( Y Z Z Y ) [Y,Z] X + cyclic = [X, [Y, Z]] + cyclic = 0 due to the Jacobi identity (13.2). (See textbooks for proof of the 2nd Bianchi identity.) 13 For a position independent metric (e.g. Cartesian coordinates) the Riemann tensor (and thus the scalar curvature) vanishes as the Γ vanish. 14 For a plane with polar coordinates we get a position dependent metric Γ do not vanish. However the curvature vanishes. 15 The curvature does not depend on the choice of coordinates. ( ) r 2 : ds 2 = dr 2 + r 2 dθ 2 and thus the 59

61 18 RIEMANNIAN CONNECTIONS 18 Riemannian connections Metric: Let M be equipped with a pseudo-riemannian metric: a symmetric, non-degenerate tensor field: g(x, Y ) = X, Y of type ( 0 2). Non-degenerate means that, for any point p M, X, Y T p, one has g p (X, Y ) = 0 Y T p X = 0. In components, X, Y = g ik X i Y k, g ik = g ki (symmetric) and det g ik 0. With the metric we can lower and raise indices: X i = g ik X k, ω i = g ik ω k, where g ik denotes the inverse of g ik. It also works for tensor fields of different types: T i k = T lk g il = T il g lk. Given a basis (e 1,..., e n ) of T p, the covectors of the dual basis (e 1,..., e n ) become themselves vectors; indeed e i = g ij e j. Riemann connection: The metric tensor g at a point p in M is a symmetric ( 0 2) tensor. It assigns a magnitude g(x, X) to each vector X on T p (M), denoted by d(x) and defines the angle between any two vectors X, Y ( 0) on T p (M) via a(x, Y ) = arccos ( ) g(x, Y ). (18.1) d(x)d(y ) If a(x, Y ) = π 2 then X and Y are orthogonal. Further observations: The length of a curve with tangent vector X between t 1 and t 2 is L(t 1, t 2 ) = d(x) dt. t 1 If (e a ) is a basis of T p (M), the components of g with respect to this basis are g ab = g(e a, e b ). Like in special relativity we classify vectors at a point as timelike (g(x, X) > 0), null (g(x, X) = 0) and space like (g(x, X) < 0). t 2 Definition: let (M, g) be a pseudo-riemannian manifold. An affine connection is a metric connection if parallel transport along any smooth curve γ on M preserve the inner product: for autoparallel fields X(t), Y (t) (see (16.7)), g γ(t) (X(t), Y (t)) is independent of t along γ. Theorem: an affine connection is metric if and only if (no proof) g = 0. (18.2) Eq. (18.2) is equivalent for (g(y, Z)) to X g = 0 = Xg(Y, Z) g( X Y, Z) g(y, X Z) 60

62 18 RIEMANNIAN CONNECTIONS or Xg(Y, Z) = g( X Y, Z) + g(y, X Z). (18.3) Theorem: For every pseudo-riemannian manifold (M, g), there exists a unique affine connection such that (a) has vanishing torsion ( is symmetric), (b) is metric. Proof: T = 0 (vanishing torsion) means X Y = Y X + [X, Y ]. Inserting this into (18.3) (and the linearity of g) gives Xg(Y, Z) = g( Y X, Z) + g([x, Y ], Z) + g(y, X Z). (18.4) By cyclic permutations one obtains as well Y g(z, X) = g( Z Y, X) + g([y, Z], X) + g(z, Y X), (18.5) Zg(X, Y ) = g( X Z, Y ) + g([z, X], Y ) + g(x, Z Y ). (18.6) Taking the linear combination (18.5) + (18.6) - (18.4), we get (Koszul formula): 2g( Z Y, X) = Xg(Y, Z) + Y g(z, X) + Zg(X, Y ) g([z, X], Y ) g([y, Z], X) + g([x, Y ], Z). (18.7) The right hand side is independent of. Since g is non-degenerate, the uniqueness of follows from (18.7). Definition: the unique connection on (M, g) from the above theorem is called the Riemannian or Levi-Civita connection. We determine the Christoffel symbols for the Riemannian connection in a given chart (U, x 1,..., x n ). For this purpose we take X = k, Y = j, Z = i in (18.7) and we use [ i, j ] = 0 as well as i, j = g ij. The result is i j, k = Γ l ij l, k, }{{} g lk 2Γ l ijg lk = k i, j + j i, k + i k, j }{{}}{{}}{{} g ij g ik g kj, 61

63 18 RIEMANNIAN CONNECTIONS or g lk Γ l ij = 1 2 (g kj,i + g ik,j g ji,k ). (18.8) g ij denoting the inverse matrix of g ij, we obtain which is exactly equation (9.11). Γ l ij = 1 2 glk (g kj,i + g ik,j g ji,k ), (18.9) Properties of the Riemannian connection: (i) The inner product of any two vectors remains constant upon parallel transporting them along any curve γ (g(x, Y ) γ(t) = g(x, Y ) γ(0) ). (ii) The covariant derivative commutes with raising or lowering indices, e.g. Tk;l i = (g km T im ) ;l = g km T im ;l, because g km;l = 0. Riemann tensor: the curvature tensor of a Riemannian connection has the following additional symmetry properties (without proof): R(X, Y )Z, U = R(X, Y )U, Z, (18.10) R(X, Y )Z, U = R(Z, U)X, Y. (18.11) In coordinate expression the Riemann tensor satisfies the following symmetries: R i jkl = R i jlk is always the case, (18.12) R i jkl = 0 1st Bianchi identity, (18.13) (jkl) R i jkl;m = 0 2nd Bianchi identity. (18.14) (klm) Eqs. (18.13) and (18.14) are valid for vanishing torsion. Here denotes the cyclic sum. Additionally, R ijkl = R jikl, (18.15) R ijkl = R klij, (18.16) for the Riemannian connection with R ijkl = g is R s jkl. (jkl) Ricci and Einstein tensor 62

64 18 RIEMANNIAN CONNECTIONS R ik = R j ijk Ricci tensor (18.17) R = R i i scalar curvature (18.18) G ik = R ik 1 2 Rg ik Einstein tensor (18.19) By symmetry, R ik = R ki, G ik = G ki and R k i;k = 1 2 R ;i, (18.20) which are the contracted 2nd Bianchi identity. G k i;k = 0, (18.21) Proof: R ik = g jl R lijk = g jl R jkli, 2nd Bianchi identity gives: Then we take the (ik)-trace: R i jkl;m + R i jlm;k + R i jmk;l = 0. R jl;m + R i jlm;i R jm;l = 0, }{{} g ik R jklm;i R j l;m g ik R j klm;i R j m;l = 0, (jm)-trace: For (18.21): R j l;j + g ik R kl;i R ;l = 0. }{{} 2R j l;j (18.20) G k i = R k i 1 2 gk ir = R k i 1 2 δk ir G k i;k = R k i;k 1 2 (δk i R) ;k = 1 2 R ;i 1 }{{} 2 R ;i = 0 (18.20) Without proof in n dimensions, the Riemann tensor has c n = n2 (n 2 1) 12 independent components (c 1 = 0, c 2 = 1, c 3 = 6, c 4 = 20). 63

65 19 PHYSICAL LAWS WITH GRAVITATION Part V General Relativity 19 Physical laws with gravitation 19.1 Mechanics The physical laws are relations among tensors (scalars and vectors being tensors of rank 0 and 1 respectively). Thus the physical laws read the same in all coordinate systems (provided the physical quantities are transformed suitably) and satisfy general covariance (same form). Practically, this means that from the special relativity laws that hold in absence of gravitation, we have to replace η αβ by g αβ and replace derivation by covariant derivation. In an inertial system, we have the equation of motion (see (4.3)) m duα dτ = f α. (19.1) According to the equivalence principle, (19.1) holds in a local IS. f α does not contain gravitational forces as they would vanish in a local IS. We transform it to general KS (coordinate system), then the Lorentz vector f α gets transformed to f µ = xµ ξ f α (ξ α is in local IS, x µ is in KS). Equation (19.1) α holds in a local IS reads then m Du µ } dτ {{} covariant derivative given in (16.8) = f µ, (19.2) ( X D dτ ) with ξµ u µ (dt dτ and dxi dτ Then equation (19.2) reads Du µ dτ = ui ) and thus = duµ dτ + Γµ νλ uν u λ. m duµ dτ = f µ mγ µ νλ uν u λ. (19.3) We see that on the right hand side there are now gravitational forces appearing explicitly (via Γ µ νλ ). Equation (19.3) (or (19.2)) is covariant (it has the same form in all coordinate systems) and reduces for g µν η αβ (thus Γ µ νλ = 0) to equation (19.1) (in a local IS). The components of uµ are not independent but satisfy the condition g µν u µ u ν = c Electrodynamics According to the equivalence principle, Maxwell s equations (see (6.4) and (6.5)) α F αβ = 4π c jβ and ɛ αβγδ β F γδ = 0 are valid in a local IS. Applying the covariance principle, they become as follows in a general KS: F µν ;ν = 4π c jµ and ɛ µνλκ F λκ;ν = 0, (19.4) 64

66 19 PHYSICAL LAWS WITH GRAVITATION provided that going from coordinates ξ α in a local IS to the KS coordinates x µ we have j α j µ = xµ ξ α jα and F µν = xµ ξ α x ν ξ β F αβ. Gravity enters via the Γ µ νλ in the covariant derivative. The continuity equation αj α = 0 translates to j µ ;µ = 0. It can be shown that in the homogeneous equation the terms with Γ vanish. Thus the covariant derivative reduces to the ordinary derivative (,) Energy-momentum tensor For an ideal fluid, given by (in a local IS) with u µ : four-velocity, ρ: proper energy density, p: pressure of the fluid. T µν = (ρ + p c 2 ) u µ u ν η µν p, (19.8) In a KS this becomes T µν = (ρ + p c 2 ) u µ u ν g µν p. (19.9) In the IS the conservation law implies T µν,ν = 0 and in the KS T µν ;ν = 0 (explicitely, T µν ;ν = T µν,ν + Γ µ νλ T νλ + Γ ν νλ T µλ = 0). With (19.5), Γ ν νλ = 1 g g, we get instead x λ T µν ;ν = 1 g gt µν x ν + Γ µ νλ T νλ = 0. (19.10) This is no longer a conservation law, as we cannot form any constant of motion from (19.10). This should also not be expected, since the system under consideration can exchange energy and momentum with the gravitational field. 16 g = det(g ik ) = ɛ i 1...i n g 1i1... g nin. Consider g x l = n g km x l ɛ i 1...i n g 1i1... g ki k... g x l nin and use g ki k x l k=1 g mr g rik. Due to the antisymmetry of ɛ, only the term r = k survives. Thus g x l = g km x l g mk g. Plugging this into the definition of Γ k kl (one contraction): ( Γ k kl = gkm gmk g ml 2 x l + x k g ) kl x }{{ m = gkm 2 } vanish by interchanging (m k) g mk x l With (19.5) one can show that the inhomogeneous Maxwell equation in KS can be written as = g km x l δ m i k = = ln g x l = 1 g g x l. (19.5) 1 ( gf µν ) g x ν = 4π c 2 jµ, (19.6) and the continuity equation: j µ ;µ = 0 becomes ( gj µ ) x µ = 0. (19.7) 65

67 20 EINSTEIN S FIELD EQUATIONS 20 Einstein s field equations The field equations cannot be derived by using the covariance principle, since there is no equivalent equation in a local IS. We have to make some requirements/assumptions. Requirements: The Newtonian limit is well confirmed through all observations: φ = 4πGρ. From the Newtonian limit of the equation of motion for a particle we derived (equation (9.17)) g φ c 2. The non-relativistic limit should then be with T 00 ρc 2 (other T ij are small). g 00 = 8πG c 4 T 00, (20.1) Thus a generalization should lead to something of type G µν = 8πG c T 4 µν where G µν has to satisfy the following requirements: (1) G µν is a tensor (T µν is tensor). (2) G µν has the dimension of a second derivative. If we assume that no new dimensional constant enter in G µν then it has to be a linear combination of terms which are either second derivatives of the metric g µν or quadratic in the first derivative of g µν. (3) Since T µν is symmetric, G µν also has to be symmetric and due to the fact that T µν is covariantly conserved, i.e. T µν ;ν = 0, it follows that G µν must satisfy G µν = G νµ and G µν ;ν = 0. (4) For a weak stationary field we shall get (20.1), thus G 00 g 00. Conditions (1)-(4) determine G µν uniquely. (1) and (2) imply that G µν has to be a linear combination G µν = ar µν + brg µν (20.2) of R µν, the Ricci tensor, and R, the Ricci scalar 17. The symmetry of G µν is automatically satisfied. The contracted Bianchi identity (18.20), (18.21) suggests that G µν ;ν = 0 on the Einstein tensor, what implies b = a 2. Thus we find G µν = a(r µν 1 2 g µνr) = 8πG c 4 T µν. (20.3) The constant a has to be determined by performing the Newtonian limit. Consider weak fields: g µν = η µν + h µν, h µν 1 (non relativistic velocities: v i c), then T ik T 00 G ik G 00. Compute the trace of G µν : = a(r 2R) = ar ( ) from (20.3) g µν G µν G 00 = a R 00 R 2 00 = a(r }{{} 00 R/2). (20.4) η 00 =1 17 It can be shown that indeed the Ricci tensor is the only tensor made of the metric tensor and first and second deivatives of it, and which is linear in the second derivative. 66

68 20 EINSTEIN S FIELD EQUATIONS Compairing the two results gives R 2R 00, thus ( G 00 a R 00 R ) 2aR 00. (20.5) 2 For weak fields all terms quadratic in h µν can be neglected in the Riemann tensor; we get to leading order: R µν = R ρ µρν Γρ µν x ρ Γρ ρµ x ν ( h µν 1). For weak stationary fields we find: R 00 = Γi 00 x i with Γ i 00 = 1 g 00 2 x i. Thus G 00 2a Γi 00 x i = a g 00! = g 00, therefore a = 1. Einstein s field equations are 18 (found 1915 by Albert Einstein): R µν R 2 g µν = 8πG c 4 T µν (20.6) Together with the geodesic equation ((16.12) or (19.3)), these are the fundamental equations of general relativity. By contraction of (20.6), we find also R µ µ R 2 δµ µ }{{} =4 = R = 8πG T. (20.7) c4 R can be expressed in (20.6) in terms of T, and we get: R µν = 8πG c 4 ( T µν T ) 2 g µν (20.8) an equivalent version of the field equations. For the vacuum case where T µν = 0 we have R µν = 0. (20.9) Significance of the Bianchi identity Einstein s equation constitutes a set of non-linear coupled partial differential equations whose general solution is not known. Usually one makes some assumptions, for instance spherical symmetry. Because the Ricci tensor is symmetric, the Einstein equations constitute a set of 10 algebraically independent second order differential equations for g µν. The Einstein equations are generally covariant, so that they can at best determine the metric up to coordinate transformation ( 4 functions). Therefore we expect only 6 independent generally covariant equations for the metric. Indeed the (contracted) Bianchi identities tell us that (equation (18.21)) G ν µ;ν = 0 and hence there are 4 differential relations among the Einstein s equations. Bianchi identities can also be understood as a consequence of the general covariance of the Einstein equations. 18 Depending on the convention used for the Riemann tensor, one could also encounter a minus in front of the energymomentum tensor, as for example in Weinberg. 67

69 21 THE EINSTEIN-HILBERT ACTION 20.1 The cosmological constant As a generalization, one can relax condition (2) and have a linear term in g µν 19. The field equations become R µν R 2 g µν Λg µν = 8πG c 4 T µν, (20.10) where Λ is a constant: the cosmological constant ([Λ] = L 2 ). For point (4) the Newtonian limit of (20.10) leads to The right-hand side can also be written as 4πG(ρ ρ vacuum ), with φ = 4πρG c 2 Λ. (20.11) c 2 ρ vac = c4 Λ. (20.12) 4πG Λ corresponds to the (constant) energy density of empty space (vacuum). Λ 1/2 (distance) has to be much larger than the dimension of the solar system. 21 The Einstein-Hilbert action The field equations (20.6) can be obtained from a covariant variational principle. The action for the metric g is S D [g] = R(g) dv, (21.1) where D M is a compact region space-time, R is a scalar curvature and dv a volume element: D dv = g d 4 x (21.2) (g = det g ik, d 4 x in 4 dimensions). The Euler-Lagrange equations are the field equations in vacuum: We have δ R(g) dv = δ(g µν R µν g) d 4 x = D Consider first δr µν : D D δs D [g] = 0. (δr µν )g µν g d 4 x + D R µν δ(g µν g) d 4 x. (21.3) R µν = α Γ α µν ν Γ α µα + Γ ρ µνγ α ρα Γ ρ ναγ α ρµ. (21.4) Let us compute the variation of R µν at any point p in normal coordinates, whose center is in p itself (x(p) = 0, then Γ α βγ (0) = 0). Thus δr µν reduces (at any such p) to δr µν = ( δγ α ) µν ( δγ α ),α µα. 20 (21.5),ν Without proof one finds that δγ α µν is indeed a tensor although Γ α µν is not a tensor. (21.5) is thus a tensor equation, it holds in every coordinate system and we can also take the covariant derivative: δr µν = ( δγ α ) µν ( δγ α ) ;α µα (21.6) ;ν 19 Note that g µν;σ = variation (with respect to g) δ and normal derivative commute 68

70 21 THE EINSTEIN-HILBERT ACTION (Palatini identity). Since g µν;σ = 0 we can write (21.6) as g µν δr µν = ( g µν δγ α ) µν ( g µν δγ α ) ;α µα ;ν (21.7) = ω α ;α = ω α,α + Γ α αµ }{{} 1 g g x µ ω µ = 1 g ( g ω µ ) x µ. (21.8) Inserting this into the integral (21.3) and applying (15.21) (Gauss theorem), we get (div g ω)η = i ω η, D D where div g ω = ω α ;α and thus (δr µν )g µν g d 4 x = d0 α is the coordinate normal to D and D D ω α g d0 α. ω α = g µν δγ α µν g µα δγ ν µν (21.9) is a vector field. If the variations of δg µν vanish outside a region contained in D, then the boundary term vanishes as well. As for the second term in (21.3) ( D R µν δ(g µν g) d 4 x), we recall that for an n n matrix A(λ) we have (see linear algebra): d i) dλ det A = det A tr ( ) 1 da A dλ, ii) d dλ (A 1 1 da )A = A dλ. Thus (δg µν )g νσ = g µν δg νσ comes from ii) and δg = gg µν δg νµ comes from i) with A 1 = g µν. Hence we find the desired expressions δ g = 1 g g µν δg νµ = 1 g gαβ δg αβ, 2 2 δ(g µν g) = g δg µν 1 2 g g µν g αβ δg αβ. (21.10) And thus 0 = D R µν δ(g µν g) d 4 x = D = D g d 4 x }{{} R µνδg µν 1 2 R µνg µν g αβ δg αβ }{{}}{{} dv ( dv R µν 1 ) 2 Rg µν }{{} =G µν=0 δg µν. R α µ β ν 69

71 21 THE EINSTEIN-HILBERT ACTION Therefore δs D [g µν ] = 0 G µν = R µν 1 2 Rg µν = 0. Since δ g d 4 x = g d 4 x 1 2 gµν δg µν = D D g d 4 x g µν δg µν, it follows that if we have a cosmological constant, the Einstein s vacuum 1 2 D equations are obtained from the action principle applied on S D [g] = (R 2Λ) g d 4 x. (21.11) D The variational principle extends to matter described by any field ψ = (ψ A ) (A = 1,..., N), (we include also the electromagnetic field among the ψ A ) transforming as a tensor under change of coordinates. Consider an action of the form S D [ψ] = L(ψ, g ψ) g d 4 x, (21.12) D where g is the Riemannian connection of the metric g. If we know L in flat space, the equivalence principle prescribes to replace η αβ by g αβ and replace ordinary derivatives by covariant ones. Example: electromagnetic field L = 1 16π F µνf µν = 1 16π F µν F σρ g σµ g ρν, and the Euler-Lagrange equations in this case (F µν = A ν;µ A µ;ν = A ν,µ A µ,ν ) for the basic 4-potential A µ field read: L L µ = 0, with µ A ν = A ν;µ ; A ν µ A ν in this case L A ν = 0, and L µa ν = 1 4π F µν. The Euler-Lagrange equations are then F µν ;ν = 0, which are the Maxwell equations for vanishing current j µ (F µν ;ν = 4π c jµ and L = 1 16π F µνf µν 1 c jµ A µ with j µ A µ = g µν j ν A µ ). Variations in (21.12) with respect to the fields ψ A lead to the Euler-Lagrange equations, whereas variations with respect to the metric (which is also a function and is determined by solving Einstein s equations) gives (without proof) δ g L(ψ, g ψ) g d 4 x = 1 2 T µν δg µν g d 4 x. (21.13) D This term has to be added to the one proportional to δg µν in Einstein s action: D g d 4 c x (G 4 µν 16πG 1 ) 2 T µν }{{} =0 ( and thus G µν = 8πG c T 4 µν. For electrodynamics: T µν = 1 4π Fµσ Fν σ 1 4 F ) σρf σρ g µν (or T αβ = Fµ α F µβ Lg αβ ). And similarly for other matter fields. D δg µν 70

72 22 STATIC ISOTROPIC METRIC 22 Static isotropic metric 22.1 Form of the metric For the gravity field of Earth and Sun we assume a spherically symmetric distribution of the matter (rotation velocities v i c). Thus we need a spherically symmetric and static solution for the metric g µν (x). We first give the general form of such a metric (static and isotropic) which we then use as an ansatz to solve the field equations. For r, the Newtonian gravitational potential ϕ = GM r goes to zero. Thus, asymptotically, the metric should be Minkowskian: ds 2 = c 2 dt 2 dr 2 r 2 (dθ 2 + r sin 2 θdφ 2 ), in spherical coordinates r, θ, φ and t. Thus, ds 2 = B(r)c 2 dt 2 A(r)dr 2 C(r)r 2 (dθ 2 + sin 2 θdφ 2 ). (22.1) Due to isotropy and time independence, A, B and C cannot depend on θ, φ and t (and no linear terms in dθ and dφ). Freedom in the choice of coordinates allows to introduce a new radial coordinate in (22.1): C(r)r 2 r 2, thus C(r) can be absorbed into r. We get the standard form: ds 2 = B(r)c 2 dt 2 A(r)dr 2 r 2 (dθ 2 + sin 2 θdφ 2 ) (22.2) (θ and φ have the same significance as in Minkowski coordinates). Due to our asymptotic requirements (r ) we can assume that B(r) 1 and A(r) Robertson expansion Even without knowing the solution to the field equations, we can give an expansion of the metric for weak fields outside the mass distribution. The metric can only depend on the total mass of the considered object (Earth or Sun for instance), on the distance from it and on the constants G, c. Since A and B are dimensionless, they can only depend on a combination of the dimensionless quantity GM For GM c 2 r 1 we can then have the following expansion: B(r) = 1 2 GM c 2 r + 2(β γ) A(r) = 1 + 2γ GM c 2 r +... ( ) 2 GM c r c 2 r. (22.3) which is the Robertson expansion. The linear term in B(r) has no free parameter since it is constrained by the Newtonian limit: g φ c, φ = GM 2 r (Newtonian potential), therefore B g 00. The coefficient 2(β γ) comes from historical reasons, β and γ are independent coefficients. In the solar system, GM c 2 r GM c 2 R , then only linear terms in γ and β play a role. For general relativity: γ = β = 1 (Newtonian gravity: γ = β = 0). 71

73 22 STATIC ISOTROPIC METRIC 22.3 Christoffel symbols and Ricci tensor for the standard form The metric tensor g µν is diagonal. g 00 = B(r) g 11 = A(r) g 22 = r 2 g 33 = r 2 sin 2 θ (22.4) g 00 = 1 B(r) g 11 = 1 A(r) The non-vanishing components of Γ σ λµ = gσν 2 g 22 = 1 r 2 g 33 1 = r 2 sin 2 θ x λ + g λν x µ g ) µλ x ν are ( gµν (22.5) Γ 0 01 = Γ 0 10 = B 2B Γ 1 00 = B 2A Γ 1 11 = A 2A Γ 2 12 = Γ 2 21 = 1 r Γ 1 22 = r A Γ 1 33 = r sin2 θ A (22.6) Γ 3 13 = Γ 3 31 = 1 r Γ 3 23 = Γ 3 32 = cot θ Γ 2 33 = sin θ cos θ where stands for r. With g = r 4 AB sin 2 θ (22.7) we get ( ) ) ( ) Γ ρ ln g 2r A µρ = x µ = (0, + 2A + B, cot θ, 0. (22.8) 2B The Ricci tensor can then be calculated as and we get as a result R µν = Γρ µν x ρ Γρ ρµ x ν + Γσ µνγ ρ ρσ Γ σ ρµγ ρ νσ, (22.9) R 00 = B 2A A B ( ) 2A 2 B 2 2AB + B 2 2A r + A 2A + B, 2B ( ) = B 2A B A 4A A + B + B B ra, (22.10) ( ) R 11 = B 2B + B A 4B A + B + A B ra, (22.11) R 22 = 1 + r ( ) A 2A A B 1 B A, (22.12) R 33 = R 22 sin 2 θ, (22.13) The non-diagonal components R µν with µ ν vanish. 72

74 22 STATIC ISOTROPIC METRIC 22.4 Schwarzschild metric We assume a static, spherically symmetric, mass distribution with finite extension: 0 r r 0 ρ(r) (22.14) = 0 r > r 0 Similarly, the pressure P (r) is thought to vanish for r > r 0. The four velocity vector within the mass distribution in the static case is u µ = (u 0 = constant, 0, 0, 0). This way, the energy-momentum tensor (describing matter) does not depend on time. We then adopt the ansatz for the metric elaborated in (22.2): g µν = diag(b(r), A(r), r 2, r 2 sin 2 θ). Outside the mass distribution (r r 0 ), the Ricci tensor vanishes: R µν = 0. We have already calculated the coefficients R µν in equations (22.10) (22.13). For µ ν, R µν = 0 is trivially satisfied while the diagonal components should be set to zero: R 00 = R 11 = R 22 = R 33 = 0 (r r 0 ). Consider R 00 B + R 11 A = 1 ( ) B ra B + A = 0 and thus d (ln AB) = 0 (since ra 0) or AB = constant A dr (or ln AB = constant). For r we require A = B = 1, therefore AB = 1 A(r) = 1 B(r). Introducing this into R 22 (22.12) and R 11 (22.11) leads to R 22 = 1 rb B = 0, (22.15) R 11 = B 2B B rb = + 2B rb = 1 dr 22 = 0. (22.16) 2rB 2rB dr With (22.15), (22.16) is automatically satisified (since R 22 = 0 also its derivative vanishes). We write (22.15) as d (rb) = 1. (22.17) dr We integrate it and get rb = r + constant }{{} = r 2a. Then 2a B(r) = 1 2a r, A(r) = 1 1 2a r, (22.18) for r r 0. This solution for the vacuum Einstein equations was found in 1916 by Schwarzschild. The Schwarzschild solution is ds 2 = ( 1 2a ) c 2 dt 2 dr2 r 1 2a r The constant can be determined by considering the Newtonian limit: g 00 = B(r) r Thus one introduces the so called Schwarzschild radius: r 2 (dθ 2 + sin 2 θdϕ 2 ) (22.19) φ c 2 = 1 2GM c 2 r = 1 2a r. r S = 2a = 2GM c 2 73

75 23 GENERAL EQUATIONS OF MOTION The Schwarzschild radius of the Sun is r s, = 2GM 3 km (M kg, R = km) so r S, R c 2 = 2GM c 2 R A clock at rest in r has the proper time dτ = B dt, thus dt dτ diverges at r r S. This implies that a photon emitted at r = r S will be infinitely redshifted (t is not a good coordinate either for events taking place at r r S ). A star, whose radius r star is smaller than r S, is a black hole since photons emitted at its surface cannot reach regions with r > r S. Expanding the Schwarzschild metric in power of r S r and comparing it with the Robertson expansion (22.3), one finds β = γ = 1 for general relativity. 23 General equations of motion We now consider the motion of a freely falling material particle or photon in a static isotropic gravitational field (e.g. motion of planets around the Sun). For the relativistic orbit x k (λ) of a particle in a gravitational field we have: and g µν dx µ dλ dx ν dλ = ( ds dλ d 2 x k dλ 2 = dx µ dx ν Γk µν dλ dλ ) 2 ( ) 2 dτ = c 2 = dλ (23.1) c 2 m 0, λ = τ. (23.2) 0 m = 0 For a massive particle we can take the proper time as a parameter for the trajectory or orbit (dλ = dτ). For massless particles one has to choose another parameter. For the spherically symmetric gravitational field, we use the metric (r > r, radius of the star) with the coordinates (x 0, x 1, x 2, x 3 ) = (ct, r, θ, φ). ds 2 = B(r)c 2 dt 2 dr 2 A(r) r 2 (dθ 2 + sin 2 θdϕ 2 ), (23.3) Equations (23.1) (23.3) define the relativistic Kepler problem. Using the Christoffel symbols given in (22.6), we get for (23.1): d 2 x 0 dλ 2 = dx 0 B B dλ dr dλ, (23.4) d 2 r dλ 2 = B 2A ( dx 0 dλ ) 2 A 2A ( ) 2 dr + r dλ A ( ) 2 dθ + r sin2 θ dλ A ( ) 2 dφ, (23.5) dλ d 2 ( ) 2 θ dλ 2 = 2 dθ dr dφ + sin θ cos θ, (23.6) r dλ dλ dλ d 2 φ dλ 2 = 2 r Equation (23.6) can be solved by dφ dr dθ dφ 2 cot θ dλ dλ dλ dλ. (23.7) θ = π = constant. (23.8) 2 Without loss of generality we can choose the coordinate system such that θ = π 2, this way the trajectory lies on the plane with θ = π 2. d 2 θ dλ = 0 corresponds to angular momentum conservation. With (23.8) 2 21 Apparently it seems that the Schwarzschild metric is singular for r = r s, but this is not the case. It is only an artefact of the coordinate choice. To be discussed later. 74

76 23 GENERAL EQUATIONS OF MOTION we get for (23.7) : ( 1 d r 2 r 2 dφ ) = 0, (23.9) dλ dλ which leads to r 2 dφ = constant = l. (23.10) dλ l can be interpreted as the (orbital) angular momentum (per unit mass). Equations (23.8) and (23.10) follow from angular momentum conservation, which is a consequence of spherical symmetry (rotation invariance). Equation (23.4) can be written as (B = B(r(λ)) ( ( ) ) d dx 0 ln + ln B = 0, (23.11) dλ dλ [( ) ] dx which can be integrated as ln 0 dλ B = constant or B dx0 dλ In (23.5) we use (23.8), (23.10) and (23.12) and get: We multiply it with 2A ( dr dλ) and get Integration gives d 2 r dλ 2 + F 2 B 2AB 2 + A 2A A d dλ [ A = constant = F. (23.12) ( ) 2 dr l2 = 0. (23.13) dλ Ar3 ( ) ] 2 dr + l2 dλ r 2 F 2 22 = 0. (23.14) B ( ) 2 dr + l2 dλ r 2 F 2 = ɛ = constant. (23.15) B Integrating it once more we get r = r(λ). Inserting then this result into (23.10) and (23.12), we obtain with one more integration φ = φ(λ) and t = t(λ). Next we eliminate λ and get r = r(t) and φ = φ(t). Together with θ = π 2, this is then a complete solution (generally it has to be done numerically). Equation (23.2) becomes g µν dx µ dλ dx ν dλ = B ( ) dx 0 2 A dλ ( dr dλ ) 2 ( ) 2 dθ r 2 r 2 sin 2 θ dλ using (23.8), (23.10), (23.12) and (23.15). On the other hand c 2 (m 0) ɛ =. 0 (m = 0) We are left with two integration constants, F and l. 22 Notice: da dr = A dλ dλ ( ) 2 dφ = ɛ, (23.16) dλ 75

77 23 GENERAL EQUATIONS OF MOTION 23.1 Trajectory From (23.15) we get and with (23.10), Thus, dφ dr = dφ dλ φ(r) = F dr 2 dλ = B l2 r 2 A dλ dr = l r 2 F 2 B ɛ, (23.17) A l2 r ɛ. (23.18) 2 dr A(r) r 2. (23.19) F 2 B(r)l 1 2 r ɛ 2 l 2 With this we can find the trajectory φ = φ(r) in the orbital plane. (Massive particles: 2 integration constants F 2 l 2 and ɛ l 2, massless particles: only F 2 l 2 ). Trajectory in Schwarzschild metric: Insert Schwarzschild metric: B(r) = A 1 (r) = 1 r S r = 1 2a r ṫ = dt dλ, ṙ = dr dλ, φ = dφ dλ. and write: Then with (23.8), (23.10), (23.12) and (23.15) we get θ = π ( 2, cṫ 1 2a ) = F, r 2 φ = l. (23.20) r Multiplying (23.15) with B and using AB = 1, we have The radial component can be written as with the effective potential (2a = 2GM c, ɛ = {c 2, 0}) 2 GM r V eff (r) = l 2 ṙ 2 2 aɛ r + l2 2r 2 al2 r 3 = F 2 ɛ = constant. (23.21) 2 ṙ V eff(r) = constant, (23.22) 2r 2 + l2 2r GMl2 2 c 2 r (m 0) 3 GMl2 c 2 r 3 (m = 0). (23.23) A formal solution r = r(λ) of (23.22) is given through the following integral : dr λ = ± 2(constant Veff (r)). (23.24) Due to the 1 r term (relativistic), this is an elliptical integral which has to be solved numerically. 3 For small values of r, centrifugal potential term dominates (as long as l is not too small), then for even smaller values of r the attractive relativistic term takes over: 23 l r v l2 r 2 v 2 24 v c 10 4 v2 c

78 23 GENERAL EQUATIONS OF MOTION V eff collision circular unstable scattering r ellipse with precession circular stable Figure 11: Effective potential for massive particles in Schwarzschild metric GM r l 2 c 2 r 2 GM r 1 Eq. (23.22) differs from the non-relativistic case by an additional r 3 by terms of order v2 c. 2 Observations: v 2 c 2 (23.25) term and ṙ = dr dτ dr differs from dt Where V eff has a minimum there are bounded solutions, however due to the relativistic effects there will be small deviations from the elliptical orbits (precession of the perihelion). As a special case, with ṙ = 0, the circular orbit is a possible solution (in which case the constant in (23.22) is equal to the value of V eff at its minimum). The solution at the maximum of V eff is an unstable circular orbit. If the constant is positive one gets non-bounded trajectories (corresponding to hyperbolic solutions in the non-relativistic case). 77

79 23 GENERAL EQUATIONS OF MOTION If the constant is larger than the maximum value of the potential, the particle falls into the center. At minimum and maximum we have dv eff dr = 0. For m 0 we get c 2 l 2 r2 2 r r S + 3 = 0. (23.26) In order to have two real solutions, we need 3c2 l < 1. That means 2 r 2 S l l crit = 3 r S c. (23.27) For l l crit, the angular momentum barrier gets smaller and smaller until the maximum and minimum fall together for l = l crit. For l < l crit, the potential decreases monotonically for r 0. V eff collision circular unstable scattering r Figure 12: Effective potential for massless particles in Schwarzschild metric 78

80 23 GENERAL EQUATIONS OF MOTION Here both terms are proportional to l 2, thus the shape of V eff does not depend on l. At r max = 3 2 r S the potential has a maximum. At r max the photons can move on a circular orbit, which is unstable. If the constant in (23.22) is smaller than V eff (r max ) then the incoming photon will be scattered, whereas if the constant is bigger the photon will be absorbed at the center for r r S the Schwarzschild solution is not applicable 79

81 24 LIGHT DEFLECTION Part VI Applications of General Relativity 24 Light deflection The trajectory r = r(φ) of a photon in the gravitational field is given by (23.19) (ɛ = 0): r φ(r) = φ(r 0 ) + r 0 d r r 2 A( r). (24.1) F 2 B( r)l 1 r 2 2 φ r 0 φ r Sun light ray Figure 13: Deflection of light by the Sun We will now show that light is deflected by a massive body, carrying through calculations for the Sun. In fig. 13, the following quantities are defined: light is deflected by φ and r 0 is the minimal distance (or impact parameter) from the Sun. For simplification we assume also r 0 r S. As starting point of the integration we choose the minimum distance r 0, where we set φ(r 0 ) = 0. Going from r 0 till r the angle changes by φ( ). Along the drawn trajectory the radial vector turns by 2φ( ). If the trajectory would be a straight line, then 2φ( ) = π. Thus φ = π π = 0 for a straight line and in general (φ(r 0 ) = 0): φ = 2φ( ) π. (24.2) At r 0, r(φ) is a minimum, thus ( dr dφ) r 0 = 0. (24.3) 80

82 24 LIGHT DEFLECTION light ray φ(r) φ( ) = π 2 Sun φ(r) φ( ) = π 2 Figure 14: Non-deflected ray of light From (24.3) we get with (23.17) and (23.18) the condition F 2 l 2 = B(r 0). (24.4) This way we can eliminate the constants F and l in terms of r 0 with (24.1): dr φ( ) = A(r) r B(r 0) r 2 B(r) 1. (24.5) r0 2 r 0 r 2 0 Let us compute the integral by inserting the Robertson expansion A(r) = 1+γ 2a r, B(r) = 1 2a r a = rs 2 = GM c ). We keep terms up to a 2 r with B(r 0 ) r 2 [ ( B(r) r0 2 1 r2 1 r a r 1 )] 1 r 0 [ r 2 = r 2 0 ] [ 1 1 ] 2ar. r 0 (r + r 0 ) (with We get using 1 + x = 1 + x 2, φ( ) = r 0 [ dr r2 r0 2 arccos r 0 r ( 1 + γ a ) r + ar r 0 (r + r 0 ) ( r0 ) + γ a r2 r0 2 + a ] r r0 r r 0 r r 0 r + r 0 r 0 = π 2 + γ a r 0 + a r 0. (24.6) With (24.2) we get φ = 4a r 0 ( ) 1 + γ 2 = 2r S r 0 ( 1 + γ 2 ). (24.7) 81

83 24 LIGHT DEFLECTION For general relativity, γ = 1, r S = 2GM c 2 and thus φ = 2r S r 0 For a light ray which just grazes the surface of the Sun (r 0 = R = km) we get (π = ): ( ) 1 + γ φ = (24.8) 2 On May 29, 1919, an eclipse allowed experimental confirm of this result. Figure 15: Gravitational lensing in the Abel 2218 galaxy cluster 26 cheating with Newton s theory gives half this result that is