Lectures on Gravitation

IFUP–TH/2017 LECTURES ON GRAVITATION Pietro Menotti Dipartimento di Fisica, Universitàdi Pisa, Italy March 2017 arXiv:1703.05155v1 [gr-qc] 15 Mar 2017 2 3 INDEX Foreword 7 Acknowledgments 7 Chapter 1: The Lorentz group 9 1. The group of inertial transformations: the three solutions 2. Lorentz transformations 3. SL(2,C) is connected 4. SL(2,C) is the double covering of the restricted Lorentz group 5. The restricted Lorentz group is simple 6. SL(2,C) is simply connected 7. Little groups 8. There are no non-trivial unitary finite dimensional representation of the Lorentz group 9. The conjugate representation 10. Relativistically invariant field equations 11. The Klein-Gordon equation 12.The Weyl equation 13. The Majorana mass 14. The Dirac equation 15. The Rarita- Schwinger equation 16. Transformation properties of quantum fields Chapter 2: Equivalence principle and the path to general relativity 35 1. Introduction 2. Motion of a particle in a gravitational field 3. Hamiltonian of a particle in a gravitational field Chapter 3: Manifolds 41 1. Introduction 2. Mappings, pull-back and push-forward 3. Vector fields and diffeomorphisms 4. The Lie derivative 5. Components of the Lie derivative 4 6. Commuting vector fields 7. Stokes’s theorem Chapter 4: The covariant derivative 49 1. Introduction 2. Transformation properties of the connection 3. Parallel transport 4. Geodesics 5. Curvature 6. Torsion 7. The structure equations 8. Non-abelian gauge fields on differential manifolds 9. Non compact electrodynamics 10. Meaning of torsion 11. Vanishing of the torsion 12. Metric structure 13. Isometries 14. Metric compatibility 15. Contracted Bianchi identities 16. Orthonormal reference frames 17. Compact electrodynamics 18. Symmetries of the Riemann tensor 19. Sectional curvature and Schur theorem 20. Spaces of constant curvature 21. Geometry in diverse dimensions 22. Flat and conformally flat spaces 23. The non abelian Stokes theorem 24. Vanishing of torsion and of the Riemann tensor 25. The Weyl-Schouten theorem 26. Hodge * operation 27. The current 28. Action for gauge fields Chapter 5: The action for the gravitational field 79 1. The Hilbert action 2. Einstein Γ Γ action − 5 3. Palatini first order action 4. Einstein-Cartan formulation 5. The energy momentum tensor 6. Coupling of matter to the gravitational field 7. Dimensions of the Hilbert action 8. Coupling of the Dirac field to the gravitational field 9. The field equations 10. The generalized energy- momentum tensor 11. Einstein-Eddington affine theory Chapter 6: Submanifolds 101 1. Introduction 2. Extrinsic curvature 3. The trace-K action 4. The boundary terms in the Palatini formulation 5. The boundary terms in the Einstein-Cartan formulation 6. The Gauss and Codazzi relations Chapter 7: Hamiltonian formulation of gravity 111 1. Introduction 2. Constraints and dynamical equations of motion 3. The Cauchy problem for the Maxwell and Einstein equations in vacuum 4. The action in canonical form 5. The Poisson algebra of the constraints 6. Fluids 7. The energy conditions 8. The cosmological constant Chapter 8: The energy of a gravitational system 125 1. The non abelian charge conservation 2. Background Bianchi identities 3. The superpotential 4. The positive energy theorem Chapter 9: The linearization of gravity 135 6 1. Introduction 2. The Fierz-Pauli equation 3. The massless case 4. The VDVZ discontinuity 5. Classical general relativity from field theory in Lorentz space Chapter 10: Quantization in external gravitational fields 149 1. Introduction 2. Scalar product of two solutions 3. The scalar product for general stationary metric 4. Schwarzschild solution and its maximal extension 5. Proper acceleration of stationary observers in Schwarzschild metric and surface gravity 6. The accelerated detector 7. Elements of causal structure of space-time 8. Rindler metric 9. Eigenfunctions of K in the Rindler metric 10. The Bogoliubov transformation 11. Quantum field theory in the Rindler 12. Rindler space with a reflecting wall 13. Resolution in discrete wave packets 14. Hawking radiation in 4-dimensional Schwarzschild space Chapter 11: N = 1 Supergravity 189 1. The Wess-Zumino model 2. Noether currents 3. N = 1 supergravity Appendix: Derivation of the hamiltonian equations of motion and ofthePoissonalgebraoftheconstraints 201 7 FOREWORD The suggested textbook for this course are, in the order R.M. Wald, General Relativity, The Universit of Chicago Press, Chicago and London, 1984 referred to as [Wald] S.W.Hawking and G.F.R. Ellis, The large scale structure of space-time, Cambridge Uni- versity Press, 1974 referred to as [HawkingEllis] S. Weinberg, Gravitation and Cosmology, John Wiley and Sons, New York, 1972, referred to as [WeinbergGC] S. Weinberg, The quantum theory of fields, Cambridge University Press, 1995, referred to as [WeinbergQFT] Reference to other books and papers are given at the end of each section. ACKNOWLEDGMENTS I am grateful to Damiano Anselmi, Giovanni Morchio, Carlo Rovelli and Domenico Sem- inara for useful discussions. 8 Chapter 1 The Lorentz group 1.1 The group of inertial transformations: the three solutions In this section we examine a group theoretical argument which shows that the Lorentz transformations are a solution of a symmetry problem in which the independence of the velocity of light from the inertial frame does not play a direct role (see [1],[2] and reference therein). Actually there might be no signal with such a property and still Lorentz transformation would maintain their validity. As we shall see the solutions of the symmetry problem are three: One is Galilei transformations, the other is Lorentz transformations while the third solution has to be discarded of physical grounds. We work for simplicity in 1 + 1 dimensions (x, t). Extension to 1 + n dimensions is trivial. We accept Galilean invariance as a symmetry group in the following sense: in addition to space-time translations we assume that the equivalent frames form a continuous family which depends on a single continuous parameter α. In setting up the rods and synchronizing the clocks we must respect homogeneity; e.g. for synchronizing the clocks we can use any signal (elastic waves, particles etc.) provided we divide by 2 the time of come back. The most general transformation for the coordinates of the events is x′ = f(x, t, α) t′ = g(x, t, α). We require: 9 10 CHAPTER 1. THE LORENTZ GROUP 1) Homogeneity of space time under space time translations i.e. in ∂f ∂f dx′ = dx + dt ∂x ∂t ∂g ∂g dt′ = dx + dt ∂x ∂t the partial derivatives do not depend on x and t. Thus we have a linear transformation. Choosing the origin of space and time properly we have x′ = a(α)x + b(α)t t′ = c(α)x + d(α)t. If v is the speed of O′ in the frame L we have 0= a(α)x + b(α)t = a(α)vt + b(α)t; i.e. b(α)= va(α). − From now on we shall use as parameter v instead of α i.e. write x′ = a(v)x + b(v)t t′ = c(v)x + d(v)t. and the equivalent frames move one with respect to the other with constant speed. 2) Isotropy of space Inverting x y = x and x′ y′ = x′ we admit to have also a transformation of the → − → − group with some speed u y′ = a(u)y + b(u)t t′ = c(u)y + d(u)t. Substituting the previous into the above we get a(v)= a(u); b(v)= b(u); c(v)= c(u); d(v)= d(u) − − and thus u = v; a = a( v ); d = d( v ); c( v)= c(v). − | | | | − − 3) Group law The equivalence of all inertial frames implies that the transformation form a group. Thus there must exist the inverse (unique) a(w) wa(w) a(v) va(v) − − = I c(w) d(w) ! c(v) d(v) ! 1.1. THE GROUP OF INERTIAL TRANSFORMATIONS: THE THREE SOLUTIONS11 which provides a(v) w = v = vρ(v) − d(v) − where a(v) ρ(v) . ≡ d(v) We obtain also a(v) d(w) = . c(v) − c(w) 1 1 But as A− A = I = AA− we must also have a(w) d(v) = c(w) − c(v) and dividing a(v) d(w) 1 = ; i.e. ρ(w)= d(v) a(w) ρ(v) that is we have reached the functional equation 1 ρ( vρ(v)) = = ρ(vρ(v)) − ρ(v) where in the last passage we took into account that a and d and thus ρ are even functions. To solve the equation set ζ(x) xρ(x)= ζ(x); ρ(x)= x and we have 1 ρ(ζ(v)) = ρ(v) i.e. v ρ((ζ(v)) = ζ(v) or ζ(ζ(v)) = v. (1.1) As a(0) = d(0) = 1 we have the additional information ζ(0) = 0 and ζ′(0) = 1. Expand in power series 2 3 ζ(x)= x + a2x + a3x + ... Eq.(1.1) becomes x + a x2 + a x3 + + a (x + a x2 + a x3 + ... )2 + a (x + a x2 + a x3 + ... )3 + = x 2 3 ··· 2 2 3 3 2 3 ··· which gives 2a2 =0. 12 CHAPTER 1. THE LORENTZ GROUP Then the equation becomes a x3 + + a (x + a x3 + ... )3 + a (x ++a x3 + ... )4 + =0 3 ··· 3 3 4 3 ··· which gives 2a3 =0 and so on. Finally we have ζ(v)= v, with the consequences a(v)= d(v) and w = v. − For a proof of ζ(v)= v without recourse to a power series expansion see [1]. We now exploit the relation a(v) va(v) a(v) va(v) − = I c(v) a(v) ! c(v) a(v) ! − obtaining a2(v)+ va(v)c(v)=1. We notice that a(v) va(v) 1 a2(v) − SL(2, R).

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