Universit`a degli Studi di Napoli “Federico II” Dipartimento di Fisica “Ettore Pancini” PhD Thesis in “Fundamental and Applied Physics” Extreme Regimes in Quantum Gravity Supervisor: Candidate: Dr. Giampiero Esposito Emmanuele Battista Abstract The thesis is divided into two parts. In the first part the low-energy limit of quantum gravity is analysed, whereas in the second we deal with the high-energy domain. Inthefirstpart,byapplyingtheeffectivefieldtheorypointofviewtothequantizationofgeneral relativity, detectable, thoughtiny, quantumeffects intheposition ofNewtonian Lagrangian points oftheEarth-Moonsystemarefound. Inordertomakemorerealisticthequantumcorrectedmodel proposed, the full three-body problem where the Earth and the Moon interact with a generic massive bodyand the restricted four-bodyproblem involving the perturbativeeffects producedby thegravitational presenceoftheSunintheEarth-Moon systemarealsostudied. Afterthat, anew quantum theory having general relativity as its classical counterpart is analysed. By exploiting this framework, an innovative interesting prediction involving the position of Lagrangian points within the context of general relativity is described. Furthermore, the new pattern provides quantum corrections to the relativistic coordinates of Earth-Moon libration points of the order of few millimetres. The second part of the thesis deals with the Riemannian curvature characterizing the boosted form assumed by the Schwarzschild-de Sitter metric. The analysis of the Kretschmann invariant and the geodesic equation shows that the spacetime possesses a “scalar curvature singularity” within a 3-sphere and that it is possible to define what we here call “boosted horizon”, a sort of elastic wall where all particles are surprisingly pushed away, suggesting that such “boosted geometries” are ruled by a sort of “antigravity effect”. Eventually, the equivalence with the coordinate shift method is invoked in order to demonstrate that all δ2 terms appearing in the Riemann curvature tensor give vanishing contribution in distributional sense. iii ¨ı¿¼ Contents List of Figures vii List of Tables ix Introduction xi Part I: the low-energy limit 1 General relativity as an effective field theory 3 1.1 The quantization of general relativity . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.1.1 Three approaches to quantizing gravity . . . . . . . . . . . . . . . . . . . . 4 1.1.2 Feynman rules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6 1.1.3 Ultraviolet divergences . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11 1.2 The effective field theory of gravitation . . . . . . . . . . . . . . . . . . . . . . . . . 18 1.2.1 The energy expansion of the gravitational action . . . . . . . . . . . . . . . 19 1.2.2 The path integral . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 1.3 The leading quantum corrections to the Newtonian potential . . . . . . . . . . . . 23 1.3.1 Three ways to define a potential . . . . . . . . . . . . . . . . . . . . . . . . 24 1.3.2 One-particle reducible potential . . . . . . . . . . . . . . . . . . . . . . . . . 27 1.3.3 Scattering and bound-states potential . . . . . . . . . . . . . . . . . . . . . 34 2 The restricted three-body problem in effective field theories of gravity 43 2.1 Restricted three-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 2.1.1 Quantum corrected Lagrangian . . . . . . . . . . . . . . . . . . . . . . . . . 45 2.1.2 Lyapunov definition of stability . . . . . . . . . . . . . . . . . . . . . . . . . 50 2.1.3 Derivatives of the full potential . . . . . . . . . . . . . . . . . . . . . . . . . 53 2.1.4 Non-collinear Lagrangian points . . . . . . . . . . . . . . . . . . . . . . . . 60 2.1.5 Alternative route to quintic equations . . . . . . . . . . . . . . . . . . . . . 64 2.1.6 Collinear Lagrangian points . . . . . . . . . . . . . . . . . . . . . . . . . . . 78 2.2 Motion in the neighbourhood of a given motion . . . . . . . . . . . . . . . . . . . . 82 2.2.1 Variational equations . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 82 2.2.2 Solution of variational equations . . . . . . . . . . . . . . . . . . . . . . . . 84 2.2.3 The case of constant coefficients and the concept of first-order stability . . 87 2.2.4 The case of periodic coefficients . . . . . . . . . . . . . . . . . . . . . . . . . 88 2.2.5 The equation defining the characteristic exponents . . . . . . . . . . . . . . 94 2.2.6 An important theorem by Poincar´e . . . . . . . . . . . . . . . . . . . . . . . 96 2.2.7 Variation from Hamiltonian equations . . . . . . . . . . . . . . . . . . . . . 97 2.2.8 Stability analysis of Lagrangian points . . . . . . . . . . . . . . . . . . . . . 100 iv Contents 2.2.9 Displaced periodic orbits for a solar sail . . . . . . . . . . . . . . . . . . . . 102 3 Quantum description of more detailed Newtonian models 113 3.1 Full three-body problem in effective field theories of gravity . . . . . . . . . . . . . 113 3.1.1 The classical integrals . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114 3.1.2 Reduced form of the equations of motion . . . . . . . . . . . . . . . . . . . 117 3.1.3 Periodic solutions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 120 3.1.4 General solution of the quantum corrected variational equations . . . . . . 123 3.1.5 A scheme for the resolution of variational equations . . . . . . . . . . . . . 130 3.2 Restricted four-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 131 3.2.1 Equations of motion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 132 3.2.2 The solar radiation pressure and the linear stability at L . . . . . . . . . . 136 4 4 Towards a new quantum theory 143 4.1 Theoretical predictions of general relativity . . . . . . . . . . . . . . . . . . . . . . 143 4.1.1 Post-Newtonian approximation . . . . . . . . . . . . . . . . . . . . . . . . . 144 4.1.2 Corrections on the position of Lagrangian points . . . . . . . . . . . . . . . 146 4.2 The new quantum theory . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156 4.2.1 Quantum effects on Lagrangian points . . . . . . . . . . . . . . . . . . . . . 157 4.2.2 A possible choice of the quantum potential . . . . . . . . . . . . . . . . . . 160 4.3 Laser ranging techniques . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 161 Part II: the high-energy limit 5 Boosted spacetimes 167 5.1 The boosting procedure . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167 5.1.1 Aichelburg and Sexl method . . . . . . . . . . . . . . . . . . . . . . . . . . 168 5.1.2 Boosted Schwarzschild-de Sitter solution . . . . . . . . . . . . . . . . . . . . 170 5.1.3 four-dimensional form of the boosted metric . . . . . . . . . . . . . . . . . . 175 5.1.4 Coordinate transformations . . . . . . . . . . . . . . . . . . . . . . . . . . . 176 5.2 Riemann curvature of the boosted Schwarzschild-de Sitter spacetime . . . . . . . . 177 5.2.1 The Riemann curvature tensor . . . . . . . . . . . . . . . . . . . . . . . . . 177 5.2.2 Spacetime singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180 5.2.3 The Kretschmann invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . 182 5.2.4 Boosted horizon . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 184 5.3 The coordinate shift method . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187 5.3.1 Formal aspects . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 189 5.3.2 de Sitter and Schwarzschild-de Sitter backgrounds . . . . . . . . . . . . . . 191 Conclusions and open problems 199 Appendices 205 A Summary of Feynman rules for quantum gravity 207 B Useful integrals 211 C Asymptotic expansions 217 v Contents D Notes on the system x˙ = Ax 221 E The tetrad formalism 231 Bibliography 232 vi List of Figures 1.1 The four-graviton vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23 1.2 The one-loop correction to the four-graviton vertex . . . . . . . . . . . . . . . . . . 24 1.3 Vertex correction diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 25 1.4 Graviton vacuum polarization diagrams . . . . . . . . . . . . . . . . . . . . . . . . 25 1.5 The set of corrections included in the one-particle reducible potential . . . . . . . . 26 1.6 The gravitational interaction of two particles without vacuum polarization . . . . . 29 1.7 The Newtonian potential. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 1.8 Vertex correction diagram 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.9 Vertex correction diagram 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32 1.10 The box diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35 1.11 The crossed-box diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36 1.12 The triangle diagrams . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37 1.13 The double-seagull diagram . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38 1.14 The set of vertex correction diagrams contributing to the scattering potential . . . 39 1.15 The vacuum polarization diagrams contributing to the scattering potential . . . . . 41 2.1 The synodic coordinate system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46 2.2 The Lagrangian points of the Earth-Moon system in Newtonian theory . . . . . . . 50 ∂U(x,y) 2.3 Plot of . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ∂x (cid:12)λ=0 ∂2U(x,y)(cid:12) 2.4 Plot of (cid:12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55 ∂x2 (cid:12) (cid:12)λ=0 ∂2U(x,y)(cid:12) 2.5 Plot of (cid:12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 (cid:12) ∂x∂y (cid:12)λ=0 ∂2U(x,y)(cid:12) 2.6 Plot of (cid:12) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56 ∂y2 (cid:12) (cid:12)λ=0 2.7 Plot of the potent(cid:12)ial U(x,0) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57 (cid:12) 2.8 Schematic geomet(cid:12)ry of the restricted three-body problem involving a solar sail . . 104 2.9 Time evolution of the function ξ(t) defined in Eq. (2.362) for L in the Newtonian 4 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.10 Time evolution of the function η(t) defined in Eq. (2.363) for L in the Newtonian 4 case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107 2.11 Periodic orbits at linear order around the Lagrangian point L in Newtonian theory 108 4 2.12 Time evolution of the function ξ(t) defined in Eq. (2.362) for L in the quantum 4 corrected model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 108 2.13 Time evolution of the function η(t) defined in Eq. (2.363) for L in the quantum 4 corrected model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 vii List of Figures 2.14 Periodic orbits at linear order around the Lagrangian point L in the quantum 4 corrected model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109 2.15 Periodic orbits at linear order around the Lagrangian point L in Newtonian theory 110 2 2.16 Periodic orbits at linear order around the Lagrangian point L in the quantum 2 corrected model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111 3.1 Schematic set-up of the full three-body problem . . . . . . . . . . . . . . . . . . . . 118 3.2 Parametric plot of the spacecraft motion about L in the classical restricted four- 4 body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.3 Parametric plot of the spacecraft motion aboutL in the z-direction in the classical 4 restricted four-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 134 3.4 Parametric plot of the spacecraft motion about L in the quantum corrected re- 4 stricted four-body problem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.5 ParametricplotofthespacecraftmotionaboutL inthez-direction inthequantum 4 corrected restricted four-body problem . . . . . . . . . . . . . . . . . . . . . . . . . 136 3.6 Parametric plot in the classical restricted four-body problem of the spacecraft mo- tion about L in the presence of the solar radiation pressure . . . . . . . . . . . . . 137 4 3.7 Parametricplotinthequantumcorrectedrestrictedfour-bodyproblemofthespace- craft motion about L in the presence of the solar radiation pressure . . . . . . . . 138 4 3.8 PlotsofthespacecraftmotionaboutL intheclassicalrestrictedfour-bodyproblem 4 and with different initial velocities . . . . . . . . . . . . . . . . . . . . . . . . . . . 139 3.9 Plots of the spacecraft motion about L in the quantum corrected restricted four- 4 body problem and with different initial velocities . . . . . . . . . . . . . . . . . . . 140 3.10 Plot of the force required to induce stability at L both in the classical and in the 4 quantum corrected restricted four-body problem . . . . . . . . . . . . . . . . . . . 142 4.1 Schematic set-up of Satellite/Lunar Laser Ranging technique . . . . . . . . . . . . 162 5.1 LIGO detection of the gravitational wave event “GW150914” . . . . . . . . . . . . 168 5.2 Hyperboloid illustrating de Sitter space . . . . . . . . . . . . . . . . . . . . . . . . 172 5.3 Contour plot of the Kretschmann invariant of the boosted Schwarzschild-de Sitter metric . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 183 5.4 Numerical solution of the geodesic equation I . . . . . . . . . . . . . . . . . . . . . 185 5.5 Numerical solution of the geodesic equation II . . . . . . . . . . . . . . . . . . . . . 185 5.6 Numerical solution of the geodesic equation III . . . . . . . . . . . . . . . . . . . . 186 5.7 Numerical solution of the geodesic equation IV . . . . . . . . . . . . . . . . . . . . 187 5.8 Numerical solution of the geodesic equation V . . . . . . . . . . . . . . . . . . . . . 188 5.9 Numerical solution of the geodesic equation VI . . . . . . . . . . . . . . . . . . . . 188 A.1 The scalar propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 A.2 The graviton propagator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 207 A.3 The three-graviton vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 208 A.4 The two scalar-one graviton vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 A.5 The two scalar-two graviton vertex . . . . . . . . . . . . . . . . . . . . . . . . . . . 209 viii List of Tables 1.1 The values assumed by κ and κ in the three different potentials . . . . . . . . . . 42 1 2 2.1 Quantum details concerning non-collinear Lagrangian points . . . . . . . . . . . . . 63 2.2 Quantum corrections on the position of Newtonian non-collinear Lagrangian points 64 2.3 Values of λ , λ , and λ for Eq. (2.81) in the case of one-particle reducible potential 73 3 2 1 2.4 Values of λ , λ , and λ for Eq. (2.81) in the case of scattering potential . . . . . . 73 3 2 1 2.5 Values of λ , λ , and λ for Eq. (2.81) in the case of bound-states potential . . . . 73 3 2 1 2.6 Values of λ , λ , and λ for Eq. (2.89) in the case of one-particle reducible potential 74 3 2 1 2.7 Values of λ , λ , and λ for Eq. (2.89) in the case of scattering potential . . . . . . 74 3 2 1 2.8 Values of λ , λ , and λ for Eq. (2.89) in the case of bound-states potential . . . . 74 3 2 1 2.9 Quantum details concerning collinear Lagrangian points . . . . . . . . . . . . . . . 81 2.10 Quantum corrections on the position of Newtonian collinear Lagrangian points . . 81 4.1 GeneralrelativitycorrectionsonthepositionofNewtoniannon-collinearLagrangian points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 152 4.2 General relativity details of Lagrangian points . . . . . . . . . . . . . . . . . . . . . 154 4.3 General relativity corrections on the position of Newtonian collinear Lagrangian points . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 155 4.4 Distances from the Earth and planar coordinates of the planetoid at all Lagrangian points in the new quantum regime . . . . . . . . . . . . . . . . . . . . . . . . . . . 158 4.5 Quantum corrections on the relativistic position of Lagrangian points . . . . . . . 159 5.1 Location of the “boosted horizon” as a function of the boost velocity v . . . . . . . 186 5.2 General relativity corrections on the position of Newtonian Lagrangian points for the Sun-Earth system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201 5.3 General relativity corrections on the position of Newtonian Lagrangian points for the Sun-Jupiter system . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 202 ix