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This book presents basic General Relativity and provides a basis for understanding and using the fundamental theory. General Relativity is a beautiful geometric theory, simple in its mathematical formulation. It leads to numerous consequences with striking physical interpretations: gravitational waves, black holes, cosmological models, and so on. The first part of the book outlines the fundamentals of the subject. Chapters in this part look at Riemannian and Lorentzian geometry, Special and General Relativity, the Einstein equations, the Schwarzschild spacetime, black holes, and cosmology. The second part presents a number of more advanced topics such as general Einstein spacetimes, the Cauchy problem, relativistic fluids, and Relativistic Kinetic Theory.

We review recent progress in massive gravity. We start by showing how different theories of massive gravity emerge from a higher-dimensional theory of general relativity, leading to the Dvali-Gabadadze-Porrati model (DGP), cascading gravity, and ghost-free massive gravity. We then explore their theoretical and phenomenological consistency, proving the absence of Boulware-Deser ghosts and reviewing the Vainshtein mechanism and the cosmological solutions in these models. Finally, we present alternative and related models of massive gravity such as new massive gravity, Lorentz-violating massive gravity and non-local massive gravity.

The Laser Interferometer Space Antenna (LISA) has the potential to reveal wonders about the fundamental theory of nature at play in the extreme gravity regime, where the gravitational interaction is both strong and dynamical. In this white paper, the Fundamental Physics Working Group of the LISA Consortium summarizes the current topics in fundamental physics where LISA observations of gravitational waves can be expected to provide key input. We provide the briefest of reviews to then delineate avenues for future research directions and to discuss connections between this working group, other working groups and the consortium work package teams. These connections must be developed for LISA to live up to its science potential in these areas.

A bstract We compute the scattering amplitude for classical black-hole scattering to third order in the Post-Minkowskian expansion, keeping all terms needed to derive the scattering angle to that order from the eikonal formalism. Our results confirm a conjectured relation between the real and imaginary parts of the amplitude by Di Vecchia, Heissenberg, Russo, and Veneziano, and are in agreement with a recent computation by Damour based on radiation reaction in general relativity.

A bstract We propose a method to compute the scattering angle for classical black hole scattering directly from two massive particle irreducible diagrams in a heavy-mass effective field theory approach to general relativity, without the need of subtracting iteration terms. The amplitudes in this effective theory are constructed using a recently proposed novel colour-kinematic/double copy for tree-level two-scalar, multi-graviton amplitudes, where the BCJ numerators are gauge invariant and local with respect to the massless gravitons. These tree amplitudes, together with graviton tree amplitudes, enter the construction of the required D -dimensional loop integrands and allow for a direct extraction of contributions relevant for classical physics. In particular the soft/heavy-mass expansions of full integrands is circumvented, and all iterating contributions can be dropped from the get go. We use this method to compute the scattering angle up to third post-Minkowskian order in four dimensions, including radiation reaction contributions, also providing the expression of the corresponding integrand in D dimensions.

A bstract We extend the Post-Minkowskian (PM) effective field theory (EFT) approach to incorporate conservative and dissipative radiation-reaction effects in a unified framework. This is achieved by implementing the Schwinger-Keldysh “in-in” formalism and separating conservative and non-conservative terms according to the formulation in [ 1 ], which we show promotes Feynman’s i 0-prescription and cutting rules to a prominent role at the classical level. The resulting integrals, involving both Feynman and retarded propagators, can be bootstrapped to all orders in the velocity via differential equations with boundary conditions including potential and radiation modes. As a paradigmatic example we provide an ab initio derivation of the classical solution to the scattering problem in general relativity to O ( G 3 ). For the sake of completeness, we also reproduce the leading order radiation-reaction effects in classical electrodynamics.

A bstract It was shown recently that the static tidal response coefficients, called Love numbers, vanish identically for Kerr black holes in four dimensions. In this work, we confirm this result and extend it to the case of spin-0 and spin-1 perturbations. We compute the static response of Kerr black holes to scalar, electromagnetic, and gravitational fields at all orders in black hole spin. We use the unambiguous and gauge-invariant definition of Love numbers and their spin-0 and spin-1 analogs as Wilson coefficients of the point particle effective field theory. This definition also allows one to clearly distinguish between conservative and dissipative response contributions. We demonstrate that the behavior of Kerr black hole responses to spin-0 and spin-1 fields is very similar to that of the spin-2 perturbations. In particular, static conservative responses vanish identically for spinning black holes. This implies that vanishing Love numbers are a generic property of black holes in four-dimensional general relativity. We also show that the dissipative part of the response does not vanish even for static perturbations due to frame-dragging.

A bstract We obtain the quadratic-in-spin terms of the conservative Hamiltonian describing the interactions of a binary of spinning bodies in General Relativity through O ( G 2 ) and to all orders in velocity. Our calculation extends a recently-introduced framework based on scattering amplitudes and effective field theory to consider non-minimal coupling of the spinning objects to gravity. At the order that we consider, we establish the validity of the formula proposed in [ 1 ] that relates the impulse and spin kick in a scattering event to the eikonal phase.

A bstract We exhibit an initial-value formulation of the worldline quantum field theory (WQFT) approach to the classical two-body problem in general relativity. We show that the Schwinger-Keldysh (in-in) formalism leads to purely retarded propagators in the evaluation of observables in the WQFT. Integration technology for retarded master integrals is introduced at third post-Minkowskian (3PM) order. As an application we compute the complete radiation-reacted impulse and radiated four momentum for the scattering of two non-spinning neutron stars including tidal effects at 3PM order, as well as the leading (2PM) far-field gravitational waveform.

A bstract We compute the next-to-leading order term in the scattering waveform of uncharged black holes in classical general relativity and of half-BPS black holes in N = 8 supergravity. We propose criteria, generalizing explicit calculations at next-to-leading order, for determining the terms in amplitudes that contribute to local observables. For general relativity, we construct the relevant classical integrand through generalized unitarity in two distinct ways, (1) in a heavy-particle effective theory and (2) in general relativity minimally-coupled to scalar fields. With a suitable prescription for the matter propagator in the former, we find agreement between the two methods, thus demonstrating the absence of interference of quantum and classically-singular contributions. The classical N = 8 integrand for massive scalar fields is constructed through dimensional reduction of the known five-point one-loop integrand. Our calculation exhibits novel features compared to conservative calculations and inclusive observables, such as the appearance of master integrals with intersecting matter lines and the appearance of a classical infrared divergence whose absence from classical observables requires a suitable definition of the retarded time.