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The MICROSCOPE experiment was designed to test the weak equivalence principle in space, by comparing the low-frequency dynamics of cylindrical "free-falling" test masses controlled by electrostatic forces. We use data taken during technical sessions aimed at estimating the electrostatic stiffness of MICROSCOPE's sensors to constrain a short-range Yukawa deviation from Newtonian gravity. We take advantage of the fact that in the limit of small displacements, the gravitational interaction (both Newtonian and Yukawa-like) between nested cylinders is linear, and thus simply characterised by a stiffness. By measuring the total stiffness of the forces acting on a test mass as it moves, and comparing it with the theoretical electrostatic stiffness (expected to dominate), it is a priori possible to infer constraints on the Yukawa potential parameters. However, we find that measurement uncertainties are dominated by the gold wires used to control the electric charge of the test masses, though their related stiffness is indeed smaller than the expected electrostatic stiffness. Moreover, we find a non-zero unaccounted for stiffness that depends on the instrument's electric configuration, hinting at the presence of patch-field effects. Added to significant uncertainties on the electrostatic model, they only allow for poor constraints on the Yukawa potential. This is not surprising, as MICROSCOPE was not designed for this measurement, but this analysis is the first step to new experimental searches for non-Newtonian gravity in space., Accepted for publication in CQG's MICROSCOPE special issue

We consider the entropy associated with the large-scale structure of the Universe in the linear regime, where the Universe can be described by a perturbed Friedmann-Lema\^itre spacetime. In particular, we compare two different definitions proposed in the literature for the entropy using a spatial averaging prescription. For one definition, the entropy of the large-scale structure for a given comoving volume always grows with time, both for a CDM and a $\Lambda$CDM model. In particular, while it diverges for a CDM model, it saturates to a constant value in the presence of a cosmological constant. The use of a light-cone averaging prescription in the context of the evaluation of the entropy is also discussed., Comment: 10 pages, 4 figures. Presentation improved, typos corrected, previous subsection III.B merged with subsection II.C, comments, clarifications and a reference added. Version accepted for publication in GRG

This article is dedicated to the use the MICROSCOPE mission's data to test chameleon theory of gravity. We take advantage of the technical sessions aimed to characterize the electrostatic stiffness of MICROSCOPE's instrument intrinsic to its capacitive measurement system. Any discrepancy between the expected and measured stiffness may result from unaccounted-for contributors, i.e., extra forces. This work considers the case of chameleon gravity as a possible contributor. It was previously shown that, in situations similar to these measurement sessions, a chameleon fifth force appears and acts as a stiffness for small displacements. The magnitude of this new component of the stiffness is computed over the chameleon's parameter space. It allows us to derive constraints by excluding any force inconsistent with the MICROSCOPE data. As expected---since MICROSCOPE was not designed for the purpose of such an analysis---these new bounds are not competitive with state-of-the-art constraints, but they could be improved by a better estimation of all effects at play in these sessions. Hence, our work illustrates this novel technique as a new way of constraining fifth forces.

Nuclear physics experiments give reaction rates that, via modelling and comparison with primordial abundances, constrain cosmological parameters. The error bars of a key reaction, \dpg, were tightened in 2020, bringing to light discrepancies between different analyses and calling for more accurate measurements of other reactions., 2 pages

This article describes the dynamics of a charge particle in a electromagnetic field in presence of a scalar fifth force. Focusing to the fifth force that would be induced by a chameleon field, the profile of which can be designed properly in the laboratory, it draws its physical effects on the cyclotron motion of a particle in a static and uniform magnetic field. The fifth force induces a drift of the trajectory that is estimated analytically and compared to numerical computations for profiles motivated by the ones of a chameleon field within two nested cylinders. The magnitude of the effect and the detectability of this drift are discussed to argue that this may offer a new experimental design to test small fifth force in the laboratory. More important, at the macroscopic level it induces a current that can in principle also be measured, and would even allow one to access the transverse profile of the scalar field within the cavity. In both cases, aligning the magnetic field with the local gravity field suppresses the effects of Newtonian gravity that would be several order larger than the ones of the fifth force otherwise and the Newtonian gravity of the cavity on the particle is also argued to be negligible. Given this insight, this experimental set-up, with its two effects -- on a single particle and at the macroscopic level -- may require attention to demonstrate its actual feasability in the laboratory., 16 pages, 21 figures

This article presents a new model-independent constraint for bouncing black hole geometries. Using the thin shell formalism, this constraint sets a bound on the minimal allowed radius of the time-like surface of the collapsing star at the bounce. It follows from this bound that the shell always bounces in an untrapped region or precisely on a trapping horizon. This constraint is of purely kinematical origin as it descends from the continuity of the metric between the geometries describing the interior and exterior of the collapsing object so that it is completely model-independent. The second part of this work investigates the conditions under which an effective extension of the Oppenheimer-Snyder collapse can describe consistent bouncing black hole solutions. We show that on top of the previous kinematical constraint, an additional dynamical condition has to be satisfied in order for the model to admit well-defined black-to-white hole solutions. As expected, this second condition turns out to be model-dependent. The resulting class of models describing bouncing compact objects are characterized by three parameters for the star (mass, initial radius and density) and two quantum parameters descending from the UV-completion of the exterior and interior geometries. The solution space contains (1) bouncing stars and (2) bouncing black holes and (3) a new class of astrophysical objects which alternate between a bouncing star and a bouncing black hole. Finally, we provide an explicit construction of a black-to-white hole bounce using the techniques of spatially closed LQC and discuss the novel properties induced by the presence of the inner horizon in this new framework. This generic framework lays down the foundation for interesting phenomenological investigations concerning the astrophysical properties of these objects, as well as a novel platform for further developments., V2 - 29 pages, 4 figures, Clarifications on the construction of the model added, typos corrected, matched the version accepted in JCAP

This article presents an effective quantum extension of the seminal Oppenheimer-Snyder (OS) collapse in which the singularity resolution is modeled using the effective dynamics of the spatially closed loop quantum cosmology. Imposing the minimal junction conditions, namely the Israel-Darmois conditions, we glue this bouncing LQC geometry to the classical vacuum exterior Schwarzschild geometry across a time-like thin-shell. Consistency of the construction leads to several major deviations from the classical OS collapse model. Firstly, no trapped region can form and the bounce occurs always above, or at most at the Schwarzschild radius. Secondly, the bouncing star discussed here admits an IR cut-off, additionally to the UV cut-off and corresponds therefore to a pulsating compact object. Thirdly, the scale at which quantum gravity effects become non-negligible is encoded in the ratio between the UV cut-off of the quantum theory and the IR cut-off, which in turn, encodes the minimal energy density $\rho_{\text{min}}$ of the star prior to collapse. This energy density is no more fixed by the mass and maximal radius as in the classical OS model, but is now a free parameter of the model. In the end, while the present model cannot describe a black-to-white hole bounce as initially suggested by the Planck star model, it provides a concrete realization of a pulsating compact object based on LQC techniques. Consistency of the model shows that its regime of applicability is restricted to Planckian relics while macroscopic stellar objects are excluded. This first minimal construction should serve as a platform for further investigations in order to explore the physics of bouncing compact objects within the framework of loop quantum cosmology., Comment: 30 pages, 7 figures, notation changed, clarifications added in Appendix, published version in PRD

This article proposes a generalization of the Oppenheimer-Snyder model which describes a bouncing compact object. The corrections responsible for the bounce are parameterized in a general way so as to remain agnostic about the specific mechanism of singularity resolution at play. It thus develops an effective theory based on a thin shell approach, inferring generic properties of such a UV complete gravitational collapse. The main result comes in the form of a strong constraint applicable to general UV models : if the dynamics of the collapsing star exhibits a bounce, it always occurs below, or at most at the energy threshold of horizon formation, so that only an instantaneous trapping horizon may be formed while a trapped region never forms. This conclusion relies solely on i) the assumption of continuity of the induced metric across the time-like surface of the star and ii) the assumption of a classical Schwarzschild geometry describing the (vacuum) exterior of the star. In particular, it is completely independent of the choice of corrections inside the star which leads to singularity-resolution. The present model provides thus a general framework to discuss bouncing compact objects, for which the interior geometry is modeled either by a classical or a quantum bounce. In the later case, our no-go result regarding the formation of trapped region suggests that additional structure, such as the formation of an inner horizon, is needed to build consistent models of matter collapse describing black-to-white hole bounces. Indeed, such additional structure is needed to keep quantum gravity effects confined to the high curvature regime, in the deep interior region, providing thus a new challenge for current constructions of quantum black-to-white hole bounce models., 20 pages, Published in JCAP

Recent measurements of the D(p,$\gamma)^3$He, nuclear reaction cross-section and of the neutron lifetime, along with the reevaluation of the cosmological baryon abundance from cosmic microwave background (CMB) analysis, call for an update of abundance predictions for light elements produced during the big-bang nucleosynthesis (BBN). While considered as a pillar of the hot big-bang model in its early days, BBN constraining power mostly rests on deuterium abundance. We point out a new $\simeq1.8\sigma$-tension on the baryonic density, or equivalently on the D/H abundance, between the value inferred on one hand from the analysis of the primordial abundances of light elements and, on the other hand, from the combination of CMB and baryonic oscillation data. This draws the attention on this sector of the theory and gives us the opportunity to reevaluate the status of BBN in the context of precision cosmology. Finally, this paper presents an upgrade of the BBN code PRIMAT., Comment: 10 pages, 4 figures. Accepted in MNRAS

This article investigates the properties of a scalar fifth force that arises in a scalar tensor-theory with a chameleon screening mechanism in the context of gravity space missions like the MICROSCOPE experiment. In such an experiment, the propagation of the chameleon field inside the nested cylinders of the experiment causes a fifth force when the cylinders are not perfectly co-axial. We propose a semi-analytic method to compute the field distribution and the induced fifth force and compare it to a full numerical simulation, in settings where the cylindrical symmetry is broken. The scaling of the fifth force with both the parameters of the model and the geometry of the experiment is discussed. We show that the fifth force is repulsive, hence adds a destabilizing stiffness that should be included in the force budget acting on the detector. This opens the way to a new method to constrain a scalar fifth force in screened models., Comment: Submitted to Phys. Rev. D

In the literature different approaches have been proposed to compute the anisotropies of the astrophysical gravitational wave background. The different expressions derived, although starting from our work Cusin, Pitrou, Uzan, Phys.Rev.D96, 103019 (2017) [1], seem to differ. This article compares the various theoretical expressions proposed so far and provides a separate derivation based on a Boltzmann approach. We show that all the theoretical formula in the literature are equivalent and boil down to the one of Ref. [1] when a proper matching of terms and integration by parts are performed. The difference between the various predictions presented for anisotropies in a cosmological context can only lie in the astrophysical modeling of sources, and neither in the theory nor in the cosmological description of the large scale structures. Finally we comment on the gauge invariance of expressions., Comment: 6 pages

We speculate on the development and availability of new innovative propulsion techniques in the 2040s, that will allow us to fly a spacecraft outside the Solar System (at 150 AU and more) in a reasonable amount of time, in order to directly probe our (gravitational) Solar System neighborhood and answer pressing questions regarding the dark sector (dark energy and dark matter). We identify two closely related main science goals, as well as secondary objectives that could be fulfilled by a mission dedicated to probing the local dark sector: (i) begin the exploration of gravitation's low-acceleration regime with a man-made spacecraft and (ii) improve our knowledge of the local dark matter and baryon densities. Those questions can be answered by directly measuring the gravitational potential with an atomic clock on-board a spacecraft on an outbound Solar System orbit, and by comparing the spacecraft's trajectory with that predicted by General Relativity through the combination of ranging data and the in-situ measurement (and correction) of non-gravitational accelerations with an on-board accelerometer. Despite a wealth of new experiments getting online in the near future, that will bring new knowledge about the dark sector, it is very unlikely that those science questions will be closed in the next two decades. More importantly, it is likely that it will be even more urgent than currently to answer them. Tracking a spacecraft carrying a clock and an accelerometer as it leaves the Solar System may well be the easiest and fastest way to directly probe our dark environment., White Paper submitted to ESA's Voyage 2050 call for papers. Reference updated

We show that the anisotropies of the astrophysical stochastic gravitational wave background in the mHz band have a strong dependence on the modelling of galactic and sub-galactic physics. We explore a wide range of self-consistent astrophysical models for stellar evolution and for the distribution of orbital parameters, all calibrated such that they predict the same number of resolved mergers to fit the number of detections during LIGO/Virgo O1+O2 observations runs. We show that different physical choices for the process of black hole collapse and cut-off in the black hole mass distribution give fractional differences in the angular power spectrum of anisotropies up to 50\% on all angular scales. We also point out that the astrophysical information which can be extracted from anisotropies is complementary to the isotropic background and individual mergers. These results underline the interest in the anisotropies of the stochastic gravitational wave background as a new and potentially rich field of research, at the cross-road between astrophysics and cosmology., Comment: 6 pages, 4 figures; accepted in MNRAS

While most strong-gravitational-lensing systems may be roughly modelled by a single massive object between the source and the observer, in the details all the structures near the light path contribute to the observed images. These additional contributions, known as line-of-sight effects, are non-negligible in practice. This article proposes a new theoretical framework to model the line-of-sight effects, together with very promising applications at the interface of weak and strong lensing. Our approach relies on the dominant-lens approximation, where one deflector is treated as the main lens while the others are treated as perturbations. The resulting framework is technically simpler to handle than the multi-plane lensing formalism, while allowing one to consistently model any sub-critical perturbation. In particular, it is not limited to the usual external-convergence and external-shear parameterisation. As a first application, we identify a specific notion of line-of-sight shear that is not degenerate with the ellipticity of the main lens, and which could thus be extracted from strong-lensing images. This result supports and improves the recent proposal that Einstein rings might be powerful probes of cosmic shear. As a second application, we investigate the distortions of strong-lensing critical curves under line-of-sight effects, and more particularly their correlations across the sky. We find that such correlations may be used to probe, not only the large-scale structure of the Universe, but also the dark-matter halo profiles of strong lenses. This last possibility would be a key asset to improve the accuracy of the measurement of the Hubble-Lema\^itre constant via time-delay cosmography., Comment: 39+14 pages, 15 figures. v2: discussion improved in sec. 3.2; figs. 4 and 15 updated. v3: minor typos corrected, matches published version; v4, v5: other typos corrected

When a luminous source is extended, its distortions by weak gravitational lensing are richer than a mere combination of magnification and shear. In a recent work, we proposed an elegant formalism based on complex analysis to describe and calculate such distortions. The present article further elaborates this finite-beam approach, and applies it to a realistic cosmological model. In particular, the cosmic correlations of image distortions beyond shear are predicted for the first time. These constitute new weak-lensing observables, sensitive to very-small-scale features of the distribution of matter in the Universe. While the major part of the analysis is performed in the approximation of circular sources, a general method for extending it to noncircular sources is presented and applied to the astrophysically relevant case of elliptic sources., 14+2 pages, 6 figures

This article explores the properties (amplitude and shape) of the angular power spectrum of the anisotropies of the astrophysical gravitational wave background (AGWB) focusing on the signatures of the astrophysical models describing sub-galactic physics. It demonstrates that while some parameters have negligible impact others, and in particular the stellar evolution models, the metallicity and the merger time delay distribution can result in relative differences of order 40% in the angular power spectrum of anisotropies in both the LIGO/Virgo and LISA frequency bands. It is also shown that the monopole and the anisotropic components of the AGWB are complementary and sensitive to different astrophysical parameters. It follows that AGWB anisotropies are a new observable with the potential to provide new astrophysical information that can not be accessed otherwise., 24 pages, 19 figures

The standard theory of weak gravitational lensing relies on the approximation that light beams are infinitesimal. Our recent work showed that the finite size of sources, and the associated light beams, can cause nonperturbative corrections to the weak-lensing convergence and shear. This article thoroughly investigates these corrections in a realistic cosmological model. The continuous transition from infinitesimal to finite beams is understood, and reveals that the previous results overestimated finite-beam effects due to simplistic assumptions on the distribution of matter in the Universe. In a KiloDegree Survey-like setting, finite-beam corrections to the cosmic shear remain subpercent, while percent-level corrections are only reached on subarcmin scales. This article thus demonstrates the validity of the infinitesimal-beam approximation in the interpretation of current weak-lensing data., Comment: 14+2 pages, 10 figures

This chapter reviews the basic ideas of electrostatics (Coulomb’s law) and magnetostatics (the Biot–Savart law). It studies the motion of a charge in a Coulomb field in detail. The chapter shows how the motion of a charge q in the Coulomb field of a charge Q held fixed at the origin of an inertial frame is governed by the Lorentz equation. Moreover, it can be solved like the Kepler problem discussed in the previous Book. Next, the chapter discusses the Rutherford scattering formula which established the ‘planetary’ model of the atom, the Bohr–Sommerfeld quantization which displayed the limits of the theory, and finally the spin coupling explaining the atomic fine structure.

This chapter shows how the Newtonian law of motion of a particle subject to a gradient force derived from a ‘potential energy’ can always be obtained from an extremal principle, or ‘principle of least action’. According to Newton’s first law, the trajectory representing the motion of a free particle between two points p1 and p2 is a straight line. In other words, out of all the possible paths between p1 and p2, the trajectory effectively followed by a free particle is the one that minimizes the length. However, even though the use of the principle of extremal length of the paths between two points gives the straight line joining the points, this does not mean that the straight-line path is traced with constant velocity in an inertial frame. Moreover, the trajectory describing the motion of a particle subject to a force is not uniform and rectilinear.

This chapter embarks on the study of Newton’s law of gravitation. It first discusses gravitational mass and inertial mass, a measure of the ‘resistance’ of the point particle to an applied force. The numerical value of the inertial mass of a body can in principle be obtained from collision experiments by assigning to a reference body a unit inertial mass of one kilogram or, more rigorously, one ‘inertial kilogram’. Next, the chapter considers the ratio of gravitational and inertial masses. It considers that, in the absence of friction, all objects, no matter what their inertial mass, or the nature of their constituents, or the internal energy or cohesive forces of their constituents, fall in the same way in an external gravitational field. Finally, this chapter studies Newton’s gravitational force and field, as well as the Poisson equation and the gravitational Lagrangian.

This chapter deals with the Schwarzschild metric. To find the gravitational potential U produced by a spherically symmetric object in the Newtonian theory, it is necessary to solve the Poisson equation ΔU = 4πGρ. Here, the matter density ρ and U depend only on the radial coordinate r and possibly on the time t. Outside the source the solution is U = –GM/r, where M = 4π ∫ ρr2dr is the source mass. In general relativity the problem is to find the ‘spherically symmetric’ spacetime solutions of the Einstein equations, and the analog of the vacuum solution U = –GM/r is the Schwarzschild metric.

This chapter recalls several relevant aspects of Newton’s theory of gravity, as well as Maxwell’s theory of electromagnetism, to describe the conceptual path that Albert Einstein followed in going from the theory of special relativity to general relativity. Looking at 1907 and beyond, the chapter shows that the ambition of Einstein was to construct a theory in which all reference frames (and therefore none) were privileged. Moreover, there were no longer any inertial forces, no ordering of Newton’s absolute space. Therefore, Einstein’s theory was one in which the laws of physics had the same form in all frames, inertial or not, so that no frame could be regarded as being privileged. In brief, he sought a theory of general relativity.

This chapter discusses the Schwarzschild black hole. It demonstrates how, by a judicious change of coordinates, it is possible to eliminate the singularity of the Schwarzschild metric and reveal a spacetime that is much larger, like that of a black hole. At the end of its thermonuclear evolution, a star collapses and, if it is sufficiently massive, does not become stabilized in a new equilibrium configuration. The Schwarzschild geometry must therefore represent the gravitational field of such an object up to r = 0. This being said, the Schwarzschild metric in its original form is singular, not only at r = 0 where the curvature diverges, but also at r = 2m, a surface which is crossed by geodesics.

This chapter considers Newton’s 1665 explanations of the dynamics in the laws governing the motion of a planet around the Sun, which were established by Johannes Kepler in 1618. The first law states that the motion is planar and the trajectories are ellipses. The second states that the area swept out by the radius vector per unit time is constant. Finally, the cube of the semi-major axis a is proportional to the square of the period P, a3 = (const)P2. The chapter begins with the reduced equations of motion before turning to the ellipses of Kepler. It then illustrates the Kepler problem in the Lagrangian formalism, as well as central forces.

This chapter covers the Kerr metric, which is an exact solution of the Einstein vacuum equations. The Kerr metric provides a good approximation of the spacetime near each of the many rotating black holes in the observable universe. This chapter shows that the Einstein equations are nonlinear. However, there exists a class of metrics which linearize them. It demonstrates the Kerr–Schild metrics, before arriving at the Kerr solution in the Kerr–Schild metrics. Since the Kerr solution is stationary and axially symmetric, this chapter shows that the geodesic equation possesses two first integrals. Finally, the chapter turns to the Kerr black hole, as well as its curvature singularity, horizons, static limit, and maximal extension.

This chapter observes the reaction force acting on a charge due to the radiation it emits. It also considers the related questions of renormalization and physical interpretation. Modifying the Lorentz equation introduced in Chapter 11 by including a radiation reaction force provides a heuristic method of describing the expected slowing of an accelerated charge in response to the radiation it emits. The chapter then goes on to describe the Abraham–Lorentz–Dirac reaction force, the counter-effect of the radiation of an accelerated charge on its motion. In addition, the chapter shows that a hydrogen atom, this time described by the Thomson model, is unstable in Maxwell theory.

This chapter presents the laws of motion of an ensemble of point masses forming a solid body whose shape is invariant, or a fluid whose shape can vary with time. It argues that an ensemble of point masses constitutes a solid if the distances between the points can be assumed constant. The chapter then provides examples of the motions of a solid. Finally, it demonstrates the Euler equations of fluid motion. Here, it states that a perfect fluid is characterized by its (inertial) mass density ρ(t, xⁱ), its pressure p(t, xⁱ) which phenomenologically describes its internal collisions, and a velocity field v(t, xⁱ) giving its velocity at xⁱ at time t.

This chapter continues the discussion from the previous chapter. The motion of a charge (m,q) in the field of another charge (m′,q′) and in its own field can now be studied in the lowest orders of the velocities directly using the equation of motion obtained in the preceding chapter. However, the features of this motion are revealed more easily by deriving them from the Darwin Lagrangian. This allows for a rigorous establishment of a balance between the energy radiated by the system and the mechanical energy lost by the system. The chapter concludes this general study of the electromagnetic radiation of a system of charges by outlining the ‘post-Minkowski’ approach based on iteration in the ‘coupling constant’ qq′ rather than in the velocities.

This chapter addresses the problem of fine-tuning the initial conditions of the previous chapter’s hot Big Bang model, so that the universe has the observed properties, as well as the problem of the origin of large-scale structure. It shows that these problems are related to each other, and can be solved by assuming a period of accelerated expansion in the earliest history of the universe. Since the 1980s, the general acceptance of this idea of a primordial inflationary phase can be considered as the third phase in the history of the development of relativistic cosmology. The chapter first outlines the issues with the hot Big Bang model: the flatness problem; the Big Bang horizon, and monopole problems; and the problem of the origin of the large-scale structure. It then provides a solution in the form of inflation, and goes on to discuss ‘chaotic’ inflation.

This chapter focuses on Cartan structure equations. It first introduces a 1-form and its exterior derivative, before turning to a study of the connection and torsion forms, thereby expressing the torsion as a function of the connection forms and establishing the torsion differential 2-forms. It then turns to the curvature forms drawn from Chapter 23 and Cartan’s second structure equation, along with the curvature 2-forms. It also studies the Levi-Civita connection. The components of the Riemann tensor are then studied, with a Riemannian manifold, or a metric manifold with a torsion-less connection. The Riemann tensor of the Schwarzschild metric are finally discussed.

This chapter presents some elements of differential geometry, the ‘vector’ version of Euclidean geometry in curvilinear coordinates. In doing so, it provides an intrinsic definition of the covariant derivative and establishes a relation between the moving frames attached to a trajectory introduced in Chapter 2 and the moving frames of Cartan associated with curvilinear coordinates. It illustrates a differential framework based on formulas drawn from Chapter 2, before discussing cotangent spaces and differential forms. The chapter then turns to the metric tensor, triads, and frame fields as well as vector fields, form fields, and tensor fields. Finally, it performs some vector calculus.

This chapter describes the first steps toward an understanding of large structures, which are observed in the universe at all scales—galaxies, groups of galaxies, and galactic clusters. It does so by studying the evolution of perturbations at linear order in Friedmann–Lemaître spacetimes. To simplify the discussion, the chapter limits the scope to the textbook case where the spatial sections of the background space are Euclidean (K = 0), and anisotropic perturbations and entropy perturbations are absent. This basically means that the matter reduces to a single fluid. The relativistic and Newtonian theories of cosmological perturbations differ. Finally, the chapter discusses the limit in which they converge.

This chapter begins by examining p-forms and the exterior product, as well as the dual of a p-form. Meanwhile, the exterior derivative is an operator, denoted d, which acts on a p-form to give a (p + 1)-form. It possesses the following defining properties: if f is a 0-form, df(t) = t f (where t is a vector of Eₙ), which coincides with the definition of differential 1-forms. Moreover, d(α + β) = dα + dβ, where α and β are forms of the same degree. Moreover, the exterior calculus can be used to obtain a compact and elegant formulation of Maxwell’s equations.

This chapter turns to the description of the interaction of a scalar field with particles which ‘feel’—that is, ‘charged’ particles. If the field is massless, and therefore long-range, and if the particle charge corresponds to its inertial mass, we have what is known as Nordström theory, a coherent theory of gravity which, however, disagrees with experiment. Nordström theory describes gravity by means of a massless scalar field φ. According to the ‘weak equivalence principle’, gravitational masses are equal to inertial masses, m = mg. When velocities are small, the gravitational field created is also weak.

This chapter turns to the gravitational radiation produced by a system of massive objects. The discussion is confined to the linear approximation of general relativity, which is compared with the Maxwell theory of electromagnetism. In the first part of the chapter, the properties of gravitational waves, which are the general solution of the linearized vacuum Einstein equations, are studied. Next, it relates these waves to the energy–momentum tensor of the sources creating them. The chapter then turns to the ‘first quadrupole formula’, giving the gravitational radiation field of these sources when their motion is due to forces other than the gravitational force.

This chapter shows that fluctuations of quantum origin are generated during inflation and that this process supplies initial conditions compatible with the observations. These fluctuations are therefore an important prediction of inflationary models. The chapter thus begins with a study of perturbations during inflation, proceeding in a similar manner to the previous chapter by finding the perturbation of the energy–momentum tensor of the scalar field. Another method of deriving the equations of motion of the perturbations is to start from the action of general relativity coupled to a scalar field, and expand to second order in the metric and scalar field perturbations. The chapter then proceeds with the determination of the initial conditions and the slow-roll inflation.

This chapter examines solutions to the Maxwell equations in a vacuum: monochromatic plane waves and their polarizations, plane waves, and the motion of a charge in the field of a wave (which is the principle upon which particle detection is based). A plane wave is a solution of the vacuum Maxwell equations which depends on only one of the Cartesian spatial coordinates. The monochromatic plane waves form a basis (in the sense of distributions, because they are not square-integrable) in which any solution of the vacuum Maxwell equations can be expanded. The chapter concludes by giving the conditions for the geometrical optics limit. It also establishes the connection between electromagnetic waves and the kinematic description of light discussed in Book 1.

This chapter attributes an inertial ‘mass–energy’ to particles. It also distinguishes between the action of an external field and of long-range and short-range internal forces, which is useful for establishing the laws of dynamics of an interacting body—that is, the equations determining its world line. The chapter also presents the 4-momentum conservation law for massive particles and light particles in inertial reference frames. It then gives some examples which illustrate the role played by this law in collisions. Finally, the chapter illustrates the conservation law by the Compton experiment, that is, the collision of a light corpuscle with a particle, and the concept of the quantum of action that can be derived from it.

This chapter provides a few examples of representations of the universe on a large scale—a first step in constructing a cosmological model. It first discusses the Copernican principle, which is an approximation/hypothesis about the matter distribution in the observable universe. The chapter then turns to the cosmological principle—a hypothesis about the geometry of the Riemannian spacetime representing the universe, which is assumed to be foliated by 3-spaces labeled by a cosmic time t which are homogeneous and isotropic, that is, ‘maximally symmetric’. After a discussion on maximally symmetric space, this chapter considers spacetimes with homogenous and isotropic sections. Finally, this chapter discusses Milne and de Sitter spacetimes.

This chapter presents the basics of the ‘effective-one-body’ approach to the two-body problem in general relativity. It also shows that the 2PN equations of motion can be mapped. This can be done by means of an appropriate canonical transformation, to a geodesic motion in a static, spherically symmetric spacetime, thus considerably simplifying the dynamics. Then, including the 2.5PN radiation reaction force in the (resummed) equations of motion, this chapter provides the waveform during the inspiral, merger, and ringdown phases of the coalescence of two non-spinning black holes into a final Kerr black hole. The chapter also comments on the current developments of this approach, which is instrumental in building the libraries of waveform templates that are needed to analyze the data collected by the current gravitational wave detectors.

This chapter continues the discussion of the laws of relativistic dynamics for systems of point particles, beginning with the law of angular momentum conservation in collisions. It considers an ensemble of free particles each characterized by its (constant) momentum pa. The total momentum p = Σapa does not depend on the inertial frame used, but the angular momentum will depend on the frame, because its definition involves radius vectors between an event reference point and points qa on the particle world lines. Furthermore, these are chosen to be simultaneous in a given frame. The chapter also formulates the equations of motion for particles possessing an internal rotation or ‘spin’.

This chapter presents Maxwell equations determining the electromagnetic field created by an ensemble of charges. It also derives these equations from the variational principle. The chapter studies the equation’s invariances: gauge invariance and invariance under Poincaré transformations. These allow us to derive the conservation laws for the total charge of the system and also for the system energy, momentum, and angular momentum. To begin, the chapter introduces the first group of Maxwell equations: Gauss’s law of magnetism, and Faraday’s law of induction. It then discusses current and charge conservation, a second set of Maxwell equations, and finally the field–energy momentum tensor.