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This paper studies the Bondi-Metzner-Sachs group in homogeneous projective coordinates, because it is then possible to write all transformations of such a group in a manifestly linear way. The 2-sphere metric, Bondi-Metzner-Sachs metric, asymptotic Killing vectors, generators of supertranslations, as well as boosts and rotations of Minkowski spacetime, are all re-expressed in homogeneous projective coordinates. Last, the integral curves of vector fields which generate supertranslations are evaluated in detail. This work prepares the ground for more advanced applications of the differential geometry of asymptotically flat spacetimes in projective coordinates., Comment: 26 pages, Latex, 9 figures. Misprints corrected and presentation improved

This paper considers the possibility that abstract thinking and advanced synthesis skills might encourage extraterrestrial civilizations to accept communication with mankind on Earth. For this purpose, a notation not relying upon the use of alphabet and numbers is proposed, in order to denote just some basic geometric structures of current physical theories: vector fields, one-form fields, and tensor fields of arbitrary order. An advanced civilization might appreciate the way here proposed to achieve a concise description of electromagnetism and general relativity, and hence it might accept the challenge of responding to our signals. The abstract symbols introduced in this paper to describe the basic structures of physical theories are encoded into black and white bitmap images that can be easily converted into short bit sequences and modulated on a carrier wave for radio transmission., Comment: 15 pages, 2 figures

This paper points out that, in a four-dimensional spacetime manifold, the four coordinates of any point may be viewed as arising from five homogeneous coordinates according to the well-established recipes of projective geometry. Moreover, if the homogeneous coordinates change by a linear transformation, ruled by the coefficients of a matrix of the group GL(5,R), this induces a fractional linear transformation of spherically symmetric spacetime coordinates which can be exploited to bring infinity down to a finite distance. The boundary of spherically symmetric spacetimes here studied is the disjoint union of three points: future timelike infinity, past timelike infinity, spacelike infinity, and the three-dimensional products of half-lines with a 2-sphere. For the single points, the underlying structure is a projective map which destroys the 2-sphere, whereas the two products of a half-line with the 2-sphere arise from a projective map which preserves the 2-sphere. Geodesics are then studied in the projectively transformed (t',r',theta',phi') coordinates for Schwarzschild spacetime, with special interest in their way of approaching our points at infinity. Next, Nariai and de Sitter spacetimes are studied with our projective method. Since the kinds of infinity here defined depend only on the symmetry of interest in a spacetime manifold, they have a broad range of applications, which motivate the innovative analysis of Schwarzschild, Nariai and de Sitter spacetime., Comment: 16 pages in double-column format; 18 plots in 14 figures

This paper investigates first the four branches of BMS transformations, motivated by the classification into elliptic, parabolic, hyperbolic and loxodromic proposed a few years ago in the literature. We first prove that to each normal elliptic transformation of the complex variable zeta used in the metric for cuts of null infinity there corresponds a BMS supertranslation. We then study the conformal factor in the BMS transformation of the u variable as a function of the squared modulus of zeta. In the loxodromic and hyperbolic cases, the conformal factor is either monotonically increasing or monotonically decreasing as a function of the real variable given by the modulus of zeta. The Killing vector field of the Bondi metric is also studied in correspondence with the four admissible families of BMS transformations. Eventually, all BMS transformations are re-expressed in the homogeneous coordinates suggested by projective geometry. It is then found that BMS transformations are the restriction to a pair of unit circles of a more general set of transformations. Within this broader framework, the geometry of such transformations is studied by means of its Segre manifold., Comment: In the final version, the presentation has been improved

DeWitt's suggestion that the wave function of the universe should vanish at the classical big-bang singularity is here considered within the framework of one-loop quantum cosmology. For pure gravity at one loop about a flat four-dimensional background bounded by a 3-sphere, three choices of boundary conditions are considered: vanishing of the linearized magnetic curvature when only transverse-traceless gravitational modes are quantized; a one-parameter family of mixed boundary conditions for gravitational and ghost modes; diffeomorphism invariant boundary conditions for metric perturbations and ghost modes. A positive zeta(0) value in these cases ensures that, when the 3-sphere boundary approaches zero, the resulting one-loop wave function approaches zero. This property may be interpreted by saying that, in the limit of small three-geometry, the resulting one-loop wave function describes a singularity-free universe. This property holds for one-loop functional integrals, which are not necessarily equivalent to solutions of the quantum constraint equations., Comment: In the final version, the presentation has been improved

This paper exploits the power of the Cayley-Dickson algebra to generate stationary rotating black hole solutions in one fell swoop. Specifically, we derive the nine-dimensional Myers-Perry solution with four independent angular momenta by using the Janis-Newman algorithm and Giampieri's simplification method, exploiting the octonion algebra. A general formula relating the dimension of the Cayley-Dickson algebra with the maximum number of angular momenta in each dimension is derived. Finally, we discuss the cut-off dimension for using the Cayley-Dickson construction along with the Janis-Newman algorithm for producing the rotating solutions., Comment: 27 pages, Latex. In the final version, the presentation has been improved, misprints no longer occur, and 3 references have been added

This paper builds two detailed examples of generalized normal in non-Euclidean spaces, i.e. the hyperbolic and elliptic geometries. In the hyperbolic plane we define a n-sided hyperbolic polygon P, which is the Euclidean closure of the hyperbolic plane H, bounded by n hyperbolic geodesic segments. The polygon P is built by considering the unique geodesic that connects the n+2 vertices (tilde z),z0,z1,...,z(n-1),z(n). The geodesics that link the vertices are Euclidean semicircles centred on the real axis. The vector normal to the geodesic linking two consecutive vertices is evaluated and turns out to be discontinuous. Within the framework of elliptic geometry, we solve the geodesic equation and construct a geodesic triangle. Also in this case, we obtain a discontinuous normal vector field. Last, the possible application to two-dimensional Euclidean quantum gravity is outlined., Comment: 27 pages, 1 figure. In the final version, the presentation has been improved, and new References have been added

This paper applies the Numerov and phase-integral methods to the stationary Schrodinger equation that studies bound states of charm anti-charm quarks. The former is a numerical method well suited for a matrix form of second-order ordinary di erential equations, and can be applied whenever the stationary states admit a Taylor-series expansion. The latter is an analytic method that provides, in principle, even exact solutions of the stationary Schrodinger equation, and well suited for applying matched asymptotic expansions and higher order quantization conditions. The Numerov method is found to be always in agreement with the early results of Eichten et al., whereas an original evaluation of the phase-integral quantization condition clarifies under which conditions the previous results in the literature on higher-order terms can be obtained., Comment: 23 pages, 2 tables and 2 figures. In the final version, the presentation and the numerical analysis have been improved, and a misprint in Eq. (2.3) has been amended

This paper performs a systematic investigation of geodesic motion in Euclidean Schwarzschild geometry, which is studied in the equatorial plane. The explicit form of geodesic motion is obtained in terms of incomplete elliptic integrals of first, second and third kind. No elliptic-like orbits exist in Euclidean Schwarzschild geometry, unlike the corresponding Lorentzian pattern. Among unbounded orbits, only unbounded first-kind orbits are allowed, unlike general relativity where unbounded second-kind orbits are always allowed., Comment: v3: version accepted for publication in EPJ C

This paper studies the nature of fractional linear transformations in a general relativity context as well as in a quantum theoretical framework. Two features are found to deserve special attention: the first is the possibility of separating the limit-point condition at infinity into loxodromic, hyperbolic, parabolic and elliptic cases. This is useful in a context in which one wants to look for a correspondence between essentially self-adjoint spherically symmetric Hamiltonians of quantum physics and the theory of Bondi-Metzner-Sachs transformations in general relativity. The analogy therefore arising, suggests that further investigations might be performed for a theory in which the role of fractional linear maps is viewed as a bridge between the quantum theory and general relativity. The second aspect to point out is the possibility of interpreting the limit-point condition at both ends of the positive real line, for a second-order singular differential operator, which occurs frequently in applied quantum mechanics, as the limiting procedure arising from a very particular Kleinian group which is the hyperbolic cyclic group. In this framework, this work finds that a consistent system of equations can be derived and studied. Hence one is led to consider the entire transcendental functions, from which it is possible to construct a fundamental system of solutions of a second-order differential equation with singular behavior at both ends of the positive real line, which in turn satisfy the limit-point conditions., Comment: 68 pages, 12 figures. All figures are now of better quality. Misprints have been amended

In this paper the Feynman Green function for Maxwell's theory in curved space-time is studied by using the Fock-Schwinger-DeWitt asymptotic expansion; the point-splitting method is then applied, since it is a valuable tool for regularizing divergent observables. Among these, the stress-energy tensor is expressed in terms of second covariant derivatives of the Hadamard Green function, which is also closely linked to the effective action; therefore one obtains a series expansion for the stress-energy tensor. Its divergent part can be isolated, and a concise formula is here obtained: by dimensional analysis and combinatorics, there are two kinds of terms: quadratic in curvature tensors (Riemann, Ricci tensors and scalar curvature) and linear in their second covariant derivatives. This formula holds for every space-time metric; it is made even more explicit in the physically relevant particular cases of Ricci-flat and maximally symmetric spaces, and fully evaluated for some examples of physical interest: Kerr and Schwarzschild metrics and de Sitter space-time., Comment: 42 pages, Latex

This paper proposes a toy model where, in the Einstein equations, the right-hand side is modified by the addition of a term proportional to the symmetrized partial contraction of the Ricci tensor with the energy-momentum tensor, while the left-hand side remains equal to the Einstein tensor. Bearing in mind the existence of a natural length scale given by the Planck length, dimensional analysis shows that such a term yields a correction linear in h to the classical term, that is instead just proportional to the energy-momentum tensor. One then obtains an effective energy-momentum tensor that consists of three contributions: pure energy part, mechanical stress and thermal part. The pure energy part has the appropriate property for dealing with the dark sector of modern relativistic cosmology. Such a theory coincides with general relativity in vacuum, and the resulting field equations are here solved for a Dunn and Tupper metric, for departures from an interior Schwarzschild solution as well as for a Friedmann-Lemaitre-Robertson-Walker universe., Comment: 3 figures, Revtex4

In Cosmology and in Fundamental Physics there is a crucial question like: where the elusive substance that we call Dark Matter is hidden in the Universe and what is it made of?, that, even after 40 years from the Vera Rubin seminal discovery does not have a proper answer. Actually, the more we have investigated, the more this issue has become strongly entangled with aspects that go beyond the established Quantum Physics, the Standard Model of Elementary particles and the General Relativity and related to processes like the Inflation, the accelerated expansion of the Universe and High Energy Phenomena around compact objects. Even Quantum Gravity and very exotic DM particle candidates may play a role in framing the Dark Matter mystery that seems to be accomplice of new unknown Physics. Observations and experiments have clearly indicated that the above phenomenon cannot be considered as already theoretically framed, as hoped for decades. The Special Topic to which this review belongs wants to penetrate this newly realized mystery from different angles, including that of a contamination of different fields of Physics apparently unrelated. We show with the works of this ST that this contamination is able to guide us into the required new Physics. This review wants to provide a good number of these "paths or contamination" beyond/among the three worlds above; in most of the cases, the results presented here open a direct link with the multi-scale dark matter phenomenon, enlightening some of its important aspects. Also in the remaining cases, possible interesting contacts emerges., Comment: 64 pages 16 Figures. In print on Frontiers in Astronomy and Space Sciences, for the Research Topic: "When Planck, Einstein and Vera Rubin Meet. Dark Matter: What is it ? Where is it?"

The concept of boundary plays an important role in several branches of general relativity, e.g., the variational principle for the Einstein equations, the event horizon and the apparent horizon of black holes, the formation of trapped surfaces. On the other hand, in a branch of mathematics known as geometric measure theory, the usefulness has been discovered long ago of yet another concept, i.e., the reduced boundary of a finite-perimeter set. This paper proposes therefore a definition of finite-perimeter sets and their reduced boundary in general relativity. Moreover, a basic integral formula of geometric measure theory is evaluated explicitly in the relevant case of Euclidean Schwarzschild geometry, for the first time in the literature. This research prepares the ground for a measure-theoretic approach to several concepts in gravitational physics, supplemented by geometric insight. Moreover, such an investigation suggests considering the possibility that the in-out amplitude for Euclidean quantum gravity should be evaluated over finite-perimeter Riemannian geometries that match the assigned data on their reduced boundary. As a possible application, an analysis is performed of the basic formulae leading eventually to the corrections of the intrinsic quantum mechanical entropy of a black hole., Comment: v2: Section 4 is completely new, and important references have been added. v3: Sub-section 4.1 has been added, with new original calculations. v4: The figure has been amended

This paper applies the phase-integral method to the stationary theory of alpha-decay. The rigorous form of the connection formulae, and their one-directional nature that was not widely known in the physical literature, are applied. The condition for obtaining s-wave metastable states affects the stationary state at large distance from the nucleus, which is dominated by the cosine of the phase integral minus (pi over 4). Accurate predictions for the lowest s-wave metastable state and mean life of the radioactive nucleus are obtained in the case of Uranium. The final part of the paper describes the phase-integral algorithm for evaluating stationary states by means of a suitable choice of freely specifiable base function. Within this framework, an original approximate formula for the phase integrand with arbitrary values of the angular momentum quantum number is obtained., Comment: 20 pages, latex

Given the Thomas-Fermi equation sqrt(x)phi''=phi*(3/2), this paper changes first the dependent variable by defining y(x)=sqrt(x phi(x)). The boundary conditions require that y(x) must vanish at the origin as sqrt(x), whereas it has a fall-off behaviour at infinity proportional to the power (1/2)(1-chi) of the independent variable x, chi being a positive number. Such boundary conditions lead to a 1-parameter family of approximate solutions in the form sqrt(x) times a ratio of finite linear combinations of integer and half-odd powers of x. If chi is set equal to 3, in order to agree exactly with the asymptotic solution of Sommerfeld, explicit forms of the approximate solution are obtained for all values of x. They agree exactly with the Majorana solution at small x, and remain very close to the numerical solution for all values of x. Remarkably, without making any use of series, our approximate solutions achieve a smooth transition from small-x to large-x behaviour. Eventually, the generalized Thomas-Fermi equation that includes relativistic, non-extensive and thermal effects is studied, finding approximate solutions at small and large x for small or finite values of the physical parameters in this equation., Comment: 18 pages, 6 figures

When the Schr\"{o}dinger equation for stationary states is studied for a system described by a central potential in $n$-dimensional Euclidean space, the radial part of stationary states is an even function of a parameter $\lambda$ which is a linear combination of angular momentum quantum number $l$ and dimension $n$, i.e., $\lambda=l+{(n-2)\over 2}$. Thus, without setting a priori $n=3$, complex values of $\lambda$ can be achieved, in particular, by keeping $l$ real and complexifying $n$. For suitable values of such an auxiliary complexified dimension, it is therefore possible to obtain results on scattering amplitude and phase shift that are completely equivalent to the results obtained in the sixties for Yukawian potentials in $\mathbb{R}^3$. Moreover, if both $l$ and $n$ are complexified, the possibility arises of recovering the partial wave amplitude from residues of a function of two complex variables. Thus, the complex angular momentum formalism can be imbedded into a broader framework, where a correspondence exists between the scattering amplitude and a skew curve in $\mathbb{R}^3$., Comment: 11 pages; the line after Eq. (1.1), and two misprints, have been amended

An Ermakov-Pinney-like equation associated with the scalar wave equation in curved space-time is here studied. The example of Schwarzschild space-time considered in the present work shows that this equation can be viewed more as a model equation, with interesting applications in black hole physics. Other applications studied involve cosmological space-times (de Sitter) and pulse of plane gravitational waves. In all these cases the evolution of the Ermakov-Pinney field seems to be consistent with a rapid blow-up, unlike the Schwarzschild case where spatially damped oscillations are allowed. Eventually, the phase function is also evaluated in many of the above space-time models., Comment: 20 pages, 10 figures. The presentation has been improved

Within the solar system, approximate realizations of the three-body problem occur when a comet approaches a planet while being affected mainly by such a planet and the Sun, and this configuration was investigated by Tisserand within the framework of Newtonian gravity. The exact relativistic treatment of the problem is not an easy task, but the present paper develops an approximate calculational scheme which computes for the first time the tiny effective-gravity correction to the equation of the surface for all points of which it is equally advantageous to regard the heliocentric motion as being perturbed by the attraction of Jupiter, or the jovicentric motion as being perturbed by the attraction of the Sun. This analysis completes the previous theoretical investigations of effective-gravity corrections to the Newtonian analysis of three-body systems, and represents an intermediate step towards relativistic effects on cometary motions., Comment: Published version

The first part of the paper proves that a subset of the general set of Ermakov-Pinney equations can be obtained by differentiation of a first-order non-linear differential equation. The second part of the paper proves that, similarly, the equation for the amplitude function for the parametrix of the scalar wave equation can be obtained by covariant differentiation of a first-order non-linear equation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary 1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi)=0, whereas the 1-form rho fulfills the non-linear equation div(rho)+rho**2=0. The auxiliary 1-forms (psi,rho) are evaluated explicitly in Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained. Thus, the novel method developed in this paper can be used with profit in physical applications., Comment: 13 pages. Sections 6, 7, 8 are entirely new

In this work we consider an equation for the Riemann zeta-function in the critical half-strip. With the help of this equation we prove that finding non-trivial zeros of the Riemann zeta-function outside the critical line would be equivalent to the existence of complex numbers for which equation (5.1) in the paper holds. Such a condition is studied, and the attempt of proving the Riemann hypothesis is found to involve also the functional equation (6.26), where t is a real variable bigger than or equal to 1 and n is any natural number. The limiting behavior of the solutions as t approaches 1 is then studied in detail., Comment: 30 pages, 2 figures. In the final version, the presentation has been improved

In the eighties, Schroder studied a quantum mechanical model where the stationary states of Schrodinger's equation obey nonlocal boundary conditions on a circle in the plane. For such a problem, we perform a detailed one-loop calculation for three choices of the kernel characterizing the nonlocal boundary conditions. In such cases, the zeta(0) value is found to coincide with the one resulting from Robin boundary conditions. The detailed technique here developed may be useful for studying one-loop properties of quantum field theory and quantum gravity if nonlocal boundary conditions are imposed., Comment: 17 pages, Revtex4. In the final version, the presentation in section 5 has been improved, and important References have been added

A new proof of the Weyl limit point-limit circle criterion is obtained, with systematic emphasis on Sobolev-space methods., Comment: Plain Tex, 24 pages. In the final version, the presentation has been improved, with a new section and new references, and misprints have been amended

Half of the Bondi-Metzner-Sachs (BMS) transformations consist of orientation-preserving conformal homeomorphisms of the extended complex plane known as fractional linear (or Mobius) transformations. These can be of 4 kinds, i.e. they are classified as being parabolic, or hyperbolic, or elliptic, or loxodromic, depending on the number of fixed points and on the value of the trace of the associated 2x2 matrix in the projective version of the SL(2,C) group. The resulting particular forms of SL(2,C) matrices affect also the other half of BMS transformations, and are used here to propose 4 realizations of the asymptotic symmetry group that we call, again, parabolic, or hyperbolic, or elliptic, or loxodromic. In the second part of the paper, we prove that a subset of hyperbolic and loxodromic transformations, those having trace that approaches infinity, correspond to the fulfillment of limit-point condition for singular Sturm-Liouville problems. Thus, a profound link may exist between the language for describing asymptotically flat space-times and the world of complex analysis and self-adjoint problems in ordinary quantum mechanics., Comment: 16 pages, 1 figure. In the final version, a missing symbol in Eq. (1.1) has been restored

We study trapped surfaces from the point of view of local isometric embedding into three-dimensional Riemannian manifolds. When a two-surface is embedded into three-dimensional Euclidean space, the problem of finding all surfaces applicable upon it gives rise to a non-linear partial differential equation of Monge-Ampere type, first discovered by Darboux, and later reformulated by Weingarten. Even today, this problem remains very difficult, despite some remarkable results. We find an original way of generalizing the Darboux technique, which leads to a coupled set of 6 non-linear partial differential equations. For the 3-manifolds occurring in Friedmann-(Lemaitre)-Robertson-Walker cosmologies, we show that the local isometric embedding of trapped surfaces into them can be proved by solving just one non-linear equation. Such an equation is here solved for the three kinds of Friedmann model associated with positive, zero, negative curvature of spatial sections, respectively., Comment: 21 pages, 3 figures

This paper discusses three matter-of-principle methods for measuring the general relativity correction to the Newtonian values of the position of collinear Lagrangian points L1 and L2 of the Sun-Earth-satellite system. All approaches are based on time measurements. The first approach exploits a pulsar emitting signals and two receiving antennas located at L1 and L2, respectively. The second method is based on a relativistic positioning system based on the Lagrangian points themselves. These first two methods depend crucially on the synchronization of clocks at L1 and L2. The third method combines a pulsar and an artificial emitter at the stable points L4 or L5 forming a basis for the positioning of the collinear points L1 and L2. Further possibilities are mentioned and the feasibility of the measurements is considered., Comment: 8 pages, 2 figures

This work is a pedagogical review dedicated to a modern description of the Bondi-Metzner-Sachs group. The curved space-times that will be taken into account are the ones that suitably approach, at infinity, Minkowski space-time. In particular we will focus on asymptotically flat space-times. In this work the concept of asymptotic symmetry group of those space-times will be studied. In the first two sections we derive the asymptotic group following the classical approach which was basically developed by Bondi, van den Burg, Metzner and Sachs. This is essentially the group of transformations between coordinate systems of a certain type in asymptotically flat space-times. In the third section the conformal method and the notion of asymptotic simplicity are introduced, following mainly the works of Penrose. This section prepares us for another derivation of the Bondi-Metzner-Sachs group which will involve the conformal structure, and is thus more geometrical and fundamental. In the subsequent sections we discuss the properties of the Bondi-Metzner-Sachs group, e.g. its algebra and the possibility to obtain as its subgroup the Poincar\'e group, as we may expect. The paper ends with a review of the Bondi-Metzner-Sachs invariance properties of classical gravitational scattering discovered by Strominger, that are finding application to black hole physics and quantum gravity in the literature., Comment: 62 pages, 9 figures. Misprints have been amended and two important references have been added

Towards the end of nineteenth century, Celestial Mechanics provided the most powerful tools to test Newtonian gravity in the solar system, and led also to the discovery of chaos in modern science. Nowadays, in light of general relativity, Celestial Mechanics leads to a new perspective on the motion of satellites and planets. The reader is here introduced to the modern formulation of the problem of motion, following what the leaders in the field have been teaching since the nineties. In particular, the use of a global chart for the overall dynamics of N bodies and N local charts describing the internal dynamics of each body. The next logical step studies in detail how to split the N-body problem into two sub-problems concerning the internal and external dynamics, how to achieve the effacement properties that would allow a decoupling of the two sub-problems, how to define external-potential-effacing coordinates and how to generalize the Newtonian multipole and tidal moments. The review paper ends with an assessment of the nonlocal equations of motion obtained within such a framework, a description of the modifications induced by general relativity of the theoretical analysis of the Newtonian three-body problem, and a mention of the potentialities of the analysis of solar-system metric data carried out with the Planetary Ephemeris Program., Comment: 26 pages, 1 figure

Trapped surface formation in general relativity can be studied through a coupled set of nonlinear equations, where various terms can be neglected, as was proved by a rigorous mathematical analysis of Christodoulou. This paper is devoted to a pedagogical synthesis of the mathematical formalism employed in this analysis, i.e. the optical structure of general relativity investigated by Christodoulou, Klainerman and other authors after them., Comment: 11 pages. Misprints in Eqs. (1.5), (1.6), (3.24), (3.25) have been amended

We examine quantum corrections of time delay arising in the gravitational field of a spinning oblate source. Low-energy quantum effects occurring in Kerr geometry are derived within a framework where general relativity is fully seen as an effective field theory. By employing such a pattern, gravitational radiative modifications of Kerr metric are derived from the energy-momentum tensor of the source, which at lowest order in the fields is modelled as a point mass. Therefore, in order to describe a quantum corrected version of time delay in the case in which the source body has a finite extension, we introduce a hybrid scheme where quantum fluctuations affect only the monopole term occurring in the multipole expansion of the Newtonian potential. The predicted quantum deviation from the corresponding classical value turns out to be too small to be detected in the next future, showing that new models should be examined in order to test low-energy quantum gravity within the solar system., Comment: 12 pages, 1 figure

In this paper we study the behavior of the Casimir energy of a "multi-cavity" across the transition from the metallic to the superconducting phase of the constituting plates. Our analysis is carried out in the framework of the ARCHIMEDES experiment, aiming at measuring the interaction of the electromagnetic vacuum energy with a gravitational field. For this purpose it is foreseen to modulate the Casimir energy of a layered structure composing a multi-cavity coupled system by inducing a transition from the metallic to the superconducting phase. This implies a thorough study of the behavior of the cavity, where normal metallic layers are alternated with superconducting layers, across the transition. Our study finds that, because of the coupling between the cavities, mainly mediated by the transverse magnetic modes of the radiation field, the variation of energy across the transition can be very large., Comment: 17 pages, 3 figures. Abstract and title slightly modified. Figures and comments added

Recent work in the literature has advocated using the Earth-Moon-planetoid Lagrangian points as observables, in order to test general relativity and effective field theories of gravity in the solar system. However, since the three-body problem of classical celestial mechanics is just an approximation of a much more complicated setting, where all celestial bodies in the solar system are subject to their mutual gravitational interactions, while solar radiation pressure and other sources of nongravitational perturbations also affect the dynamics, it is conceptually desirable to improve the current understanding of solar system dynamics in general relativity, as a first step towards a more accurate theoretical study of orbital motion in the weak-gravity regime. For this purpose, starting from the Einstein equations in the de Donder-Lanczos gauge, this paper arrives first at the Levi-Civita Lagrangian for the geodesic motion of celestial bodies, showing in detail under which conditions the effects of internal structure and finite extension get cancelled in general relativity to first post-Newtonian order. The resulting nonlinear ordinary differential equations for the motion of planets and satellites are solved for the Earth's orbit about the Sun, written down in detail for the Sun-Earth-Moon system, and investigated for the case of planar motion of a body immersed in the gravitational field produced by the other bodies (e.g. planets with their satellites). At this stage, we prove an exact property, according to which the fourth-order time derivative of the original system leads to a linear system of ordinary differential equations. This opens an interesting perspective on forthcoming research on planetary motions in general relativity within the solar system, although the resulting equations remain a challenge for numerical and qualitative studies., Comment: 26 pages, 3 figures

This paper studies first the differential inequalities that make it possible to build a global theory of pseudo-holomorphic functions in the case of one or several complex variables. In the case of one complex dimension, we prove that the differential inequalities describing pseudo-holomorphic functions can be used to define a one-real-dimensional manifold (by the vanishing of a function with nonzero gradient), which is here a 1-parameter family of plane curves. On studying the associated envelopes, such a parameter can be eliminated by solving two nonlinear partial differential equations. The classical differential geometry of curves can be therefore exploited to get a novel perspective on the equations describing the global theory of pseudo-holomorphic functions., Comment: 25 pages

Classical as well as quantum features of the late-time evolution of cosmological models with fluids obeying a Shan-Chen-like equation of state are studied. The latter is of the type $p=w_{\rm eff}(\rho)\,\rho$, and has been used in previous works to describe, e.g., a possible scenario for the growth of the dark-energy content of the present Universe. At the classical level the fluid dynamics in a spatially flat Friedmann-Robertson-Walker background implies the existence of two possible equilibrium solutions depending on the model parameters, associated with (asymptotic) finite pressure and energy density. We show that no future cosmological singularity is developed during the evolution for this specific model. The corresponding quantum effects in the late-time behavior of the system are also investigated within the framework of quantum geometrodynamics, i.e., by solving the (minisuperspace) Wheeler-DeWitt equation in the Born-Oppenheimer approximation, constructing wave-packets and analyzing their behavior., Comment: 18 pages, 6 figures, revtex4 macros; to appear in PRD

Archimedes is a feasibility study of a future experiment to ascertain the interaction of vacuum fluctuations with gravity. The experiment should measure the force that the earth's gravitational field exerts on a Casimir cavity by using a small force detector. Here we analyse the main parameters of the experiment and we present its conceptual scheme, which overcomes in principle the most critical problems., Comment: 3 pages, MG14 Conference

Archimedes is an INFN-funded pathfinder experiment aimed at verifying the feasibility of measuring the interaction of vacuum fluctuations with gravity. The final experiment will measure the force exerted by the gravitational field on a Casimir cavity whose vacuum energy is modulated with a superconductive transition, by using a balance as a small force detector. Archimedes is a two-year project devoted to test the most critical experimental aspects, in particular the balance resonance frequency and quality factor, the thermal modulation efficiency and the superconductive sample realization., Comment: 4 pages, 2 figures

In the sixties, DeWitt discovered that the advanced and retarded Green functions of the wave operator on metric perturbations in the de Donder gauge make it possible to define classical Poisson brackets on the space of functionals that are invariant under the action of the full diffeomorphism group of spacetime. He therefore tried to exploit this property to define invariant commutators for the quantized gravitational field, but the operator counterpart of such classical Poisson brackets turned out to be a hard task. On the other hand, the mathematical literature studies often an approximate inverse, the parametrix, which is, strictly, a distribution. We here suggest that such a construction might be exploited in canonical quantum gravity. We begin with the simplest case, i.e. fundamental solution and parametrix for the linear, scalar wave operator; the next step are tensor wave equations, again for linear theory, e.g. Maxwell theory in curved spacetime. Last, the nonlinear Einstein equations are studied, relying upon the well-established Choquet-Bruhat construction, according to which the fifth derivatives of solutions of a nonlinear hyperbolic system solve a linear hyperbolic system. The latter is solved by means of Kirchhoff-type formulas, while the former fifth-order equations can be solved by means of well-established parametrix techniques for elliptic operators. But then the metric components that solve the vacuum Einstein equations can be obtained by convolution of such a parametrix with Kirchhoff-type formulas. Some basic functional equations for the parametrix are also obtained, that help in studying classical and quantum version of the Jacobi identity., Comment: 27 pages

A review is made of the basic tools used in mathematics to define a calculus for pseudodifferential operators on Riemannian manifolds endowed with a connection: esistence theorem for the function that generalizes the phase; analogue of Taylor's theorem; torsion and curvature terms in the symbolic calculus; the two kinds of derivative acting on smooth sections of the cotangent bundle of the Riemannian manifold; the concept of symbol as an equivalence class. Physical motivations and applications are then outlined, with emphasis on Green functions of quantum field theory and Parker's evaluation of Hawking radiation., Comment: 14 pages, paper in honour of Gaetano Vilasi

In the restricted four-body problem consisting of the Earth, the Moon and the Sun as the primaries and a spacecraft as the planetoid, we take into account the solar perturbation in the description of the motion of a spacecraft in the vicinity of the stable Earth-Moon libration points L4 and L5 both in the classical regime and in the context of effective field theories of gravity. We then evaluate the location of all Lagrangian points in the Earth-Moon system within the framework of general relativity. For the points L4 and L5, the corrections of coordinates are of order a few millimeters. After that, we set up a scheme where the theory which is quantum corrected has as its classical counterpart the Einstein theory, instead of the Newtonian one. By virtue of the effective-gravity correction to the longdistance form of the potential among two point masses, all terms involving the ratio between the gravitational radius of the primary and its separation from the planetoid get modified. Within this framework, for the Lagrangian points of stable equilibrium, we find quantum corrections of order two millimeters, whereas for Lagrangian points of unstable equilibrium we find quantum corrections below a millimeter. Finally, general relativity corrections to Newtonian position of collinear Lagrangian points turn out to be below the millimiter, whereas on stable equilibrium points they are of order of a few millimiters., Comment: 41 pages, 9 figures. Comments in the very end of Sec. VI have been amended and shortened. The last typos have been corrected

An assessment is made of recent attempts to evaluate how quantum gravity may affect the anisotropy spectrum of the cosmic microwave background. A perturbative scheme for the solution of the Wheeler-DeWitt equation has been found to allow for enhancement of power at large scales, whereas the alternative predicts a suppression of power at large scales. Both effects are corrections which, although conceptually interesting, turn out to be too small to be detected. Another scheme relies upon a Born-Oppenheimer analysis: by using a perturbative approach to the nonlinear ordinary differential equation obeyed by the two-point function for scalar fluctuations, a new family of power spectra have been obtained and studied by the authors., Comment: 4 pages, prepared for the AT1 Session of MG14 in Rome

We apply the ultrarelativistic boosting procedure to map the metric of Schwarzschild-de Sitter spacetime into a metric describing de Sitter spacetime plus a shock-wave singularity located on a null hypersurface, by exploiting the picture of the embedding of an hyperboloid in a five-dimensional Minkowski spacetime. After reverting to the usual four-dimensional formalism, we also solve the geodesic equation and evaluate the Riemann curvature tensor of the boosted Schwarzschild-de Sitter metric by means of numerical calculations, which make it possible to reach the ultrarelativistic regime gradually by letting the boost velocity approach the speed of light. Eventually, the analysis of the Kretschmann invariant (and of the geodesic equation) shows the global structure of space- time, as we demonstrate the presence of a "scalar curvature singularity" within a 3-sphere and find that it is also possible to define what we have called "boosted horizon", a sort of elastic wall where all particles are surprisingly pushed away. This seems to suggest that such "boosted geometries" are ruled by a sort of "antigravity effect" since all geodesics seem to refuse entering the "boosted horizon" and are "reflected" by it, even though their initial conditions are aimed at driving the particles towards the "boosted horizon" itself., Comment: prepared for the ES3 (Exact Solutions (Physical Aspects)) session of the MG14 Conference, Rome, July 2015

The one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., both at the Lagrangian points of stable equilibrium and at those of unstable equilibrium the Newtonian values of planetoid's coordinates are changed by a few millimetres in the Earth-Moon system. First, we find that the equations governing the position of both noncollinear and collinear quantum libration points are algebraic fifth degree and ninth degree equations, respectively. Second, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points. By taking into account the quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist., Comment: prepared for the AT1 (Extended Theories of Gravity and Quantum Cosmology) session of the MG14 Conference, Rome, July 2015

The realization of a nonlocal quantum field theory without losing unitarity, gauge invariance and causality is investigated. It is commonly retained that such a formulation is possible at tree level, but at quantum level acausality reappears at one loop. We suggest the the problem of acausality is, in a broad sense, similar to the one about anomalies in quantum field theory. By virtue of this analogy, we suggest that acausal diagrams resulting from the fermionic sector and the bosonic one might cancel each other, with a suitable content of fields and suitable symmetries. As a simple example, we show how supersymmetry can alleviate this problem in a simple and elegant way, i.e., by leading to exact cancellations of harmful diagrams, to all orders of perturbation theory, in the case of a nonlocal Wess-Zumino model. On the other hand, the same is true for a super Yang-Mills model, but in this case, other important acausal diagrams are also originating from supersymmetric D-terms. As a consequence, we conjecture that, for a generic gauge theory, an infinite number of superfields are needed in order to obtain a complete cancellation between all diagrams. This framework is naturally occurring in string theory, where infinite Kaluza-Klein towers of higher-spin fields are generally predicted., Comment: Supergraphs reformulation in non-local case are added, with a formal derivation of these. More explicit calculations, in main interesting one-loop processes, are explicitly shown. More comments and references. Main results and Conclusions remain the same

Recent work in the literature has shown that the one-loop long distance quantum corrections to the Newtonian potential imply tiny but observable effects in the restricted three-body problem of celestial mechanics, i.e., at the Lagrangian libration points of stable equilibrium the planetoid is not exactly at equal distance from the two bodies of large mass, but the Newtonian values of its coordinates are changed by a few millimeters in the Earth-Moon system. First, we assess such a theoretical calculation by exploiting the full theory of the quintic equation, i.e., its reduction to Bring-Jerrard form and the resulting expression of roots in terms of generalized hypergeometric functions. By performing the numerical analysis of the exact formulas for the roots, we confirm and slightly improve the theoretical evaluation of quantum corrected coordinates of Lagrangian libration points of stable equilibrium. Second, we prove in detail that also for collinear Lagrangian points the quantum corrections are of the same order of magnitude in the Earth-Moon system. Third, we discuss the prospects to measure, with the help of laser ranging, the above departure from the equilateral triangle picture, which is a challenging task. On the other hand, a modern version of the planetoid is the solar sail, and much progress has been made, in recent years, on the displaced periodic orbits of solar sails at all libration points, both stable and unstable. The present paper investigates therefore, eventually, a restricted three-body problem involving Earth, Moon and a solar sail. By taking into account the one-loop quantum corrections to the Newtonian potential, displaced periodic orbits of the solar sail at libration points are again found to exist., Comment: 36 pages, 8 figures. In the final version, the discussion has been improved and a new section (Sec. III) has been added

The scalar wave equation in Kasner spacetime is solved, first for a particular choice of Kasner parameters, by relating the integrand in the wave packet to the Bessel functions. An alternative integral representation is also displayed, which relies upon the method of integration in the complex domain for the solution of hyperbolic equations with variable coefficients. In order to study the propagation of wave fronts, we integrate the equations of bicharacteristics which are null geodesics, and we are able to express them, for the first time in the literature, with the help of elliptic integrals for another choice of Kasner parameters. For generic values of the three Kasner parameters, the solution of the Cauchy problem is built through a pair of integral operators, where the amplitude and phase functions in the integrand solve a coupled system of partial differential equations. The first is the so-called transport equation, whereas the second is a nonlinear equation that reduces to the eikonal equation if the amplitude is a slowly varying function. Remarkably, the analysis of such a coupled system is proved to be equivalent to building first an auxiliary covariant vector having vanishing divergence, while all nonlinearities are mapped into solving a covariant generalization of the Ermakov-Pinney equation for the amplitude function. Last, from a linear set of equations for the gradient of the phase one recovers the phase itself. This is the parametrix construction that relies upon Fourier-Maslov integral operators, but with a novel perspective on the nonlinearities in the dispersion relation. Furthermore, the Adomian method for nonlinear partial differential equations is applied to generate a recursive scheme for the evaluation of the amplitude function in the parametrix., Comment: 29 pages, Latex. Sections V and VI contain new original work on bicharacteristics and on the parametrix of the wave equation. The last misprints have been amended

Archimedes is a feasibility study to a future experiment to ascertain the interaction of vacuum fluctuations with gravity. The future experiment should measure the force that the Earth's gravitational field exerts on a Casimir cavity by using a balance as the small force detector. The Archimedes experiment analyses the important parameters in view of the final measurement and experimentally explores solutions to the most critical problems., Comment: 10 pages, 4 figures