We employ a perturbative analysis to study the evolution of large-scale peculiar velocity fields within the framework of Newtonian gravity and then compare our results to those of the corresponding relativistic treatment. In so doing, we use the same mathematical formalism and apply the same physical approach. This facilitates a direct and transparent comparison between the two treatments. Our study recovers and extends the familiar Newtonian results on the one hand, while on the other it shows that the Newtonian analysis leads to substantially weaker growth-rates for the peculiar velocity field, compared to the relativistic approach. This implies that, by using Newton's rather than Einstein's theory, one could seriously underestimate the overall kinematic evolution of cosmological peculiar motions. We are also in the position to identify the reason the two theories arrive at such considerably different results and conclusions.
Although bulk peculiar motions are commonplace in the universe, most theoretical studies either bypass them, or take the viewpoint of the idealised Hubble-flow observers. As a result, the role of these peculiar flows remains largely unaccounted for, despite the fact that relative-motion effects have led to the misinterpretation of the observations in a number of occasions. Here, we examine the implications of large-scale peculiar flows for the interpretation of the deceleration parameter. We compare, in particular, the deceleration parameters measured by the Hubble-flow observers and by their …
We consider the linear kinematics of large-scale peculiar motions in a perturbed Friedmann universe. In so doing, we take the viewpoint of the "real" observers that move along with the peculiar flow, relative to the smooth Hubble expansion. Using relativistic cosmological perturbation theory, we study the linear evolution of the peculiar velocity field, as well as the expansion/contraction, the shear and the rotation of the bulk motion. Our solutions show growth rates considerably stronger than those of the earlier treatments, which were mostly Newtonian. On scales near and beyond the Hubble r…
We give a careful general relativistic and (1+3)-covariant analysis of cosmological peculiar velocities induced by matter density perturbations in the presence of a cosmological constant. In our quasi-Newtonian approach, constraint equations arise to maintain zero shear of the non-comoving fundamental worldlines which define a Newtonian-like frame, and these lead to the (1+3)-covariant dynamical equations, including a generalized Poisson-type equation. We investigate the relation between peculiar velocity and peculiar acceleration, finding the conditions under which they are aligned. In this c…
A brief review is given of all the Hamiltonians and effective potentials calculated hitherto covering the post-Newtonian (pN) dynamics of a two body system. A method is presented to compare (conservative) reduced Hamiltonians with nonreduced potentials directly at least up to the next-to-leading-pN order.
Cosmological peculiar velocities in general relativity
Queen Mary University of London（伦敦大学玛丽女王学院）；University of the Western Cape（西开普敦大学）；National Institute for Theoretical & Computational Sciences（理论与计算科学国家研究所）
We study the peculiar motion of non-relativistic matter in a fully covariant way. The exact nonlinear equations are derived and then applied to the case of pressure-free matter, moving relatively to a quasi-Newtonian Eulerian frame. Our two-frame formalism facilitates the study of the nonlinear kinematics of the matter, as the latter decouples from the background expansion and starts to ``turn around'' and collapse. Applied to second perturbative order, our equations provide a fully covariant formulation of the Zeldovich approximation, which by construction addresses the mildly nonlinear regim…
The geometry of a light wavefront, evolving from a initial flat wavefront in the 3-space associated with a post-Newtonian relativistic spacetime, is studied numerically by means of the ray tracing method. For a discretization of the bidimensional light wavefront, a surface fitting technique is used to determine the curvature of this surface. The relationship between the intrinsic curvature of the wavefront and the change of the arrival time at different points on the Earth is also numerically discussed.
Gravitational physics of VLBI experiment conducted on September 8, 2002 and dedicated to measure the speed of gravity (a fundamental constant in the Einstein equations) is treated in the first post-Newtonian approximation. Explicit speed-of-gravity parameterization is introduced to the Einstein equations to single out the retardation effect caused by the finite speed of gravity in the relativistic time delay of light, passing through the variable gravitational field of the solar system. The speed-of-gravity 1.5 post-Newtonian correction to the Shapiro time delay is derived and compared with ou…
We determine the onset point of secular instability for the nonaxisymmetric bar mode in rigidly rotating equilibrium configurations in the Post-Newtonian approximation, in order to apply it to neutron stars. The treatment is based on a precedent Newtonian analytic energy variational method which we have extended to the Post-Newtonian case. This method, based upon Landau's theory of second-order phase transitions, provides the critical value of the ellipsoid polar eccentricity e at the onset of viscosity-driven instability and it is valid for any equation of state. The extension of this method …
Recently it has been shown that despite previous claims the cosmological constant affects light bending. In the present article we study light bending and the advance of Mercury's perihelion in the context of the Newtonian limit of Schwarzschild-de Sitter spacetime employing the special relativistic equivalence of mass and energy. In both cases, up to a constant factor, we find the same results as in the full general relativistic treatment of the same phenomena. These approximate and intuitive arguments demonstrate clearly what effects should have been expected from the presence of $Λ$ in the …
In this review we present the theoretical background for treating General Relativity as an effective field theory and focus on the concrete results of such a treatment. As a result we present the calculations of the low-energy leading gravitational corrections to the Newtonian potential between two sources.
Peculiar velocity fields from analytic solutions of General Relativity
Instituto de Ciencias Nucleares, Universidad Nacional Autónoma de México（墨西哥国立自治大学核科学研究所）；Instituto de Ciencias Físicas, Universidad Nacional Autónoma de México（墨西哥国立自治大学物理科学学院）
We study the nonlinear gravitational dynamics of a universe filled with a pressureless fluid and a cosmological constant $Λ$ in the context of Newtonian gravity, and in the relativistic post-Friedmann approach proposed in paper I [I. Milillo et al., Phys. Rev. D 92, 023519 (2015).]. The post-Friedmann approximation scheme is based on the $1/c$ expansion of the space-time metric and the energy-momentum tensor, and includes nonlinear Newtonian cosmology. Here we establish the nonlinear post-Friedmann framework in the Lagrangian-coordinates approach for structure formation. For this we first iden…
We study an effect of inhomogeneity of density distribution of the Universe. We propose a new Lagrangian perturbation theory with a backreaction effect by inhomogeneity. The inhomogeneity affects the expansion rate in a local domain and its own growing rate. We numerically analyze a one-dimensional plane-symmetric model, and calculate the probability distribution functions (PDFs) of several observed variables to discuss those statistical properties. We find that the PDF of pairwise peculiar velocity shows an effective difference from the conventional Lagrangian approach, i.e. even in one-dimen…
Lemaitre-Tolman-Bondi (LTB) solutions are used frequently to describe the collapse or expansion of spherically symmetric inhomogeneous mass distributions in the Universe. These exact solutions are obtained in the synchronous gauge where nonlinear dynamics (with respect to the FLRW background) induce large deviations from the FLRW metric. In this paper we show explicitly that this is a gauge artefact (for realistic sub-horizon inhomogeneities). We write down the nonlinear gauge transformation from synchronous to Newtonian gauge for a general LTB solution using the fact that the peculiar velocit…
The peculiarities of the inverse square law of Newtonian gravity in standard Big Bang Cosmology are discussed. It is shown that the incorporation of an additive term to Newtonian gravitation, as the inverse Yukawa-like field, allows remove the incompatibility between the flatness of the Universe and the density of matter in the Friedmann equation, provides a new approach for dark energy, and enable theoretical deduce the Hubble-Lemaitre's law. The source of this inverse Yukawa-like field is the ordinary baryonic matter and represents the large-scale contribution of gravity in accordance with t…
This work develops the dynamics of perfectly elastic solid model for application to the outer crust of a magnetised neutron star. Particular attention is given to the Noether identities responsible for energy-momentum conservation, using a formulation that is fully covariant, not only (as is usual) in a fully relativistic treatment but also (sacrificing accuracy and elegance for economy of degrees of gravitational freedom) in the technically more complicated case of the Newtonian limit. The results are used to obtain explicit (relativistic and Newtonian) formulae for the propagation speeds of …
A 250-year old Newtonian problem, first studied by Euler, turns out to share a lot of similarities with the most extreme astrophysical relativistic object, the Kerr black hole. Although the framework behind the two fields is completely different, both problems are related to gravitational fields that have quite intriguing analogies with respect to orbital motions of a test-body in them. The fundamental reason responsible for their extraordinary similarity is the integrability of both problems, as well as their common multipolar structure. In this paper we demonstrate the existence of a multitu…
We make an attempt to dynamically study, in four space-time dimensions, the classical gravitational collapse of a macroscopic circular fundamental string, by a truncation of the Einstein equations that suppresses retarded features but keeps the main self-gravity peculiarities of the relativistic string dynamics, and allows the investigation of a possible infinite red-shift. The numerical solution of the string evolution in the self-induced metric shows an infinite red shift at a macroscopic size of the string, when the string reaches the velocity of light. We further include the back-reaction …
Loop corrections to the gravitational potential are usually inferred from scattering amplitudes, which seems quite different from how the linearized Einstein equations are solved with a static, point mass to give the classical potential. In this study we show how the Schwinger-Keldysh effective field equations can be used to compute loop corrections to the potential in a way which parallels the classical treatment. We derive explicit results for the one loop correction from the graviton self-energy induced by a massless, minimally coupled scalar.
We present extensions of the treatment contained in our recent paper on nonlocal Newtonian cosmology [C. Chicone and B. Mashhoon, J. Math. Phys. 57, 072501 (2016)]. That is, the implications of the recent nonlocal generalization of Einstein's theory of gravitation are further investigated within the regime of Newtonian cosmology. In particular, we treat the nonlocal problem of structure formation for a spherically symmetric expanding dust model and show numerically that as the central density contrast grows, it tends to decrease slowly with radial distance as the universe expands. The nonlocal…
Recently a covariant approach to cold matter universes in the zero-shear hypersurfaces (or longitudinal) gauge has been developed. This approach reveals the existence of an integrability condition, which does not appear in standard non-covariant treatments. A simple derivation and generalization of the integrability condition is given, based on showing that the quasi-Newtonian models are a sub-class of the linearized `silent' models. The solution of the integrability condition implies a propagation equation for the acceleration. It is shown how the velocity and density perturbations are then o…
Aims: We study the impact of Post-Newtonian correction terms on the energetic interaction between a gravitational wave (GW)-emitting supermassive black hole (SMBH) binary system and incoming stars via three-body scattering experiments. Methods: We use the AR-chain code to simulate with high accuracy the interactions between stars and an SMBH binary at separations of$\sim 1000\,R_{sch}$. For all of the interactions, we investigate in detail the energy balance of the three-body systems, using both Newtonian and Post-Newtonian expressions for the SMBH binary orbital energy, taking into account th…
We construct analytic models of incompressible, rigidly rotating stars in PN gravity and study their stability against nonaxisymmetric Jacobi-like bar modes. PN configurations are modeled by homogeneous triaxial ellipsoids and the metric is obtained as a solution of Einstein's equations in 3+1 ADM form. We use an approximate subset of the equations well-suited to numerical integration for strong field, 3D configurations in quasi--equilibrium. These equations are exact at PN order, and admit an analytic solution for homogeneous ellipsoids. In this paper we present this solution, as well as anal…
In recent years, the idea of sub- and super-Chandrasekhar limiting mass white dwarfs (WDs), which are potential candidates to produce under- and over-luminous type Ia supernovae, respectively, has been a key interest in the scientific community. Although researchers have proposed different models to explain these peculiar objects, modified theories of Einstein's gravity, particularly $f(R)$ gravity with $R$ being the scalar curvature, seems to be one of the finest choices to explain both the regimes of these peculiar WDs. It was already shown that considering higher-order corrections to the St…
Inspiraling binary charged black holes in an external magnetic field: Application of post-Newtonian dynamics in Einstein-Maxwell theory
Shanghai Astronomical Observatory, Chinese Academy of Sciences（中国科学院上海天文台）；University of Chinese Academy of Sciences（中国科学院大学）；Beijing Normal University（北京师范大学）
During the final stages of a neutron-star binary coalescence, stellar quasi-normal modes can become resonantly excited by tidal fields. If the strain exerted by the excited modes exceeds the extent to which the crust can respond linearly, localised crustal failures may occur. In this work, we re-examine resonant $g$-mode excitations of relativistic neutron stars in the last $\sim$ 10 seconds of an inspiral. We adopt realistic equations of state that pass constraints from GW170817, include 3rd order post-Newtonian terms for the conservation orbital motion, and employ a 2.5 post-Newtonian scheme…
The basic concepts and hypotheses of Newtonian Cosmology necessary for a consistent treatment of the averaged cosmological dynamics are formulated and discussed in details. The space-time, space, time and ensemble averages for the cosmological fluid fields are defined and analyzed with a special attention paid to their analytic properties. It is shown that all averaging procedures require an arrangement for a standard measurement device with the same measurement time interval and the same space region determined by the measurement device resolution to be prescribed to each position and each mo…
We use covariant methods to analyse the nonlinear evolution of self-gravitating, non-relativistic media. The formalism is first applied to imperfect fluids, aiming at the kinematic effects of viscosity, before extended to inhomogeneous magnetised environments. The nonlinear electrodynamic formulae are derived and successively applied to electrically resistive and to highly conductive fluids. By nature, the covariant equations isolate the magnetic effects on the kinematics and the dynamics of the medium, combining mathematical transparency and physical clarity. Employing the Newtonian analogue …
Massive black-hole binaries (MBHBs) are thought to be the main source of gravitational waves (GWs) in the low-frequency domain surveyed by ongoing and forthcoming Pulsar Timing Array campaigns and future space-borne missions, such as {\it eLISA}. However, many low-redshift MBHBs in realistic astrophysical environments may not reach separations small enough to allow significant GW emission, but rather stall on (sub)pc-scale orbits. This "last-parsec problem" can be eased by the appearance of a third massive black hole (MBH) -- the "intruder" -- whose action can force, under certain conditions, …
This is the second in a series of papers on the construction of explicit solutions to the stationary axisymmetric Einstein equations which can be interpreted as counter-rotating disks of dust. We discuss the class of solutions to the Einstein equations for disks with constant angular velocity and constant relative density which was constructed in the first part. The metric for these spacetimes is given in terms of theta functions on a Riemann surface of genus 2. We discuss the metric functions at the axis of symmetry and the disk. Interesting limiting cases are the Newtonian limit, the static …
We investigate the nature of additional scalar degrees of freedom contained in extended hybrid metric-Palatini gravity, outlining the emergence of two coupled dynamical scalar modes. In particular, we discuss the weak field limit of the theory, both in the static case and from a gravitational waves perspective. In the first case, performing an analysis at the lowest order of the post parameterized Newtonian (PPN) structure of the model, we stress the settling of Yukawa corrections to the Newtonian potential. In this respect, we show that one scalar field can have long range interactions and us…
The gravitomagnetic tidal Love number of a slowly rotating body was calculated previously under the assumption that the velocity perturbation created by the tidal field consists of an induction piece proportional to the vector potential, and a rotational piece that scales with $Ω$, the body's angular velocity. The second part of this assumption is wrong: the rotational piece of the velocity perturbation scales in fact like $Ω^0 = 1$. The previous calculations are therefore incorrect, and the purpose of this paper is to repair the mistake. To keep the technical difficulties to a minimum, the tr…
The Earth's geoid is one of the most essential and fundamental concepts to provide a gravity field-related height reference in geodesy and associated sciences. To keep up with the ever-increasing experimental capabilities and to consistently interpret high-precision measurements without any doubt, a relativistic treatment of geodetic notions (including the geoid) within Einstein's theory of General Relativity is inevitable. Building on the theoretical construction of isochronometric surfaces and the so-called redshift potential for clock comparison, we define a relativistic gravity potential a…
We discuss the analytical determination of the location of the Last Stable Orbit (LSO) in circular general relativistic orbits of two point masses. We use several different ``resummation methods'' (including new ones) based on the consideration of gauge-invariant functions, and compare the results they give at the third post-Newtonian (3PN) approximation of general relativity. Our treatment is based on the 3PN Hamiltonian of Jaranowski and Schäfer. One of the new methods we introduce is based on the consideration of the (invariant) function linking the angular momentum and the angular frequenc…
We provide a fully general-relativistic treatment of cosmological perturbations in a universe permeated by a large-scale primordial magnetic field, using the Ellis-Bruni gauge-invariant formalism. The exact non-linear equations for general relativistic magnetohydrodynamic evolution are derived. A number of applications are made: the behaviour of small perturbations to Friedmann universes are studied; a comparison is made with earlier Newtonian treatments of cosmological perturbations and some effects of inflationary expansion are examined.
We present a numerical scheme that solves the initial value problem in full general relativity for a binary neutron star in quasi-equilibrium. While Newtonian gravity allows for a strict equilibrium, a relativistic binary system emits gravitational radiation, causing the system to lose energy and slowly spiral inwards. However, since inspiral occurs on a time scale much longer than the orbital period, we can adopt a quasi-equilibrium approximation. In this approximation, we integrate a subset of the Einstein equations coupled to the equations of relativistic hydrodynamics to solve the initial …
We propose an approximate approach for studying the relativistic regime of stellar tidal disruptions by rotating massive black holes. It combines an exact relativistic description of the hydrodynamical evolution of a test fluid in a fixed curved spacetime with a Newtonian treatment of the fluid's self-gravity. Explicit expressions for the equations of motion are derived for Kerr spacetime using two different coordinate systems. We implement the new methodology within an existing Newtonian Smoothed Particle Hydrodynamics code and show that including the additional physics involves very little e…
LISA is a joint space mission of the ESA and NASA for detecting low frequency gravitational radiation in the band $10^{-5} - 1$ Hz. In order to attain the requisite sensitivity for LISA, the laser frequency noise must be suppressed below the other secondary noises such as the optical path noise, acceleration noise etc. This is achieved because of the redundancy in the data, more specifically, by combining six appropriately time-delayed data streams containing fractional Doppler shifts - time delay interferometry (TDI). The orbits of the spacecraft are computed in the gravitational field of the…
We introduce a formulation for spinning gravitating objects in the effective field theory in the post-Newtonian scheme in the context of the binary inspiral problem. We aim at an effective action, where all field modes below the orbital scale are integrated out. We spell out the relevant degrees of freedom, in particular the rotational ones, and the associated symmetries. Building on these symmetries, we introduce the minimal coupling part of the point particle action in terms of gauge rotational variables, and construct the spin-induced nonminimal couplings, where we obtain the leading order …
We address issues with extant formulations of dissipative effects in the effective field theory (EFT) which describes the post-Newtonian (PN) inspiral of two gravitating bodies by (re)formulating several parts of the theory. Novel ingredients include gauge invariant spherical fields in the radiation zone; a system zone which preserves time reversal such that its violation arises not from local odd propagation but rather from interaction with the radiation sector in a way which resembles the balayage method; 2-way multipoles to perform zone matching within the EFT action; and a double-field rad…
Few recent generations of cosmologists have solved non-local newtonian equations of the gravitational instability in an expanding universe. In this approach pancaking is the predominant form of first collapsing objects. Relativistic counterparts of these equations contain the electric and magnetic parts of the Weyl tensor. In the linear theory the magnetic part is associated with gravitational waves. If the magnetic part is ignored, then the newtonian limit of the relativistic equations is reduced to the closed set of the local Lagrangian equations. Recently this fact drew much attention since…
Our recent Newtonian treatment of the smooth-averaged mutual-friction force acting on the neutron superfluid and locally induced by the pinning of quantized neutron vortices to proton fluxoids in the outer core of superfluid neutron stars is here adapted to the general-relativistic framework. We show how the local nonrelativistic motion of individual vortices can be matched to the global dynamics of the star using the fully 4D covariant Newtonian formalism of Carter & Chamel (2004). We derive all the necessary dynamical equations for carrying out realistic simulations of superfluid rotating ne…
This thesis deals with the systematic treatment of quantum-mechanical systems in post-Newtonian gravitational fields. Starting from clearly spelled-out assumptions, employing a framework of geometric background structures defining the notion of a post-Newtonian expansion, our systematic approach allows to properly derive the post-Newtonian coupling of quantum-mechanical systems to gravity based on first principles. This sets it apart from more heuristic approaches that are commonly employed, for example, in the description of quantum-optical experiments under gravity. Regarding single particle…
We present a systematic treatment of the linear theory of scalar gravitational perturbations in the synchronous gauge and the conformal Newtonian (or longitudinal) gauge. We first derive the transformation law relating the two gauges. We then write down in parallel in both gauges the coupled, linearized Boltzmann, Einstein and fluid equations that govern the evolution of the metric perturbations and the density fluctuations of the particle species. The particle species considered include cold dark matter (CDM), baryons, photons, massless neutrinos, and massive neutrinos (a hot dark matter or H…
We study the evolution of cosmological perturbations, using a hybrid approximation scheme which upgrades the weak-field limit of Einstein's field equations to account for post-Newtonian scalar and vector metric perturbations and for leading-order source terms of gravitational waves, while including also the first and second-order perturbative approximations. Our equations, which are derived in the Poisson gauge, provide a unified description of matter inhomogeneities in a Universe filled with a pressureless and irrotational fluid and a cosmological constant, ranging from the linear to the high…
The dynamical system studied in previous papers of this series, namely a bound time-like geodesic motion in the exact static and axially symmetric space-time of an (originally) Schwarzschild black hole surrounded by a thin disc or ring, is considered to test whether the often employed "pseudo-Newtonian" approach (resorting to Newtonian dynamics in gravitational potentials modified to mimic the black-hole field) can reproduce phase-space properties observed in the relativistic treatment. By plotting Poincaré surfaces of section and using two recurrence methods for similar situations as in the r…
