Many cosmological observations call for the existence of dark matter. The most direct evidence for dark matter is inferred from the measured flatness of galactic rotation curves. The latter is based on Newtonian gravity. Alternative approaches to the rotation curve problem by means of general relativity have recently been put forward. The class of models of interest is a subset of the axially symmetric and stationary solutions of Einstein's equations with rotating dust. As a step toward the understanding of general relativistic galaxy models, we analyse rigidly as well as non-rigidly rotating (Post-)Newtonian spacetimes. We find that the Newtonian limit of the considered general relativistic galaxy model leads to Post-Newtonian terms in the metric.
Newtonian cosmological perturbation equations valid to full nonlinear order are well known in the literature. Assuming the absence of the transverse-tracefree part of the metric, we present the general relativistic counterpart valid to full nonlinear order. The relativistic equations are presented without taking the slicing (temporal gauge) condition. The equations do have the proper Newtonian and first post-Newtonian limits. We also present the relativistic pressure correction terms in the Newtonian hydrodynamic equations.
A Newtonian uniform ball expanding in empty space constitutes a common heuristic analogy for FLRW cosmology. We discuss possible implementations of the corresponding general-relativistic problem and a variety of new cosmological analogies arising from them. We highlight essential ingredients of the Newtonian analogy, including that the quasilocal energy is always `Newtonian' in the sense that the magnetic part of the Weyl tensor does not contribute to it. A symmetry of the Einstein-Friedmann equations produces another one in the original Newtonian system.
We construct a self-consistent relativistic Newtonian analogue corresponding to gravitational static spherical symmetric spacetime geometries, staring directly from a generalized scalar relativistic gravitational action in Newtonian framework, which gives geodesic equations of motion identical to those of the parent metric. Consequently, the derived velocity-dependent relativistic scalar potential, which is a relativistic generalization of Newtonian gravitational potential, exactly reproduces the relativistic gravitational features corresponding to any static spherical symmetric spacetime geom…
With continuous advances in technology, future satellite gradiometry missions will be capable of performing precision relativistic experiments and imposing constraints on modern gravity theories. To this end, the full first-order post-Newtonian tidal tensor under inertially guided and Earth-pointing local frames along post-Newtonian orbits is worked out. The physical picture behind the "Mashhoon-Theiss anomaly" is explained at the post-Newtonian level. The relativistic precession of the local frame with respect to the sidereal frame will produce modulations of Newtonian tidal forces along cert…
Recent general-relativistic extensions of Newtonian rotation laws for self-gravitating stationary fluids allow one to rederive, in the first post-Newtonian approximation, the well known geometric dragging of frames, and two new weak-field effects within rotating tori. These are the recently discovered anti-dragging and a new effect that measures the deviation from the Keplerian motion and/or the contribution of the fluids selfgravity. They can be applied to the study of the existence of the (post-)Newtonian limits of solutions and in investigations of inequalities relating parameters of rotati…
With Einstein's inertial motion (free-falling and non-rotating relative to gyroscopes), geodesics for non-relativistic particles can intersect repeatedly, allowing one to compute the space-time curvature $R^{\hat{0} \hat{0}}$ exactly. Einstein's $R^{\hat{0} \hat{0}}$ for strong gravitational fields and for relativistic source-matter is identical with the Newtonian expression for the relative radial acceleration of neighboring free-falling test-particles, spherically averaged.--- Einstein's field equations follow from Newtonian experiments, local Lorentz-covariance, and energy-momentum conserva…
The post-Newtonian hydrodynamic equations for a non-perfect fluid are developed within the framework of a post-Newtonian Boltzmann equation. The post-Newtonian components of the energy-momentum tensor are determined by considering the relativistic Eckart decomposition for a viscous and heat conducting fluid. From the relativistic Grad distribution function its post-Newtonian expression is derived. The hydrodynamic equations for the mass density, mass-energy density and momentum density are determined from a post-Newtonian transfer equation and Grad's distribution function. In the non-relativis…
We show how standard Newtonian N-body simulations can be interpreted in terms of the weak-field limit of general relativity by employing the recently developed Newtonian motion gauge. Our framework allows the inclusion of radiation perturbations and the non-linear evolution of matter. We show how to construct the weak-field metric by combining Newtonian simulations with results from Einstein-Boltzmann codes. We discuss observational effects on weak lensing and ray tracing, identifying important relativistic corrections.
We present a new approximation to include fully general relativistic pressure and velocity in Newtonian hydrodynamics. The energy conservation, momentum conservation and two Poisson's equations are consistently derived from Einstein's gravity in the zero-shear gauge assuming weak gravity and action-at-a-distance limit. The equations show proper special relativity limit in the absence of gravity. Our approximation is complementary to the post-Newtonian approximation and the equations are valid in fully nonlinear situations.
The standard numerical tools for studying non-linear collapse of matter are Newtonian $N$-body simulations. Previous work has shown that these simulations are in accordance with General Relativity (GR) up to first order in perturbation theory, provided that the effects from radiation can be neglected. In this paper we show that the present day matter density receives more than 1$\%$ corrections from radiation on large scales if Newtonian simulations are initialised before $z=50$. We provide a relativistic framework in which \emph{unmodified} Newtonian simulations are compatible with linear GR …
We present a new simple relativistic model for planetary motion describing accurately the anomalous precession of the perihelion of Mercury and its origin. The model is based on transforming Newton's classical equation for planetary motion from absolute to real spacetime influenced by the gravitational potential and introducing the concept of influenced direction.
Non-equilibrium and equilibrium thermodynamics of an interacting component in a special-relativistic multi-component system is discussed by use of an entropy identity. The special case of the corresponding free component is considered. Equilibrium conditions and especially the multi-component Killing relation of the 4-temperature are discussed. Two axioms characterize the mixture: additivity of the energy momentum tensors and of the 4-entropies of the components generating those of the mixture. The resulting quantities of a component and of the mixture, energy, energy flux, momentum flux, stre…
Equation of motion for relativistic compact binaries with the strong field point particle limit : Formulation, the first post-Newtonian and multipole terms
We clarify the subtle issue of finding the correct mapping of Newtonian simulations to light-cone observables at very large distance scales. A faithful general-relativistic interpretation specifies a gauge, i.e. a chart that relates the simulation data to points of the space-time manifold. It has already been pointed out that the implicit gauge choice of Newtonian simulations is indeed different from the Poisson gauge that is commonly adopted for relativistic calculations, the difference being most significant at large scales. It is therefore inconsistent, for example, to predict weak-lensing …
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 fie…
On the Status of the Geodesic Principle in Newtonian and Relativistic Physics
Bob Geroch和Pong Soo Jang ["广义相对论中的物体运动"]的一个定理。"Journal of数学物理16（1），（1975）]提供了一种意义，其中测地线原理具有广义相对论（GR）中定理的地位。我最近证明了一个类似的定理在几何化的牛顿引力（牛顿-嘉当理论）的背景下也成立。牛顿理论中的物体运动"Journal of数学物理52（3），（2011）]。在这里，我比较这两个定理的解释。我认为，尽管有一些明显的差异之间的定理，在几何化的牛顿引力的测地线原则的地位是，作必要的修改，惊人的相对论性的情况下相似。
We establish the existence of a wide class of inhomogeneous relativistic solutions to the Einstein-Euler equations that are well approximated on cosmological scales by solutions of Newtonian gravity. Error estimates measuring the difference between the Newtonian and relativistic solutions are provided.
