Newtonian Evolution of the Weyl Tensor (original) (raw)

On the Significance of the Weyl Curvature in a Relativistic Cosmological Model

Modern Physics Letters A, 2009

The Weyl curvature includes the Newtonian field and an additional field, the so-called anti-Newtonian. In this paper, we use the Bianchi and Ricci identities to provide a set of constraints and propagations for the Weyl fields. The temporal evolutions of propagations manifest explicit solutions of gravitational waves. We see models with purely Newtonian field are inconsistent with relativistic models and obstruct sounding solutions. Therefore, both fields are necessary for the nonlocal nature and radiative solutions of gravitation.

Magnetic fields and the Weyl tensor in the early universe

General Relativity and Gravitation

We have solved the Einstein-Maxwell equations for a class of metrics with constant spatial curvature by considering only a primordial magnetic field as source. We assume a slight modification of the Tolman averaging relations so that the energy-momentum tensor of this field possesses an anisotropic pressure component. This inhomogeneous magnetic universe is isotropic and its time evolution is guided by the usual Friedmann equations. In the case of a flat universe, the space-time metric is free of singularities (except the well-known initial singularity at t = 0). It is shown that the anisotropic pressure of our model has a straightforward relation to the Weyl tensor. We then analyze the effect of this new ingredient on the motion of test particles and on the geodesic deviation of the cosmic fluid.

Cosmology in the Newtonian Limit

Numerical N -body simulations of large scale structure formation in the universe are based on Newtonian gravity. However, according to our current understanding, the most correct theory of gravity is general relativity. It is therefore important to understand which degrees of freedom and which features are lost when the relativistic universe is approximated, or rather replaced, by a Newtonian one. This is the main purpose of our investigation. We first define Newtonian cosmology and we give an overview on general relativity, both in its standard and covariant formulations. We show how the two theories deal with inhomogeneous cosmological models and we explain the role that inhomogeneities play in the dynamics of the universe on large scales. We define averaging in cosmology and we introduce the backreaction conjecture. Then we review on how Newtonian gravity and general relativity relate to each other in the fully non-linear regime. For this purpose we discuss frame theory, whose aim is to reconcile Newton's and Einstein's theories under the same formal structure. We carry out the same investigation also in the weak-field, small-velocity limit of general relativity, and we derive the Newtonian limit resorting to the framework of post-Newtonian cosmology. Finally we remark that there are solutions of Newtonian gravity which do not have any relativistic counterpart. This suggests that there are cases in cosmology in which the two theories are irreconcilable and that the reliability of the Newtonian approximation requires further theoretical investigation.

Emergence of modified Newtonian gravity from thermodynamics

arXiv: Statistical Mechanics, 2019

Being inspired by Verlinde's proposal that general relativistic gravity has a thermodynamic origin as an entropic force, Newtonian gravity is reexamined in view of nonequilibrium thermodynamics. Here, firstly, an unspecified scalar field potential is introduced and treated as a thermodynamic variable on an equal footing with the fluid variables. Then, the effects of irreversibility on the field are explored through the analysis of the entropy production rate in the linear regime. Remarkably, the second law of thermodynamics imposes a stringent constraint on the allowable field, which turns out to be of gravity. The resulting field equation for the gravitational potential contains a dissipative term originating from irreversibility. It is found that the system relaxes to the conventional theory of Newtonian gravity up to a certain spatial scale (typically the solar scale), whereas on the larger scale (such as the galaxy scale) a potential needed in Modified Newtonian Dynamics (MO...

Weyl type f(Q, T) gravity, and its cosmological implications

The European Physical Journal C

We consider an f (Q, T) type gravity model in which the scalar non-metricity Q αμν of the space-time is expressed in its standard Weyl form, and it is fully determined by a vector field w μ. The field equations of the theory are obtained under the assumption of the vanishing of the total scalar curvature, a condition which is added into the gravitational action via a Lagrange multiplier. The gravitational field equations are obtained from a variational principle, and they explicitly depend on the scalar nonmetricity and on the Lagrange multiplier. The covariant divergence of the matter energy-momentum tensor is also determined, and it follows that the nonmetricity-matter coupling leads to the nonconservation of the energy and momentum. The energy and momentum balance equations are explicitly calculated, and the expressions of the energy source term and of the extra force are found. We investigate the cosmological implications of the theory, and we obtain the cosmological evolution equations for a flat, homogeneous and isotropic geometry, which generalize the Friedmann equations of standard general relativity. We consider several cosmological models by imposing some simple functional forms of the function f (Q, T), and we compare the predictions of the theory with the standard CDM model.

On the physical consequences of a Weyl invariant theory of gravity

2020

In this paper we explore the physical consequences of assuming Weyl invariance of the laws of gravity from the classical standpoint exclusively. Actual Weyl invariance requires to replace the underlying Riemannian geometrical structure of the background spacetimes by Weyl integrable geometry (WIG). We show that gauge freedom, a distinctive feature of Weyl invariant theories of gravity, leads to very unusual consequences. For instance, within the cosmological setting in a WIG-based conformal invariant gravity theory, also known as conformal general relativity (CGR), a static universe is physically equivalent to a universe undergoing de Sitter expansion. It happens also that spherically symmetric black holes are physically equivalent to singularity-free wormholes. Another outstanding consequence of gauge freedom in the framework of CGR is that inflation is not required to explain the flatness, horizon and relict particle abundances, among other puzzles that arise in standard GR-based ...

Weyl rescaled Newton-Cartan geometry and non-relativistic conformal hydrodynamics

arXiv (Cornell University), 2015

The non-relativistic covariant framework for fields is extended to investigate fields and fluids on scale covariant curved backgrounds. The scale covariant Newton-Cartan background is constructed using the localization of spacetime symmetries of non-relativistic fields in flat space. Following this, we provide a Weyl covariant formalism which can be used to study scale invariant fluids. By considering ideal fluids as an example, we describe its thermodynamic and hydrodynamic properties and explicitly demonstrate that it satisfies the local second law of thermodynamics. As a further application, we consider the low energy description of Hall fluids. Specifically, we find that the gauge fields for scale transformations lead to corrections of the Wen-Zee and Berry phase terms contained in the effective action.

Post-Newtonian cosmological dynamics in Lagrangian coordinates

Monthly Notices of the Royal Astronomical Society, 1996

We study the non-linear dynamics of self-gravitating irrotational dust in a general relativistic framework, using synchronous and comoving (i.e. Lagrangian) coordinates. All the equations are written in terms of a single tensor variable, the metric tensor of the spatial sections orthogonal to the fluid flow. This treatment allows an unambiguous expansion in inverse (even) powers of the speed of light. To lowest order, the Newtonian approximation -in Lagrangian form -is derived and written in a transparent way; the corresponding Lagrangian Newtonian metric is obtained. Post-Newtonian corrections are then derived and their physical meaning clarified. A number of results are obtained: i) the master equation of Lagrangian Newtonian dynamics, the Raychaudhuri equation, can be interpreted as an equation for the evolution of the Lagrangian-to-Eulerian Jacobian matrix, complemented by the irrotationality constraint; ii) the Lagrangian spatial metric reduces, in the Newtonian limit, to that of Euclidean 3-space written in timedependent curvilinear coordinates, with non-vanishing Christoffel symbols, but vanishing spatial curvature (a particular example of it is given within the Zel'dovich approximation); iii) a Lagrangian version of the Bernoulli equation for the evolution of the "velocity potential" is obtained. iv) The Newtonian and post-Newtonian content of the electric and magnetic parts of the Weyl tensor is clarified. v) At the Post-Newtonian level, an exact and general formula is derived for gravitational-wave emission from non-linear cosmological perturbations; vi) a straightforward application to the anisotropic collapse of homogeneous ellipsoids shows that the ratio of these post-Newtonian terms to the Newtonian ones tends to diverge at least like the mass density. vii) It is argued that a stochastic gravitational-wave background is produced by non-linear cosmic structures, with present-day closure density Ω gw ∼ 10 −5 -10 −6 on 1 -10 Mpc scales.

A variational formulation for the Newtonian cosmology

Il Nuovo Cimento B, 1979

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General relativity as a unified fluid and field theory

Journal of Physics: Conference Series

Einstein's dream for a unified field theory of Nature is attained with a classical fluid theory founded on space, time and spin, rather than on Einstein's spacetime. An invariant quantum theory for the primordial fluid obeys the homogeneous Klein-Gordon equation, which is the same three-dimensional classical wave equation (CWE) initially tried by Schrödinger to formulate quantum mechanics (QM), but abandoned by linear superposition considerations. Primordial fluid pervades universe, obeys energy and momentum conservation, and is formed by sagions: energy-like, discrete, extended Planck-size objects of finite size, carriers of linear momentum and spin, moving in absolute 3D-curved space with speed C along straightest path. We briefly describe our novel non-harmonic and inherently quantized solutions for CWE in spherical coordinates, discovered in 1995. Solutions include a steady-state background field (possibly related to the CMB, to non-locality, and to action-at-a-distance), quantized helices, and inherently quantized functions exhibiting stable dynamic equilibrium, and isomorphism under many transformations, including the classical Doppler case, and the relativistic Lorentz, Poincaré, and Einstein transformations. Isomorphism preempts ab initio a few interpretative issues regarding relativistic and classical transformations. Mathematical fields represent the realistic physical temporal evolution of the primordial fluid in curved 3D-space.