scholarly journals F(R) GRAVITY IN PURELY AFFINE FORMULATION

2008 ◽  
Vol 23 (12) ◽  
pp. 1891-1901 ◽  
Author(s):  
NIKODEM J. POPŁAWSKI
Keyword(s):  
General Relativity ◽  
Hamiltonian Dynamics ◽  
Einstein Frame ◽  
Nonlinear Dependence ◽  
Metric Tensor ◽  
Theories Of Gravity ◽  
Metric Variation ◽  
Peculiar Behavior ◽  
Theories Of Gravitation ◽  
Affine Metric

The purely affine, metric-affine and purely metric formulation of general relativity are dynamically equivalent and the relation between them is analogous to the Legendre relation between the Lagrangian and Hamiltonian dynamics. We show that one cannot construct a dynamically equivalent, purely affine Lagrangian from a metric-affine or metric F(R) Lagrangian, nonlinear in the curvature scalar. Thus the equivalence between the purely affine picture and the two other formulations does not hold for metric-affine and metric theories of gravity with a nonlinear dependence on the curvature, i.e. F(R) gravity does not have a purely affine formulation. We also show that this equivalence is restored if the metric tensor is conformally transformed from the Jordan to the Einstein frame, in which F(R) gravity turns into general relativity with a scalar field. This peculiar behavior of general relativity, among relativistic theories of gravitation, with respect to purely affine, metric-affine and purely metric variation could indicate the physicality of the Einstein frame. On the other hand, it could explain why this theory cannot interpolate among phenomenological behaviors at different scales.

scholarly journals Investigating $f(R)$ gravity and cosmologies

2021 ◽  
Author(s):  
Vaibhav Kalvakota
Keyword(s):  
General Relativity ◽  
Lagrangian Density ◽  
Ricci Scalar ◽  
Metric Tensor ◽  
Extended Theory ◽  
Theories Of Gravity ◽  
Theory Of Gravity ◽  
Hilbert Action ◽  
Geometric Nature

The f (R) theory of gravity is an extended theory of gravity that is based on general relativity in the simplest case of $f(R) = R$. This theory extends such a function of the Ricci scalar into arbitrary functions that are not necessarily linear, i.e. could be of the form $f(R) = \alpha R^{2}$. The action for such a theory would be $S_{EH} = \frac{1}{2k} \int f(R) + L^{m}\; d^{4}x\sqrt{−g}$, where $S_{EH}$ is the Einstein-Hilbert action for our theory, $g$ is the determinant of the metric tensor $g_{\mu \nu}$ and $L^{m}$ is the Lagrangian density for matter. In this paper, we will look at some of the physical implications of such a theory, and the importance of such a theory in cosmology and in understanding the geometric nature of such f (R) theories of gravity.

scholarly journals GRAVITATION, ELECTROMAGNETISM AND THE COSMOLOGICAL CONSTANT IN PURELY AFFINE GRAVITY

2009 ◽  
Vol 18 (05) ◽  
pp. 809-829 ◽  
Author(s):  
NIKODEM J. POPŁAWSKI
Keyword(s):  
General Relativity ◽  
Electromagnetic Field ◽  
Cosmological Constant ◽  
Electromagnetic Fields ◽  
Ricci Tensor ◽  
Einstein Equations ◽  
Metric Tensor ◽  
Affine Metric ◽  
Affine Gravity

The Eddington Lagrangian in the purely affine formulation of general relativity generates the Einstein equations with the cosmological constant. The Ferraris–Kijowski purely affine Lagrangian for the electromagnetic field, which has the form of the Maxwell Lagrangian with the metric tensor replaced by the symmetrized Ricci tensor, is dynamically equivalent to the Einstein–Maxwell Lagrangian in the metric formulation. We show that the sum of the two affine Lagrangians is dynamically inequivalent to the sum of the analogous Lagrangians in the metric–affine/metric formulation. We also show that such a construction is valid only for weak electromagnetic fields. Therefore the purely affine formulation that combines gravitation, electromagnetism and the cosmological constant cannot be a simple sum of terms corresponding to separate fields. Consequently, this formulation of electromagnetism seems to be unphysical, unlike the purely metric and metric–affine pictures, unless the electromagnetic field couples to the cosmological constant.

scholarly journals Invariant quantities in the multiscalar-tensor theories of gravitation

2016 ◽  
Vol 31 (02n03) ◽  
pp. 1641003 ◽  
Author(s):  
Piret Kuusk ◽  
Laur Järv ◽  
Ott Vilson
Keyword(s):  
Scalar Fields ◽  
Einstein Frame ◽  
Current Paper ◽  
Jordan Frame ◽  
General Frame ◽  
Newtonian Approximation ◽  
Theories Of Gravity ◽  
Theories Of Gravitation

The aim of the current paper is to study the multiscalar-tensor theories of gravity without derivative couplings. We construct a few basic objects that are invariant under a Weyl rescaling of the metric and transform covariantly when the scalar fields are redefined. We introduce rules to construct further such objects and put forward a scheme that allows to express the results obtained either in the Einstein frame or in the Jordan frame as general ones. These so-called “translation” rules are used to show that the parametrized post-Newtonian approximation results obtained in the aforementioned two frames indeed are the same if expressed in a general frame.

NEW CLASS OF METRIC THEORIES OF GRAVITY NOT DESCRIBED BY THE PARAMETRIZED POST-NEWTONIAN (PPN) FORMALISM

1991 ◽  
Vol 06 (30) ◽  
pp. 5511-5532 ◽  
Author(s):  
IGNAZIO CIUFOLINI
Keyword(s):  
General Relativity ◽  
A Priori ◽  
Relativistic Effects ◽  
Weak Field ◽  
Metric Tensor ◽  
New Class ◽  
Theories Of Gravity ◽  
Experimental Values ◽  
Ppn Parameters ◽  
Infinite Set

After an introduction to theories of gravity alternative to general relativity, metric theories (Sec. 1) and the parametrized post-Newtonian (PPN) formalism (Sec. 2), we define a new class of metric theories of gravity (Sec. 3). It turns out that the post-Newtonian approximation of these new theories is not described by the PPN formalism (Sec. 4); in fact, in the limit of weak field and slow motions, the post-Newtonian expression of the metric tensor contains an, a priori, infinite set of new terms and correspondingly an, a priori, infinite set of new PPN parameters. As a consequence, the parametrized post-Newtonian formulas describing the classical relativistic tests should include these new parameters, and therefore the experimental values of the classical relativistic effects should not be used to put limits only on the standard ten PPN parameters. Finally, we note that a subset of this new class of theories has the same post-Newtonian limit and value of the PPN parameters as general relativity, and therefore is automatically in agreement with the classical general-relativistic tests (Sec. 4, theory III).

scholarly journals THE PHYSICAL FOUNDATIONS FOR THE GEOMETRIC STRUCTURE OF RELATIVISTIC THEORIES OF GRAVITATION: FROM GENERAL RELATIVITY TO EXTENDED THEORIES OF GRAVITY THROUGH EHLERS–PIRANI–SCHILD APPROACH

2012 ◽  
Vol 09 (08) ◽  
pp. 1250072 ◽  
Author(s):  
S. CAPOZZIELLO ◽  
M. DE LAURENTIS ◽  
L. FATIBENE ◽  
M. FRANCAVIGLIA
Keyword(s):  
General Relativity ◽  
Large Class ◽  
Geometric Structure ◽  
Space Time ◽  
Correct Analysis ◽  
Extended Theories Of Gravity ◽  
Theories Of Gravity ◽  
Causal Structures ◽  
Extended Theories ◽  
Theories Of Gravitation

We discuss in a critical way the physical foundations of geometric structure of relativistic theories of gravity by the so-called Ehlers–Pirani–Schild formalism. This approach provides a natural interpretation of the observables showing how relate them to General Relativity and to a large class of Extended Theories of Gravity. In particular we show that, in such a formalism, geodesic and causal structures of space-time can be safely disentangled allowing a correct analysis in view of observations and experiment. As specific case, we take into account the case of f(R)-gravity.

scholarly journals Breaking the cosmological background degeneracy by two-fluid perturbations in f(R) gravity

2015 ◽  
Vol 24 (07) ◽  
pp. 1550053 ◽  
Author(s):  
Amare Abebe
Keyword(s):  
General Relativity ◽  
Exact Solutions ◽  
Structure Formation ◽  
Background Level ◽  
Cosmological Perturbations ◽  
Gauge Invariant ◽  
First Order ◽  
Cosmological Background ◽  
Theories Of Gravity ◽  
Two Fluid

One of the exact solutions of f(R) theories of gravity in the presence of different forms of matter exactly mimics the ΛCDM solution of general relativity (GR) at the background level. In this work we study the evolution of scalar cosmological perturbations in the covariant and gauge-invariant formalism and show that although the background in such a model is indistinguishable from the standard ΛCDM cosmology, this degeneracy is broken at the level of first-order perturbations. This is done by predicting different rates of structure formation in ΛCDM and the f(R) model both in the complete and quasi-static regimes.

scholarly journals CAN f(R) GRAVITY MIMIC GENERAL RELATIVITY?

2012 ◽  
Vol 10 ◽  
pp. 63-68 ◽  
Author(s):  
JE-AN GU
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Early Time ◽  
Einstein Equations ◽  
Modified Theories Of Gravity ◽  
Cosmological Perturbations ◽  
Long Run ◽  
Theories Of Gravity ◽  
The Stability ◽  
Higher Order Differential Equations

We discuss the stability of the general-relativity (GR) limit in modified theories of gravity, particularly the f(R) theory. The problem of approximating the higher-order differential equations in modified gravity with the Einstein equations (2nd-order differential equations) in GR is elaborated. We demonstrate this problem with a heuristic example involving a simple ordinary differential equation. With this example we further present the iteration method that may serve as a better approximation for solving the equation, meanwhile providing a criterion for assessing the validity of the approximation. We then discuss our previous numerical analyses of the early-time evolution of the cosmological perturbations in f(R) gravity, following the similar ideas demonstrated by the heuristic example. The results of the analyses indicated the possible instability of the GR limit that might make the GR approximation inaccurate in describing the evolution of the cosmological perturbations in the long run.

All classical predictions of general relativity from special relativistic Hamiltonian dynamics

2018 ◽  
Vol 33 (29) ◽  
pp. 1850169
Author(s):  
J. H. Field
Keyword(s):  
General Relativity ◽  
Euclidean Space ◽  
Hamiltonian Dynamics ◽  
Gravitational Redshift ◽  
Newtonian Mechanics ◽  
Light Deflection ◽  
Schwarzschild Metric ◽  
Related Literature ◽  
General Relativistic ◽  
Relativistic Hamiltonian Dynamics

Previous special relativistic calculations of gravitational redshift, light deflection and Shapiro delay are extended to include perigee advance. The three classical, order G, post-Newtonian predictions of general relativity as well as general relativistic light-speed-variation are therefore shown to be also consequences of special relativistic Newtonian mechanics in Euclidean space. The calculations are compared to general relativistic ones based on the Schwarzschild metric equation, and related literature is critically reviewed.

scholarly journals The radiation instability in modified gravity

2021 ◽  
Vol 36 (08n09) ◽  
pp. 2150060
Author(s):  
Spiros Cotsakis ◽  
Dimitrios Trachilis
Keyword(s):  
General Relativity ◽  
General Solution ◽  
Modified Gravity ◽  
Evolution Equations ◽  
Formal Series ◽  
Series Expansions ◽  
General Polynomial ◽  
Theories Of Gravity ◽  
Polynomial Extensions

We study the problem of the instability of inhomogeneous radiation universes in quadratic Lagrangian theories of gravity written as a system of evolution equations with constraints. We construct formal series expansions and show that the resulting solutions have a smaller number of arbitrary functions than that required in a general solution. These results continue to hold for more general polynomial extensions of general relativity.

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