Proof that Einstein’s field equations are invalid: Exposition of the unimodular defect

Albert Einstein first presented his gravitational field equations in unimodular coordinates. In these coordinates, the field equations can be written explicitly in terms of the Einstein pseudotensor for the energy-momentum of the gravitational field. Since this pseudotensor produces, by contraction, a first-order intrinsic differential invariant, it violates the laws of pure mathematics. This is sufficient to prove that Einstein’s unimodular field equations are invalid. Since the unimodular form must hold in the general theory of relativity, it follows that the latter is also physically and mathematically unsound, lacking a proper mathematical foundation.

Dynamical Analysis and Cosmological Evolution in Weyl Integrable Gravity

We investigate the cosmological evolution for the physical parameters in Weyl integrable gravity in a Friedmann–Lemaître–Robertson–Walker universe with zero spatial curvature. For the matter component, we assume that it is an ideal gas, and of the Chaplygin gas, from the Weyl integrable gravity a scalar field is introduced by a geometric approach which provides an interaction with the matter component.We calculate the stationary points for the field equations and we study their stability properties. Furthermore, we solve the inverse problem for the case of an ideal gas and prove that the gravitational field equations can follow from the variation of a Lagrangian function. Finally, variational symmetries are applied for the construction of analytic and exact solutions.

Observational constraints on Tsallis modified gravity

The thermodynamics-gravity conjecture reveals that one can derive the gravitational field equations by using the first law of thermodynamics and vice versa. Considering the entropy associated with the horizon in the form of non-extensive Tsallis entropy, S ∼ Aβ here we first derive the corresponding gravitational field equations by applying the Clausius relation δQ = TδS to the horizon. We then construct the Friedmann equations of Friedmann-Lemaître-Robertson-Walker (FLRW) universe based on Tsallis modified gravity (TMG). Moreover, in order to constrain the cosmological parameters of TMG model, we use observational data, including Planck cosmic microwave background (CMB), weak lensing, supernovae, baryon acoustic oscillations (BAO), and redshift-space distortions (RSD) data. Numerical results indicate that TMG model with a quintessential dark energy is more compatible with the low redshift measurements of large scale structures by predicting a lower value for the structure growth parameter σ8 with respect to ΛCDM model. This implies that TMG model would slightly alleviate the σ8 tension.

Strings in bimetric spacetimes

We put forward a two-dimensional nonlinear sigma model that couples (bosonic) matter fields to topological Hořava gravity on a nonrelativistic worldsheet. In the target space, this sigma model describes classical strings propagating in a curved spacetime background, whose geometry is described by two distinct metric fields. We evaluate the renormalization group flows of this sigma model on a flat worldsheet and derive a set of beta-functionals for the bimetric fields. Imposing worldsheet Weyl invariance at the quantum level, we uncover a set of gravitational field equations that dictate the dynamics of the bimetric fields in the target space, where a unique massless spin-two excitation emerges. When the bimetric fields become identical, the sigma model gains an emergent Lorentz symmetry. In this single metric limit, the beta-functionals of the bimetric fields reduce to the Ricci flow equation that arises in bosonic string theory, and the bimetric gravitational field equations give rise to Einstein’s gravity.

Determination of time evolution of the gravitational constant in the framework of Brans-Dicke Theory: An easy wa

The present article demonstrates a very simple mathematical way to determine the time-dependence of the dynamical gravitational constant () in the framework of the Brans-Dicke theory of gravity. Brans-Dicke field equations, for a matter-dominated, pressure-less and spatially flat universe with homogeneous and isotropic space-time, have been used for this formulation. The gravitational constant () is the reciprocal of the Brans-Dicke scalar field (). Using a simple ansatz, which represents the Brans-Dicke scalar field () as a function of time, the possible values of a constant parameter (constituting the ansatz) have been calculated with the help of the field equations, using the values of some cosmological parameters at the present time. The values of that parameter (belonging to the ansatz) lead to the conclusion that the scalar field () decreases and consequently the gravitational constant () increases with time. The value of the relative time-rate of change of the gravitational constant (i.e., ) has also been estimated and this quantity has been found to be independent of time. Time-dependence of and has been depicted graphically for all values of the parameter belonging to the ansatz. The novel features of this study are that the gravitational field equations did not have to be solved, unlike other studies, to arrive at the results and the mathematical scheme for calculations is extremely easy in comparison to other recent studies in this regard.

Thermodynamics of traversable wormholes in f(R,T) gravity

In this paper, we discuss thermodynamics for spherically symmetric and static traversable wormholes which include Morris–Thorne wormholes and charged wormholes in the background of [Formula: see text] gravity. The local coordinates have been used to find trapping horizons of these objects and generalized surface gravity has been worked out on the trapping horizons. The expression for the unified first law has also been derived from the gradient of Misner–Sharp energy with the help of gravitational field equations and from this law the first law of wormhole dynamics has been obtained. We have done this analysis for the simplest case of [Formula: see text] gravity where [Formula: see text], [Formula: see text] and [Formula: see text] being the traces of the Ricci and stress–energy tensors. Also, we have extended these thermodynamic results to non-minimal curvature-matter coupling.

Generalization of Vaidya’s Metric for A Radiating Star to Rotating Star

Schwarzschild's external solution of Einstein’s gravitational field equations in the general theory of relativity for a static star has been generalized by Vaidya [1], taking into account the radiation of the star. Here, we generalize Vaidya’s metric to a star that is rotating and radiating. Although, there is a famous Kerr solution [2] for a rotating star, but here is a simple solution for a rotating star which may be termed as a zero approximate version of the Kerr solution. Results are discussed.

Variational Principles in Teleparallel Gravity Theories

We study the variational principle and derivation of the field equations for different classes of teleparallel gravity theories, using both their metric-affine and covariant tetrad formulations. These theories have in common that, in addition to the tetrad or metric, they employ a flat connection as additional field variable, but dthey iffer by the presence of absence of torsion and nonmetricity for this independent connection. Besides the different underlying geometric formulation using a tetrad or metric as fundamental field variable, one has different choices to introduce the conditions of vanishing curvature, torsion, and nonmetricity, either by imposing them a priori and correspondingly restricting the variation of the action when the field equations are derived, or by using Lagrange multipliers. Special care must be taken, since these conditions form non-holonomic constraints. Here, we explicitly show that all of the aforementioned approaches are equivalent, and that the same set of field equations is obtained, independently of the choice of the geometric formulation and variation procedure. We further discuss the consequences arising from the diffeomorphism invariance of the gravitational action, and show how they establish relations between the gravitational field equations.

PHYSICAL PROPERTIES OF THE THIN ACCRETION DISK OF THE TAUB-NUT BLACK HOLE

The Taub-NUT space-time metric is one of the vacuum solutions to Einstein's gravitational field equations. In this metric, the Newman-Unti-Tamburino parameter (NUT) and its effect on the physical properties of a thin accretion disk are of particular interest. In this paper, calculations are performed to determine the physical properties of a thin accretion disk around the Taub-NUT black hole based on the Page-Thorne model. The influence of the NUT parameter on the angular velocity, binding energy, angular momentum of particles, effective potential, energy flow, and temperature of the accretion disk is revealed. According to the data obtained, the temperature of the accretion disk of the Taub-NUT black hole decreases as the value of the NUT parameter increases.