Standard General Relativity Theory Research Articles

How the non-metricity of the connection arises naturally in the classical theory of gravity

Spacetime geometry is described by two–a priori independent–geometric structures: the symmetric connection Γ and the metric tensor g. Metricity condition of Γ (i.e. ∇g = 0) is implied by the Palatini variational principle, but only when the matter fields belong to an exceptional class. In case of a generic matter field, Palatini implies non-metricity of Γ. Traditionally, instead of the (first order) Palatini principle, we use in this case the (second order) Hilbert principle, assuming metricity condition a priori. Unfortunately, the resulting right-hand side of the Einstein equations does not coincide with the matter energy-momentum tensor. We propose to treat seriously the Palatini-implied non-metric connection. The conventional Einstein’s theory, rewritten in terms of this object, acquires a much simpler and universal structure. This approach opens a room for the description of the large scale effects in General Relativity (dark matter?, dark energy?), without resorting to purely phenomenological terms in the Lagrangian of gravitational field. All theories discussed in this paper belong to the standard General Relativity Theory, the only non-standard element being their (much simpler) mathematical formulation. As a mathematical bonus, we propose a new formalism in the calculus of variations, because in case of hyperbolic field theories the standard approach leads to nonsense conclusions.

Cover Picture: Astron. Nachr. 01/2023

Synthetic image of the accretion disk around the black hole at the center of the galaxy M87. The bright emission corresponds to the part of the disk approaching us. The simulation is based on the so-called pseudo-complex General Relativity (pcGR), an extension of the standard theory of General Relativity (GR) constructed to avoid singularities. At a spatial resolution of 20 μ arcsec (left image), comparable to the resolution of the Event Horizon Telescope (EHT) of 24 μ arcsec, the predictions of GR and pcGR are indistinguishable. Only at even higher spatial resolution, better than 5 μ arcsec (right image), the pcGR prediction of a dark ring followed further in by a bright ring could be revealed. For further details see the article by Maghlaoui and Hess, page e220067 in this issue.

Bianchi Type V Universe and Bulk Viscous Models with Time Dependent Gravitational Constant and Cosmological Constant in General Relativity

This paper deals with Bulk Viscous Models and Bianchi type- universe in the standard general relativity theory by assuming where constants, is the density. Einstein field equations ( ) have been solved in the presence of a variable gravitational constant as well as a variable cosmological constant together with a bulk viscous fluid. In order to find a deterministic solution of EFEs, a simple form of Hubble parameter being constant when where has been considered here that shows the signature flip from early deceleration to present acceleration. Moreover, the bulk viscous coefficient is allowed to vary with the density as . The physical and geometrical behavior are described in detail for the obtained model. The model obtained here is in good agreement with the present cosmological observations.

Nonlocal Gravitomagnetism

We briefly review the current status of nonlocal gravity (NLG), which is a classical nonlocalgeneralization of Einstein’s theory of gravitation based on a certain analogy with the nonlocalelectrodynamics of media. Nonlocal gravity thus involves integro-differential field equationsand a causal constitutive kernel that should ultimately be determined from observational data.We consider the stationary gravitational field of an isolated rotating astronomical source in the linearapproximation of nonlocal gravity. In this weak-field and slow-motion approximation of NLG,we describe the gravitomagnetic field associated with the rotating source and compare our resultswith gravitoelectromagnetism (GEM) of the standard general relativity theory. Moreover, we brieflystudy the energy-momentum content of the GEM field in nonlocal gravity.

Predictions of the pseudo‐complex general relativity

Within the pseudo‐complex general relativity (GR), a simulation of accretion disks for SgrA* is performed at the frequency of 250 GHz as used by the Event Horizon Telescope. Differences from the standard theory of GR are pointed out, such as the presence of a dark ring followed by an inner bright ring. In the second part, the interpretation of the source of the gravitational wave event GW150914, observed a year ago, is reanalyzed. Further comments are made on the observation of pulsations in NGC7793 P13.

General Relativity and the Theory of a Self-Interacting Abelian Gauge Field

The standard theory of general relativity (GR) can be written in a form proposed by Eddington using the parametric representation of the metric tensor. In this paper, the equations of the standard theory of GR using the parametric representation are first developed. Afterwards, the fundamental ideas of a new type of abelian self-interacting gauge theory are presented. Finally, it is shown that the gauge field equations of this new theory are identical to the parametric form of Einstein’s equations of general relativity. It is concluded that classical gravity can be described either by the usual theory of GR in a curved space-time or, alternatively as a self-interacting gauge theory independent of the dynamics of space-time.

Unimodular theory of gravity and inflation

We investigate inflation and its scalar perturbation driven by a massive scalar field in the unimodular theory of gravity. We introduce a parameter ξ with which the theory is invariant under general unimodular coordinate transformations. When the unimodular parameter is , the classical picture of inflation is reproduced in the unimodular theory because it recovers the background equations of the standard theory of general relativity. We show that for , the theory is equivalent to the standard theory of general relativity at the perturbation level. Unimodular gravity constrains the gauge degree of freedom in the scalar perturbation, but the perturbation equations are similar to those in general relativity. For , we derive the power spectrum and the spectral index, and obtain the unimodular correction to the tensor-to-scalar ratio. Depending on the value of ξ, the correction can either raise or lower the value of the tensor-to-scalar ratio.

RECONSTRUCTING f(R, T) GRAVITY FROM HOLOGRAPHIC DARK ENERGY

We consider cosmological scenarios based on f(R, T) theories of gravity (R is the Ricci scalar and T is the trace of the energy–momentum tensor) and numerically reconstruct the function f(R, T) which is able to reproduce the same expansion history generated, in the standard General Relativity theory, by dark matter and holographic dark energy. We consider two special f(R, T) models: in the first instance, we investigate the modification R + 2f(T), i.e. the usual Einstein–Hilbert term plus a f(T) correction. In the second instance, we consider a f(R) + λT theory, i.e. a T correction to the renown f(R) theory of gravity.

Cosmic microwave background with Brans-Dicke gravity. I. Covariant formulation

In the covariant cosmological perturbation theory, a 1+3 decomposition ensures that all variables in the frame-independent equations are covariant, gauge-invariant and have clear physical interpretations. We develop this formalism in the case of Brans-Dicke gravity, and apply this method to the calculation of cosmic microwave background (CMB) anisotropy and large scale structures (LSS). We modify the publicly available Boltzmann code CAMB to calculate numerically the evolution of the background and adiabatic perturbations, and obtain the temperature and polarization spectra of the Brans-Dicke theory for both scalar and tensor mode, the tensor mode result for the Brans-Dicke gravity are obtained numerically for the first time. We first present our theoretical formalism in detail, then explicitly describe the techniques used in modifying the CAMB code. These techniques are also very useful to other gravity models. Next we compare the CMB and LSS spectra in Brans-Dicke theory with those in the standard general relativity theory. At last, we investigate the ISW effect and the CMB lensing effect in the Brans-Dicke theory. Constraints on Brans-Dicke model with current observational data is presented in a companion paper (paper II).

Extended theories of gravity and their cosmological and astrophysical applications

Astrophysical observations are pointing out huge amounts of “dark matter” and “dark energy” needed to explain the observed large scale structure and cosmic dynamics. The emerging picture is a spatially flat, homogeneous Universe undergoing the today observed accelerated phase. Despite of the good quality of astrophysical surveys, commonly addressed as Precision Cosmology, the nature and the nurture of dark energy and dark matter, which should constitute the bulk of cosmological matter-energy, are still unknown. Furthermore, up to now, no experimental evidence has been found, at fundamental level, to explain such mysterious components. The problem could be completely reversed considering dark matter and dark energy as “shortcomings” of General Relativity in its simplest formulation (a linear theory in the Ricci scalar R, minimally coupled to the standard perfect fluid matter) and claiming for the “correct” theory of gravity as that derived by matching the largest number of observational data, without imposing any theory a priori. As a working hypothesis, accelerating behavior of cosmic fluid, large scale structure, potential of galaxy clusters, rotation curves of spiral galaxies could be reproduced by means of extending the standard theory of General Relativity. In other words, gravity could acts in different ways at different scales and the above “shortcomings” could be due to incorrect extrapolations of the Einstein gravity, actually tested at short scales and low energy regimes. After a survey of what is intended for Extended Theories of Gravity in the so called “metric” and “Palatini” approaches, we discuss some cosmological and astrophysical applications where the issues related to the dark components are addressed by enlarging the Einstein theory to more general f (R) Lagrangians, where f (R) is a generic function of Ricci scalar R, not assumed simply linear. Obviously, this is not the final answer to the problem of “dark-components” but it can be considered as an operative scheme whose aim is to avoid the addition of unknown exotic ingredients to the cosmic pie.

Einstein algebras and general relativity

A purely algebraic structure called an Einstein algebra is defined in such a way that every spacetime satisfying Einstein's equations is an Einstein algebra but not vice versa. The Gelfand representation of Einstein algebras is defined, and two of its subrepresentations are discussed. One of them is equivalent to the global formulation of the standard theory of general relativity; the other one leads to a more general theory of gravitation which, in particular, includes so-called regular singularities. In order to include other types of singularities one must change to sheaves of Einstein algebras. They are defined and briefly discussed. As a test of the proposed method, the sheaf of Einstein algebras corresponding to the spacetime of a straight cosmic string with quasiregular singularity is constructed.

Dirac's large numbers hypothesis in Einstein's theory of gravitation

A method is given for incorporating Dirac's Large Numbers hypothesis into Einstein's standard theory of general relativity. The method involves the assumption that at each point in space there exist two types of clocks, a cosmological clock measuring ephemeris timeτ and an atomic clock measuring atomic timet A . Newton's law of universal gravitation is formulated relativistically in terms of these two times and the proper distance determined by measuring rods between simultaneous events, and a method is given for operationally identifyingG. The Large Numbers hypothesis requirement thatG A ∝1/t A is then used to establish the relationship between the two times. Alternative derivations of the time relationship not involving a time-varying gravitational “constant” are obtained by intercomparison of various large numbers. It is shown that the resulting relationship betweent A andτ gives agreement with the observed natural microwave radiations. Also, the Large Numbers hypothesis leads to a time formed from the fundamental constants of Nature that is comparable to the age of the Universe.