Derivation Of Field Equations Research Articles

Covariant and manifestly projective invariant formulation of Thomas-Whitehead gravity

Thomas-Whitehead (TW) gravity is a recently formulated projectively invariant extension of Einstein-Hilbert gravity. Projective geometry was used long ago by Thomas to succinctly package equivalent paths encoded by the geodesic equation. Projective invariance in gravity has further origins in string theory through a geometric action constructed from the method of coadjoint orbits using the Virasoro algebra. A projectively invariant connection arises from this construction, a part of which is known as the diffeomorphism field. TW gravity exploits projective Gauss-Bonnet terms in the action functional to endow the diffeomorphism field with dynamics, while allowing the theory to collapse to general relativity in the limit that the diffeomorphism field vanishes and the connection becomes Levi-Civita. In the original formulation of TW gravity, the diffeomorphism field is projectively invariant but not tensorial and the connection is projectively invariant but not affine. In this paper we reformulate TW gravity in terms of projectively invariant tensor fields and a projectively invariant covariant derivative, derive field equations respecting these symmetries, and show that the field equations obtained are classically equivalent across formulations. This provides a “Rosetta Stone” between this newly constructed covariant and projective invariant formulation of TW gravity and the original formulation that was manifestly projective invariant, but not covariant. Published by the American Physical Society 2024

Ricci solitons on anisotropic fluid spacetime in f(R, T)-gravity

In this paper we study the behaviour of anisotropic fluid spacetimes in f(R, T)-gravity for admitting Ricci Soliton. We obtain the form of tangential pressure of the fluid for steady, shrinking and expanding Ricci soliton when Ricci soliton vector field becomes a Killing vector field. Modified Poisson and Liouville equations are derived from the Ricci soliton equation in f(R, T)-gravity having anisotropic fluid. Also harmonic significance of Ricci soliton on anisotropic fluid spacetime in f(R, T)-gravity with a harmonic potential function is derived. We derive field equations for perfect fluid plane symmetric static spacetime and obtain their solutions. For each solution we then obtain Ricci soliton vector fields. We observe that such spacetimes admit expanding, shrinking or steady Ricci solitons. All the obtained solitons are trivial in nature. When these Ricci soliton vector fields become Killing vector fields then our results are in full agreement with the theorems deduced from our study.

Observational constraints on the fractal cosmology

In this paper, we explore a fractal model of the universe proposed by Calcagni (J High Energy Phys 03:120, 2010) for a power-counting renormalizable field theory living in a fractal spacetime. Considering a timelike fractal profile, we derived field equations in fractal cosmology, in order to explore the structure formation and the expansion history in fractal universe. Numerical investigations based on matter power spectra diagrams report higher structure growth in fractal cosmology, being in contrast to local galaxy surveys. Additionally, according to the evolution of Hubble parameter diagrams, it can be understood that Hubble constant would decrease in fractal cosmology, which is also incompatible with low redshift estimations of H_0. So, concerning primary numerical studies, it seems that fractal cosmology is not capable to alleviate the tensions between local and global observational probes. Then, in pursuance of more accurate results, we constrain the fractal cosmology by observational data, including Planck cosmic microwave background (CMB), weak lensing, supernovae, baryon acoustic oscillations (BAO), and redshift-space distortions (RSD) data. The derived constraints on fractal dimension beta indicate that there is no considerable deviation from standard model of cosmology.

Noncommutative D=5 Chern–Simons gravity: Kaluza–Klein reduction and chiral gravitational anomaly

Actions for noncommutative (NC) gauge field theories can be expanded perturbatively in powers of the noncommutativity parameter theta using the Seiberg–Witten map between ordinary classical fields and their NC counterparts. The leading order term represents classical (theta =0) action while higher-order terms give us theta -dependent NC corrections that ought to capture some aspects of quantum gravity. Building on previous work of Aschieri and Castellani on NC Chern–Simons (CS) gauge and gravity theories, showing that non-trivial theta -dependence exists only for spacetime dimensions Dge 5, we investigate a correlated effect of these extra spatial dimensions and noncommutativity on four-dimensional physics, up to first-order in theta . Assuming that one spatial dimension is compactified into a circle, we apply the Kaluza–Klein reduction procedure on the NC D=5 CS theory for the conformal gauge group SO(4, 2), to obtain an effective, theta -dependent four-dimensional theory of gravity that has Einstein–Hilbert gravity with negative cosmological constant as its commutative limit. We derive field equations for this modified theory of gravity and study the effect of NC interactions on some classical geometries, such as the AdS-Schwarzschild black hole. We find that this NC background spacetime gives rise to chiral gravitational anomaly due to the nonvanishing theta -dependent Pontryagin density.

Information content and minimum-length metric: A drop of light

In the vast amount of results linking gravity with thermodynamics, statistics, information, a path is described which tries to explore this connection from the point of view of (non)locality of the gravitational field. First the emphasis is put on that well-known thermodynamic results related to null hypersurfaces (i.e. to lightsheets and to generalized covariant entropy bound) can be interpreted as implying an irreducible intrinsic nonlocality of gravity. This nonlocality even if possibly concealed at ordinary scales(depending on which matter is source of the gravitational field, and which matter we use to probe the latter) unavoidably shows up at the smallest scales, read the Planck length $l_p$, whichever are the circumstances we are considering. Some consequences are then explored of this nonlocality when embodied in the fabric itself of spacetime by endowing the latter with a minimum length $L$, in particular the well-known and intriguing fact that this brings to get the field equations, and all of gravity with it, as a statistical-mechanical result. This is done here probing the neighborhood of a would-be (in ordinary spacetime) generic event through lightsheets (instead of spacelike or timelike geodesic congruences as in other accounts) from it. The tools for these derivations are nonlocal quantities, and among them the minimum-length Ricci scalar stands out both for providing micro degrees of freedom for gravity in the statistical account and for the fact that intriguingly the ordinary, or `classical', Ricci scalar can not be recovered from it in the $L\to 0$ limit. Emphasis is put on that classical gravity is generically obtained this way for $\hbar\ne 0$, but not in the $\hbar\to 0$ limit (the statistically derived field equations become singular in this limit), adding to previous results in this sense. (truncated Abstract; see the paper for full Abstract)

A class of QFTs with higher derivative field equations leading to standard dispersion relation for the particle excitations

Given any (Feynman) propagator which is Lorentz and translation invariant, it is possible to construct an action functional for a scalar field such that the quantum field theory, obtained by path integral quantization, leads to this propagator. In general, such a theory will involve derivatives of the field higher than two and can even involve derivatives of infinite order. The poles of the given propagator determine the dispersion relation for the excitations of this field. I show that it is possible to construct field theories in which the dispersion relation is the same as that of standard Klein-Gordan field, even though the Lagrangian contains derivatives of infinite order. I provide a concrete example of this situation starting from a propagator which incorporates the effects of the zero-point-length of the spacetime. I compare the path integral approach with an alternative, operator-based approach, and highlight the advantages of using the former.

Traversable wormholes in f(R) massive gravity

In this work, the study of traversable wormholes in $f(R)$-massive gravity with the function $f(R)=R+\alpha_{1} R^{n}$, where $\alpha_{1}$ and $n$ are arbitrary constants, is considered. We choose the modified shape function $b(r)$. We consider a spherically symmetric and static wormhole metric and derive field equations. Moreover, we visualize the wormhole geometry using embedding diagrams. Furthermore, we check the null, weak, dominant and strong energy conditions at the wormhole throat with a radius $r_{0}$ invoking three types of redshift functions, $\Phi={\rm constant},\gamma_{1}/r,\,\log(1+\gamma_{2}/r)$ with $\gamma_{1}$ and $\gamma_{2}$ are arbitrary real constants. We also compute the volume integral quantifier to calculate the amount of the exotic matter near the constructed wormhole throat.

Resonance and stability of higher derivative theories of a derived type

We consider the class of higher derivative field equations whose wave operator is a square of another self-adjoint operator of lower order. At the free level, the models of this class are shown to admit a two-parameter series of integrals of motion. The series includes the canonical energy. Every conserved quantity is unbounded in this series. The interactions are included into the equations of motion such that a selected representative in conserved quantity series is preserved at the non-linear level. The interactions are not necessarily Lagrangian, but they admit Hamiltonian form of dynamics. The theory is stable if the integral of motion is bounded from below due to the interaction. The motions are finite in the vicinity of the conserved quantity minimum. The equations of motion for fluctuations have the derived form with no resonance. The general constructions are exemplified by the models of the Pais-Uhlenbeck oscillator with multiple frequency and Podolsky electrodynamics. The example is also considered of stable non-abelian Yang-Mills theory with higher derivatives.

Anisotropic MHRDE model in BD theory of gravitation

In this paper, in the framework of the Brans–Dicke [Phys. Rev. 124 (1961) 925] Gravitation theory, we propose to study the spatially homogeneous, anisotropic and axially symmetric model filled with dark matter and dark energy. Here, we consider the modified holographic Ricci dark energy proposed by Chen and Jing [Phys. Rev. B 679 (2009) 144] as a feasible state of darkness. To achieve a solution, we consider the time-dependent deceleration parameter, which contributes to the average scale factor of [Formula: see text], where [Formula: see text] and [Formula: see text] are arbitrary constants. We have derived field equations of Brans–Dicke theory of gravitation with the help of an axially symmetric anisotropic Bianchi-type space-time. We have determined the cosmological parameters, namely, deceleration parameter, matter energy density, anisotropic dark energy density, BD scalar field, skewness parameter, EoS parameter and jerk parameter. Here, the various phenomena like the Big Bang, expanding the universe, and shift from anisotropy to isotropy are observed in the model. A comprehensive physical debate of these dynamic parameters is provided through a graphical representation. We observe that we have a quintessence model that exhibits a smooth transition from decelerated stage to an accelerated phase of the universe. This situation is in complete agreement with the modern cosmology scenario. Some physical and geometric behaviors are also discussed and discovered to be in excellent agreement with SNe Ia Supernova’s latest observations.

Dyonic black holes in anti-de Sitter Einstein–Yang–Mills theory, characterised by an infinite set of global charges

We here derive field equations for static, spherically symmetric, dyonic Einstein–Yang–Mills theory with a negative cosmological constant . We are able to find new non-trivial black hole solutions to this system in two regimes: where the gauge fields are small; and where . We also show that some may be uniquely characterised by a countably infinite set of asymptotically-defined charges. This may have implications for Bizon’s modified ‘No Hair’ conjecture.

Gravitational waves and degrees of freedom in higher derivative gravity

We study the degrees of freedom of the metric in a general class of higher derivative gravity models, which are interesting in the context of quantum gravity as they are (super)renormalizable. First, we linearize the theory for a flat background metric in Teyssandier gauge for an arbitrary number of spacetime dimensions $D$. The higher-order derivative field equations for the metric perturbation can be decomposed into tensorial and scalar field equations resembling massless and massive wave equations. For the massive tensor field in $D$-dimensions we demonstrate that the harmonic gauge condition is induced dynamically and only the transverse modes are excited in the presence of a matter source. For the special case of quadratic gravity in four-dimensional spacetime, we show that only the quadrupole moment contributes to the gravitational radiation from an idealized binary system.

Dynamics of axially symmetric anisotropic modified holographic Ricci dark energy model in Brans-Dicke theory of gravitation

In this paper, we have derived field equations of Brans-Dicke (Phys. Rev. 124, 925 (1961)) theory of gravitation with the help of an axially symmetric anisotropic Bianchi-type space-time in the presence of dark matter and anisotropic modified holographic Ricci dark energy. We have presented a cosmological model solving the field equations. We have used i) the hybrid expansion law, ii) a relation between metric potentials and iii) the modified holographic Ricci dark energy defined by Chen and Jing (Phys. Lett. B 679, 144 (2009)) to solve the field equations. We have determined the cosmological parameters, namely, EoS parameter, matter energy density, anisotropic dark energy density, Skewness parameter, deceleration and jerk parameters. A detailed physical discussion of these dynamical parameters is presented through a graphical representation. We observe that we have a quintessence model which exhibits a smooth transition from decelerated phase to an accelerated phase of the universe. This situation is quite in agreement with the scenario of modern cosmology.

Graded Lie Algebroids: A Framework for Geometrization of Matter and Forces Unification

In this paper, we introduce a geometric structure that is capable of describing matter and forces simultaneously. This structure can be established by using the notion of $$Z_{2}$$ -graded Lie algebroid structures and graded semi-Riemannian metrics on them. Using calculus of variations, we derive field equations from the extended Hilbert–Einstein action. The derived equations contain Yang–Mills and Einstein field equations simultaneously. The even part of the graded Lie algebroid describes forces and its odd part is related to matter and particles.

Localization of interacting fields in five-dimensional braneworld models

We study the localization properties of fundamental fields which are coupled to one another through the gauge mechanism both in the original Randall–Sundrum (RS) and in the modified Randall–Sundrum (MRS) braneworld models: scalar–vector, vector–vector, and spinor–vector configuration systems. For this purpose, we derive conditions of localization, namely, the finiteness of integrals over the extra coordinate in the action of the system considered. We also derive field equations for each of the systems and then obtain their solutions corresponding to the extra dimension by a separation of variable method for every field involved in each system. We then insert the obtained solutions into the conditions of localization to seek whether or not the solutions are in accordance with the conditions of localization. We obtain that not all of the configuration systems considered are localizable on the brane of the original RS model while, on the contrary, they are localizable on the MRS braneworld model with some restrictions. In terms of field localizability on the brane, this result shows that the MRS model is much better than the original RS model.

Static Axially Symmetric Models and Structure Scalars in Self-Interacting Brans–Dicke Gravity

This paper investigates static axially symmetric models in self-interacting Brans–Dicke gravity. We discuss physically feasible sources of models, derive field equations as well as evolution equations from Bianchi identities and construct structure scalars. Using these scalars and evolution equations, the inhomogeneity factors of the system are evaluated. It is found that structure scalars related to double dual of Riemann tensor control the density inhomogeneity. Finally, we obtain exact solutions of homogenous isotropic and inhomogeneous anisotropic spheroid models. It turns out that homogenous solutions reduce to Schwarzschild type interior solutions for a spherical case. We conclude that homogenous models involve homogenous distribution of scalar field whereas inhomogeneous correspond to inhomogeneous scalar field.

К ВЫВОДУ УРАВНЕНИЙ ГРАВИТАЦИОННОГОПОЛЯ В f (R) ГРАВИТАЦИИ С КИНЕТИЧЕСКИМ СКАЛЯРОМ КРИВИЗНЫ

К ВЫВОДУ УРАВНЕНИЙ ГРАВИТАЦИОННОГОПОЛЯ В f (R) ГРАВИТАЦИИ С КИНЕТИЧЕСКИМ СКАЛЯРОМ КРИВИЗНЫ

The Geometry of Space and Its Application to Theory of General Relativity, Derivation of Field Equations

In this paper we study the geometry of  space and applications of this space to general theory of relativity. In  space we obtained analog Ricci - Jacobi identity; the geodesic lines equation have been researched; we introduced analog of Darboux theory in case of  space, so it was shown the tensor  can be presented as the sum of two tensors symmetrical and antisymmetrical with property  We discussed some partial cases gravitational and electromagnetic interaction, and their connection to geometry structure; we considered stronger electromagnetic field in  space. We derived the general field equations (electromagnetic and gravitational) from the variation principle.

Separate Einstein‐Eddington spaces and the cosmological constant

Based on Eddington affine variational principle on a locally product manifold, we derive the separate Einstein space described by its Ricci tensor. The derived field equations split into two field equations of motion that describe two maximally symmetric spaces with two cosmological constants. We argue that the invariance of the bi‐field equations under projections on the separate spaces, may render one of the cosmological constants to zero. We also formulate the model in the presence of a scalar field. The resulted separate Einstein‐Eddington spaces maybe considered as two states that describe the universe before and after inflation. A possibly interesting affine action for a general perfect fluid is also proposed. It turns out that the condition which leads to zero cosmological constant in the vacuum case, eliminates here the effects of the gravitational mass density of the perfect fluid, and the dynamic of the universe in its final state is governed by only the inertial mass density of the fluid. image

Unified field equations coupling four forces and principle of interaction dynamics

The main objective of this article is to postulate a principle of interaction dynamics (PID) and to derive field equations coupling the four fundamental interactions based on first principles. PID is a least action principle subject to div$_A$-free constraints for the variational element with $A$ being gauge potentials. The Lagrangian action is uniquely determined by 1) the principle of general relativity, 2) the $U(1)$, $SU(2)$ and $SU(3)$ gauge invariances, 3) the Lorentz invariance, and 4) the principle of representation invariance (PRI), introduced in [11]. The unified field equations are then derived using PID. The field model spontaneously breaks the gauge symmetries, and gives rise to a new mass generation mechanism.The unified field model introduces a natural duality between the mediators and their dual mediators, and can be easily decoupled to study each individual interaction when other interactions are negligible. The unified field model, together with PRI and PID applied to individual interactions, provides clear explanations and solutions to a number of outstanding challenges in physics and cosmology, including e.g. the dark energy and dark matter phenomena, the quark confinement, asymptotic freedom, short-range nature of both strong and weak interactions, decay mechanism of sub-atomic particles, baryon asymmetry, and the solar neutrino problem.

Derivation of Field Equations in Space with the Geometric Structure Generated by Metric and Torsion

This paper is devoted to the derivation of field equations in space with the geometric structure generated by metric and torsion tensors. We also study the geometry of the space generated jointly and agreed on by the metric tensor and the torsion tensor. We showed that in such space the structure of the curvature tensor has special features and for this tensor we obtained analog Ricci-Jacobi identity and evaluated the gap that occurs at the transition from the original to the image and vice versa, in the case of infinitely small contours. We have researched the geodesic lines equation. We introduce the tensor παβ which is similar to the second fundamental tensor of hypersurfaces Yn-1, but the structure of this tensor is substantially different from the case of Riemannian spaces with zero torsion. Then we obtained formulas which characterize the change of vectors in accompanying basis relative to this basis itself. Taking into considerations our results about the structure of such space we derived from the variation principle the general field equations (electromagnetic and gravitational).