Gravitational Equations Research Articles

Existence of traversable wormholes in the minimally coupled gravity model

The objective of this study is to examine the possible existence of traversable wormhole geometries within the context of [Formula: see text] gravity. To meet this objective, we employ the Karmarkar condition to construct the shape function that aids in identifying the wormhole configurations. This developed function is found to satisfy the essential conditions and provides a link between two asymptotically flat spacetime regions. We then assume the Morris–Thorne line element that expresses the wormhole configuration and formulate the anisotropic gravitational equations for a particular minimal matter-spacetime coupled model of the modified theory. Afterward, we develop three solutions and determine their viability by analyzing whether they violate the null energy conditions. Different stability methods are applied to the resulting geometries to explore the acceptance of the considered modified model. We conclude that the developed wormhole structures potentially fulfill the required criteria and thus exist in this modified gravity under all choices of the matter Lagrangian density.

Image formation near hyperbolic umbilic in strong gravitational lensing

Hyperbolic umbilic (HU) is a point singularity of the gravitational lens equation, giving rise to a ring-shaped image formation made of four highly magnified images, off-centred from the lens centre. Recent observations have revealed new strongly lensed image formations near HU singularities, and many more are expected in ongoing and future observations. Like fold/cusp, image formations near HU also satisfy magnification relation (), i.e., the signed magnification sum of the four images equals zero. Here, we study how deviates from zero as a function of area () covered by the image formation near HU and the distance ( d) of the central maxima image (which is part of the HU image formation) from the lens centre for ideal single- and double-component cluster-scale lenses. For lens ellipticity values ≥0.3, the central maxima image will form sufficiently far from the lens centre ( d≳5″), similar to the observed HU image formations with . We also find that, in some cases, double-component and actual cluster-scale lenses can lead to large cross-sections for HU image formations for sources at z≳5, effectively increasing the chances to observe HU image formation at high redshifts. Finally, we study the time delay distribution in the observed HU image formation, finding that not only are these images highly magnified, but the relative time delay between various pairs of HU characteristic image formation has a typical value of ∼100 days, an order of magnitude smaller than generic five-image formations in cluster lenses, making such image formations optimal targets for time delay cosmography studies.

Gravitational spinoptics in a curved space-time

In this paper we discuss propagation of the weak high-frequency gravitational waves in a curved spacetime background. We develop a so-called spinoptics approximation which takes into account interaction of the spin of the field with the curvature of the background metric. This is achieved by modifying the standard geometric optics approximation by including the helicity sensitive terms of the order 1/ω in the eikonal equation. The novelty of the approach developed in this paper is that instead of study of the high-frequency expansion of the equations for the gravitational field perturbations we construct the effective action for the gravitational spinoptics. The gravitational spinoptics equations derived by variation of the effective action correctly reproduce the earlier obtained results. However, the proposed effective action approach is technically more simple and transparent. It allows one to reduce the study of the high-frequency gravitational waves to study classical dynamics of massless particles with internal discrete degree of freedom (helicity). The formalism is covariant and it can be applied for arbitrary vacuum space-time background.

Gravitational lensing of spherically symmetric black holes in dark matter halos

The gravitational lensing of supermassive black holes surrounded by dark matter halo has attracted a great number of interests in recent years. However, many studies employed simplified dark matter density models, which makes it very hard to give a precise prediction on the dark matter effects in real astrophysical galaxies. In this work, to more accurately describe the distribution of dark matter in real astrophysical galaxies, we study the gravitational lensing of black holes in astrophysical dark matter halo models (Beta, Burkert, Brownstein, and Moore). The deflection angle is obtained using a generalized Gibbons-Werner approach. The visual angular positions and the Einstein rings are also calculated by adopting the gravitational lens equation. Specifically, we choose the supermassive black holes in Milky Way Galaxy, Andromeda galaxy (M31), Virgo galaxy (M87), and ESO138-G014 galaxy as examples, including the corresponding fitted value of dark matter halos. The results suggest that the dark matter halo described by the Beta model has non-negligible influences on the gravitational deflection angle and gravitational lensing observations. However, the Burkert, Brownstein, and Moore models have relatively small influences on angular position of images and the Einstein ring.

Parametric resonance of gravitational waves in general scalar-tensor theories

Gravitational waves offer a potent mean to test the underlying theory of gravity. In general theories of gravity, such as scalar-tensor theories, one expects modifications in the friction term and the sound speed in the gravitational wave equation. In that case, rapid oscillations in such coefficients, e.g. due to an oscillating scalar field, may lead to narrow parametric resonances in the gravitational wave strain. We perform a general analysis of such possibility within DHOST theories. We use disformal transformations to find the theory space with larger resonances, within an effective field theory approach. We then apply our formalism to a non-minimally coupled ultra-light dark matter scalar field, assuming the presence of a primordial gravitational wave background, e.g., from inflation. We find that the resonant peaks in the spectral density may be detectable by forthcoming detectors such as LISA, Taiji, Einstein Telescope and Cosmic Explorer.

On the viscoelastic-electromagnetic-gravitational analogy

The analogy between electromagnetism and gravitation was achieved by linearizing the tensorial gravitational equations of general relativity and converting them into a vector form corresponding to Maxwell’s electromagnetic equations. On this basis, we use the equivalence with viscoelasticity and propose a theory of gravitational waves. We add a damping term to the differential equations, which is equivalent to Ohm’s law in electromagnetism and Maxwell’s viscosity in viscoelasticity, to describe the attenuation of the waves. The differential equations in viscoelasticity are those of cross-plane shear waves, commonly referred to as SH waves. A plane-wave analysis gives the phase velocity, the energy velocity, the quality factor and the attenuation factor of the field as well as the energy balance. To obtain these properties, we use the analogy with viscoelasticity; the properties of electromagnetic and gravitational waves are similar to those of shear waves. The presence of attenuation means that the transient field is generally a composition of inhomogeneous (non-uniform) plane waves, where the propagation and attenuation vectors do not point in the same direction and the phase velocity vector and the energy flux (energy velocity) are not collinear. The polarization of cross-plane field is linear and perpendicular to the propagation-attenuation plane, while the polarization of the field within the plane is elliptical. Transient wave fields in the space–time domain are analyzed with the Green function (in homogeneous media) and with a grid method (in heterogeneous media) based on the Fourier pseudospectral method for calculating the spatial derivatives and a Runge–Kutta scheme of order 4 for the time stepping. In the examples, wave propagation at the Sun–Earth and Earth–Moon distances using quadrupole sources is considered in comparison to viscoelastic waves. The Green and grid solutions are compared to test the latter algorithm. Finally, an example of propagation in heterogeneous media is presented.

Applicability of modified Gauss–Bonnet gravity models on the existence of stellar structures

In this paper, we explore the existence of spherically symmetric strange quark configurations coupled with anisotropic fluid setup in the framework of modified Gauss–Bonnet theory. In this regard, we adopt two models such as (i)f(G)=βG2, and (ii)f(G)=δ1Gx(δ2Gy+1), and derive the field equations representing a static sphere. We then introduce bag constant in the gravitational equations through the use of MIT bag model, so that the quarks’ interior can be discussed. Further, we work out the modified equations under the use of Tolman IV ansatz to make their solution possible. Junction conditions are also employed to find the constants involved in the considered metric potentials. Afterwards, different values of model parameters and bag constant are taken into account to graphically exploring the resulting solutions. This analysis is done by considering five strange quark objects like Her X-I, LMC X-4, 4U 1820-30, PSR J 1614-2230, and Vela X-I. Certain tests are also applied on the developed models to check their physical feasibility. It is much interesting that this modified gravity under its both considered functional forms yield physically viable and stable results for certain parametric values.

Quantum charged black holes

Within the framework of braneworld holography, we construct a quantum charged black hole localized on a three-dimensional anti-de Sitter (AdS) brane that intersects the asymptotic boundary of the four-dimensional AdS spacetime at the conformal defects and incorporates quantum backreaction effects from the conformal field theory (CFT) on the brane. This quantum charged black hole is an exact solution of the semiclassical gravitational equation corresponding to a theory with higher curvature gravity and nonminimally coupled nonlinear gauge field. In the framework of double holography, we investigate the thermodynamics of the quantum charged black hole from three perspectives: a pure bulk perspective, in which four-dimensional classical Einstein gravity couples to Maxwell electrodynamics and a codimension-one tensional brane; a brane perspective, where semiclassical higher curvature gravity is subject to quantum backreaction from the holographic CFT on the brane, yielding a quantum charged black hole; and a boundary perspective, where the defect CFT is coupled to a boundary CFT at the asymptotic boundary and the degrees of freedom for defect quantum conformal matter is considered. In so doing, we obtain doubly holographic formulations of both the first law of thermodynamics and the Smarr (energy) relations for the quantum charged black holes.

Reentrant phase transitions of quantum black holes

We show that backreaction of quantum fields on black hole geometries can trigger new thermal phase transitions. Specifically, we study the phase behavior of the three-dimensional quantum-corrected static Bañados-Teitelboim-Zanelli black hole, an exact solution to specific semiclassical gravitational equations due to quantum conformal matter, discovered through braneworld holography. Focusing on the canonical ensemble, for large backreaction, we find novel reentrant phase transitions as the temperature monotonically increases, namely, from thermal anti–de Sitter space to the black hole and back to thermal anti–de Sitter. The former phase transition is first-order, a quantum analog of the classical Hawking-Page phase transition, while the latter is zeroth order and has no classical counterpart. Published by the American Physical Society 2024

Gauge-invariant formulation for the gravitational wave equations

A gauge-invariant formulation for the gravitational wave equations is presented. Using this approach, weak, plane wave solutions in a vacuum are derived in various theories. These include general relativity with two modes of polarization with helicity ±2, Yang’s theory with three modes of polarization with helicity ±2 and 0, and so-called ‘general metric theories’ with six modes of polarization with helicity ±2, ±1, and two 0s. To identify the polarizations of gravitational waves, it is explicitly demonstrated how the gauge-invariant approach reproduces the earlier results.

Newtonian gravitational waves from a continuum

Gravitational waves are being shown to derive directly from Newtonian dynamics for a continuous mass distribution, e.g. compressible fluids or equivalent. It is shown that the equations governing a continuous mass distribution, i.e. the inviscid Navier–Stokes equations for a general variable gravitational field g ( t , x ) , are equivalent to a form identical to the Maxwell equations from electromagnetism, subject to a specified condition. The consequence of this equivalence is the creation of gravity waves that propagate at a finite speed. The latter implies that Newtonian gravitation as presented in this paper is not ‘spooky action at a distance’ but rather is similar to electromagnetic waves propagating at finite speed, despite the apparent form appearing in the integrated field formula. In addition, this proves that, in analogy to the Maxwell equations, the Newtonian gravitation equations are Lorentz invariant for waves propagating at the speed of light. Since gravitational waves were so far derived only from Einstein’s general relativity theory, it becomes appealing to check if there is a connection between the Newtonian waves presented in this paper and the general relativity type of waves at least in a certain limit of overlapping validity, i.e. as a flat-space approximation. The latter is left for follow-up research.

Little rip and pseudo rip cosmological models with coupled dark energy based on a new generalized entropy

In this paper, we study Little Rip (LR) and Pseudo Rip (PR) cosmological models containing two coupled fluids: dark energy and dark matter. We assume a spatially flat Friedmann–Robertson–Walker (FRW) universe. The interaction between the dark energy and the dark matter fluid components is described in terms of the parameters in the generalized Equation of State (EoS) in presence of the bulk viscosity. We consider entropic cosmology and use a description based on a new generalized entropy function, which was proposed by Nojiri–Odintsov–Faraoni [From nonextensive statistics and black hole entropy to the holographic dark universe, Phys. Rev. D 105(4) (2022) 044042]. Conditions for the appearance of the (LR) and the (PR) in terms of the parameters of the (EoS) are obtained. Introducing an energy density [Formula: see text] corresponding to a specified entropy function [Formula: see text], together with an interaction term [Formula: see text] in the gravitational equations of motion, we derive modified forms of the EoS parameters. We discuss the corrections of the thermodynamic parameters associated with the generalized entropy function. Properties of the late universe as well as in the early universe in this formalism are pointed out.

Gravitational radiation of a spherically symmetric source in f(R)-gravitation

It is shown that Birkhoff’s theorem for the general theory of relativity is overcome in the f(R)-theory of gravitation. That means, the f(R)-theory of gravitation, unlike Einstein’s general theory of relativity, does not forbid gravitational radiation from a spherically symmetric source (whether stationary or non-stationary). As a consequence, in the f(R)-theory a spherically symmetric gravitational deformation (e.g., collapse/expansion or pulsation) could emit gravitational waves (of tensor- and scalar polarization modes), a phenomenon impossible in the general relativity. A test model is examined and it turns out that the gravitational radiation is strongest when the surface of the deforming object is in the vicinity of the (modified) event horizon, even suddenly flares up just outside the latter. In this letter, within the f(R)-theory of gravitation, a gravitational wave equation and a formula for the gravitational emission power are derived. These formulae, along with searching for signals, can be used for the experimental test of the f(R)-theory. In general, including the spherically symmetry case, gravitational radiation of both tensor- and scalar polarization modes are allowed, although under some circumstance the contribution of scalar modes is strongly suppressed.

On graviton–photon conversions in magnetic environments

Graviton–photon conversions in a given external electric or magnetic field, known as the Gertsenshtein mechanism, are usually treated using the four-potential for photons. In terms of the electric and magnetic (EM) fields, however, proper identification of the fields in curved spacetime is important. By misidentifying the fields in Minkowski form, as is often practiced in the literature, we show that the final equation for photon conversion is correct in transverse-tracefree gauge only for planar gravitational waves in a uniform and constant external field. Even in the former method, to recover the EM fields from the four-potential in curved spacetime, one should properly take into account the metric involved in the relation. By including the metric perturbation in the graviton conversion equation, we show that a magnetic environment can cause tachyonic instability term in gravitational wave equation.

Minimal Action Principle for Gravity and Electrodynamics, Einstein Lambda, and Lagrange Points

The relativistic equations of gravitation and electromagnetism in the form of Vlasov – Einstein – Maxwell equations are proposed and analyzed. For weakly relativistic equations we get an analog of Mealn – McCree solution. We also study Lagrange points in non-relativistic case with Einstein lambda- term.

More on the fact that Rastall = GR

Rastall gravity is the same as General Relativity, with a simple algebraic redefinition of what is called the energy–momentum tensor. Despite it having been very clearly explained by M. Visser several years go, there are still many papers claiming big differences between the two formulations of gravitational equations and trying to use them for problems of physics. When going this way, the totally ignored task is to explain why the conserved energy–momentum quantities and the quantities used for other purposes are different from each other. Moreover, when researchers are using the non-conserved energy density and pressure for determining the sound speed, it is just inconsistent with the Rastall gravity. I carefully explain all this, and also show how one could construct a variational principle for producing equations in the Rastall form.

Viscous exotic fluid interacting with dark matter and future infinite singularities

By making use of two viable Lagrangian [Formula: see text] models, we investigate the behavior of the state parameter of the interacting viscous dark energy. We, from constant deceleration parameter, explore the cosmological implications of the viscosity and interaction between the dark energy and dark matter in terms of redshift. We then construct the viscosity and the interaction parameters that are respectively parameterized by [Formula: see text] and [Formula: see text] and obtain interesting results. Later, we investigate the behavior of some bulk viscosity models describing little rip and pseudo rip future singularities within [Formula: see text] modified gravity. We obtain interesting gravitational equations of motion for viscous dark energy coupled with dark matter. The solution of these equations gives us an analytic expression for characteristic properties of these cosmological models.

Gravitational lensing of Schwarzschild and charged black holes immersed in perfect fluid dark matter halo

Dark matter and dark energy dominate the behavior of our universe. The dark matter usually forms halo structures in large number of galaxies. Properties of dark matter halo can be revealed and understood from the gravitational lensing observations. In this work, a comprehensive study on the gravitational lensing of black holes immersed in dark matter halos is presented. To effectively model the supermassive black hole in a galaxy center (which is surrounded by dark matter halo) in a simple way, we investigate the Schwarzschild black hole and charged Reissner-Nordström black hole immersed in a perfect fluid dark matter halo. In the present work, several basic quantities in gravitational lensing (the gravitational deflection angle of light, photon sphere, black hole shadow radius, gravitational lens equation and Einstein ring) are calculated and analyzed analytically and numerically. A second order analytical expansion of gravitational deflection angle is obtained in the weak deflection limit, and the full gravitational deflection angle (including all order perturbation contributions applicable to both weak and strong deflection limits) is also calculated numerically as comparisons. It enables us to analyze the perfect fluid dark matter influences on gravitational deflection angle and gravitational lensing beyond the leading order, which were not sufficiently studied in previous works. Assuming M ∼ λDM ∼ Q, our results show that dark matter can greatly influence the gravitational lensing of central black holes.

On Derivation of Equations of Gravitation from the Principle of Least Action, Relativistic Milne–McCrea Solutions, and Lagrange Points

On Derivation of Equations of Gravitation from the Principle of Least Action, Relativistic Milne–McCrea Solutions, and Lagrange Points

Constrain the time variation of the gravitational constant via the propagation of gravitational waves

The gravitational constant variation means the breakdown of the strong equivalence principle. As the cornerstone of general relativity, the validity of general relativity can be examined by studying the gravitational constant variation. Such variations have the potential to affect both the generation and propagation of gravitational waves. Here, our focus lies on the effect of gravitational constant variation specifically on the propagation of gravitational waves. In a previous paper (An et al., 2023 [13]) we have studied such effect based on a Maxwell-like equation. That work admits two drawbacks. The assumption that describing gravitational waves with Maxwell-like equation is not solid. And more such treatment can only deal with space dependent gravitational constant. In the current paper we will remedy these two issues. We employ two analytical methods, namely based on the Fierz-Pauli action and the perturbation of Einstein-Hilbert action around Minkowski spacetime, both leading to the same gravitational wave equation. By solving this equation, we find the effects of gravitational constant variation on gravitational wave propagation. The result is consistent with previous investigations based on Maxwell-like equations for gravitational waves for space dependent gravitational constant cases. In addition we can constrain the time variation of the gravitational constant via the propagation of gravitational waves based on GW170817 data. And more, we find that small variations in the gravitational constant result in an amplitude correction at the leading order and a phase correction at the sub-leading order for gravitational waves. These results provide valuable insights for probing gravitational constant variation and can be directly applied to gravitational wave data analysis.