Exact Wormhole Solutions Research Articles

Deflection of light by wormholes and its shadow due to dark matter within modified symmetric teleparallel gravity formalism

Abstract We explore the possibility of traversable wormhole formation in the dark matter halos in the context of $f(Q)$ gravity. We obtain the exact wormhole solutions with anisotropic matter source based on the Bose-Einstein condensate, Navarro-Frenk-White, and pseudo-isothermal matter density profiles. Notably, we present a novel wormhole solution supported by these dark matters using the expressions for the density profile and rotational velocity along with the modified field equations to calculate the redshift and shape functions of the wormholes. With a particular set of parameters, we demonstrate that our proposed wormhole solutions fulfill the flare-out condition against an asymptotic background. Additionally, we examine the energy conditions, focusing on the null energy conditions at the wormhole's throat, providing a graphical representation of the feasible and negative regions. Our study also examines the wormhole's shadow in the presence of various dark matter models, revealing that higher central densities result in a shadow closer to the throat, whereas lower values have the opposite effect. Moreover, we explore the deflection of light when it encounters these wormholes, particularly noting that light deflection approaches infinity at the throat, where the gravitational field is extremely strong.

Wormholes and energy conditions in f(R,T) gravity

In this paper, we explore the existence of wormholes in the context of [Formula: see text] gravity. The [Formula: see text] theory is a curvature-matter coupled modified gravity that depends on an arbitrary function of the Ricci scalar [Formula: see text] and the trace of the stress-energy tensor [Formula: see text]. In this work, we adopt two different choices for the matter Lagrangian density ([Formula: see text] and [Formula: see text]) and investigate the impact of each one on wormhole structure. By adequately specifying the redshift function and the shape function, we found a variety of exact wormhole solutions in the theory. Our finding indicates that for both classes of wormholes, the energy density is always positive throughout the spacetime, while the radial pressure is negative. This means exotic matter is necessary for the existence of wormholes in [Formula: see text] gravity.

Traversable wormholes with charge and non-commutative geometry in the [formula omitted] gravity

We consider modified symmetric teleparallel gravity (STG), in which gravitational Lagrangian is given by the arbitrary function of non-metricity scalar Q to study static and spherically symmetric charged traversable wormhole solutions with non-commutative background geometry. The matter source at the wormhole throat is acknowledged to be anisotropic, and the redshift function has a constant value (thus, our wormhole solution is non-tidal). We study the obtained field equations with the two functional forms of f(Q) STG models, such as linear f(Q)=αQ+β and non-linear f(Q)=Q+mQn models under Gaussian and Lorentzian distributions. Our analysis found the exact wormhole solutions for the linear STG model only. Also, for the non-linear model, we derived numerically suitable forms of wormhole shape functions directly from the modified Einstein Field Equations (EFEs). Besides, we probed these models via Null, Dominant, and Strong energy conditions with respect to free Modified gravity (MOG) parameters α, β, m, and n. We also used Tolman–Oppenheimer–Vokloff (TOV) equation to investigate the stability of wormhole anisotropic matter in considered MOG. Finally, we plot the equation of state.

Wormholes in [formula omitted] gravity satisfying the null energy condition with isotropic pressure

We consider the f(R,T) theory of gravity, in which the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar and the trace of the energy–momentum tensor, to study static spherically symmetric wormhole geometries sustained by matter sources with isotropic pressure. According to restrictions on the wormhole geometries, we carefully adopt different strategies to construct solutions with the properties and characteristics of wormholes. Using an utterly general procedure, we provide several examples of wormholes in which the matter threading the wormhole throat satisfies all of the energy conditions and discuss general mechanisms for finding them. Finally, we postulate a smooth transformation for simplifying the nonlinear field equations and have more consistent results than the other ones to conclude that the results can be viewed as specific exact wormhole solutions without exotic matter.

Two different shape functions for wormholes in f(R) theory with non-commutative geometry and Lorentzian distribution

In this work, the solutions of traversable wormholes are investigated inside modified [Formula: see text] gravity under non-commutative geometry since matter possesses Lorentzian density distribution of a particle-like gravitation source. To find the exact wormhole solutions, two different shape functions [Formula: see text], [Formula: see text], and [Formula: see text], [Formula: see text], are considered. The first shape function was proposed by Mishra and Sharma [A new shape function for wormholes in [Formula: see text] gravity and General Relativity, preprint (2020), arXiv:2003.00298v1 [physics.gen-ph]], however the second is newly defined in this paper. The behaviors of both shape functions are analyzed with the throat radius [Formula: see text]. The equation-of-state (EoS) parameter energy conditions, and anisotropy parameter are discussed with graphical point of view.

Conformally symmetric traversable wormholes in modified teleparallel gravity

In this paper, we consider wormhole geometries in the context of teleparallel equivalent of general relativity (TEGR) as well as $f(T)$ gravity. The TEGR is an alternative geometrical formulation of Einstein's general relativity, where modified teleparallel gravity or $f(T)$ gravity has been invoked as an alternative approach for explaining an accelerated expansion of the universe. We present the analytical solutions under the assumption of spherical symmetry and the existence of a conformal Killing vectors to proceed a more systematic approach in searching for exact wormhole solutions. More preciously, the existence of a conformal symmetry places restrictions on the model. Considering the field equations with a diagonal tetrad and anisotropic distribution of the fluid, we study the properties of traversable wormholes in TEGR that violates the weak and the null energy conditions at the throat and its vicinity. In the second part, wormhole solutions are constructed in the framework of $f(T)$ gravity, where $T$ represents torsion scalar. As a consistency check, we also discuss the behavior of energy conditions with a viable power-law $f(T)$ model and the corresponding shape functions. In addition, a wide variety of solutions are deduced by considering a linear equation of state relating the density and pressure, for the isotropic and anisotropic pressure, independently of the shape functions, and various phantom wormhole geometries are explored.

Study on wormhole geometry with rho (R, R^{'}) matter in modified gravity

In this work, static traversable wormholes are investigated in R^2 gravity with logarithmic trace term T, where R denotes the Ricci scalar, and T=-rho +p_r+2p_t>0, the trace of the energy momentum tensor. The connection between energy density of the matter component and the Ricci scalar is taken into account. Exact wormhole solutions are determined for three different novel forms of energy density: rho =alpha _1 R+beta _1 R^{'}e^{xi _1 R}, rho =alpha _2 R e^{xi _2 R} and rho =alpha _3 R^2+beta _2 R^{'} e^{xi _3 R^{'}}, where prime denotes derivative with respect to r. The parameters alpha _1, beta _1, xi _1, alpha _2, xi _2, alpha _3, xi _3 and beta _2 play an important role for the absence of exotic matter inside the wormhole geometry. The parameter space is separated into numerous regions where the energy conditions are obeyed.

Wormhole models in f(R,T) gravity

Models of static wormholes within the [Formula: see text] extended theory of gravity are investigated, in particular the family [Formula: see text], with [Formula: see text] being the trace of the energy–momentum tensor. Models corresponding to different relations for the pressure components (radial and lateral), and several equations-of-state (EoS), reflecting different matter content, are worked out explicitly. The solutions obtained for the shape functions of the generated wormholes obey the necessary metric conditions, as manifested in other studies in the literature. The respective energy conditions reveal the physical nature of the wormhole models thus constructed. It is found, in particular, that for each of those considered, the parameter space can be divided into different regions, in which the exact wormhole solutions fulfill the null energy conditions (NEC) and the weak energy conditions (WEC), respectively, in terms of the lateral pressure. Moreover, the dominant energy condition (DEC) in terms of both pressures is also valid, while [Formula: see text]. A similar solution for the theory [Formula: see text] is found numerically, where [Formula: see text] and [Formula: see text] are either constant or functions of [Formula: see text], leading to the result that the NEC in terms of the radial pressure is also valid. For nonconstant [Formula: see text] models, attention is focused on the behavior [Formula: see text]. To finish, the question is addressed, how [Formula: see text] will affect the wormhole solutions corresponding to fluids of the form [Formula: see text], in the three cases such as NEC, WEC and DEC. Issues concerning the nonconservation of the matter energy–momentum tensor, the stability of the solutions obtained, and the observational possibilities for testing these models are discussed in Sec. 6.

Model of wormholes under Krori-Barua spacetime in the presence of matter

We intend to investigate particular models of wormholes with ordinary matter source. The exact wormhole solutions are deduced by numerically solving the Einstein’s field equations with the stress-energy-tensor components. A particular form of feasible wormhole shape function is utilized. We study the solutions considering the Krori-Barua (KB) ansatz [K.D. Krori and J. Barua, J. Phys. A: Math. Gen. 8, 508 (1975)] in presence of perfect fluid matter. The obtained solutions are free from central singularity. The wormhole model thus obtained under Krori-Barua spacetime satisfies the flare-out conditions. Some physical features are briefly discussed.

Static traversable wormholes in [formula omitted] gravity

Traversable wormholes, studied by Morris and Thorne [3] in general relativity, are investigated in this research paper in f(R, T) gravity by introducing a new form of non-linear f(R, T) function. By using this novel function, the Einstein’s field equations in f(R, T) gravity are derived. To obtain the exact wormhole solutions, the relations pt=ωρ and pr=sinh(r)pt, where ρ is the energy density, pr is the radial pressure and pt is the tangential pressure, are used. Other than these relations, two forms of shape function defined in literature are used, and their suitability is examined by exploring the regions of validity of null, weak, strong and dominant energy conditions. Consequently, the radius of the throat or the spherical region, with satisfied energy conditions, is determined and the presence of exotic matter is minimized.

Asymptotically flat wormhole solutions with variable equation-of-state parameter

This paper discusses exact wormhole solutions in the framework of general relativity with a general equation of state that reduced to a linear equation of state asymptotically. By considering a special shape function, we find classes of solutions which are asymptotically flat. We study the violation of the null energy condition as the main ingredient in wormhole physics. The possibility of finding wormhole solutions with asymptotically different state parameter is investigated. We show that in principle, a wormhole with a vanishing redshift function and the selected shape function, cannot satisfy the null energy condition at a large distance from the wormhole. Our solutions have the positive total amount of matter in the ``volume integral quantifier'' method. For this class of solutions, fluid near the wormhole throat is in the phantom regime which at some $r={r}_{2}$, the phantom regime is connected to a distribution of dark energy regime. Thus, we need a small amount of exotic matter to construct wormhole solutions.

Charged wormhole solutions in Einstein-Cartan gravity

Static solutions representing wormhole configurations in Einstein-Cartan theory ({{\sf ECT}}) in the presence of electric charge are obtained. The solutions are described by a constant redshift function with matter content consisting of a Weyssenhoff fluid along with an anisotropic matter and energy momentum tensor ({\sf EMT}) of the electric field which together generalize the anisotropic energy momentum tensor in Einstein-Maxwell theory in order to include the effects of the intrinsic angular momentum (spin) of the particles. Assuming the equation of state ({{\sf EoS}}) $p_r={\sf w}_1\rho$ and $p_t={\sf w}_2\rho$, we derive exact wormhole solutions satisfying weak and null energy conditions. Depending on the value of the spin square density at the wormhole throat these solutions can be asymptotically flat, de-Sitter or anti de-Sitter. Observational aspects of the wormhole solutions are also studied, using gravitational lensing effects. It is found that the throat can act as a photon sphere near which the light deflection angle has arbitrarily large values. Moreover, for a particular class of solutions, when ${\sf w}_1\rightarrow-{\sf w}_2$ the lensing features of the present model mimic those of the Ellis wormhole in the weak field limit.

Wormholes with ρ(R, R′) matter in f(R, T) gravity

Models of static wormholes are investigated in the framework of $f(R,T)$ gravity (where $R$ is the curvature scalar and $T$ is the trace of the energy-momentum tensor). An attempt to link the energy density of the matter component to the Ricci scalar is made, which for the Morris and Thorne wormhole metric with constant redshift function yields $R(r)=2{b}^{\ensuremath{'}}(r)/{r}^{2}$. Exact wormhole solutions are obtained for three particular cases when $f(R,T)=R+2\ensuremath{\lambda}T$: $\ensuremath{\rho}(r)=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{\ensuremath{'}}(r)$, $\ensuremath{\rho}(r)=\ensuremath{\alpha}{R}^{2}(r)+\ensuremath{\beta}{R}^{\ensuremath{'}}(r)$, and $\ensuremath{\rho}(r)=\phantom{\rule{0ex}{0ex}}\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{2}(r)$. Additionally, traversable wormhole models are obtained for the two first cases. However, when the wormhole matter energy density is of the third type, only solutions with constant shape correspond to traversable wormholes. Exact wormhole solutions possessing the same properties can be constructed when $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{\ensuremath{-}2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}r{R}^{2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{r}^{\ensuremath{-}1}{R}^{2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\phantom{\rule{0ex}{0ex}}\ensuremath{\beta}{r}^{2}{R}^{2}(r)$, $\ensuremath{\rho}=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{r}^{3}{R}^{2}(r)$, and $\ensuremath{\rho}=\ensuremath{\alpha}{r}^{m}R(r)\mathrm{log}(\ensuremath{\beta}R(r))$, as well. On the other hand, for $f(R,T)=R+\ensuremath{\gamma}{R}^{2}+2\ensuremath{\lambda}T$ gravity, two wormhole models are constructed, assuming that the energy density of the wormhole matter is $\ensuremath{\rho}(r)=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{R}^{2}(r)$ and $\ensuremath{\rho}(r)=\ensuremath{\alpha}R(r)+\ensuremath{\beta}{r}^{3}{R}^{2}(r)$, respectively. In this case, the functional form of the shape function is taken to be $b(r)=\sqrt{{\stackrel{^}{r}}_{0}r}$ (where ${\stackrel{^}{r}}_{0}$ is a constant) and the possible existence of appropriate static traversable wormhole configurations is proven. The explicit forms of the pressures ${P}_{r}$ and ${P}_{l}$ leading to this result are found in both cases. As a general feature, the parameter space can be divided into several regions according to which of the energy conditions are valid. These results can be viewed as an initial step towards using specific properties of the new exact wormhole solutions to propose new functional forms for describing the matter content of wormholes.

Charged wormholes supported by 2-fluid immiscible matter particle

We provide a 2-fluid immiscible matter source that supplies fuel to construct wormhole spacetime. The exact wormhole solutions are found in the model having, besides real matter or ordinary matter, some quintessence matter along with charge distribution. We intend to derive a general metric of a charged wormhole under some density profiles of galaxies that is also consistent with the observational profile of rotation curve of galaxies. We have shown that the effective mass remains positive as well as the wormhole physics violate the null energy conditions. Some physical features are briefly discussed.

КРОТОВЫЕ НОРЫ В ТЕОРИИ ГРАВИТАЦИИ С НЕМИНИМАЛЬНОЙ КИНЕТИЧЕСКОЙ СВЯЗЬЮ И КОСМОЛОГИЧЕСКОЙ ПОСТОЯННОЙ

КРОТОВЫЕ НОРЫ В ТЕОРИИ ГРАВИТАЦИИ С НЕМИНИМАЛЬНОЙ КИНЕТИЧЕСКОЙ СВЯЗЬЮ И КОСМОЛОГИЧЕСКОЙ ПОСТОЯННОЙ

Novel third-order Lovelock wormhole solutions

In this work, we consider wormhole geometries in third-order Lovelock gravity and investigate the possibility that these solutions satisfy the energy conditions. In this framework, by applying a specific equation of state, we obtain exact wormhole solutions, and by imposing suitable values for the parameters of the theory, we find that these geometries satisfy the weak energy condition in the vicinity of the throat, due to the presence of higher order curvature terms. Finally, we trace out a numerical analysis, by assuming a specific redshift function, and find asymptotically flat solutions that satisfy the weak energy condition throughout the spacetime.

Rotating cylindrical wormholes and energy conditions

We seek wormholes among rotating cylindrically symmetric configurations in general relativity. Exact wormhole solutions are presented with such sources of gravity as a massless scalar field, a cosmological constant, and a scalar field with an exponential potential. However, none of these solutions are asymptotically flat, which excludes the existence of wormhole entrances as local objects in our Universe. To overcome this difficulty, we try to build configurations with flat asymptotic regions using the cut-and-paste procedure: on both sides of the throat, a wormhole solution is matched to a properly chosen region of flat space-time at some surfaces [Formula: see text] and [Formula: see text]. It is shown, however, that if the source of gravity in the throat region is a scalar field with an arbitrary potential, then one or both thin shells appearing on [Formula: see text] and [Formula: see text] inevitably violate the null energy condition. Thus, although rotating wormhole solutions are easily found without exotic matter, such matter is still necessary for obtaining asymptotic flatness.

Exact wormhole solutions with nonminimal kinetic coupling

We consider static spherically symmetric solutions in the scalar-tensor theory of gravity with a scalar field possessing the nonminimal kinetic coupling to the curvature. The lagrangian of the theory contains the term $(\varepsilon g^{\mu\nu}+\eta G^{\mu \nu})\phi_{,\mu}\phi_{,\nu}$ and represents a particular case of the general Horndeski lagrangian, which leads to second-order equations of motion. We use the Rinaldi approach to construct analytical solutions describing wormholes with nonminimal kinetic coupling. It is shown that wormholes exist only if $\varepsilon=-1$ (phantom case) and $\eta>0$. The wormhole throat connects two anti-de Sitter spacetimes. The wormhole metric has a coordinate singularity at the throat. However, since all curvature invariants are regular, there is no curvature singularity there.

Wormholes supported by two non-interacting fluids

We provide a new matter source that supplies fuel to construct wormhole spacetime. The exact wormhole solutions are found in the model having, besides real matter, an anisotropic dark energy. We have shown that the exotic matters that are the necessary ingredients for wormhole physics violate null and weak energy conditions but obey strong energy condition marginally. Though the wormhole comprises of exotic matters yet the effective mass remains positive. We have calculated the effective mass of the wormhole up to 8 km from the throat (assuming throat radius as 4 km) as 1.3559M ⊙. Some physical features are briefly discussed.

EXACT SOLUTIONS OF BRANS–DICKE WORMHOLES IN THE PRESENCE OF MATTER

A fundamental ingredient in wormhole physics is the presence of exotic matter, which involves the violation of the null energy condition. Although a plethora of wormhole solutions have been explored in the literature, it is useful to find geometries that minimize the usage of exotic matter. In this work, we find exact wormhole solutions in Brans–Dicke theory where the normal matter threading the wormhole satisfies the null energy condition throughout the geometry. Thus, the latter implies that it is the effective stress–energy tensor containing the scalar field, that plays the role of exotic matter, that is responsible for sustaining the wormhole geometry. More specifically, we consider a zero redshift function and a particular choice for the scalar field and determine the remaining quantities, namely, the stress–energy tensor components and the shape function. The solution found is not asymptotically flat, so that this interior wormhole spacetime needs to be matched to an exterior vacuum solution.