Comparative analysis of Einstein gravity and Rastall gravity for the compact objects

Wormholes are non-trivial topological structures that arise as exact solutions to Einstein’s field equations, theoretically connecting distinct regions of spacetime via a throat-like geometry. While static traversable wormholes necessarily require exotic matter that violates the classical energy conditions, subsequent studies have sought to minimize such violations by introducing time-dependent geometries embedded within cosmological backgrounds. This review provides a comprehensive survey of evolving wormhole solutions, emphasizing their formulation within both general relativity and alternative theories of gravity. We explore key developments in the construction of non-static wormhole spacetimes, including those conformally related to static solutions, as well as dynamically evolving geometries influenced by scalar fields. Particular attention is given to the wormholes embedded into Friedmann–Lemaître–Robertson–Walker (FLRW) universes and de Sitter backgrounds, where the interplay between the cosmic expansion and wormhole dynamics is analyzed. We also examine the role of modified gravity theories, especially in hybrid metric–Palatini gravity, which enable the realization of traversable wormholes supported by effective stress–energy tensors that do not violate the null or weak energy conditions. By systematically analyzing a wide range of time-dependent wormhole solutions, this review identifies the specific geometric and physical conditions under which wormholes can evolve consistently with null and weak energy conditions. These findings clarify how such configurations can be naturally integrated into cosmological models governed by general relativity or modified gravity, thereby contributing to a deeper theoretical understanding of localized spacetime structures in an expanding universe.

Locally-rotationally-symmetric Bianchi type-I viscous and non -viscous\ncosmological models are explored in general relativity (GR) and in f(R,T)\ngravity. Solutions are obtained by assuming that the expansion scalar is\nproportional to the shear scalar which yields a constant value for the\ndeceleration parameter (q=2). Constraints are obtained by requiring the\nphysical viability of the solutions. A comparison is made between the viscous\nand non-viscous models, and between the models in GR and in f(R,T) gravity. The\nmetric potentials remain the same in GR and in f(R,T) gravity. Consequently,\nthe geometrical behavior of the $f(R,T)$ gravity models remains the same as the\nmodels in GR. It is found that f(R,T) gravity or bulk viscosity does not affect\nthe behavior of effective matter which acts as a stiff fluid in all models. The\nindividual fluids have very rich behavior. In one of the viscous models, the\nmatter either follows a semi-realistic EoS or exhibits a transition from stiff\nmatter to phantom, depending on the values of the parameter. In another model,\nthe matter describes radiation, dust, quintessence, phantom, and the\ncosmological constant for different values of the parameter. In general, f(R,T)\ngravity diminishes the effect of bulk viscosity.\n

Several recent developments have highlighted the significance of the vanishing complexity factor formalism in understanding the structure and evolution of stellar relativistic compact objects. This formalism, introduced through a novel definition proposed by Herrera (Phys. Rev. D 97:044010, 2018), offers valuable insights into the dynamics of such systems. In this manuscript, we explored a class of realistic solutions to the static and spherically symmetric field equations characterized by two fluid distributions: ordinary stellar matter and dark energy, within the framework of this formalism. Utilizing the well-known Tolman-VI solution as the seed ansatz for the metric coefficient grr, we employed the complexity-free format to derive an analytic solution for the other metric coefficient, gtt. Subsequently, we obtained the solutions of gravitational field equations for our proposed spacetime model by incorporating the linear dark energy equation of state. These results were applied to the astrophysical compact star candidate LMC X-4, with M=1.04M⊙ and R=8.4km. The potential viability and credibility of the proposed dark star solutions were thoroughly analyzed by examining key constraints, including the regularity of metric functions, physical adequacy through matter variables, state parameter behavior, energy conditions, stability tests (such as pressure anisotropy and hydrostatic equilibrium), the speed of sound, and the mass–radius relation for this compact star candidate. Notably, the estimated values of the dark energy coupling factor, presented in Table 1, highlight the exotic nature of the fluid distribution and effectively quantify the contribution of dark energy to the structure and evolution of an ultra-relativistic dark compact star. These findings strongly support our model solutions and demonstrate improvements over previously reported results in Rej et al. (Chin J Phys 87:608, 2024).

Rastall gravity is a modification of Einstein's general relativity in which the energy-momentum conservation is not satisfied and depends on the gradient of the Ricci curvature. It is currently in dispute whether Rastall gravity is equivalent to general relativity (GR). In this work, we constrain the theory using the rotation curves of low surface brightness (LSB) spiral galaxies. By fitting the rotation curves of LSB galaxies, we obtain parameter of the Rastall gravity. The values of LSB galaxies satisfy the weak energy condition (WEC) and strong energy condition (SEC). Combining the values of type Ia supernovae and the gravitational lensing of elliptical galaxies on Rastall gravity, we conclude that Rastall gravity may be equivalent to general relativity.

The main focus of this paper is to explore the possibility of providing a new family of exact solutions for suitable anisotropic spherically symmetric systems in the realm of general relativity involving the embedding spherically symmetric static metric into the five-dimensional pseudo-Euclidean space. In this regard, we ansatz a new metric potential lambda (r), and we obtained the other metric potential nu (r) by mains of embedding class one approach. The unknown constants are determined by the matching of interior space-time with the Schwarzschild exterior space-time. The physical acceptability of the generating celestial model for anisotropic compact stars is approved via acting several physical tests of the main salient features viz., energy density, radial and tangential pressures, anisotropy effect, dynamical equilibrium, energy conditions, and dynamical stability, which are well-compared with experimental statistics of four different compact stars: PSR J1416-2230, PSR J1903+327, 4U 1820-30 and Cen X-3. Conclusively, all the compact stars under observations are realistic, stable, and are free from any physical or geometrical singularities. We find that the embedding class one solution for anisotropic compact stars is viable and stable, plus, it provides circumstantial evidence in favor of super-massive pulsars.

In this paper, we generate a new generalized solution for modeling of compact anisotropic astrophysical configurations by using Karmarkar condition of embedded class 1 spacetime manifold. We demonstrate that the new solution satisfies all required physical conditions. We investigate several physical properties of compact star models, i.e. Vela X-1 (Mass [Formula: see text][Formula: see text], radius = [Formula: see text][Formula: see text]km), PSRJ [Formula: see text] (Mass [Formula: see text][Formula: see text], radius = [Formula: see text][Formula: see text]km) and PSRJ [Formula: see text] (Mass [Formula: see text][Formula: see text], radius = [Formula: see text][Formula: see text]km) in conformity with the observational data. The proposed solution is free from singularities, satisfies causality condition and displays well-behaved nature inside the anisotropic configurations. All energy conditions and hydrostatic equilibrium condition are well defined inside the anisotropic fluid spheres. The adiabatic index throughout the stellar interior is greater than [Formula: see text] and the compactification factor lies within the Buchdahl limit [Formula: see text]. We study the physical features of the solution in detail, analytically as well as graphically for compact star Vela X-1 with [Formula: see text] ranging from [Formula: see text] to [Formula: see text].

Relativistic models for anisotropic compact stars: A review

Seven new solutions to the interior static and spherically symmetric Einstein's field equations (EFE) are found and investigated. These new solutions are a generalisation of the quadratic density fall-off profile of the Tolman VII solution. The generalisation involves the addition of anisotropic pressures and electric charge to the density profile. Of these new solutions three are found to obey all the necessary conditions of physical acceptability, including linear stability under radial perturbations, and causality of the speed of pressure waves inside the object. Additionally an equation of state can be found for all the physically viable solutions. The generalised pulsation equation for interior solutions to the EFE that include both electric charge and pressure anisotropy is derived and used to determine the stability of the solutions. However the pulsation equation found is general and can be used for all new solutions that contain these ingredients.

Science sets itself apart from other paths to truth by recognizing that even its greate practitioners sometimes err.

Late-time acceleration with a scalar field source: Observational constraints and statefinder diagnostics

Readers of this volume will notice that it contains only a few papers on general relativity. This is because most papers documenting the genesis and early development of general relativity were not published in Annalen der Physik.After Einstein took up his new prestigious position at the PrussianAcademy of Sciences in the spring of 1914, the Sitzungsberichte of the Berlin academy almost by default became the main outlet for his scientific production. Two of the more important papers on general relativity, however, did find their way into the pages of the Annalen [35,41].Although I shall discuss both papers in this essay, the main focus will be on [35], the first systematic exposition of general relativity, submitted in March 1916 and published in May of that year. Einstein’s first paper on a metric theory of gravity, co-authored with his mathematician friend Marcel Grossmann, was published as a separatum in early 1913 and was reprinted the following year in Zeitschrift fur Mathematik und Physik [50,51]. Their second (and last) joint paper on the theory also appeared in this journal [52]. Most of the formalism of general relativity as we know it today was already in place in this Einstein-Grossmann theory. Still missing were the generally-covariant Einstein field equations. As is clear from research notes on gravitation from the winter of 1912–1913 preserved in the so-called “Zurich Notebook,” Einstein had considered candidate field equations of broad if not general covariance, but had found all such candidates wanting on physical grounds. In the end he had settled on equations constructed specifically to be compatible with energy-momentum conservation and with Newtonian theory in the limit of weak static fields, even though it remained unclear whether these equations would be invariant under any non-linear transformations. In view of this uncertainty, Einstein and Grossmann chose a fairly modest title for their paper: “Outline (“Entwurf”) of a Generalized Theory of Relativity and of a Theory of Gravitation.” The Einstein-Grossmann theory and its fields equations are therefore also known as the “Entwurf” theory and the “Entwurf” field equations. Much of Einstein’s subsequent work on the “Entwurf” theory went into clarifying the covariance properties of its field equations. By the following year he had convinced himself of three things. First, generallycovariant field equations are physically inadmissible since they cannot determine the metric field uniquely. This was the upshot of the so-called “hole argument” (“Lochbetrachtung”) first published in an appendix to [51]. Second, the class of transformations leaving the “Entwurf” field equations invariant was as broad ∗ E-mail: janss011@tc.umn.edu 1 An annotated transcription of the gravitational portion of the “Zurich Notebook” is published as Doc.10 in [11]. For facsimile reproductions of these pages, a new transcription, and a running commentary, see [89]. 2 See Sect. 2 for further discussion of the hole argument.

The theoretical basis of dark energy remains unknown and could signify a need to modify the laws of gravity on cosmological scales. In this study, we investigate how the clustering and motions of galaxies can be used as probes of modified gravity theories, using galaxy and direct peculiar velocity auto- and cross-correlation functions. We measure and fit these correlation functions in simulations of ΛCDM, DGP, and f(R) cosmologies and, by extracting the characteristic parameters of each model, we show that these theories can be distinguished from General Relativity (GR) using these measurements. We present forecasts showing that with sufficiently large data samples, this analysis technique is a competitive probe that can help place limits on allowed deviations from GR. For example, a peculiar velocity survey reaching to z = 0.5 with $20{{\ \rm per\ cent}}$ distance accuracy would constrain model parameters to 3-σ confidence limits log10|fR0| < −6.45 for f(R) gravity and $r_\mathrm{ c} \gt 2.88 \, \mathrm{ c}/H_0$ for nDGP, assuming a fiducial GR model.

In the present article, we explore a new static, spherically symmetric charged anisotropic fluid model of compact star in curvature coordinates. We consider metric potential g44 of Durgapal’s fifth solution [1] with a specific choice of electric field intensity E and physically acceptable expression of anisotropy factor Δ , which involve parameters K (charge) and Δ (anisotropy) respectively. The solution so obtained is utilized to construct the model for superdense star like neutron star. We have analysed that corresponding to X=0.1, K=2.8, a=1.6 and by assuming surface density , the mass of the compact star comes out to be with radius 14.51 kms, which closely resembles to that of PSRJ0348 + 0432. The solution is well behaved for the values of K satisfying 1≤K5. Our model is described analytically as well as with the help of graphical representations. Our solution is well behaved and free from any central singularity. It also satisfies all the energy conditions as well as the causality condition thus reconfirming the stability of our model.

Exact general solutions for spatially flat isotropic and homogeneous cosmologies with a nonminimally coupled ghost scalar field that has polynomial potentials of the fourth degree are obtained in the framework of the Einstein–Cartan theory (ECT) and general relativity (GR). The special values of coupling constant \(\xi \) and restrictions on \(\xi \) are found for the above solutions. Some effects of torsion and scalar field potential are elucidated. It is shown that solutions can describe the bouncing models with the late-time accelerated expansion. It is demonstrated that some models admit the unified scenario for dark matter and dark energy: (i) both singular models in ECT with a de Sitter-like asymptotic and with the power-law \((t^{4/3})\) asymptotic at late times, (ii) singular and nonsingular models in GR with a de Sitter-like asymptotic at late times.

This article investigates the physical properties and stability of an isotropic compact star in F(Q,T) gravity, where T denotes the trace of the energy-momentum tensor and Q represents the nonmetricity scalar. In the present study, the impact of F(Q,T) gravity on the stability and structure of compact stars is investigated, considering a perfect fluid distribution and adopting the functional form F(Q,T)=ξ1Q+ξ2T. The field equations are solved using the two-step method proposed by Gupta and Jasim [Astrophys. Space Sci. 283 (2003) 337], along with the Buchdahl ansatz given by eφ=ϒ(1+ϵr2)ϒ+ϵr2. To investigate the physical properties of the LMC X-4 compact star, a static spherically symmetric metric is employed in the interior region, while a Reissner-Nordstrom exterior metric is adopted for the exterior region. In the interior region of the proposed stellar object, we analyze the behaviour of the spacetime metric functions (eφ and eϑ), energy density (ρ), pressure (P), charge (q), ratio of pressure to density (Ω) and the energy conditions. The equilibrium state of the star is analysed using the Tolman-Oppenheimer-Volkoff (TOV) equation, expressed as FG+FH+F(Q,T)+FQ=0. To assess the stability of the configuration, we examine the adiabatic index (Γ), the causality condition, regularity conditions, the well-behaved condition and the stability against convection criterion.