Einstein Equations Research Articles

Closed causal curves in f(ℛ,𝒜,AμνAμν) gravity

In this paper, we investigate a [Formula: see text] solution of Einstein field equations, an extension of [Formula: see text] Misner space in the context of Ricci-inverse ([Formula: see text])-gravity theory. For that, we consider two different class, namely, Class-II model which is defined by the function [Formula: see text] and Class-III model defined by the function [Formula: see text]. Here, [Formula: see text] is the anti-curvature tensor with [Formula: see text] as its scalar, and [Formula: see text] is the Ricci scalar. Using this [Formula: see text] metric as background, the modified field equations for Ricci-inverse gravity were derived and obtain the results. These derived field equations are then solvable by choosing the pure radiation field as the energy-momentum tensor, including a cosmological constant as does in the general relativity. This implies that the chosen [Formula: see text] space-time is a valid solution in this novel gravity theory, and thereby violate the causality condition, analogous to general relativity.

Geometric Relational Framework for General‐Relativistic Gauge Field Theories

AbstractIt is recalled how relationality arises as the core insight of general‐relativistic gauge field theories from the articulation of the generalized hole and point‐coincidence arguments. Hence, a compelling case for a manifestly relational framework ensues naturally. A formulation for such a framework is proposed, based on a significant development of the dressing field method of symmetry reduction. A version for the group of automorphisms of a principal bundle over a manifold is first developed, as it is the most natural and elegant, and as hosts all the mathematical structures relevant to general‐relativistic gauge field theory. However, as the standard formulation is local, on , the relational framework for local field theory is then developed. The generalized point‐coincidence argument is manifestly implemented, whereby the physical field‐theoretical degrees of freedoms co‐define each other and define, coordinatize, the physical spacetime itself. Applying the framework to General Relativity, relational Einstein equations are obtained, encompassing various notions of “scalar coordinatization” à la Kretschmann–Komar and Brown–Kuchař.

Well-balanced High-order Finite Difference Weighted Essentially Nonoscillatory Schemes for a First-order Z4 Formulation of the Einstein Field Equations

We develop a new class of high-order accurate well-balanced finite difference (FD) weighted essentially nonoscillatory (WENO) methods for numerical general relativity (GR), which can be applied to any first-order reduction of the Einstein field equations, even if nonconservative terms are present. We choose the first-order nonconservative Z4 formulation of the Einstein equations, which has a built-in cleaning procedure that accounts for the Einstein constraints and that has already shown its ability in keeping stationary solutions stable over long timescales. By introducing auxiliary variables, the vacuum Einstein equations in first-order form constitute a partial differential equation system of 54 equations that is naturally nonconservative. We show how to design FD-WENO schemes that can handle nonconservative products. Different variants of FD WENO are discussed, with an eye to their suitability for higher-order accurate formulations for numerical GR. We successfully solve a set of fundamental tests of numerical GR with up to ninth-order spatial accuracy. Due to their intrinsic robustness, flexibility, and ease of implementation, FD-WENO schemes can effectively replace traditional central finite differencing in any first-order formulation of the Einstein field equations, without any artificial viscosity. When used in combination with well-balancing, the new numerical schemes preserve stationary equilibrium solutions of the Einstein equations exactly. This is particularly relevant in view of the numerical study of the quasi-normal modes of oscillations of relevant astrophysical sources. In conclusion, general relativistic high-energy astrophysics could benefit from this new class of numerical schemes and the ecosystem of desirable capabilities built around them.

Height-function-based 4D reference metrics for hyperboloidal evolution

Hyperboloidal slices are spacelike slices that reach future null infinity. Their asymptotic behaviour is different from Cauchy slices, which are traditionally used in numerical relativity simulations. This work uses free evolution of the formally-singular conformally compactified Einstein equations in spherical symmetry. One way to construct gauge conditions suitable for this approach relies on building the gauge source functions from a time-independent background spacetime metric. This background reference metric is set using the height function approach to provide the correct asymptotics of hyperboloidal slices of Minkowski spacetime. The present objective is to study the effect of different choices of height function on hyperboloidal evolutions via the reference metrics used in the gauge conditions. A total of 10 reference metrics for Minkowski are explored, identifying some of their desired features. They include 3 hyperboloidal layer constructions, evolved with the non-linear Einstein equations for the first time. Focus is put on long-term numerical stability of the evolutions, including small initial gauge perturbations. The results will be relevant for future (puncture-type) hyperboloidal evolutions, 3D simulations and the development of coinciding Cauchy and hyperboloidal data, among other applications.

Wheeler-DeWitt canonical quantum gravity in hydrogenoid atoms according to de Broglie-Bohm and the geodesic hypothesis. Einstein’s Quantum Field Equation

We explore the geodesic hypothesis of orbital trajectories of the electrons in hydrogenoid atoms, in the frame of de Broglie-Bohm quantum theory. It is intended that the space-time can be curved, at very short distances, by the effect of the joint action of the energy content of the atomic system and the contribution of the electric and quantum potentials. The geodesic hypothesis would explain the non-lose of energy in the electron orbital trajectories. So we explore a conception where particles and waves interact in a closed system: the waves guide the particles and the particles generate the spacetime perturbation that acts as a wave, beyond the pilot-wave theory. We establish the equivalence, in a local neighborhood, between the electron trajectory of an hydrogenoid atom in the Minkowskian space where the de Broglie-Bohm can be cast with its movement in a Lorentzian manifold, according to the concept of metric tangent. Through the geodesic condition and the invariance of the elemental length, we establish a relationship between some components of the metrics. But as the particles in microphysics do not follow the Einstein’s field equation, we consider the 3+1 decomposition according to ADM and the quantization in the Wheeler De Witt theory and with a so-called quantum Einstein field equation, with a decomposition of spacetime into three-dimensional sheets of a spatial character. Then we derive from the moment-energy tensor a further equation between the components of the metrics. It opens the avenue to characterize the metric by an exact solution of the Einstein’s field equations.

Generalized Schwarzschild Spacetimes with a Linear Term and a Cosmological Constant

Particular Kottler spacetimes are analytically investigated. The investigated spacetimes are spherically symmetric nonrotating spacetimes endowed with a Schwarzschild solid-angle element. SchwarzschildNairiai spacetimes, Schwarzschild spacetimes with a linear term, and Schwarzschild spacetimes with a linear term and a cosmological constant are studied. The infinite-redshift surfaces are analytically written. To this aim, the parameter spaces of the models are analytically investigated, and the conditions for which the analytical radii are reconducted to the physical horizons are used to set and to constrain the parameter spaces. The coordinate-singularity-avoiding coordinate extensions are newly written. Schwarzschild spacetimes with a linear term and a cosmological constant termare analytically studied, and the new singularity-avoiding coordinate extensions are detailed. The new roles of the linear term and of the cosmological constant term in characterizing the Schwarzschild radius are traced. The generalized Schwarzschild–deSitter case and generalized Schwarzschild–anti-deSitter case are characterized in a different manner. The weak field limit is newly recalled. The embeddings are newly provided. The quantum implementation is newly envisaged. The geometrical objects are newly calculated. As a result, for the Einstein field equations, the presence of quintessence is newly excluded. The Birkhoff theorem is newly proven to be obeyed.

Existence of remnants of regular black holes with de Sitter centers

We address the question of the end point of the quantum evaporation of regular black holes with the de Sitter centers specified by the algebraic structure of their stress–energy tensors [Formula: see text]. Most of the presented in the literature regular solutions belong to this class. In the case of a nonzero background cosmological constant stress–energy tensors with [Formula: see text] connect smoothly two de Sitter vacua, in the center and at infinity, and generate regular solutions to the Einstein equations, which describe the spacetime structure of regular black holes and their remnants. We show that for this class black holes their spacetime symmetry prevents a complete evaporation and provides the existence of thermodynamically stable remnants.

Analogy of space-time as an elastic medium - Estimation of a creep coefficient of space from space data via the MOND theory and the gravitational lensing effect - the ball cluster and via time data from the GPS effect- comparison, discussion and implication of the results for dark matter and Einstein's field equation

Analogy of space-time as an elastic medium - Estimation of a creep coefficient of space from space data via the MOND theory and the gravitational lensing effect - the ball cluster and via time data from the GPS effect- comparison, discussion and implication of the results for dark matter and Einstein's field equation

Complexity characterization in modeling anisotropic compact stellar structures

In this study, we present an exact solution to the Einstein field equations, featuring a static, spherically symmetric, anisotropic compact stellar fluid sphere with non-zero complexity. The model is constructed using a specific metric potential for the gtt\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$g_{tt}$$\\end{document} component, along with a defined complexity profile. The model parameters are fixed by ensuring the continuity of the interior and exterior metrics across the surface of the star and by setting the radial pressure to zero across the boundary. The stability of the model has been analyzed by verification of different standard conditions. Validation of our model is supported by recent observational data from well-known pulsars.

Hybrid star model with Tolman–Buchdahl metric potentials in non-conservative theory of gravity

In this study, we propose a streamlined approach to analyze the characteristics of compact stars with a unique equation of state resulting from the hybridization of dark matter and baryonic matter within the framework of Rastall gravity. The impact of dark energy on the star’s interior is investigated by introducing a linear coupling parameter β (called the Rastall parameter). The interior space–time model is developed using metric potentials eν=A+Br2 and eλ(r)=a(1+br2)a−br2 for temporal and spatial coordinates, respectively. The parameter b is set to 0.005 km−2 for the specific stellar candidate PSRJ1614-2230. Physical features derived from the exact solutions of the Einstein field equations are analyzed by considering various choices of the Rastall parameter. The results indicate that the proposed model satisfies the necessary conditions for a stable and physically plausible stellar structure.

Self-gravitating matter in stationary and axisymmetric black hole spacetimes

Abstract All black holes (BHs) in nature are expected to be described by the Kerr vacuum solution of general relativity. However, the Kerr BH interior contains several problematic features such as a Cauchy horizon, a curvature singularity, and a causality-violating region. Non-Kerr BH models, which are used to examine the genericity of these features, typically contain nontrivial matter content. When such self-gravitating matter is minimally-coupled to Einstein–Hilbert gravity, the Einstein equations can be directly used to investigate its physical properties. We examine here the properties of matter in a broad class of stationary and axisymmetric, geodesically-integrable BH spacetimes, and how they are linked to various features of the spacetime geometry. In these spacetimes, we find the matter to typically flow along timelike Killing orbits in the BH exterior, usually exhibiting differential rotation but sometimes additionally also non-rigid rotation. At a horizon, the matter rest-frame energy density, ε, and principal normal pressure, pn , are shown to necessarily satisfy p n = − ϵ , implying that only specific types of matter can thread stationary event horizons (e.g. electromagnetic fields but not massless real scalar fields). Furthermore, we introduce Boyer–Lindquist-like coordinates for the nonstationary regions in the BH interior, which show the matter to be comoving with the interior cosmology. We also obtain simple expressions for the expansions of the ingoing and outgoing zero angular momentum null congruences and comment on the light-focussing behavior of the cosmology. Finally, we verify above results explicitly by working with a representative set of well-known BH spacetimes which contain various types of matter—scalar fields, electromagnetic fields, anisotropic fluids. Some spacetimes have singularities while others have regular interiors. In the exterior, the matter satisfies the weak energy condition. The framework developed here can be extended to cover more general spacetimes.

Quantum geometrical properties of topological materials

The momentum space of topological insulators and topological superconductors is equipped with a quantum metric defined from the overlap of neighboring valence band states or quasihole states. We investigate the quantum geometrical properties of these materials within the framework of Dirac models and differential geometry. Their momentum space is found to be always a maximally symmetric space with a constant Ricci scalar, and the vacuum Einstein equation is satisfied in 3D with a finite cosmological constant. For linear Dirac models, several geometrical properties are found to be independent of the band gap, including a peculiar straight line geodesic, constant volume of the curved momentum space, and the exponential decay form of the nonlocal topological marker, indicating the peculiar yet universal quantum geometrical properties of these models.

Boosted Kerr–Newman black holes

Abstract In this paper we obtain a new solution of Einstein field equations which describes a boosted Kerr–Newman black hole relative to a Lorentz frame at future null infinity. To simplify our analysis we consider a particular configuration in which the boost is aligned with the black hole angular momentum. The boosted Kerr–Newman black hole is obtained considering the complete asymptotic Lorentz transformations of Robinson–Trautman coordinates to Bondi–Sachs, including the perturbation term of the boosted Robinson–Trautman metric. To verify that the final form of the metric is indeed a solution of Einstein field equations, we evaluate the corresponding energy–momentum tensor the boosted Kerr–Newman solution. To this end, we consider the electromagnetic energy–momentum tensor built with the Kerr boosted metric together with its timelike killing vector. We show that the Papapetrou field thus obtained engender an energy–momentum tensor which satisfies Einstein field equations up to 4th order for the Kerr–Newman metric. To proceed, we examine the causal structure of the boosted Kerr–Newman black hole in Bondi–Sachs coordinates as in a preferred timelike foliation. We show that the ultimate effect of a nonvanishing charge is to shrink the overall size of the event horizon and ergosphere areas when compared to the neutral boosted Kerr black holes. Considering the preferred timelike foliation we obtain the electromagnetic fields for a proper nonrotating frame of reference. We show that while the electric field displays a pure radial behavior, the magnetic counterpart develops an involved structure with two intense lobes of the magnetic field observed in the direction opposite to the boost.

Metric reconstruction in Kerr spacetime

Abstract Metric reconstruction is the general problem of parameterizing GR in terms of its two “true degrees of freedom” e.g., by a complex scalar “potential”—in practice mostly with the aim of simplifying the Einstein equation (EE) within perturbative approaches. In this paper, we re-analyze the metric reconstruction procedure by Green, Hollands, and Zimmerman (GHZ) [Class. Quant. Grav. 37, 075001 (2020)], which is a generalization of the Chrzanowski-Cohen-Kegeles (CCK) approach. Contrary to the CCK method, that by GHZ is applicable not only to the vacuum, but also to the sourced linearized Einstein equation (EE) on Kerr. Our main innovation is a version of the GHZ integration scheme that is suitable for the initial value problem of the sourced linear EE. By iteration, our scheme gives the metric to as high an order in perturbation theory as one might wish, in principle. At each order, the metric perturbation is a sum of a corrector, obtained by solving a triangular system of transport equations, a reconstructed piece, obtained from a Hertz potential as in the CCK approach, and an algebraically special perturbation, determined by the ADM quantities. As a byproduct, we determine the precise relations between the asymptotic tail of the Hertz potential in the GHZ and CCK schemes, and the quantities relevant for gravitational radiation, namely, the energy flux, news- and memory tensors, and their associated BMS-supertranslations. We also discuss ways of transforming the metric perturbation to Lorenz gauge.

Quark Stars in $f(R,T)$ Gravity: Mass-to-Radius Profiles and Observational Data

Abstract In this paper, we explore the $f(R,T)$ gravity theory, which introduces a coupling between matter and curvature, through the simplest linear functional form $f(R,T)= R + 2\beta T$. We derive the modified Einstein field equations and conservation equations for this theory. We then apply this framework to study the structural properties of quark stars (QSs) composed of interacting quark matter, considering perturbative QCD corrections and color superconductivity. By solving the modified Tolman-Oppenheimer-Volkoff equations, we investigate the mass-radius relation, stability criteria, and energy conditions of QSs. Our results indicate that the $f(R,T)$ gravity significantly influences the properties of QSs, leading to deviations from General Relativity. The analysis shows consistency with recent observational data, suggesting that the modified gravity framework could provide viable models for the study of compact stars.

A connection between massive electrodynamics and the Einstein-Maxwell equations

Quaternions are the best mathematical construct for creatingvarious equations in electrodynamics, which has led to the emergence of new terms with unique physical implications. Since quaternions also have noncommutative properties that are reflected in curved space-time too, a formulation of a theory using quaternions can be compared with that formulated in some curved space-time. Furthermore, we calculate the Maxwell equations in curved space-time and observe the presence of extra terms that are not present in flat space-time. An electric current arises because of the coupling between the magnetic field and curvature. Upon comparing the outcomes of the two methods, we discovered a correlation between mass and gravity, indicating their similarity. Equations formulated via quaternions are equivalent to those formulated in curved space-time. The optical chirality and its flux are generalized to massive electrodynamics.

A natural model for curved inflation

Abstract Inflationary models with a non-zero background curvature require additional hy- pothesis or parameters compared to flat inflation and the procedure to construct them cannot be as simple as in the flat case. For this reason, there is no consen- sus on the primordial power spectrum that should be considered at large scales in a curved Universe. In this letter, we propose a model of curved inflation in which the usual canonical quantization and Bunch–Davies vacuum choice of the flat case can be considered. The framework is a recently proposed modification of general relativity in which a non-dynamical topological term is added to the Einstein equation. The model is universal as it is the same for any background curvature, and no additional parameters or hypothesis on the initial spatial curvature are introduced. This gives a natural and simple solution to the problem of constructing curved inflation, and at the same time provides an additional argument for this topological modification of general relativity.

Morris–Thorne-type wormholes with global monopole charge and the energy conditions

In this paper, we investigate Morris–Thorne-type wormholes with global monopole charge using various shape function forms known in the literature. We solve the Einstein field equations incorporating an anisotropic energy–momentum tensor and obtain different physical quantities associated with the matter-content. A crucial aspect of this study is the non-exotic matter distribution, examined through the evaluation of energy conditions, and exploring how different shape functions impact these conditions. Additionally, the anisotropy parameter is calculated to quantify the extent of attractive or repulsive behavior. Our study demonstrates that for different types of shape function forms, the energy conditions are influenced by the global monopole parameter. Our findings provide valuable insights for further theoretical explorations of these fascinating hypothetical structures in the realm of general relativity and beyond.

Anisotropic model observing pulsars from Neutron Star Interior Composition with modified Van der Waals equation of state

This paper is designed for heavy pulsars coming from the Neutron Star Interior Composition Explorer. The research model is describe by Einstein field equations for anisotropic fluid configuration with spherical symmetry. As per present perceptiveness, modified non-linear Van der Waals equation of state is used to relate physical variables. The continuity of inner and outer matter is obtained by comparing inner spacetime to outer Schwarzschild metric. The physical viability of this model is evaluated and further it is compared with observational data of pulsars PSR J0348+0432, PSR J0740+6620 and PSR J0030+0451. The model fulfils all physical and mathematical characteristics of the dense structure studies. It offers the factual proofs carried by evolution of celestial configurations. The working model presented here is physically viable and shows stable behaviour.

Classification of the non-null electrovacuum solution of Einstein–Maxwell equations with three-parameter abelian group of motions

The classification of the Stackel spaces of the electrovacuum of the type (3.0) has been done. These spaces are invariant under the action of the three-parameter abelian group of motions and belong to the first type Bianchi spaces. In the case of a non-zero cosmological term, the metrics and potentials contain solutions of a nonlinear ordinary differential equation of the second order. When the cosmological term equals zero, the metrics and the components of the electromagnetic field tensor are expressed through elementary functions. Thus the classification of the electrovacuum stackel spaces of all types is completed and complete list of these spaces is constructed.