Schwarzschild Metrics and Quasi-Universes

The exterior and interior Schwarzschild solutions are rewritten replacing the usual radial variable with an angular one. This allows us to obtain some results that otherwise are less apparent or even hidden in other coordinate systems.     

Some Further Properties of the Accelerated Kerr-Schild Metrics

We extend the previously found accelerated Kerr-Schild metrics for Einstein-Maxwell-null dust and Einstein-Born-Infeld-null dust equations to the cases including the cosmological constant. This way we obtain the generalization of the charged de Sitter metrics in static space-times. We also give a generalization of the zero acceleration limit of our previous Einstein-Maxwell and Einstein-Born-Infeld solutions.     

Rigidly Rotating Dust in General Relativity

A solution to the Einstein field equations that represents a rigidly rotating dust accompanied by a thin matter shell of the same type is found.     

Non‐Volkov solutions for a charge in a plane wave

As it is known, a set of solutions of the Klein‐Gordon and Dirac equations with a plane‐wave field was found for the first time by Volkov. We construct new solutions of these equations different from the Volkov ones. In particular, the new solutions are characterized by quantum numbers different from Volkov solutions. In fact, our result is based on the demonstration that the transversal charge motion in a plane wave can be mapped by a special quantum transformation to transversal free particle motion. Similarly, we find new sets of solutions of the Klein‐Gordon and Dirac equations with the combined electromagnetic field.     

Cosmological Expansion and Its Effect on Small Systems

In order to study the effect of large scale cosmological expansion on small systems, we assume a FriedmannRobertson-Walker type coordinate system in presence of a nonzero cosmological constant and derive a non-static Reissner-Nrdstr(o)m metric. It is an analytic function of r for all values except at r = 0, which is singular. By determining the equation of motion in this metric we can estimate how expansion of the universe may affect Pioneer‘smotion. Because the metric does not have any event horizon and so high potential regions are accessible, this may help us in better understanding AGN phenomenon.     

Cosmological Expansion and Its Effect on Small Systems

In order to study the effect of large scale cosmological expansion on small systems, we assume a Friedmann- Robertson-Walker type coordinate system in presence of a nonzero cosmological constant and derive a non-static Reissner-Nrdstr6m metric. It is an analytic function of r for all values except at r = O, which is singular. By determining the equation of motion in this metric we can estimate how expansion of the universe may affect Pioneer＇s motion. Because the metric does not have any event horizon and so high potential regions are accessible, this may help us in better understanding AGN phenomenon.     

On a Framework for Generating Nonstatic Solutions of the Einstein Field Equations

In this paper we present a simple generalisation of all spherically symmetric static solutions. We present a framework to obtain both anisotropic and isotropic models with and without a barotropic equation of state of the form p = . The nonstaticity in some models necessarily requires a nonzero heat flux which dictates the thermodynamics of our models.     

Relativistic polytropic models of charged anisotropic compact objects

In this paper, we introduce new viable solutions to the Einstein-Maxwell field equations by incorporating the features of anisotropic matter distributions within the realm of the general theory of relativity (\begin{document}${\rm GR}$\end{document}). To obtain these solutions, we employed the Finch-Skea spacetime, along with a generalized polytropic equation of state (\begin{document}${\rm EoS}$\end{document}). We constructed various models of generalized polytropes by assuming different values of the polytropic index, i.e., \begin{document}$\eta= \dfrac{1}{2},~ \dfrac{2}{3},~ 1$\end{document}, and \begin{document}$ 2 $\end{document}. Next, numerous physical characteristics of these considered models were studied via graphical analysis, and they were found to obey all the essential conditions for astrophysical compact objects. Furthermore, such outcomes of charged anisotropic compact star models could be reproduced in various other cases including linear, quadratic, and polytropic \begin{document}${\rm EoS}$\end{document}     

The Gravitational Field of a Circulating Light Beam

Exact solutions of the Einstein field equations are found for the exterior and interior gravitational field of an infinitely long circulating cylinder of light. The exterior metric is shown to contain closed timelike lines.     

Static fluid spheres admitting Karmarkar condition

We explore a new relativistic anisotropic solution of the Einstein field equations for compact stars based on embedding class one condition.For this purpose,we use the embedding class one methodology by employing the Karmarkar condition.Employing this methodology,we obtain a particular differential equation that connects both the gravitational potentials e^λ and e^ν.We solve this particular differential equation choosing a simple form of generalized gravitational potential grr to describe a complete structure of the space-time within the stellar configuration.After determining this space-time geometry for the stellar models,we discuss thermodynamical observables including radial and tangential pressures,matter density,red-shift,velocity of sound,etc.,in the stellar models.We also perform a complete graphical analysis,which shows that our models satisfy all the physical and mathematical requirements of ultra-high dense collapsed structures.Further,we discuss the moment of inertia and M-R curve for rotating and non-rotating stars.     

Toroidal source around a static black hole

A sequence of exact solutions of Einstein equations is obtained describing the fields of a Schwarzschild black hole surrounded by static, axially symmetric, thick toroids. Its properties are illustrated by plotting the shape of the gravitational field, the deformation of the black-hole horizon and several types of time-like geodesics, in dependence on parameters of the toroid.Dedicated to Professor Jií Horáek on the occasion of his 60th birthday.     

Birkhoff Strata of Sato Grassmannian and Algebraic Curves

Algebraic and geometric structures associated with Birkhoff strata of Sato Grassmannian are analyzed. It is shown that each Birkhoff stratum Σ_S contains a subset W_? of points for which each fiber of the corresponding tautological subbundle TB_WS is closed with respect to multiplication. Algebraically TB_WS is an infinite family of infinite-dimensional commutative associative algebras and geometrically it is an infinite tower of families of algebraic curves. For the big cell the subbundle TBW_? represents the tower of families of normal rational (Veronese) curves of all degrees. For W₁ such tautological subbundle is the family of coordinate rings for elliptic curves. For higher strata, the subbundles TBW_1,2,..,n represent families of plane (n + 1, n + 2) curves (trigonal curves at n = 2) and space curves of genus n. Two methods of regularization of singular curves contained in TBW_?, namely, the standard blowing-up and transition to higher strata with the change of genus are discussed.     

Mach's principle and higher‐dimensional dynamics

We briefly discuss the current status of Mach's principle in general relativity and point out that its last vestige, namely, the gravitomagnetic field associated with rotation, has recently been measured for the earth in the GP‐B experiment. Furthermore, in his analysis of the foundations of Newtonian mechanics, Mach provided an operational definition for inertial mass and pointed out that time and space are conceptually distinct from their operational definitions by means of masses. Mach recognized that this circumstance is due to the lack of any a priori connection between the inertial mass of a body and its Newtonian state in space and time. One possible way to improve upon this situation in classical physics is to associate mass with an extra dimension. Indeed, Einstein's theory of gravitation can be locally embedded in a Ricci‐flat 5D manifold such that the 4D energy‐momentum tensor appears to originate from the existence of the extra dimension. An outline of such a 5D Machian extension of Einstein's general relativity is presented.     

New exact solutions of Einstein—Maxwell equations for magnetostatic fields

The symmetry reduction method based on the Fréchet derivative of differential operators is applied to investigate symmetries of the Einstein-Maxwell field equations for magnetostatic fields, which is a coupled system of nonlinear partial differential equations of the second order. The technique yields invariant transformations that reduce the given system of partial differential equations to a system of nonlinear ordinary differential equations. Some of the reduced systems are further studied to obtain the exact solutions.     

A class of stationary rotating string cosmological models

We obtain a one parameter class of stationary rotating string cosmological models of which the well-known Gödel universe is a particular case. By suitably choosing the free parameter function, it is always possible to satisfy the energy conditions. The rotation of the model hinges on the cosmological constant which turns out to be negative. String-dust distribution in Gödel-type universes is also briefly discussed.     

Odd-Soliton Solutions to the Einstein Equations

Using the Inverse Scattering Method (ISM) of Belinskii and Zakharov a new odd-soliton solutions to the Einstein's field equations for an axially symmetric space-time in general relativity are obtained in the determinant form and shown to include Weyl's half-integral delta static solutions in a special case.     

The Weyl Tensor and Equilibrium Configurations of Self-Gravitating Fluids

It is shown that (except for two well defined cases), the necessary and sufficient condition for any spherically symmetric distribution of fluid to leave the state of equilibrium (or quasi-equilibrium), is that the Weyl tensor changes with respect to its value in the state of equilibrium (or quasi-equilibrium).     

The gravitational energy‐momentum pseudo‐tensor of higher‐order theories of gravity

We derive the gravitational energy momentum tensor for a general Lagrangian of any order and in particular for a Lagrangian such as . We prove that this tensor, in general, is not covariant but only affine, then it is a pseudo‐tensor. Furthermore, the pseudo‐tensor is calculated in the weak field limit up to a first non‐vanishing term of order h² where h is the metric perturbation. The average value of the pseudo‐tensor over a suitable spacetime domain is obtained. Finally we calculate the power per unit solid angle Ω carried by a gravitational wave in a direction for a fixed wave number under a suitable gauge. These results are useful in view of searching for further modes of gravitational radiation beyond the standard two modes of General Relativity and to deal with nonlocal theories of gravity where terms involving are present. The general aim of the approach is to deal with theories of any order under the same standard of Landau pseudo‐tensor.