Applications of Lie Symmetries to Higher Dimensional Gravitating Fluids

We consider a radiating shear-free spherically symmetric metric in higher dimensions. Several new solutions to the Einstein’s equations are found systematically using the method of Lie analysis of differential equations. Using the five Lie point symmetries of the fundamental field equation, we obtain either an implicit solution or we can reduce the governing equations to a Riccati equation. We show that known solutions of the Einstein equations can produce infinite families of new solutions. Earlier results in four dimensions are shown to be special cases of our generalised results.     

Spherically Symmetric Static Solutions of the Einstein Equations with Elastic Matter Source

The Einstein equations for a spherically symmetric static distribution of elastic matter are examined. The existence of regular solutions near the center is proven under a fairly mild hypothesis on the constitutive equation. These solutions are uniquely determined by the choice of central pressure and constitutive equation. It is also shown for a Hookean elastic material that these solutions can be integrated outward till the radial pressure vanishes, thus one can join an exterior Schwarzschild metric to obtain a maximal solution of the Einstein equations.     

New shear-free relativistic models with heat flow

We study shear-free spherically symmetric relativistic models with heat flow. Our analysis is based on Lie’s theory of extended groups applied to the governing field equations. In particular, we generate a five-parameter family of transformations which enables us to map existing solutions to new solutions. All known solutions of Einstein equations with heat flow can therefore produce infinite families of new solutions. In addition, we provide two new classes of solutions utilising the Lie infinitesimal generators. These solutions generate an infinite class of solutions given any one of the two unknown metric functions.     

On the generation of new metrics containing arbitrary functions

We present simple methods by which certain solutions of the Einstein vacuum equation (and some other equations) can be used to generate further solutions of these equations. In some cases the new solutions admit a smaller number of metric automorphisms than the original ones and are, in this sense, more general.Supported by N.R.C. Grant No. A-5205.Supported by N.R.C. Grant No. A-3993.     

Infinite hierarchies of exact solutions of the einstein and einstein-maxwell equations for interacting waves and inhomogeneous cosmologies

For space-times with two spacelike isometries, we present infinite hierarchies of exact solutions of the Einstein and Einstein-Maxwell equations as represented by their Ernst potentials. This hierarchy contains three arbitrary rational functions of an auxiliary complex parameter. They are constructed using the so-called "monodromy transform" approach and our new method for the solution of the linear singular integral equation form of the reduced Einstein equations. The solutions presented, which describe inhomogeneous cosmological models or gravitational and electromagnetic waves and their interactions, include a number of important known solutions as particular cases.     

Higher dimensional Vaidya metric in Einstein and de Sitter background

Spherically symmetric non-static higher dimensional metrics are considered in connection with Einstein’s field equations. Two exact solutions are derived. One of them corresponds to a mixture of perfect fluid and pure radiation field and represents higher dimensional Vaidya metric in the cosmological background of Einstein static universe. The other corresponds to a pure radiation field and represents higher dimensional Vaidya metric in the background de Sitter universe. For both of these solutions, the cosmological constant is taken to be non-zero. Many known solutions are derived as particular cases.     

Warp factors and extended sources in two transverse dimensions

We study the solutions of the Einstein equations in (d+2)-dimensions, describing parallel p-branes (p=d−1) in a space with two transverse dimensions of positive gaussian curvature. These solutions generalize the solutions of Deser and Jackiw of point particle sources in (2+1)-dimensional gravity with cosmological constant. Determination of the metric is reduced to finding the roots of a simple algebraic equation. These roots also determine the nontrivial “warp factors” of the metric at the positions of the branes. We discuss the possible role of these solutions and the importance of “warp factors” in the context of the large extra dimensions scenario.     

The series solution to the metric of stationary vacuum with axisymmetry

The multipole moment method not only conduces to the understanding of the deformation of the space--time, but also serves as an effective tool to approximately solve the Einstein field equation with. However, the usual multipole moments are recursively determined by a sequence of symmetric and trace-free tensors, which is inconvenient for practical resolution. In this paper, we develop a simplified procedure to generate the series solutions to the metric of the stationary vacuum with axisymmetry, and show its validity. In order to understand the free parameters in the solution, we propose to take the Schwarzschild metric as a standard ruler, and some well- known examples are analysed and compared with the series solutions in detail.     

Non-minimally coupled dark fluid in Schwarzschild spacetime

If one assumes a particular form of non-minimal coupling, called the conformal coupling, of a perfect fluid with gravity in the fluid–gravity Lagrangian then one gets modified Einstein field equation. In the modified Einstein equation the effect of the non-minimal coupling does not vanish if one works with spacetimes for which the Ricci scalar vanishes. In the present work we use the Schwarzschild metric in the modified Einstein equation, in the presence of non-minimal coupling with a fluid, and find out the energy–density and pressure of the fluid. In the present case the perfect fluid is part of the solution of the modified Einstein equation. We also solve the modified Einstein equation, using the flat spacetime metric and show that in the presence of non-minimal coupling one can accommodate a perfect fluid of uniform energy–density and pressure in the flat spacetime. In both the cases the fluid pressure turns out to be negative. Except these non-trivial solutions it must be noted that the vacuum solutions also remain as trivial valid solutions of the modified Einstein equation in the presence of non-minimal coupling.     

On General Solutions for Field Equations in Einstein and?Higher Dimension Gravity

We prove that the Einstein equations can be solved in a very general form for arbitrary spacetime dimensions and various types of vacuum and non-vacuum cases following a geometric method of anholonomic frame deformations for constructing exact solutions in gravity. The main idea of this method is to introduce on (pseudo) Riemannian manifolds an alternative (to the Levi-Civita connection) metric compatible linear connection which is also completely defined by the same metric structure. Such a canonically distinguished connection is with nontrivial torsion which is induced by some nonholonomy frame coefficients and generic off-diagonal terms of metrics. It is possible to define certain classes of adapted frames of reference when the Einstein equations for such an alternative connection transform into a system of partial differential equations which can be integrated in very general forms. Imposing nonholonomic constraints on generalized metrics and connections and adapted frames (selecting Levi-Civita configurations), we generate exact solutions in Einstein gravity and extra dimension generalizations.     

Einstein vacuum field equations with a single non-null Killing vector

It is shown that, in the case where there is a single non-null Killing vector, the vacuum Einstein field equations imply that there is a Ricci collineation in the quotient 3-space. Using coordinates adapted to the collineation vector, we derive a fourth order partial differential equation involving the metric of the quotient 3-space and we show that if this equation is satisfied, the Ernst potential may be obtained by integrating a total Riccati equation and a straightforward set of total differential equations. We also show that if the collineation vector is null, the metric of the quotient 3-space may be expressed in terms of two real Clebsch potentials. Finally in the special case where the collineation vector is the generator of a timelike homothetic motion we reduce the field equations to a single second order partial differential equation of non-Painlevé type in two independent variables and obtain Petrov type III solution of Robinson-Trautman type.     

Further remark concerning spherically symmetric nonstatic separable solutions to the Einstein equations in the comoving frame

We consider nonstatic spherically symmetric fluid solutions to the Einstein equations which, in the comoving frame, have metric coefficients that are separable functions of their arguments and that have an origin. Subject to the vanishing of the heat flux, we show that all such solutions with shear and non-vanishing shear viscosity have a scalar polynomial singularity at the origin if the fluid satisfies both the weak and strong energy conditions. When combined with previous results [1] we conclude that for the metric forms under consideration, the only fluid solutions to the Einstein equations with vanishing heat flux which satisfy the energy conditions and are free of singularities at the origin are the Robertson-Walker solutions.     

Anisotropic fluid spheres in general relativity

A procedure is developed to find static solutions for anisotropic fluid spheres from known static solutions for perfect fluid spheres. The method is used to obtain four exact analytical solutions of Einstein’s equations for spherically symmetric self-gravitating distribution of anisotropic matter. The solutions are matched to the Schwarzschild exterior metric. The physical features of one of the solutions are briefly discussed. Many previously known perfect fluid solutions are derived as particular cases.     

Quadratic metric‐affine gravity

We consider spacetime to be a connected real 4‐manifold equipped with a Lorentzian metric and an affine connection. The 10 independent components of the (symmetric) metric tensor and the 64 connection coefficients are the unknowns of our theory. We introduce an action which is (purely) quadratic in curvature and study the resulting system of Euler–Lagrange equations. In the first part of the paper we look for Riemannian solutions, i.e. solutions whose connection is Levi‐Civita. We find two classes of Riemannian solutions: 1) Einstein spaces, and 2) spacetimes with pp‐wave metric of parallel Ricci curvature. We prove that for a generic quadratic action these are the only Riemannian solutions. In the second part of the paper we look for non‐Riemannian solutions. We define the notion of a “Weyl pseudoinstanton” (metric compatible spacetime whose curvature is purely of Weyl type) and prove that a Weyl pseudoinstanton is a solution of our field equations. Using the pseudoinstanton approach we construct explicitly a non‐Riemannian solution which is a wave of torsion in a spacetime with Minkowski metric. We discuss the possibility of using this non‐Riemannian solution as a mathematical model for the neutrino.     

Radiating Kerr particle in Einstein universe

A generalized Kerr-NUT type metric is considered in connection with Einstein field equations corresponding to perfect fluid plus a pure radiation field. A general scheme for obtaining the exact solutions of these field equations is developed. Two physically meaningful particular cases are investigated in detail. One gives the field of a radiating Kerr particle embedded in the Einstein universe. The other solution may probably represent a deSitter-like universe pervaded by a pure radiation field.     

Generation and construction of statistical spherically symmetric solutions of the gravitational equations

We show that the possibility of reducing the Einstein equations in isotropic Bondi coordinates for a spherically symmetric statistical case to two forms of a linear differential equation allows one to introduce procedures for generating a fortiori exact solutions of the gravitational equations from the known ones. A superposition of solutions is defined in a special way. Examples are given of the known solutions obtained in this way from flat space-time. The use of the proposed generating procedures allows one to find all exact solutions of the gravitational equations for neutral sources in the statistical, spherically symmetric case.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 6, pp. 5–9, June, 1990.     

Use of conformal,mapping to construct new solutions of the Einstein equations

A method is obtained for constructing new solutions of the Einstein equations from known solutions, beginning with the structure of the Einstein tensor and using conformai mapping. The sollowing cases are considered: transformation of vacuum solutions into hydrodynamic solutions and the transformation of hydrodynamic solutions into solutions of the same type. A necessary and sufficient condition, imposed on the vacuum metric, is found which guarantees that solutions with a hydrodynamic energy-momentum tensor will be produced from vacuum solutions.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 11, pp. 110–114, November, 1973.The author is deeply grateful to Ya. I. Pugachev and to the members of the seminar directed by him for discussion of the results of this paper.     

All flow-stationary cylindrically symmetric solutions of the Einstein field equations for a rotating isentropic perfect fluid

In previous papers by the author a class of flow-stationary cylindrically symmetric solutions of the Einstein field equations for a rotating isentropic perfect fluid was found. The present paper shows how all such solutions may be obtained by methods very similar to those used previously. The solutions depend on one variable and contain one completely arbitrary function f of that variable. The choice of a definite form of f corresponds to fixing the equation of state. After this is fixed, the enthalpy per unit rest-energy of the fluid, H, is determined by a linear homogeneous differential equation of second order, and all the other components of the metric are algebraically determined in terms of f and H.     

A Finsler generalisation of Einstein's vacuum field equations

This paper gives a generalisation of Einstein's vacuum field equations for Finsler metrics. The given generalised field equation reproduces the Einstein equations for Riemannian metrics, and also admits non-Riemannian solutions. This is shown in detail by deriving a first order Finsler perturbation, solving the new field equation, of the Schwarzschild metric. This perturbation turns out to be time independent. The effects of the perturbation on the three Classical Tests of General Relativity are derived, and used to give limits on the size of the perturbation parameter involved.     

The Boulware–Deser class of spacetimes radiates

We establish the result that the standard Boulware–Deser spacetime can radiate. This allows us to model the dynamics of a spherically symmetric radiating dynamical star in five-dimensional Einstein–Gauss–Bonnet gravity with three spacetime regions. The local internal region is a two-component system consisting of standard pressure-free, null radiation and an additional string fluid with energy density and nonzero pressure obeying all physically realistic energy conditions. The middle region is purely radiative which matches to a third region which is the vacuum Boulware–Deser exterior. Our approach allows for all three spacetime regions to be modeled by the same class of metric functions. A large family of solutions to the field equations are presented for various realistic equations of state. A comparison of our solutions with earlier well known results is undertaken and we show that Einstein–Gauss–Bonnet analogues of these solutions, including those of Husain, are contained in our family. We also generalise our results to higher dimensions.