A rotating three component perfect fluid source and its junction with empty space-time

The Kerr solution for empty space-time is presented in an ellipsoidally symmetric coordinate system and it is used to produce generalised ellipsoidal metrics appropriate for the generation of rotating interior solutions of Einstein’s equations. It is shown that these solutions are the familiar static perfect fluid cases commonly derived in curvature coordinates but now endowed with rotation. These are also shown to be potential fluid sources for not only Kerr but also Kerr-de Sitter empty space-time. The approach is further discussed in the context of T-solutions of Einstein’s equations and the vacuum T-solution outside a rotating source is presented. The interior source for these solutions is shown not to be a perfect fluid but rather an anisotropic three component perfect fluid for which the energy momentum tensor is derived. The Schwarzschild interior solution is given as an example of the approach.     

Matching in a class of stationary axisymmetric perfect fluid solutions of Einstein's equations

The class of previously found stationary axisymmetric perfect fluid solutions of Einstein's equations is written inh-orthogonal coordinates,h being a space-like coordinate. Matching of a big number of solutions of the class with each other seems to be possible for a proper choice of some parameters. The exterior solutions of the class are matched explicitly with interior solutions. Also, interior solutions are matched explicitly with each other.     

Comments on a paper by Collinson

The demonstration of the uniqueness of the Schwarzschild interior metric within conformally flat axisymmetric stationary spacetimes is revised. A complete proof containing the three possible branches of interior fluid solutions is given.     

Exact Interior Solutions for Charged Fluid Spheres

A new method is discussed to obtain the interior solution of Einstein-Maxwell equations for a charged static sphere from a known particular solutions of a similar kind. Beginning with a charged fluid interior solution reported by Patel and Pandya [11], a new interior Reissner-Nördstrom metric is obtained using this method and physical aspects of it are extensively discussed.     

Charged static fluid spheres in Einstein-Cartan theory

We give exact interior solutions of the Einstein-Cartan equations describing charged perfect fluid distribution in general relativity. Results previously unknown for the uncharged case are deduced and we find that the pressure is discontinuous at the boundary of the fluid sphere.     

Cosmological shock waves in general relativity

The problem of finding spherically symmetric self-similar solutions of Einstein's field equations with a barotropic perfect fluid, which can be joined through a shock wave to some cosmological models, is considered. It is found that such solutions comprise an expanding shell of matter surrounding a horizon with an interior singularity.     

Stars in five-dimensional Kaluza–Klein gravity

In the five-dimensional Kaluza–Klein (KK) theory there is a well known class of static and electromagnetic-free KK-equations characterized by a naked singularity behavior, namely the Generalized Schwarzschild solution (GSS). We present here a set of interior solutions of five-dimensional KK-equations. These equations have been numerically integrated to match the GSS in the vacuum. The solutions are candidates to describe the possible interior perfect fluid source of the exterior GSS metric and thus they can be models for stars for static, neutral astrophysical objects in the ordinary (four-dimensional) spacetime.     

Spinning null fluid in general relativity

Exterior and interior solutions of Einstein's equations are given for fluid moving with the speed of light and having a superposed spin. The spin is microscopic and does not refer to the rotation of world lines, which are straight. A strange feature is that the exterior solution is in every case locally isometric to an exterior solution for a non-spinning null fluid.     

Singular Limits in Liouville-Type Equations with Mixed Interior and Boundary Singular Sources

Motivated by the planar stationary turbulent Euler flows with the Tur-Yanovsky vortex pattern in fluid mechanics, we use the Lyapunov-Schmidt finite dimensional reduction method to prove the existence of mixed interior-boundary concentrating solutions for a class of Liouville-type equations with mixed interior and boundary singular sources.     

On matching LTB and Vaidya spacetimes through a null hypersurface

In this work the matching of a LTB interior solution representing dust matter to the Vaidya exterior solution describing null fluid through a null hypersurface is studied. Different cases in which one is able to smoothly match these two solutions to Einstein equations along a null hypesurface are discussed.     

Bifurcation phenomena in a convection problem with temperature dependent viscosity at low aspect ratio

In this paper we analyse convective solutions of a two dimensional fluid layer in which viscosity depends exponentially on temperature. This problem takes in features of mantle convection, since large viscosity variations are to be expected in the Earth’s interior. These solutions are compared with solutions obtained at constant viscosity. Special attention is paid to the influence of the aspect ratio in the solutions presented. The analysis is assisted by bifurcation techniques such as branch continuation, which has proven to be a useful, systematic method for gaining insight into the possible stationary solutions satisfied by the basic equations. One feature presented by the fluid with non constant viscosity is the presence of pitchfork and saddle-node subcritical bifurcations and the presence of convective solutions below the linear critical threshold. The analysis also provides limits of existence of stationary solutions and draws the boundaries for time dependent convection.     

Computational soliton solutions to $$$$-dimensional generalised Kadomtsev–Petviashvili and $$(2+1)$$(2+1)-dimensional Gardner–Kadomtsev–Petviashvili models and their applications

In this paper, we present a formalism to generate a family of interior solutions to the Einstein–Maxwell system of equations for a spherically symmetric relativistic charged fluid sphere matched to the exterior Reissner–Nordström space–time. By reducing the Einstein–Maxwell system to a recurrence relation with variable rational coefficients, we show that it is possible to obtain closed-form solutions for a specific range of model parameters. A large class of solutions obtained previously are shown to be contained in our general class of solutions. We also analyse the physical viability of our new class of solutions.     

Symmetry constraints in general relativity and the Schwarzchild interior metric

Einstein's equations with a perfect fluid source are subjected to compatibility conditions in the context of a space-time that contains symmetric subspaces. These conditions constitute, in some cases, a powerful tool for exhibiting the solutions to a given problem. The Schwarzchild interior metric in conformally flat coordinates is derived using these methods.     

Innere Reissner-Weyl-Lösung

With the aid of an invariance transformation of the Lagrangian we obtain a class of exact static solutions of the Einstein-Maxwell equations including perfect fluid. Application of the method to the interior and exterior Schwarzschild solution yields a corresponding solution with electromagnetic field (Reissner-Weyl solution). The boundary conditions of the resulting metric are automatically fulfilled.     

A class of solutions of Einstein's equations for the interior of a rigidly rotating perfect fluid

A general class of solutions of Einstein's equations for the interior of a rigidly rotating axisymmetric perfect fluid is presented, which depends on an arbitrary function. To get solutions explicitly one has to calculate two integrals involving the arbitrary function. The equipressure surfaces of all solutions of the class are spheres or planes. A family of solutions, which depend on four arbitrary real constants, is calculated explicitly. The solution of the family, which is obtained if we assign a specific value to one of its parameters, and which was found before, is futher generalized with the addition of one more parameter.     

理想流体球的爱因斯坦场方程内部严格解研究

当静态的具有球对称性的理想流体的密度是径向坐标的函数时，Oppenheimer-Volkoff(OV) 方程成为Riccati方程-根据OV方程的一个已知特解，能将它变换成可积分的Bernoulli方程 ，严格地求得OV方程的通解和另一特解，进一步得到理想流体球的爱因斯坦场方程的内部严 格解，即度规分量的解析表示式- 关键词： 爱因斯坦场方程 OV方程 理想流体球内部严格解     

Hamilton-Jacobi separable,axisymmetric, perfect-fluid solutions of Einstein's equations

In this paper we examine the Einstein equations with a perfect fluid source under the assumptions of (i) axial symmetry and time-independence, (ii) uniform rotation of the fluid about the symmetry axis, and (iii) separability of the Hamilton-Jacobi equation for the null geodesics of the space. These assumptions are made in an attempt to generalize the results of a similar investigation by Carter for the source-free case.We first extend Carter's results by showing that his additional assumption of separability of the wave equation is unnecessary, it being a consequence of the field equations.When the density of the fluid is non-zero, we are led to a particular solution discovered by Wahlquist, or to more symmetrical interior solutions with spherical equipressure surfaces. Except for the case of no rotation, these solutions cannot be matched to asymptotically flat exteriors.     

On fluid spheres of uniform density in general relativity

It is shown that for spherically symmetric perfect fluid solutions, with spatial isotropy and uniform density, the free gravitational fieldproduces a singularity at the centre of the sphere and when this singularity is removed the space-time is conformally flat.It is pointed out that the interior geometry of the fluid spheres with uniform density given by Thompson and Whitrow, and others, is conformally flat, and hence the spacetime is of class one.     

New stiff matter solutions to Einstein equations

New exact solutions are presented to the Einstein field equations which are spherically symmetric and static, with a perfect fluid distribution of matter satisfying the equation of state=p. One of the obtained solutions may only be used locally, the other represents the stellar interior globally and is singularity-free.     

Relativistic Charged Star Solutions in Higher Dimensions

We present a class of relativistic solutions of the Einstein-Maxwell equations for a spherically symmetric charged static fluid sphere in higher dimensions. The interior space at t=constant considered here possess (D?1) dimensional spheroidal geometry described by a higher dimensional Vaidya-Tikekar metric. A class of new static solutions of coupled Einstein-Maxwell equations is obtained in a D-dimensional space-time by prescribing the geometry of a (D?1) dimensional hyper spheroid in hydrostatic equilibrium. The solutions of the Einstein-Maxwell field equations are employed to obtain relativistic models for charged compact stars with a suitable law for variation of electric field in terms of the charged fluid content in the interior of the sphere. The central density is found to depend on the space-time dimensions and a physically realistic model is permitted for (D≥4). The validity of both Strong Energy Condition (SEC), Weak Energy Condition (WEC) are studied for a given configuration and compactness of compact objects. We found new class of solutions with interesting stellar models where it permits a star with a core having different property than the rest which however disappears in higher dimensions. The effect of dimensions on the Electric charge of the compact object is studied. We note that the upper limit of the electric field is determined by the space-time dimensions which are determined.