Investigation of singularities in general relativity

The problem of singularities for a non-symmetric and isentropic motion of a perfect fluid under the assumption of adiabatic thermodynamic processes is investigated from the standpoint of a local observer. It is shown that, whatever the distribution of matter might be, there occurred a singularity in the past in the non-rotating parts of the universe and it must occur again in the future if the universe is closed. It is further shown that the occurrence of a singularity in a rotating fluid seems inevitable, when the relativistic equation of state is considered, because of extremely high concentration of rest mass, though the possibility of its avoidance may not be ignored.     

Investigation of the problem of singularities in general relativity

Recently Addy and Datta have obtained a linearized solution for isentropic motions of a perfect fluid by assigning Cauchy data on the hypersurfacex ⁴=0 and by imposing a restriction on the equation of state. In the present paper we pursue this study and discuss the problem of singularities from the standpoint of a local observer for which a singularity is defined as a state with an infinite proper rest mass density. It is shown that for a closed universe with any distribution of matter whatsoever there occurred a singularity in the past in the nonrotating parts of the universe and it must recur in the future. Furthermore, the collapse of a rotating fluid to a singularity seems inevitable when the relativistic equation of state is considered.     

Neutrino in matter and external fields

A technique for describing various processes proceeding in matter and involving neutrinos and electrons is discussed. This technique is based on “the method of exact solutions,” which implies the use of solutions to proper Dirac equations for particle wave functions in matter. Exact solutions for the neutrino and the electron in the cases of uniform nonmoving and rotating matter are discussed. On studying relativistic neutrino motion and associated neutrino-energy quantization in rotating matter, a semiclassical interpretation of particle finite motion is developed. In the general case of neutrino and electron motion in matter with varying parameters, the corresponding effective force acting on the particles is determined. The possibility of electromagnetic-wave radiation by an electron that moves in a dense neutrino flux of varying density and which is accelerated by this kind of force is predicted.     

Horizonless,singularity-free,compact shells satisfying NEC

Gravitational collapse singularities are undesirable, yet inevitable to a large extent in General Relativity. When matter satisfying null energy condition (NEC) collapses to the extent a closed trapped surface is formed, a singularity is inevitable according to Penrose’s singularity theorem. Since positive mass vacuum solutions are generally black holes with trapped surfaces inside the event horizon, matter cannot collapse to an arbitrarily small size without generating a singularity. However, in modified theories of gravity where positive mass vacuum solutions are naked singularities with no trapped surfaces, it is reasonable to expect that matter can collapse to an arbitrarily small size without generating a singularity. Here we examine this possibility in the context of a modified theory of gravity with torsion in an extra dimension. We study singularity-free static shell solutions to evaluate the validity of NEC on the shell. We find that with sufficiently high pressure, matter can be collapsed to arbitrarily small size without violating NEC and without producing a singularity.     

Cyclic integrals and reduction of rotational relativistic Birkhoffian system

The order reduction method of the rotational relativistic Birkhoffian is studied.For a rotational relativistic Birkhoffian system.the cyclic integrals can be found by using the perfect differential method.Through these cyclic integrals,the order of the system can be reduced.If the rotational relativistic Birkhoffian system has a cyclic integral,then the Birkhoffian equations can be reduced at least two degrees and the Birkhoffian form can be kept.An example is given to illustrate the application of the results.     

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.     

A family of Jordan-Brans-Dicke Kerr solutions

A family of solutions of the vacuum Jordan-Brans-Dicke or scalar-tensor gravitational field equations is given. This family reduces to the Kerr rotating solution of the vacuum Einstein equations when the scalar field is constant. The family does not have spherical symmetry when the rotation is zero and the scalar field is not constant. The method used to generate the new solutions can also be used to obtain vacuum Jordan-Brans-Dicke solutions from any given vacuum stationary, axisymmetric solution.     

An approximate global solution of Einstein’s equations for a differentially rotating compact body

We obtain an approximate global stationary and axisymmetric solution of Einstein’s equations which can be thought of as a simple star model: a self-gravitating perfect fluid ball with a differential rotation motion pattern. Using the post-Minkowskian formalism (weak-field approximation) and considering rotation as a perturbation (slow-rotation approximation), we find approximate interior and exterior (asymptotically flat) solutions to this problem in harmonic coordinates. Interior and exterior solutions are matched, in the sense described by Lichnerowicz, on the surface of zero pressure, to obtain a global solution. The resulting metric depends on four arbitrary constants: mass density; rotational velocity at \(r=0\); a parameter that accounts for the change in rotational velocity through the star; and the star radius in the non-rotation limit. The mass, angular momentum, quadrupole moment and other constants of the exterior metric are determined in terms of these four parameters.     

The Initial Singularity of Ultrastiff Perfect Fluid Spacetimes Without Symmetries

We consider the Einstein equations coupled to an ultrastiff perfect fluid and prove the existence of a family of solutions with an initial singularity whose structure is that of explicit isotropic models. This family of solutions is ‘generic’ in the sense that it depends on as many free functions as a general solution, i.e., without imposing any symmetry assumptions, of the Einstein-Euler equations. The method we use is a that of a Fuchsian reduction.     

Gravitational Collapse of Self-Similar Perfect Fluid in 2 + 1 Gravity

Perfect fluid with kinematic self-similarity is studied in 2+1 dimensional spacetimes with circular symmetry, and various exact solutions to the Einstein field equations are given. These include all the solutions of dust and stiff perfect fluid with self-similarity of the first kind, and all the solutions of perfect fluid with a linear equation of state and self-similarity of the zeroth and second kinds. It is found that some of these solutions represent gravitational collapse, and the final state of the collapse can be either a black hole or a null singularity. It is also shown that one solution can have two different kinds of kinematic self-similarity.     

A Geometrical Approach to Hojman Theorem of a Rotational Relativistic Birkhoffian System

A geometrical approach to the Hojman theorem of a rotational relativistic Birkhoffian system is presented.The differential equations of motion of the system are established. According to the invariance of differential equations under infinitesimal transformation, the determining equations of Lie symmetry are constructed. A new conservation law of the system, called Hojman theorem, is obtained, which is the generalization of previous results given sequentially by Hojman, Zhang, and Luo et al. In terms of the theory of modern differential geometry a proof of the theorem is given.     

A Geometrical Approach to Hojman Theorem of a Rotational Relativistic Birkhoffian System

A geometrical approach to the Hojman theorem of a rotational relativistic Birkhoffian system is presented.The differential equations of motion of the system are established. According to the invariance of differential equations under infinitesimal transformation, the determining equations of Lie symmetry are constructed. A new conservation law of the system, called Hojman theorem, is obtained, which is the generalization of previous results given sequentially by Hojman, Zhang, and Luo et al. In terms of the theory of modern differential geometry a proof of the theorem is given.     

Einige Lösungen der Einsteinschen Feldgleichungen mit idealer Flüssigkeit,die sich in einen fünfdimensionalen flachen Raum einbetten lassen

All solutions of the Einstein equations for a perfect fluid are given, which are invariantly characterized by: embedding class one, Petrov typeD, zero acceleration of matter. Among these solutions are inhomogeneous cosmological models and special solutions with spherical symmetry.     

Spherical Gravitational Collapse: Tangential Pressure and Related Equations of State

We derive an equation for the acceleration of a fluid element in the spherical gravitational collapse of a bounded compact object made up of an imperfect fluid. We show that non-singular as well as singular solutions arise in the collapse of a fluid initially at rest and having only a tangential pressure. We obtain an exact solution of the Einstein equations, in the form of an infinite series, for collapse under tangential pressure with a linear equation of state. We show that if a singularity forms in the tangential pressure model, the conditions for the singularity to be naked are exactly the same as in the model of dust collapse.     

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.     

A class of algebraically special perfect fluid space-times

Solutions of the Einstein field equations are considered subject to the assumptions that (1) the source of the gravitational field is a perfect fluid, (2) the Weyl tensor is algebraically special, (3) the corresponding repeated principal null congruence is geodesic and shearfree. If in addition, the repeated principal null congruence is non-expanding, it follows that the twist of this congruence must be non-zero (for a physically reasonable fluid). The general line element subject to this additional restriction is derived. Furthermore, it is shown that all solutions of the Einstein field equations which satisfy (1) and exhibit local rotational symmetry, necessarily satisfy (2) and (3).This work was supported in part by the National Research Council of Canada.     

Null geodesies,caustics and apparent motion of galaxies in a finite rotating universe

Null solutions of the geodesic equation are presented for a universe which can be regarded as a rotating and shearing generalization of the static Einstein universe (the Ozsváth class I model). It is shown how the closest caustic, which in the static case just consists of one point at the antipode, grows to form two interwoven closed surfaces when motion is introduced. It is further shown how they prevent causality violating null-like curves. The observed transversal motion of matter (the global rotation) is calculated. The conclusions concerning the causality of the universe which an observer might draw from the global rotation are discussed.     

The analysis of the flow of a micropolar fluid between two orthogonally moving porous disks with counter rotating directions

The present work investigates the unsteady, imcompressible flow of a micropolar fluid between two orthogonally moving porous coaxial disks. The lower and upper disks are rotating with the same angular speed in counter directions. The flows are driven by the contraction and the rotation of the disks. An extension of the Von Kármán type similarity transformation is proposed and is applied to reduce the governing partial differential equations (PDEs) to a set of non-linear coupled ordinary differential equations (ODEs) in dimensionless form. These differential equations with appropriate boundary conditions are responsible for the flow behavior between large but finite coaxial rotating disks. The analytical solutions are obtained by employing the homotopy analysis method. The effects of some various physical parameters like the expansion ratio, the rotational Reynolds number, the permeability Reynolds number, and micropolar parameters on the velocity fields are observed in graphs and discussed in detail.     

A Non-geometric Relativistic Theory of Gravitation

A non-geometric relativistic theory of gravitation is developed by defining a semi-metric to replace the metric tensor as gravitational vector potential. The theory show that the energy-momentum tensor of the gravitational field belong to the gravitational source, gravitational radiation is contained in Einstein’s field equations that including the contribution of gravitational field, the real physical singularity in the gravitational field can be eliminated, and the dark matter in the universe is interpreted as the matter of pure gravitational field.     

Unified regularization of Bianchi cosmology

The Einstein equations for a perfect fluid filled spatially homogeneous space-time are expressed in a reduced form which is as close as possible to a compactified regularized first order system of differential equations, while still respecting both the scale invariance and spatial gauge symmetry of those equations. The present work is a generalization to all Bianchi types of the regularization procedure introduced by Rosquist for Bianchi types III and VI. Jantzen's unified Lie algebra automorphism formalism is used to reduce the gravitational phase space to a subspace parametrized by a minimal number of variables. Although motivated by the qualitative theory of differential equations, the reduced spatially homogeneous Einstein system has also proved useful in the search for new exact solutions.