A New Solution to Einstein's Field Equations

We construct a new exact solution to the vacuum Einstein field equations. This solution possesses a naked physical singularity. The norm of the Riemann curvature tensor of the solution takes infinity at some points and the solution does not have any event horizon around the singularity. A detailed analysis of this new singularity is also presented.     

Power Asymptotes of Homogeneous Cosmological Models in Einstein's Theory of Gravitation

Completing the discussion of an earlier treatment (Borzeszkowski, Müller (1978)) homogeneous cosmological models of Bianchi-types II, III, IV, VI and the models of Kantowski-Sachs-type are investigated. All asymptotes of the metric which have a power like dependence on the synchronized time near the cosmological singularity are listed. We assume an ideal fluid as source with the flow velocity orthogonal to the slices of homogeneity, and an equation of state π = β? 0 ? β ? 1. We utilize the fact that certain Bianchi types can be obtained by a limiting process from other ones. Special anisotropic matter-dominated solutions have been found besides the well-known Kasner asymptote.     

Power Asymptotes of Homogeneous Cosmological Models in Einstein's Theory of Gravitation

For the Bianchi-types I, V, VII, VIII, and IX, the asymptotes of the metric, which show a power-like dependence on the synchronized time t near the cosmological singularity, are investigated. Comparing the influence of the terms which in Einstein's equations describe time-derivatives, spatial curvature, and matter, on the behaviour of the asymptotes, different assumptions and conjectures usually used in literature are discussed for the equations of state p = β? (0 ? β ? 1). For Bianchi-type-IX models both the outgoing power-asymptotes and the oscillatory phase for t → 0 are considered.     

On a Generalization of Hilbert's Metrical Matter Tensor in Einstein's Field Equations

We construct a new source tensor T?_ik in Einstein's equations of gravitation. This matter tensor is a generalization of Hilbert's “metrical energy-momentum tensor” T_ik.     

Exact Solutions of Einstein's Field Equations

We examine various well known exact solutions available in the literature to investigate the recent criterion obtained in Negi and Durgapal [Gravitation and Cosmology 7, 37 (2001)] which should be fulfilled by any static and spherically symmetric solution in the state of hydrostatic equilibrium. It is seen that this criterion is fulfilled only by (i) the regular solutions having a vanishing surface density together with pressure, and (ii) the singular solutions corresponding to a non-vanishing density at the surface of the configuration. On the other hand, the regular solutions corresponding to a non-vanishing surface density do not fulfill this criterion. Based upon this investigation, we point out that the exterior Schwarzschild solution itself provides necessary conditions for the types of the density distributions to be considered inside the mass, in order to obtain exact solutions or equations of state compatible with the state of hydrostatic equilibrium in general relativity. The regular solutions with finite centre and non-zero surface densities which do not fulfill the criterion given by Negi and Durgapal (2001), in fact, cannot meet the requirement of the‘actual mass’, set up by exterior Schwarzschild solution. The only regular solution which could be possible in this regard is represented by uniform (homogeneous) density distribution. This criterion provides a necessary and sufficient condition for any static and spherical configuration (including core-envelope models) to be compatible with the structure of general relativity [that is, the state of hydrostatic equilibrium in general relativity]. Thus, it may find application to construct the appropriate core-envelope models of stellar objects like neutron stars and may be used to test various equations of state for dense nuclear matter and the models of relativistic star clusters with arbitrary large central redshifts. PACS :04.20.Jd; 04.40.Dg; 97.60.Jd.     

On a Stationary Cosmology in the Sense of Einstein's Theory of Gravitation

A world is to be considered stationary in the sense of general relativity if the coefficients of its metric are independent of time in a coordinate system in which the masses are at rest on average. The remark on the system of coordinates is important because time itself is no invariant notion but is taken only in the sense of proper time. Our definition is unique, in the form given above. On the other hand it is also possible to have points where no matter is present. At such points we may place a test body of infinitesimally small mass and analyse whether it remains at rest in our coordinate system. A necessary and sufficient condition for this is that the time lines of our coordinate system are geodesics. Therefore the static solution given by de Sitter is not an example of a stationary world. The Schwarzschild line element which, from a cosmological point of view, is a world with a single central body can also not be considered a stationary solution. Indeed, there are no stationary solutions which are also spherically symmetric for the original field equations. The only such solution for the cosmological equations is Einstein's cylinder world. It is, to my knowledge, the only stationary world known so far. In that case the average matter density and the total mass of the world has to have a well defined value given by the cosmological constant which doubtless would be purely coincidental and is thus not a satisfactory assumption. In the following we shall discuss a new solution which is in accord with the original field equations without the need of an a priori relation between mass and cosmological constant. However, we shall find that its mass cannot be less than the mass of the cylinder world.     

Bianchi type-II String Cosmological Model with Magnetic Field in Scale-Covariant Theory of Gravitation

The spatially homogeneous and totally anisotropic Bianchi type-II cosmological solutions of massive strings have been investigated in the presence of the magnetic field in the framework of scale-covariant theory of gravitation formulated by Canuto et al. (Phys. Rev. Lett. 39, 429, 1977). With the help of special law of variation for Hubble^’s parameter proposed by Berman (Nuovo Cimento 74, 182, 1983) string cosmological model is obtained in this theory. We use the power law relation between scalar field ? and scale factor R to find the solutions. Some physical and kinematical properties of the model are also discussed.     

Cosmological Solutions of Bi-metric Theories of Gravitation

The evidence in solar system tests of gravitation theories shows that in cosmological models the propagation velocities of gravitation and light may differ only by action of the local gravitational potential. The condition of approximately equal propagation velocities is shown to be consistent with cosmological models of some bi-metric tetrad theories. The way to analyze the situation is explained. We find a connection between the linear approximation of the theories with those properties of cosmological models, which are responsible for the local propagation of gravitation.     

Exact Plane-Wave Solution of the Equations of a Theory of Gravitation with a Massive Graviton

Russian Physics Journal - A theory of gravitation with a massive graviton which was proposed by Visser is considered. An exact solution of this theory is found for the case when the source of the...     

A Solution to Einstein's Equations for the Mixmaster Universe in Complex General Relativity

Starting from Einstein's equations of the Classical General Relativity, new kinds of solutions for the Mixmaster model are explored. By dispensing with the extension to the complex variable field, which is usual in problems such as the Laplace equation or the harmonic oscillator, in a similar manner to that of Quantum Mechanics, the equations appear to have solutions that belong to the complex General Relativity. A first integral is performed by establishing a separation of the first derivatives. Then a second integral is obtained once the respective equations with separate variables are found and whose integrals provide a family of complex solutions. However, reality conditions do not seem to be easily imposed at this stage. Above all, it is significant that the classical Einstein's equations for the debatably integrable Mixmaster model present complex solutions.     

On Some Exponential Potentials for a Cosmological Scalar Field as Quintessence

We present general exact solutions for two classes of exponential potentials in a scalar field model for quintessence. The coupling is minimal and we consider only dust and scalar field. To some extent, it is possible to reproduce experimental results from supernovae.