A conformal mapping and isothermal perfect fluid model

Instead of the metric conformal to flat spacetime, we take the metric conformal to a spacetime which can be thought of as minimally curved in the sense that free particles experience no gravitational force yet it has non-zero curvature. The base spacetime can be written in the Kerr-Schild form in spherical polar coordinates. The conformal metric then admits the unique three-parameter family of perfect fluid solutions which are static and inhomogeneous. The density and pressure fall off in the curvature radial coordinates asR ^–2, for unbounded cosmological model with a barotropic equation of state. This is the characteristic of an isothermal fluid. We thus have an ansatz for an isothermal perfect fluid model. The solution can also represent bounded fluid spheres.     

Sources for the Reissner-Nordström metric

New, physically motivated static sources for the Reissner-Nordström metric are found. One is a generalization of the Schwarzschild interior solution representing a sphere of constant nongravitational energy density. The other is a family of solutions for which the mass is electromagnetic in origin. Some general results are found. For a charged fluid sphere in equilibrium with pressure,m ² >q ². For a charged body with equation of state = (p), where (0)=0, the body is under tension at every point when the charge density has the same sign throughout.Supported by the National Research Council of Canada, Grant No. A5340.     

Relativistic quarks bound with a one-body tensor potential

A dipole fit to electromagnetic form factors is used to determine a quark density in the nucleon. A radial tensor potential is used to bind the quarks into states of goodJ, J _z, and parity. The tensor potential radial component is taken to satisfy the equationT = T ₀, whereT ₀ is a parameter of the model. This linear divergence equation can be simultaneously solved with the Dirac equation for the bound quark wave functions. A self-consistent solution is possible where the mass density used as the source for the binding potential is the same as that determined from the solution for the quark wave functions.     

Einstein-Proca Model, Micro Black Holes, and Naked Singularities

The Einstein-Proca equations, describing a spin-1 massive vector field in general relativity, are studied in the static spherically-symmetric case. The Proca field equation is a highly nonlinear wave equation, but can be solved to good accuracy in perturbation theory, which should be very accurate for a wide range of mass scales. The resulting first order metric reduces to the Reissner-Nordström solution in the limit as the range parameter goes to zero. The additional terms in the g ₀₀ metric coefficient are positive, as in Reissner-Nordström, in agreement with previous numerical solutions, and hence involve naked singularities.     

Static perfect fluid in Brans-Dicke theory

The static perfect fluid in Brans-Dicke theory with spherical symmetry and conformal flatness leads to a differential equation in terms of the scalar field only. We obtain a unique exact solution for the casep=, but density and pressure are singular at the center. We further consider the metric corresponding to a static nonrotating space-time with two mutually orthogonal spacelike Killing vectors in Brans-Dicke theory. We obtain a differential equation involving only the scalar field for the equation of statep= The general solution is found as a transcendental function. Finally, we generalize a theorem given by Bronnikov and Kovalchuk (1979) for perfect fluid in Einstein's theory.On leave from Jadavpur University, Calcutta-32, India.     

Relations between the wave solutions and the jost solutions off the energy shell for local central potentials

Using the Green function techniques we express the wave solutions of the radial inhomogeneous Schrödinger equation by means of the on-shell Jost and regular solutions. Making use of their boundary behaviour atr = andr = 0 we reexpress them alternatively in terms of the off-shell Jost and regular solutions. Relations among the different generalized (fully off the energy shell) Jost functions are derived and the radial matrix elements of the transition and reaction (reactance) operators are given in terms of these Jost functions. The relations reflect the principle of detailed balance.     

A point-particle model universe

The observable cosmos is modeled as a set of point-particles, representing the galaxies, which perturb a dust-filled, Robertson-Walker space-time. The analysis proceeds only to first order in=8G/c ² and employs a metric suggested by McVittie [General Relativity and Cosmology (Chapman and Hall, London, 1965)], whose original work this paper seeks to develop. Necessary and sufficient conditions are found for the metric to give rise to an energy tensor of a chosen form appropriate to the modeling. In particular, a second-order equation is found which governs a certain time-independent potential. A class of solutions to this equation is established, and the associated singularities of the mass density are shown to be of a Dirac type.     

Cosmological models in a Kaluza-Klein theory with variable rest mass

A general class of solutions is obtained for a homogeneous, spatially isotropic five-dimensional (5D) Kaluza-Klein theory with variable rest mass. These solutions generalize in the algebraic and physical sense the previously found solutions in the literature. The 4D spacetime sections of the solutions reduce to the Minkowski metric, K=0 Robertson-Walker metric with the equation of statep=np (p=pressure,n=constant sound speed,=energy density), and to the Robertson-Walker spacetime with steady-state metric. Some of the solutions, in different limits, show compactification of the fifth dimension. Some extensions of the model are discussed.     

Static circularly symmetric perfect fluid solutions with an exterior BTZ metric

In this work we study static perfect fluid stars in 2+1 dimensions with an exterior BTZ spacetime. We found the general expression for the metric coefficients as a function of the density and pressure of the fluid. We found the conditions to have regularity at the origin throughout the analysis of a set of linearly independent invariants. We also obtain an exact solution of the Einstein equations, with the corresponding equation of state p = p(), which is regular at the origin.     

Solutions for static gas spheres in general relativity

A method for constructing solutions for a relativistic static gaseous sphere is examined. Gaseous here means that the density vanishes at the outer boundary together with the pressurep. Two different classes of solutions are investigated in detail. The models of both these classes have the property that the density gradient is zero both at the center and at the surface. It is further shown that both classes yield models which are physically acceptable, i.e. both the pressure and the density are positive and finite inside the outer boundary of the sphere, and their respective gradients are negative. The trace of their energy-momentum tensors are positive, and the adiabatic sound speeds are decreasing outwards throughout the sphere. The relativistic adiabatic indices are examined, and it is found that they are decreasing functions of radial coordinate. It is shown that for the first class this index is 3/2 at the surface, while for the second class it is 4/3 at the boundary. We find that the models of the first class arestable with respect to small radial disturbances. Putting the density at the center equal to 10¹⁶g cm^–3, the maximum mass for the stable class is found to be 0.87 solar masses.     

Kerr-Schild-Vaidya fields with axial symmetry

This paper contains the Kerr-Schild-Vaidya fields with axial symmetry (all metric functions independent of Vaidya's coordinate) in closed form. The general problem of Kerr-Schild pure radiation fields without any symmetry can be reduced to a single partial differential equation by means of Kerr's theorem.     

Quantum mechanics as relativistic statistics III. A relativistic particle in two-dimensional space-time

Statistics of random world lines in a fixed electromagnetic field are considered. The equation for a vectorj ⁱ is obtained. This vector describes the density of random world lines in a pure ensemble. It is shown that in the two-dimensional space-time this equation coincides with the Dirac equation to within the terms of the order of magnitude of (/L)² ( is Compton's wavelength,L is a typical length of the system).     

Radiating gravitational collapse with shear viscosity revisited

A new model is proposed to a collapsing radiating star consisting of an isotropic fluid with shear viscosity undergoing radial heat flow with outgoing radiation. In a previous paper we have introduced a function time dependent into the g _rr , besides the time dependent metric functions and . The aim of this work is to generalize this previous model by introducing shear viscosity and compare it to the non-viscous collapse. The behavior of the density, pressure, mass, luminosity and the effective adiabatic index is analyzed. Our work is compared to the case of a collapsing shearing fluid of a previous model, for a star with 6 . The pressure of the star, at the beginning of the collapse, is isotropic but due to the presence of the shear the pressure becomes more and more anisotropic. The black hole is never formed because the apparent horizon formation condition is never satisfied. An observer at infinity sees a radial point source radiating exponentially until reaches the time of maximum luminosity and suddenly the star turns off. The effective adiabatic index has a very unusual behavior because we have a non-adiabatic regime in the fluid due to the heat flow.     

Die Zurückführung der Gleichungen eines rotationssymetrischen,statischen, inkompressiblen Plasmas von unendlicher Leitfähigkeit auf ein Randwertproblem

The equations of a rotationally symmetric, static, incompressible plasma with infinite conductivity are equivalent to an elliptic differential equation for the function where p means the pressure, ? the density, and Φ the potential of the external forces. Moreover this differential equality contains two arbitrary functions of ξ. When ξ_?² + ξ_?² < 0, both arbitrary functions can be computed from the boundary values of Hφ und H⊥ (the component of H , which is perpendicular to the boundary).     

Adiabatic Models of the Cosmological Radiative Era

We consider a generalization of the Lemaitre-Tolman-Bondi (LTB) solutions by keeping the LTB metric but replacing its dust matter source by an imperfect fluid with anisotropic pressure _ab . Assuming that total matter-energy density is the sum of a rest mass term, ^(m), plus a radiation ^(r) = 3p density where p is the isotropic pressure, Einstein's equations are fully integrated without having to place any previous assumption on the form of _ab . Three particular cases of interest are contained: the usual LTB dust solutions (the dust limit), a class of FLRW cosmologies (the homogeneous limit) and of the Vaydia solution (the vacuum limit). Initial conditions are provided in terms of suitable averages and contrast functions of the initial densities of ^(m), ^(r) and the 3-dimensional Ricci scalar along an arbitrary initial surface t = t _i . We consider the source of the models as an interactive radiation-matter mixture in local thermal equilibrium that must be consistent with causal Extended Irreversible Thermodynamics (hence _ab is shear viscosity). Assuming near equilibrium conditions associated with small initial density and curvature contrasts, the evolution of the models is qualitatively similar to that of adiabatic perturbations on a matter plus radiation FLRW background. We show that initial conditions exist that lead to thermodynamically consistent models, but only for the full transport equation of Extended Irreversible Thermodynamics. These interactive mixtures provide a reasonable approximation to a dissipative tight coupling characteristic of radiation-matter mixtures in the radiative pre-decoupling era.     

Geometric Conditions on the Type of Matter Determining the Flat Behavior of the Rotational Curves in Galaxies

In an arbitrary axisymmetric stationary spacetime, we determine the expression for the tangential velocity of test objects following a circular stable geodesic motion in the equatorial plane, as function of the metric coefficients. Next, we impose the condition, observed in large samples of disks galaxies, that the magnitude of such tangential velocity be radii independent in the dark matter dominated region, obtaining a constraint equation among the metric coefficients, and thus arriving to an iff (iff means: if and only if.) condition: The tangential velocity of test particles is radii independent iff the metric coefficients satisfied the mentioned constraint equation. Furthermore, for the static case, the constraint equation can be easily integrated, leaving the spacetime at the equatorial plane essentially with only one independent metric coefficient. With the geometry thus fixed, we compute the Einstein tensor and equate it to an arbitrary stress energy tensor, in order to determine the type of energy-matter which could produce such a geometry. Within an approximation, we deduce a constraint equation among the components of the stress energy tensor. We test in that constraint equation several well known types of matter, which have been proposed as dark matter candidates and are able to point for possible right ones. Finally, we also present the spherically symmetric static case and apply the mentioned procedure to perfect fluid stress energy tensor, recovering the Newtonian result as well as the one obtained in the axisymmetric case. We also present arguments on the need to use GR to study types of matter different than the dust one.     

Equation of state of condensed media

A new choice is proposed for the generating functional which is used to obtain an integral equation for the radial distribution function that is valid in the domain of dense fluids. The corresponding equation of state for giving the intramolecular potential in the form (r)r ^–s agrees in the case of dense fluids with the known Tait empirical equation of state. The Tait equation is extended to the high-pressure case. Processing the experimental data for water exhibits good agreement between the equation of state obtained and experiment in the pressure range from 10⁵–10⁹ Pa.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 12, pp. 43–47, December, 1981.     

Field of an ideal irrotational fluid in general relativity

General-relativistic covariant relations are found for the pressure and density of a fluid in the gravitational fields of an ideal fluid with no rotation of the fluid particles. It is shown that the equation of state of the fluid =(P) imposes specific restrictions on the metric. The result is used to analyze known solutions of spherically symmetric fields of an ideal fluid [2, 3]. It is shown that these solutions are physically unrealistic.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 3, pp. 112–115, March, 1995.     

Energy and Angular Momentum ofthe Weak Gravitational Waves on the Schwarzschild Background — Quasilocal Gauge-invariant Formulation

A four-dimensional spherically covariantgauge-invariant quasilocal framework for theperturbation of the Schwarzschild metric is given. Animportant ingredient of the analysis is the concept ofquasilocality, which does duty for the separation ofangular variables in the usual approach. A precise andfull analysis for the mono-dipole part ofthe theory is presented. Direct construction (from theconstraints) of the reduced canonical structure for theinitial data and explicit formulae for thegaugeinvariants are proposed. The reduced symplecticstructure explains the origin of the axial and polarinvariants. This enables one to introduce an energy andangular momentum for the gravitational waves, which isinvariant with respect to the gauge transformations. Anexplicit expression for the energy and new proposition for angular momentum is introduced, inparticular, compatibility of theChristodoulou-Klainerman S.A.F. condition withwell-possedness of our functionals is checked. Bothgenerators (energy and angular momentum) represent quadratic approximation ofthe adm nonlinear formulae in terms of the perturbationsof the Schwarzschild metric. The previously knownresults are presented in a new geometric andself-consistent way. Both degrees of freedom fulfill thegeneralized scalar wave equation. For the axial degreeof freedom the radial part of the equation correspondsto the Regge-Wheeler result and for the polar one we get the Zerilli result.     

Quantization of closed mini-superspace models as bound states

The Wheeler-DeWitt equation is applied to closedk>0 Friedmann-Robertson-Walker metric with various combination of cosmological constant and matter (e.g., radiation or pressureless gas). It is shown that if the universe ends in the matter dominated era (e.g., radiation or pressureless gas) with zero cosmological constant, then the resulting Wheeler-DeWitt equation describes a bound state problem. As solutions of a nondegenerate bound state system, the eigen-wave functions are real (Hartle-Hawking). Furthermore, as a bound state problem, there exists a quantization condition that relates the curvature of the three space with the various energy densities of the universe. If we assume that our universe is closed, then the quantum number of our universe isN(Gk)^–110¹²². The largeness of this quantum number is naturally explained by an early inflationary phase which resulted in a flat universe we observe today. It is also shown that if there is a cosmological constant >0 in our universe that persists for all time, then the resulting Wheeler-DeWitt equation describes a non-bound state system, regardless of the magnitude of the cosmological constant. As a consequence, the wave functions are in general complex (Vilenkin).