On the equilibrium figures of an ideal rotating fluid in the post-newtonian approximation of general relativity

A stability criterion is given for the equilibrium form of an ideal rotating fluid in the post-Newtonian approximation. This generalizes the known Lyapunov criterion in classical dynamics. The sphere stability is also investigated and it is shown that it is stable only whenR>22.2R _g (R is the relativistic sphere radius,R _g the Schwarzschild radius).     

The problem of motions up to the second post-newtonian approximation in general relativity theory

In this paper we give the Hamiltonian function for aN-body system up to the 2-P.N.A. Then as an example, from the LagrangianL _m of a test particle we derive the equations of its motion up to the 2-P.N.A. in the field of a heavy bodym ₂at rest.     

Static spherically-symmetric scalar-field theory in general relativity

A spherically-symmetric static scalar field in general relativity is considered. The field equations are defined by $$\begin{gathered} R_{ik} = - \mu \varphi _i \varphi _k ,\varphi _i = \frac{{\partial \varphi }}{{\partial x^i }}, \varphi ^i = g^{ik} \varphi _k , \hfill \\ \hfill \\ \end{gathered} $$ where ?=?(r,t) is a scalar field. In the past, the same problem was considered by Bergmann and Leipnik (1957) and Buchdahl (1959) with the assumption that ?=?(r) be independent oft and recently by Wyman (1981) with the assumption ?=?(r, t). The object of this paper is to give explicit results with a different approach and under a more general condition $$\phi _{;i}^i = ( - g)^{ - 1/2} \frac{\partial }{{\partial x^i }}\left[ {( - g)^{1/2} g^{ik} \frac{\partial }{{\partial x^k }}} \right] = - 4\pi ( -g )^{ - 1/2} \rho $$ where ?=?(r, t) is the mass or the charge density of the sources of the field.     

On rotating isothermal configurations in the post-Newtonian approximation of general relativity

The equations which govern the structure of a rotating, truncated isothermal sphere in the post-Newtonian approximation of general relativity are derived and solved numerically. Each model is parameterized by both a rotation and a relativity parameter. The density inside the configurations is tabulated and graphed as a function of both distance from the center and co-latitude. Relativistic gravitational effects are found to pull the models into states which are considerably more centrally condensed than one predicts classically. Rotation tends to flatten the isothermal configurations into oblate spheroids, though for even the largest rotation parameters the degree of flattening is only a few percent. The computed models may be similar to the cores of relativistic star clusters.     

Magnetohydrodynamic equilibrium conditions in the post-Newtonian approximation of general relativity

A set of equations, which are magnetohydrodynamic (MHD) equilibrium conditions in the post-Newtonian approximation of general relativity (PNA of GR), is obtained. The given system generalizes the previously obtained magnetohydrodynamic equilibrium conditions of classical mechanics and the hydrodynamic equilibrium conditions in the PNA of GR.     

Cosmological mass model in general relativity

Relativistic cosmological field equations are obtained for a non-static stationary Bertotti-Robinson-type space-time for interacting perfect fluid and electromagnetic field. The cosmological solution to the field equations are obtained and the nature of the electromagnetic field as well the perfect fluid are studied. The electromagnetic field generated here corresponds to a special generic case and the perfect fluid distribution degenerates into a barotropic perfect fluid with equation of statep+=0, >0. It is shown here that the interacting barotropic fluid can generate gravitation only when the cosmological constant being a function ofx in a dynamic field.     

Rapidly rotating polytropes in the post-Newtonian approximation to general relativity

The study of uniformly polytropes with axial symmetry is extended to include all rotational terms of order ⁴, where is the angular velocity, consistently within the first post-Newtonian approximation to general relativity. The equilibrium structure is determined by treating the effects of rotation and post-Newtonian gravitation as independent perturbations on the classical polytropic structure. The perturbation effects are characterized by a rotation parameter = ²/2G _c and a relativity parameter, =p _c / _c C ² , wherep _c and _c are the central pressure and density respectively. The solution to the structural problem is obtained by following Chandrasekhar's series expansion technique and is complete to the post-Newtonian rotation terms of order ². The critical rotation parameterv _c , which characterizes the configuration with maximum uniform rotation, is accurately evaluated as a function of . Numerical values for all the structural parameters needed to determine the equilibrium configurations are presented for polytropes with indicesn=1, 1.5, 2, 2 5, 3, and 3.5.     

On the equilibrium figures of an ideal rotating liquid in the post-Newtonian approximation of general relativity

Maclaurin's P-ellipsoid, which is an equilibrium figure in the post-Newtonian approximation of general relativity, is constructed in the neighbourhood of Maclaurin's classical ellipsoid. Its shape and rotation velocity are investigated. It is shown that in the case of a P-ellipsoid with the mass and the angular momentum of a basic ellipsoid the effects of general relativity reduce the rotation velocity and decrease its volume.     

A static axisymmetric anisotropic fluid solution in general relativity

Einstein's interior field equations in general relativity are considered when spacetime is static and axisymmetric and the energy-momentum tensor represents an anisotropic fluid. After imposing a set of simplifying assumptions a two-parameter solution is derived and its properties are discussed. The solution is found to be physically reasonable in a certain range of the parameters in which case the metric could represent a core of anisotropic matter.     

The periastron advance in the second post-Newtonian approximation in general relativity

In this paper we present a second order post-Newtonian approximation to the Hamiltonian of theN-body system. Subsequently we improve the well-known Robertson's formula for the perihelion advancement by a correction term of orderc, wherec ^–4 is the velocity of light.     

Some static relativistic compact charged fluid spheres in general relativity

In this work a new family of relativistic models of electrically charged compact star has been obtained by solving Einstein–Maxwell field equations with preferred form of one of the metric potentials and a suitable form of electric charge distribution function. The resulting equation of state (EOS) has been calculated. The relativistic stellar structure for matter distribution obtained in this work may reasonably models an electrically charged compact star whose energy density associated with the electric fields is on the same order of magnitude as the energy density of fluid matter itself (e.g. electrically charged bare strange stars). Based on the analytic model developed in the present work, the values of the relevant physical quantities have been calculated by assuming the estimated masses and radii of some well known strange star candidates like X-ray pulsar Her X-1, millisecond X-ray pulsar SAX J 1808.4-3658, and 4U 1820-30.     

Rapidly rotating supermassive stars in the first post-Newtonian approximation to general relativity

The structure and stability of rapidly uniformly rotating supermassive stars is investigated using the full post-Newtonian equations of hydrodynamics. The standard model of a supermassive star, a polytrope of index three, is adopted. All rotation terms up to and including those of order ⁴, where is the angular velocity, are retained. The effects of rotation and post-Newtonian gravitation on the classical configuration are explicitly evaluated and shown to be very small. The dynamical stability of the model is treated by using the binding energy approach. The most massive objects are found to be dynamically unstable when =1/c ².p _c / _c 2.2 × 10^–3, wherep _c and _c are the central pressure and density, respectively. Hence, the higher-order terms considered in this analysis do not appreciably alter the previously known stability limits.The maximum mass that can be stabilized by uniform rotation in the hydrogen-burning phase is found to be 2.9×10⁶ M , whereM is the solar mass. The corresponding nuclear-generated luminosity of 6×10⁴⁴ erg/sec^–1 is too small for the model to be applicable to the quasi-stellar objects. The maximum kinetic energy of a uniformly rotating supermassive star is found to be 3×10^–5 Mc ², whereM is the mass of the star. Masses in excess of 10¹⁰ M are required if an adequate store of kinetic energy is to be made available to a pulsar like QSO. However such large masses have rotation periods in excess of 100 yr and thus could not account for any short term periodic variability. It is concluded then that the uniformly rotating supermassive star does not provide a suitable base for a model of a QSO.     

Generalised eddington approximation for radiative transfer problems in spherically-symmetric moving media

A moment method with three stream division of the radiation field was suggested by Wilson, Wan and Sen (1980) for solving radiative transfer problems in stationary, non-grey extended shells surrounding a central star. Use was made of the generalised Eddington relations as the closure conditions of the moment equations. In the present paper the same method has been utilised to study the radiative transfer problems in a non-grey, expanding gaseous spherical shells surrounding a central star. The transfer equation has been set in comoving frame in spherical geometry. The radiation and material quantities, angles and frequencies have been expressed in comoving frame. The mean intensity, flux and K-integrals have been calculated for extensive atmospheres in the presence of different velocity fields.     

Coordinate gauge conditions in theN-body equations of motion in the first-order post-Newtonian approximation of general relativity

The explicit forms of the metric as well as the equations of motion in the first-order post-Newtonian approximation are worked out under several gauge conditions. It is noted that the so-called EIH (Einstein, Infeld, and Hoffman) equation of motion for an assembly ofN finite mass points mutually interacting via gravitation is identically obtained under three different gauge conditions, namely the harmonic gauge, Chandrasekhar gauge and a composite Chandrasekhar gauge used by Misneret al. (1970), even though the solutions for the metric are found to be all different. In one case the metric has a component apparently diverging, but finally generates regular affine connections so that the equations of motions become free from any singularity. By use of the Chandrasekhar gauge and his formulation, the second-order contribution to the acceleration of planets in the limit of test particle motion around the Sun has been calculated, the inclusion of which in the EIH set of the equations of motion would extend the relative accuracy of computing the total acceleration of any planet to better than one part in 10¹⁷.     

On slowly-rotating cosmological viscous-fluid model universes in general relativity

The dynamics of a slowly-rotating cosmological viscous-fluid universe is investigated and the rotational perturbations of such models are studied in order to substantiate the possibility that the Universe is endowed with slow rotation, in the course of presentation of some new analytic solutions. Three different cases are taken up in which the nature and role of the metric rotation (r, t) as well as that of the matter rotation (r, t) are discussed. The periods of physical validity of some of the models and the effect of viscosity on the rotational motion are also found out. Rotating models which are expanding as well are obtained, where in all the cases the rotational velocities are found to decay with the time; and these models may be taken as good examples of real astrophysical situations.     

Soliton solutions to the Einstein gravitational field equations in the presence of a spherically-symmetric static background field

Using a differential geometry approach, we have found two sets of new solutions to Einstein's equation of gravity in the presence of a spherically-symmetrical gravitational background, like the Earth. The transverse and longitudinal components of the metric tensor representing the gravity waves are all soliton solutions, propagating towards the origin of the Earth. If we consider the situation where the static background field is absent, the solutions still remain soliton-like in nature. The difference between our result and Einstein's is attributed to the two approximations taken previously — weak field and harmonic condition.On study leave of absence from Zhong Shan University, China     

On the solution space of differentially rotating neutron stars in general relativity

A highly accurate, multidomain spectral code is used in order to construct sequences of general relativistic, differentially rotating neutron stars in axisymmetry and stationarity. For bodies with a spheroidal topology and a homogeneous or an   N = 1  polytropic equation of state, we investigate the solution space corresponding to broad ranges of degree of differential rotation and stellar densities. In particular, starting from static and spherical configurations, we analyse the changes of the corresponding surface shapes as the rate of rotation is increased. For a sufficiently weak degree of differential rotation, the sequences terminate at a mass-shedding limit, while for moderate and strong rates of differential rotation they exhibit a continuous parametric transition to a regime of toroidal fluid bodies. In this article, we concentrate on the appearance of this transition, analyse in detail its occurrence and show its relevance for the calculation of astrophysical sequences. Moreover, we find that the solution space contains various types of spheroidal configurations, which were not considered in previous work, mainly due to numerical limitations.