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.     

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.     

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.     

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.     

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.     

A study of massive classical polytropes in general relativity

A detailed study of classical polytropes in general relativity has been presented for _O ((dP/dE)_O) 1.0 and _O((P/E _O)_O. The behaviour of various structural parameters with _O/_O, _O and _O are the values ofP/E and dP/dE at the centre) has been studied. The most important result of this study is the fact the qualitative behaviour of all the structural parameters depends only on the value of µ_O for the various assigned _O values. The maximum value of surface red shift occurs when µ_O=0.6 and for _O=1.0 it equals 0.618. These structures are gravitationally bound for µ_O0.8 and most so for µ_O=0.4. The maximum value of binding coefficient comes out to be 0.181 when _O=1.0. These structures have been used to model neutron stars. The maximum mass of neutron star based upon such a model comes out to be 2.55M (for µ_O=0.4 and _O=1.0) and maximum size comes out to be 15.0 km (for µ_O=0.2 and _O=1.0). It is also seen that the structures are pulsationally stable for µ0.6.     

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).     

Quasi-radial modes of rotating stars in general relativity

By using the Cowling approximation, quasi-radial modes of rotating general relativistic stars are computed along equilibrium sequences from non-rotating to maximally rotating models. The eigenfrequencies of these modes are decreasing functions of the rotational frequency. The eigenfrequency curve of each mode as a function of the rotational frequency has discontinuities, which arise from the avoided crossing with other curves of axisymmetric modes.     

Two-body motion in the post-Newtonian approximation

The motion of two massive particles is considered within the framework of the first post-Newtonian approximation. The system Hamiltonian is constructed and normalized through first order using a canonical transformation method of implicit variables. Closed-form solutions for the Delaunay elements in the phase space are obtained. The bridge between the phase space and the state space of the Lagrangian of the motion is provided by a velocity-dependent Legendre transformation. By explicit inversion of this transformation, expressions for the Keplerian elements in the state space are obtained from the Delaunay element solutions.     

The motion of rotating spheroidal bodies in general relativity

The motion of two rotating spheroidal bodies, constituting the components of a binary system in a weak gravitational field, has been considered up to terms of the second order in the small parameterV/c, whereV denotes the velocity of the bodies andc is the velocity of light.The following simplifying assumptions, consistent with a problem of astronomical interest, have been made: (1) the dimensions of the bodies are small compared with their mutual distance; (2) the bodies consist of matter in the fluid state with internal hydrostatic pressure and their oblateness is due to their own rotation; (3) there exist axial symmetry about the axis of rotation and symmetry with respect to the equatorial plane, the same symmetry properties apply to mass densities and stress tensors.The Fock-Papapetrou method was used to ascertain those terms in the equations of motion which are due to the rotation and to the oblateness of each component. Approximate solutions to the Poisson and wave equations were obtained to express the potential and retarded potential at large distances from the bodies generating them. The explicit evaluation of certain integrals has necessitated the use of the Laplace-Clairaut theory for the equibrium configuration of rotating bodies. The final expressions require the knowledge of the mass density as a function of the mean radius of the equipotential surfaces.As an interpretation of the results, the Lagrangian perturbation equations were employed to evaluate the secular motion of the nodal line for the relative orbit of the two components. The results constitute a generalization of Fock's work and furnish the contribution of the mass distribution to the rotation effect of general relativity.     

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.     

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.     

Spin down of rotating compact magnetized strange stars in general relativity

We find that in general relativity slow down of the pulsar rotation due to the magnetodipolar radiation is more faster for the strange star with comparison to that for the ordinary neutron star of the same mass. Comparison with astrophysical observations on pulsars spindown data may provide an evidence for the strange star existence and, thus, serve as a test for distinguishing it from the neutron star.     

On the energy integral for first post-Newtonian approximation

The post-Newtonian approximation for general relativity is widely adopted by the geodesy and astronomy communities. It has been successfully exploited for the inclusion of relativistic effects in practically all geodetic applications and techniques such as satellite/lunar laser ranging and very long baseline interferometry. Presently, the levels of accuracy required in geodetic techniques require that reference frames, planetary and satellite orbits and signal propagation be treated within the post-Newtonian regime. For arbitrary scalar W and vector gravitational potentials \(W^j (j=1,2,3)\), we present a novel derivation of the energy associated with a test particle in the post-Newtonian regime. The integral so obtained appears not to have been given previously in the literature and is deduced through algebraic manipulation on seeking a Jacobi-like integral associated with the standard post-Newtonian equations of motion. The new integral is independently verified through a variational formulation using the post-Newtonian metric components and is subsequently verified by numerical integration of the post-Newtonian equations of motion.     

The configuration of a thin,rotating self-similar disk in general relativity

Einstein's equations for a rotating pressure-free space-time are reduced to a system of four first-order non-linear ordinary differential equations in one self-similar dimensionless variable. Numerical results are given for the vacuum solution. A compatible thin disk can be specified by a surface density and an angular velocity . Self-similarity as a statement of the absence of scales implies that and can be written as=c ²/4Gr, =c/r, and demands that and be pure numbers.