On static spherically symmetric solutions of the vacuum Brans-Dicke theory

It is shown that among the four classes of the static spherically symmetric solutions of the vacuum Brans-Dicke theory of gravity only two are really independent. Further, by matching exterior and interior (due to physically reasonable spherically symmetric matter source) scalar fields it is found that only the Brans class I solution with a certain restriction on the solution parameters may represent an exterior metric for a nonsingular massive object. The physical viability of the black hole nature of the solution is investigated. It is concluded that no physical black hole solution different from the Schwarzschild black hole is available in the Brans-Dicke theory.     

A scalar Euclidean theory of gravitation: motion in a static spherically symmetric field

The motion of a particle in a static, spherically symmetric gravitational field is investigated in Euclidean space. The gravitational effects are described as due to a scalar field: To every point in space there is assigned a refractive index deciding the velocity of light in that point. The motion of light in the vacuum is described by the equation of classical optics. An equation of motion for material test particles is then derived by employing the usual Lagrangian formalism. The motion of the planets around the sun is explained, in particular the perihelion motion of Mercury. The present theory fully explains the four classical tests of general relativity in a mathematically far simpler way, and it can be equivalent to the Schwarzschild solution. It is also found that the effect of gravitation depends on the velocity of the particle, becoming repulsive for radial velocities larger thanc/ (c is the velocity of light). This seemingly odd result can also be obtained from the equations of general relativity, as was shown by Cavalleri and Spinelli.     

Issue of the spherically symmetric static vacuum metric in the relativistic theory of gravity

The spherically symmetric vacuum metric in the relativistic theory of gravity is analyzed numerically. It is found that there is no deviation of the numerical solution from that of general relativity except in the near-horizon range. The solution obtained has the well-established analytical asymptotics for both the near-and far-horizon limits. It satisfies the causality principle and does not impose a lower limit on the graviton mass.      

Point charge in a static,spherically symmetric Brans-Dicke field

We determine in closed form the electrostatic potential of a point test charge held at rest in a static, spherically symmetric Brans-Dicke field. This result is a generalization of the previously obtained expression for the potential of a test charge at rest near a Schwarzschild black hole. Moreover, our solution is valid for the coupled gravitational and massless scalar fields.     

Velocity and velocity bounds in static spherically symmetric metrics

We find simple expressions for velocity of massless particles with dependence on the distance, r, in Schwarzschild coordinates. For massive particles these expressions give an upper bound for the velocity. Our results apply to static spherically symmetric metrics. We use these results to calculate the velocity for different cases: Schwarzschild, Schwarzschild-de Sitter and Reissner-Nordström with and without the cosmological constant. We emphasize the differences between the behavior of the velocity in the different metrics and find that in cases with naked singularity there always exists a region where the massless particle moves with a velocity greater than the velocity of light in vacuum. In the case of Reissner-Nordström-de Sitter we completely characterize the velocity and the metric in an algebraic way. We contrast the case of classical naked singularities with naked singularities emerging from metric inspired by noncommutative geometry where the radial velocity never exceeds one. Furthermore, we solve the Einstein equations for a constant and polytropic density profile and calculate the radial velocity of a photon moving in spaces with interior metric. The polytropic case of radial velocity displays an unexpected variation bounded by a local minimum and maximum.     

Horizons and singularities in static,spherically symmetric spacetimes

We make a thorough study of the regions near finite-order metric-singularity boundaries of static, spherically symmetric spacetimes. After distinguishing curvature singularities from other types of metric breakdown, we examine the eigenvalues of the energy tensor near the singularities for positivity and energy dominance, find the causal class of the t-translation (static) Killing field, and ascertain the presence or absence of timelike, null, and spacelike geodesic incompleteness for each spacetime. For a certain subclass of spacetimes, we also show the completeness of all timelike and spacelike curves despite the superficial failure of the metric.     

Radiative transfer through a spherically symmetric cloud of extincting particles — exact solution of the inverse absorption problem

We derive an exact solution to the inverse absorption problem to calculate the density distribution in spherical symmetry of absorbing particles from the intensity pattern obtained for homogeneous illumination. We illustrate the capabilities of the method by the simple example of a constant density core and find the required numerical effort to be negligible. The applicability is discussed for physical problems where unknown absorption coefficients, particle size or density distributions can be determined from multi-frequency measurements of the transmission coefficient. The applications range from targets being evaporated by laser pulses to Bok globules in astrophysics.     

Singularities in solutions of the Einstein equations for a static spherically symmetric field

Russian Physics Journal - The nature of the singularities in the solutions of Einstein's equations for a static spherically symmetric field is investigated as a function of the choice of...     

Thermodynamic quantities for gases in static spherically symmetric backgrounds possessing a horizon

The brick-wall method is used to study the thermodynamic quantities for perfect relativistic gases in generic spherically symmetric and static background spacetimes possessing a horizon. The Wentzel-Kramers-Brillouin (WKB) approximation is employed for the Teukolsky-type master equation. It is shown that the entropy density, energy density, pressure, and state equation all contain a subleading term, which depends on the spins of the particles of the gases. When particularizing to the Schwarzschild and Reissner-Nördstrom geometries, the results previously found in the literature for those geometries are recovered.     

Complete affine connection in the causal boundary: static,spherically symmetric spacetimes

The boundary at \(\mathcal {I}^+\), future null infinity, for a standard static, spherically symmetric spactime is examined for possible linear connections. Two independent methods are employed, one for treating \(\mathcal {I}^+\) as the future causal boundary, and one for treating it as a conformal boundary (the latter is subsumed in the former, which is of greater generality). Both methods provide the same result: a constellation of various possible connections, depending on an arbitrary choice of a certain function, a sort of gauge freedom in obtaining a natural connection on \(\mathcal {I}^+\); choosing that function to be constant (for instance) results in a complete connection. Treating \(\mathcal {I}^+\) as part of the future causal boundary, the method is to impute affine connections on null hypersurfaces going out to \(\mathcal {I}^+\), in terms of a transverse vector field on each null hypersurface (there is much gauge freedom on choice of the transverse vector fields). Treating \(\mathcal {I}^+\) as part of a conformal boundary, the method is to make a choice of conformal factor that makes the boundary totally geodesic in the enveloping manifold (there is much gauge freedom in choice of that conformal factor). Similar examination is made of other boundaries, such as timelike infinity and timelike and spacelike singularities. These are much simpler, as they admit a unique connection from a similar limiting process (i.e., no gauge freedom); and that connection is complete.     

Half-range moment method for solution of the transport equation in a spherically symmetric geometry

A half-range moment method is presented for solving, in various orders of approximation, a multi-group transport equation subject to generalized boundary conditions in a spherically symmetric geometry. The results for the plane-parallel geometry are obtainable from the present analysis as a special case. The equations and the boundary conditions considered are sufficiently general to characterize a variety of problems in radiative transfer, neutron transport and phonon transport if various coefficients appearing in the equations are properly specified.     

Representation theory and integration of nonlinear spherically symmetric equations to gauge theories

A constructive proof of complete integrability of spherically symmetric self-dual equations in Euclidean spaceR ₄ for an arbitrary embedding of SU(2) in an arbitrary gauge groupG is given on the base of Lax-type representation and representation theory. The equations are solved explicitly for the case of simple Lie groupsG.     

Radiative transfer in a diurnally insolated mechanically static grey atmosphere

A numerical analysis scheme is presented for a previously proposed theory of the temperature structure versus time of a diurnally insolated atmosphere of constant density wherein heat transfer occurs by both radiation and conduction. The approach is one of linearizing the difference equations corresponding to the analytical forms (rather than vice versa). Limitations of available computers and time there-on, coupled with an ill-conditioning characteristic of the one physical (lunar) situation attacked, limit the conclusions of the present paper to an affirmation of the method and an indication that a more approximate approach to the insolation of the lunar soil would be adequate. Explicit inclusion of the transfer equation, as done here, would be desirable for more rapidly rotating bodies such as asteroids, Galilean satellites and Saturn's rings.     

Equations of motion of test bodies in a spherically symmetric,static gravitational field

We consider a class of theories of gravity in which the motion of test particles is governed by the path equation (with respect to a general affine connection Г). When we restrict attention to a spherically symmetric, static gravitational field, the path equation is characterized by three arbitrary functions of the gravitational field (as opposed to two in the case of metric theories where Г={ }). We find that there are essentially only two constraints on the three functions by appealing to solar system experiments. Therefore, we must supplement the equation of motion with other physical laws to obtain a value for the third arbitrary function (this, of course, is not necessary in the case of metric theories). If we consider theories in which both Г andg play a physical role we find in certain circumstances that this “third” condition is sufficient to prove that the theories under investigation reduce to their “metric” form.     

Existence and stability of circular orbits in general static and spherically symmetric spacetimes

The existence and stability of circular orbits (CO) in static and spherically symmetric (SSS) spacetime are important because of their practical and potential usefulness. In this paper, using the fixed point method, we first prove a necessary and sufficient condition on the metric function for the existence of timelike COs in SSS spacetimes. After analyzing the asymptotic behavior of the metric, we then show that asymptotic flat SSS spacetime that corresponds to a negative Newtonian potential at large r will always allow the existence of CO. The stability of the CO in a general SSS spacetime is then studied using the Lyapunov exponent method. Two sufficient conditions on the (in)stability of the COs are obtained. For null geodesics, a sufficient condition on the metric function for the (in)stability of null CO is also obtained. We then illustrate one powerful application of these results by showing that three SSS spacetimes whose metric function is not completely known will allow the existence of timelike and/or null COs. We also used our results to assert the existence and (in)stabilities of a number of known SSS metrics.     

The regularity of static spherically cylindrically and plane symmetric spacetimes at the origin

We find the necessary and sufficient conditions for the regularity of all scalar invariants polynomial in the Riemann tensor at the origin of spherically, cylindrically and plane symmetric static spacetimes under the assumption that the metric functions are sufficiently smooth there. These conditions turn out to be simple enough to allow a check for regularity by inspection.     

Universal time delay in static spherically symmetric spacetimes for null and timelike signals

A perturbative method of computing the total travel time of both null and lightlike rays in arbitrary static spherically symmetric spacetimes in the weak field limit is proposed.The resultant total time takes a quasi-series form of the impact parameter.The coefficient of this series at a certain order n is shown to be determined by the asymptotic expansion of the metric functions to the order n+1.For the leading order(s),the time delay,as well as the difference between the time delays of two types of relativistic signals,is shown to take a universal form for all SSS spacetimes.This universal form depends on the mass M and a post-Newtonian parameter y of the spacetime.The analytical result is numerically verified using the central black hole of galaxy M87 as the gravitational lensing center.     

Spherically symmetric static solutions in the Einstein-Cartan theory

The system of Einstein-Cartan equations has been solved in [1] in the case of zero pressure of matter and a cosmological term equal to zero. It has been shown that the gravitational spin-spin interaction exerts a stabilizing effect on the matter distribution similarly to Einstein's -term.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 9, pp. 60–64, September, 1982.In conclusion the author expresses his sincere gratitude to his own scientific supervisor for the doctorate of physicomathematical sciences, senior scientist V. N. Ponomarev, for his guidance and help in this research.