Erez-Rosen Metric Research Articles

К ОБРАТНОЙ ЗАДАЧЕ НЕБЕСНОЙ МЕХАНИКИ

The link between exterior solutions to the Einstein gravitational field equations such as the exact Erez-Rosen metric and approximate Hartle-Thorne metric is established here for the static case in the limit of linear mass quadrupole moment (Q) and second order terms in total mass (M). To this end, the Geroch-Hansen multipole moments are calculated for the Erez-Rosen and Hartle-Thorne solutions in order to find the relationship among the parameters of both metrics. The coordinate transformations are sought in a general form with two unknown functions in the corresponding limit of ~Q and ~M^2. By employing the perturbation theory, the approximate Erez-Rosen metric is written in the same coordinates as the Hartle-Thorne metric. By equating the radial and azimuthal components of the metric tensor of both solutions the sought functions are found in a straightforward way. It is shown that the approximation ~Q and ~M^2, which is used throughout the article, is physical and suitable for solving most problems of celestial mechanics in post-Newtonian physics. This approximation does not require the use of the Zipoy-Voorhees transformation, which is a necessary strict mathematical requirement in the ~Q approximation, i.e. when no other approximations are made. This implies that the explicit form of the coordinate transformations depends entirely on the approximation that is adopted in each particular case. The results obtained here are in agreement with the previous results in the literature and can be applied to different astrophysical goals. The paper pursues not only pure scientific, but also academic purposes and can be used as an auxiliary and additional material to the special courses of general theory of relativity, celestial mechanics and relativistic astrophysics.

Three-dimensional general relativistic Poynting-Robertson effect. III. Static and nonspherical quadrupolar massive source

We investigate the three-dimensional motion of a test particle in the gravitational field generated by a non-spherical compact object endowed with a mass quadrupole moment, described by the Erez-Rosen metric, and a radiation field, including the general relativistic Poynting-Robertson effect, coming from a rigidly rotating spherical emitting source located outside of the compact object. We derive the equations of motion for test particles influenced by such radiation field, recovering the two-dimensional description, and the weak-field approximation. This dynamical system admits the existence of a critical hypersurface, region where gravitational and radiation forces balance. Selected test particle orbits for different set of input parameters are displayed. The possible configurations on the critical hypersurfaces can be either latitudinal drift towards the equatorial ring or suspended orbits. We discuss about the existence of multiple hypersurface solutions through a simple method to perform the calculations. We graphically prove also that the critical hypersurfaces are stable configurations within the Lyapunov theory.

The Erez–Rosen Solution Versus the Hartle–Thorne Solution

In this work, we investigate the correspondence between the Erez–Rosen and Hartle–Thorne solutions. We explicitly show how to establish the relationship and find the coordinate transformations between the two metrics. For this purpose the two metrics must have the same approximation and describe the gravitational field of static objects. Since both the Erez–Rosen and the Hartle–Thorne solutions are particular solutions of a more general solution, the Zipoy–Voorhees transformation is applied to the exact Erez–Rosen metric in order to obtain a generalized solution in terms of the Zipoy–Voorhees parameter δ = 1 + s q . The Geroch–Hansen multipole moments of the generalized Erez–Rosen metric are calculated to find the definition of the total mass and quadrupole moment in terms of the mass m, quadrupole q and Zipoy–Voorhees δ parameters. The coordinate transformations between the metrics are found in the approximation of ∼q. It is shown that the Zipoy–Voorhees parameter is equal to δ = 1 - q with s = - 1 . This result is in agreement with previous results in the literature.

Definition of the relativistic geoid in terms of isochronometric surfaces

We present a definition of the geoid that is based on the formalism of general relativity without approximations; i.e. it allows for arbitrarily strong gravitational fields. For this reason, it applies not only to the Earth and other planets but also to compact objects such as neutron stars. We define the geoid as a level surface of a time-independent redshift potential. Such a redshift potential exists in any stationary spacetime. Therefore, our geoid is well defined for any rigidly rotating object with constant angular velocity and a fixed rotation axis that is not subject to external forces. Our definition is operational because the level surfaces of a redshift potential can be realized with the help of standard clocks, which may be connected by optical fibers. Therefore, these surfaces are also called isochronometric surfaces. We deliberately base our definition of a relativistic geoid on the use of clocks since we believe that clock geodesy offers the best methods for probing gravitational fields with highest precision in the future. However, we also point out that our definition of the geoid is mathematically equivalent to a definition in terms of an acceleration potential, i.e. that our geoid may also be viewed as a level surface orthogonal to plumb lines. Moreover, we demonstrate that our definition reduces to the known Newtonian and post-Newtonian notions in the appropriate limits. As an illustration, we determine the isochronometric surfaces for rotating observers in axisymmetric static and axisymmetric stationary solutions to Einstein's vacuum field equation, with the Schwarzschild metric, the Erez-Rosen metric, the q-metric and the Kerr metric as particular examples.

Motion4D-library extended

The new version of the Motion4D library is mainly an update to work with the four-dimensional ray tracing code GeoViS which makes full use of all the implemented metrics and geodesic integrators. New version program summaryProgram title: Motion4D-libraryCatalogue identifier: AEEX_v3_1Program summary URL:http://cpc.cs.qub.ac.uk/summaries/AEEX_v3_1.htmlProgram obtainable from: CPC Program Library, Queen’s University, Belfast, N. IrelandLicensing provisions: Standard CPC licence, http://cpc.cs.qub.ac.uk/licence/licence.htmlNo. of lines in distributed program, including test data, etc.: 189793No. of bytes in distributed program, including test data, etc.: 6808546Distribution format: tar.gzProgramming language: C++.Computer: All platforms with a C++ compiler.Operating system: Linux, Windows.RAM: 61 MBytesClassification: 1.5.Catalogue identifier of previous version: AEEX_v3_0Journal reference of previous version: Comput. Phys. Comm. 182(2011)1386External routines: GNU Scientific Library (GSL) (http://www.gnu.org/software/gsl/)Does the new version supersede the previous version?: YesNature of problem:Solve geodesic equation, parallel and Fermi–Walker transport in four-dimensional Lorentzian spacetimes. Determine gravitational lensing by integration of Jacobi equation and parallel transport of Sachs basis.Solution method:Integration of ordinary differential equations.Reasons for new version:The main reason for the new version is the update of some methods to work with the four-dimensional ray tracing code GeoViS. Furthermore, some new metrics and integrators were implemented.Summary of revisions:The four-dimensional ray tracing code GeoViS [1] is based on the Motion4D library. All of the metrics and geodesic integrators of the library can be accessed by means of GeoViS’ scheme-based scripting language. For that, some methods had to be updated.In the following, a list of newly implemented metrics is given: •AlcubierreSimple: This metric uses a simplified warp bubble function compared to the original one by Alcubierre [2], see McMonigal et al. [3].•ChazyCurzonRot: The metric of the rotational Chazy–Curzon solution is taken from Stephani et al. [4].•Curzon: The Curzon metric in cylindrical coordinates is taken from Scott and Szekeres [5].•EinsteinRosenWaveWWB: A detailed discussion of the Einstein–Rosen wave with a Weber–Wheeler–Bonnor pulse can be found in Griffiths and Micciche [6].•ErezRosenVar: The original Erez–Rosen metric is quite intricate, see e.g. Krori and Sarmah [7]. Hence, we use here a reduced version with a simpler quadrupole term.•KastorTraschen: The Kastor–Traschen metric with two black holes is taken from Griffiths and Podolský [8] Furthermore, the Dormand–Prince 5(4) and 6(5) integrators taken from Guthmann [8] were implemented.Running time:The test runs provided with the distribution require only a few seconds to run.

Metric of a Slow Rotating Body with Quadrupole Moment from the Erez-Rosen Metric

A metric representing a slowly rotating object with quadrupole moment is obtained using a perturbation method to include rotation into the weak limit of the Erez-Rosen metric. This metric is intended to tackle relativistic astrometry and gravitational lensing problems in which a quadrupole moment has to be taken into account.

Superposition of the Kerr metric with the generalized Erez-Rosen solution.

An exact asymptotically flat solution of the Einstein vacuum equations is presented. Similar to the well-known Kerr solution it reduces to the Schwarzschild metric in the absence of rotation, and in addition it possesses an arbitrary multipole structure due to the seed Erez-Rosen metric.

Geodesies in the Erez-Rosen space-time

We investigate null and time-like geodesics in the Erez-Rosen space-time, that is, in the exterior gravitational field of a mass with quadrupole moment. By using the weak-field approximation of the Erez-Rosen metric, we find the solution of the equation for equatorial time-like geodesics and determine how they differ from the corresponding Schwarzschild geodesies. For the exact form of the Erez-Rosen metric, we only draw some qualitative conclusions about the influence of the quadrupole moment on the path of test particles and on the motion of photons. We derive the relativistic contribution of the quadrupole moment to the perihelion shift and to the precession of the ascending node.

On the exterior gravitational field of a mass with a multipole moment

Recently, Gutsunaev and Manko [6] presented a procedure for obtaining new static axisymmetric solutions of Einstein's vacuum field equations from a known one. We show that this procedure is based on the property that the derivatives of a harmonic function are harmonic. The special case of a metric with mass and quadrupole moment is investigated and compared with the Erez-Rosen metric.

Exterior gravitational field of a rotating deformed mass

We obtain a new solution of the vacuum Einstein equations by means of the rank-zero Hoenselaers-Kinnersley-Xanthopoulos transformations. We take as seed metric the static axisymmetric Erez-Rosen metric, which describes the exterior field of a mass with arbitrary quadrupole moment. The Ernst potential of the new solution is presented and all metric functions are explicitly given. The new metric, which contains the Kerr and Erez-Rosen metrics as special cases, represents the exterior gravitational field of a rotating mass with arbitrary quadrupole moment.

On a class of exact geodesics of the erez-rosen metric

In a previous paper, a class of exact geodesics for the motion of a particle in a gravitational-monopole-prolate-quadrupole field was investigated, both in Newtonian mechanics and in general relativity. This paper consists of both an amplification of the analysis contained in the previous paper and an extension of the analysis to the case for negative quadrupole moment, which was not treated previously. The relativistic results are based on the monopole-quadrupole metric of Erez and Rosen, the Newtonian results on the monopole-quadrupole potential of Laplace. In the limit of vanishing quadrupole parameter (q → 0), the relativistic results reduce to those of the familiar Schwarzschild case; in the weak-field limit (r/m → ∞), the relativistic results reduce to those of the Newtonian case. The existence and stability thresholds in the relativistic case yield values that uniquely characterize the Erez-Rosen metric.