scholarly journals Approximate perfect fluid solutions with quadrupole moment

2021 ◽  
Author(s):  
Medeu Abishev ◽  
Nurzada Beissen ◽  
Farida Belissarova ◽  
Kuantay Boshkayev ◽  
Aizhan Mansurova ◽  
...  
Keyword(s):  
Perfect Fluid ◽  
Quadrupole Moment ◽  
Line Element ◽  
Approximate Solutions ◽  
Axially Symmetric ◽  
Field Equations ◽  
Matching Conditions ◽  
Fluid Source ◽  
Gravitational Fields ◽  
Spherically Symmetry

We investigate the interior Einstein’s equations in the case of a static, axially symmetric, perfect fluid source. We present a particular line element that is specially suitable for the investigation of this type of interior gravitational fields. Assuming that the deviation from spherically symmetry is small, we linearize the corresponding line element and field equations and find several classes of vacuum and perfect fluid solutions. We find some particular approximate solutions by imposing appropriate matching conditions.

The general axially symmetric static solution of Einstein’s vacuum field equations

1982 ◽  
Vol 382 (1783) ◽  
pp. 467-470 ◽  
Keyword(s):  
General Solution ◽  
Potential Function ◽  
Line Element ◽  
Static Solution ◽  
Vacuum Field ◽  
Axially Symmetric ◽  
Field Equations ◽  
Axis Of Symmetry ◽  
Axisymmetric Solutions ◽  
Associated Function

The general solution in closed form, including all the static axisymmetric solutions of Weyl, is presented in the canonical coordinates ρ and z of his line element. This general solution is constructed from an arbitrary function f ( z ), which coincides with his potential function along the axis of symmetry. To illustrate how the solution may be used, a particular function f , one resulting from a Newtonian solution, is used to find both the potential function and its associated function in the line element.

APPROXIMATE SOLUTIONS OF STATIONARY GAUGE STRINGS ON THE (r, z)-PLANE

1995 ◽  
Vol 04 (02) ◽  
pp. 267-277 ◽  
Author(s):  
R.J. SLAGTER
Keyword(s):  
Angular Momentum ◽  
Gauge Field ◽  
Approximation Scheme ◽  
Analytic Solution ◽  
Approximate Solutions ◽  
Second Order ◽  
Singular Behavior ◽  
Axially Symmetric ◽  
Field Equations ◽  
Symmetric Spacetime

We derive a class of approximate solutions of the coupled Einstein-scalar-gauge field equations on an axially symmetric spacetime. An analytic solution of the resulting elliptic PDE’s can be obtained to any desired order by constructing the Riemann functions. As an example model, a solution is presented, which resembles the Nielsen-Olesen vortex close to the z=0 hyperplane. However, the solution shows some significant deviation from the classical vortex off the z=0 plane. The singular behavior, which one usually encounters in line-mass models, manifests itself through the second-order solutions in the approximation scheme. Further, in this “toy”-model, with sufficient angular momentum of the spinning string, gφφ becomes negative for some values of r.

scholarly journals DOPPLER EFFECTS FROM BENDING OF LIGHT RAYS IN CURVED SPACE–TIMES

2006 ◽  
Vol 15 (08) ◽  
pp. 1183-1198
Author(s):  
MATTEO LUCA RUGGIERO ◽  
ANGELO TARTAGLIA ◽  
LORENZO IORIO
Keyword(s):  
Gravitational Field ◽  
Quadrupole Moment ◽  
Giant Planets ◽  
Axially Symmetric ◽  
Curved Space ◽  
Gravitational Fields ◽  
Light Rays ◽  
Doppler Effects ◽  
Electromagnetic Signals ◽  
Bending Of Light

We study Doppler effects in curved space–time, i.e. the frequency shifts induced on electromagnetic signals propagating in the gravitational field. In particular, we focus on the frequency shift due to the bending of light rays in weak gravitational fields. We consider, using the PPN formalism, the gravitational field of an axially symmetric distribution of mass. The zeroth order, i.e. the sphere, is studied then passing to the contribution of the quadrupole moment, and finally to the case of a rotating source. We give numerical estimates for situations of physical interest, and by a very preliminary analysis, we argue that analyzing the Doppler effect could lead, in principle, in the foreseeable future, to the measurement of the quadrupole moment of the giant planets of the Solar System.

scholarly journals Some special solutions of the equations of axially symmetric gravitational fields

1932 ◽  
Vol 136 (829) ◽  
pp. 176-192 ◽  
Keyword(s):  
General Problem ◽  
Approximate Solutions ◽  
Static Field ◽  
Newtonian Potential ◽  
Infinite Cylinder ◽  
Axially Symmetric ◽  
Gravitational Fields ◽  
Special Cases ◽  
Axially Symmetric Gravitational Fields ◽  
Special Case

The problem of axially symmetric fields was first treated by Weyl, who succeeded in obtaining solutions for a static field in terms of the Newtonian potential of a distribution of matter in an associated canonical space. He also solved the more general problem involving the electric field. Levi Civita, by slightly different methods, obtained solutions differing from those of Weyl in one respect, and discussed fully the case in which the field is produced by an infinite cylinder. R. Bach has discussed the special case of two spheres and has calculated their mutual attraction. Bach also considered the field of a slowly rotating sphere, and obtained approximate solutions, taking the Schwarz child solution as his zero-th approximation. The same field was discussed earlier by Leuse and Thirring, who considered, the linear terms, only, in the gravitational equation. Kornel Lanczos has also considered a special case of stationary fields and applied the results to cosmological problems. The more general case of gravitational fields produced by matter in stationary rotation has been treated by W. R. Andress and E. Akeley. Both these authors obtain approximate solutions of the general problem, and the latter treats at length the field of a rotating fluid. The object of this paper is to present some special, but exact, solutions which the author obtained some years ago and, also, two methods of successive approximation for obtaining solutions of a more general type, which behave in an assigned manner at infinity and on a surface of revolution enclosing the rotating matter to which the field is due. Our solutions include as special cases the solutions of Weyl, Levi Civita and others which pertain to static fields. Also, the approximate solutions for stationary fields obtained by Leuse and Thirring, Bach and Andress are contained in our solutions when appropriate choice of boundary conditions is made and higher order terms are neglected.

scholarly journals Exact treatment of time-dependent axisymmetric gravitation

1993 ◽  
Vol 440 (1910) ◽  
pp. 711-715 ◽  
Keyword(s):  
Exact Solution ◽  
Time Dependent ◽  
New Method ◽  
Vacuum Field ◽  
Axially Symmetric ◽  
Field Equations ◽  
Canonical Solution ◽  
Gravitational Fields ◽  
Einstein Vacuum

This paper extends an earlier treatment of time-dependent gravitational fields that are axially symmetric and non-rotating. From a consideration of the canonical solution of the Einstein vacuum field equations previously obtained as an axial expansion, a new method has been found that now provides the exact solution, whenever a certain generative key function X ( t , z ) is known.

Gravitational waves in general relativity VIII. Waves in asymptotically flat space-time

1962 ◽  
Vol 270 (1340) ◽  
pp. 103-126 ◽  
Keyword(s):  
Boundary Conditions ◽  
Gravitational Waves ◽  
Superposition Principle ◽  
Flat Space ◽  
Space Time ◽  
Axially Symmetric ◽  
Field Equations ◽  
Spatial Infinity ◽  
Metric Tensor ◽  
Gravitational Fields

Gravitational fields containing bounded sources and gravitational radiation are examined by analyzing their properties at spatial infinity. A convenient way of splitting the metric tensor and the Einstein field equations, applicable in any space-time, is first introduced. Then suitable boundary conditions are set. The group of co-ordinate transformations that preserves the boundary conditions is analyzed. Different possible gravitational fields are characterized intrinsically by a combination of (i) characteristic initial data, and (ii) Dirichlet data at spatial infinity. To determine a particular solution one must specify four functions of three variables and three functions of two variables; these functions are not subject to constraints. A method for integrating the field equations is given; the asymptotic behaviour of the metric and Riemann tensors for large spatial distances is analyzed in detail; the dynamical variables of the radiation modes are exhibited; and a superposition principle for the radiation modes of the gravitational field is suggested. Among the results are: (i) the group of allowed co-ordinate transformations contains the inhomogeneous orthochronous Lorentz group as a subgroup; (ii) each of the five leading terms in an asymptotic expansion of the Riemann tensor has the algebraic structure previously predicted from analyzing the Petrov classification; (iii) gravitational waves appear to carry mass away from the interior; (iv) time-dependent periodic solutions of the field equations which obey the stated boundary conditions do not exist. It was found that the general fields studied in the present work are in many ways very similar to the axially symmetric fields recently studied by Bondi, van der Burg & Metzner.

scholarly journals Post-Newtonian limit: second-order Jefimenko equations

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
David Pérez Carlos ◽  
Augusto Espinoza ◽  
Andrew Chubykalo
Keyword(s):  
Linear Equations ◽  
Space Time ◽  
Second Order ◽  
Field Equations ◽  
Gravitational Fields ◽  
Mass Distributions ◽  
Minkowski Space Time ◽  
Theory Of Gravitation ◽  
Gravity Probe ◽  
Gravitational Equations

Abstract The purpose of this paper is to get second-order gravitational equations, a correction made to Jefimenko’s linear gravitational equations. These linear equations were first proposed by Oliver Heaviside in [1], making an analogy between the laws of electromagnetism and gravitation. To achieve our goal, we will use perturbation methods on Einstein field equations. It should be emphasized that the resulting system of equations can also be derived from Logunov’s non-linear gravitational equations, but with different physical interpretation, for while in the former gravitation is considered as a deformation of space-time as we can see in [2–5], in the latter gravitation is considered as a physical tensor field in the Minkowski space-time (as in [6–8]). In Jefimenko’s theory of gravitation, exposed in [9, 10], there are two kinds of gravitational fields, the ordinary gravitational field, due to the presence of masses, at rest, or in motion and other field called Heaviside field due to and acts only on moving masses. The Heaviside field is known in general relativity as Lense-Thirring effect or gravitomagnetism (The Heaviside field is the gravitational analogous of the magnetic field in the electromagnetic theory, its existence was proved employing the Gravity Probe B launched by NASA (See, for example, [11, 12]). It is a type of gravitational induction), interpreted as a distortion of space-time due to the motion of mass distributions, (see, for example [13, 14]). Here, we will present our second-order Jefimenko equations for gravitation and its solutions.

scholarly journals NONUNIFORM BRANEWORLD STARS: AN EXACT SOLUTION

2009 ◽  
Vol 18 (05) ◽  
pp. 837-852 ◽  
Author(s):  
J. OVALLE
Keyword(s):  
Exact Solution ◽  
Effective Density ◽  
Interior Solution ◽  
Effective Pressure ◽  
Gravity Effect ◽  
Field Equations ◽  
Matching Conditions ◽  
Physical Behavior ◽  
Exterior Solution ◽  
Weyl Scalar

In this paper the first exact interior solution to Einstein's field equations for a static and nonuniform braneworld star with local and nonlocal bulk terms is presented. It is shown that the bulk Weyl scalar [Formula: see text] is always negative inside the stellar distribution, and in consequence it reduces both the effective density and the effective pressure. It is found that the anisotropy generated by bulk gravity effect has an acceptable physical behavior inside the distribution. Using a Reissner–Nördstrom-like exterior solution, the effects of bulk gravity on pressure and density are found through matching conditions.

Sign in / Sign up

Export Citation Format

Share Document