Behavior of elliptical objects in general theory of relativity

academicjournals.org, 2011

Cornell University - arXiv, 2022

In this paper, we solved the Einstein's field equation and obtained a line element for static, ellipsoidal objects characterized by the linear eccentricity (η) instead of quadrupole parameter (q). This line element recovers the Schwarzschild line element when η is zero. In addition to that it also reduces to the Schwarzschild line element, if we neglect terms of the order of r −2 or higher which are present within the expressions for metric elements for large distances. Furthermore, as the ellipsoidal character of the derived line element is maintained by the linear eccentricity (η), which is an easily measurable parameter, this line element could be more suitable for various analytical as well as observational studies.

Classical and Quantum Gravity, 2003

A generalization of the notion of ellipsoids to curved Riemannian spaces is given and the possibility to use it in describing the shapes of rotating bodies in general relativity is examined. As an illustrative example, stationary, axisymmetric perfect-fluid spacetimes with a so-called confocal inside ellipsoidal symmetry are investigated in detail under the assumption that the 4-velocity of the fluid is parallel to a time-like Killing vector field. A class of perfect-fluid metrics representing interior NUT-spacetimes is obtained along with a vacuum solution with a non-zero cosmological constant.

Nature, 1961

Here we present a new point of view for general relativity and/or space-time metrics that is remarkably different from the well-known viewpoint of general relativity. From this unique standpoint, we attempt to derive a new metric as an alternative to the Schwarzschild metric for any planet in the solar system. After determining the metric by means of some simple mathematical and physical manipulations, we used this alternative metric to recalculate the perihelion precession of any planet in the solar system and deflection of light that passes near the sun, as examples of this new viewpoint. While we obtained the result of classical general relativity for the perihelion procession, we found a slightly different result, relative to classical general relativity, for the deflection of light.

Hyperscience International Journals

Lagrangian method applied as well as tensor method, for a linear transformed geodesic line element of Schwarzschild-like The ‎Lagrangian method was applied for a linearly transformed geodesic line element of a Schwarzschild-like solution instead of ‎the tensor method. The solution shows that it is not only valid for spherical objects but also it is more comprehensive for ‎elliptical celestial objects. Two types of kinetic and potential energy are the basis of the calculation. Hamiltonian and ‎Lagrangian equality show that the problem has no potential energy. With this transformed geodesic line element, we obtained ‎a new coefficient for the meridional advance of an experimental particle in Schwarzschild spacetime in terms of period, ‎eccentricity, and mean distance. This new perigee equation is not only valid for the Schwarzschild metric (for a spherical ‎object), but also more accurate for the Schwarzschild-like metric (for elliptical objects).‎

In this paper, we construct a new form of a line element exactly in ellipsoidal coordinate system. Einstein field equations are based on Riemannian geometry. The Ricci curvature tensors are also broadly applicable to modern Riemannian geometry and general theory of relativity too. For solution of Einstein field equations, Ricci tensors are very essential and varying with different line elements. The line elements based on the ellipsoidal coordinates systems are difficult to work but it is more perfect than spherical coordinate systems. Because spherical coordinate systems are only in special case of ellipsoidal coordinate systems. This line element not only is valid for ellipsoidal celestial objects, but also with some simple approximation it is applicable for spherical objects too. Here we are trying to get the Ricci tensors coefficients of this new line element. The objects are Galaxies and/or stars with ellipsoidal in shape. All non-zero Ricci tensors are in ellipsoidal coordinat...

Contents 1. Special Relativity 2. Oblique Axes 3. Curvilinear Coordinates 4. Nontensors 5. Curved Space 6. Parallel Displacement 7. Christoffel Symbols 8. Geodesics 9. The Stationary Property of Geodesics 10. Covariant Differentiation 11. The Curvature Tensor 12. The Condition for Flat Space 13. The Bianci Relations 14. The Ricci Tensor 15. Einstein's Law of Gravitation 16. The Newtonian Approximation 17. The Gravitational Red Shift 18. The Schwarzchild Solution 19. Black Holes 20. Tensor Densities 21. Gauss and Stokes Theorems 22. Harmonic Coordinates 23. The Electromagnetic Field 24. Modification of the Einstein Equations by the Presence of Matter 25. The Material Energy Tensor 26. The Gravitational Action Principle 27. The Action for a Continuous Distribution of Matter 28. The Action for the Electromagnetic Field

General Relativity and Gravitation, 2012

International Journal of Modern Physics D, 2005

In this article I present a simple Newtonian heuristic for motivating a weak-field approximation for the spacetime geometry of a point particle. The heuristic is based on Newtonian gravity, the notion of local inertial frames (the Einstein equivalence principle), plus the use of Galilean coordinate transformations to connect the freely falling local inertial frames back to the "fixed stars." Because of the heuristic and quasi-Newtonian manner in which the specific choice of spacetime geometry is motivated, we are at best justified in expecting it to be a weak-field approximation to the true spacetime geometry. However, in the case of a spherically symmetric point mass the result is coincidentally an exact solution of the full vacuum Einstein field equations — it is the Schwarzschild geometry in Painlevé–Gullstrand coordinates. This result is much stronger than the well-known result of Michell and Laplace whereby a Newtonian argument correctly estimates the value of the Sch...

2019

By neglecting the cosmological constant Λ, Einstein's field equations in absence of matter and other fields read G ik = 0, which is not reasonable, since it violates the conservation law of total energy, momentum, and stress, because the gravitational field energy and momentum density cannot be represented by a vanishing Einstein tensor. In order to remedy this shortcoming, we construct a uniform metric, which allows us later to get a more general one, that is asymptotically equal to the Schwarzschild metric. This metric has a plausible energy-momentum density tensor of the gravitational field and correctly describes the effect of light deflection, but the perihelion shift of Mercury is overestimated. Because of the authors' different view in order to overcome the shortcomings, different solutions are obtained.