Ricci tensors for elliptical shaped galaxies and objects (SAO/NASA ADS Index) (original) (raw)

Ricci tensors for elliptical shaped galaxies and celestial objects

International Journal of the Physical Sciences, 2011

Einstein's field equations are based on Riemannian geometry. One of the great important in Riemannian geometry is the curves that minimize the distance between two given points. The Ricci curvature tensors are also broadly applicable to modern Riemannian geometry and general theory of relativity (GTR). The solution of Einstein's field equations, need Ricci tensors. The line elements based on the ellipsoidal coordinates systems usually are difficult to work but are more perfect than spherical coordinate systems. Here we are trying to get all non-zero Ricci tensors coefficients of a line element in an oblate spheroidal coordinate system for an elliptical shaped object. These objects can be as galaxies, stars or planets and other celestial objects that we know today.

The space-time line element for static ellipsoidal objects

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.

Behavior of elliptical objects in general theory of relativity

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Schwarzschild solution is the simplest solution of Einstein's field equations, but it has not been able to describe any non-spherical in shape as in the real are existing. Many objects like stars and/or galaxies are in the form of ellipsoidal form and consequently, the gravitational lines around the objects are different in comparison with spherical form. In this paper a line element has been constructed with the intention, not only to describe a spherical form but also to explain an ellipsoidal system in more accurate and complete form. In fact, Schwarzschild line element and its solution is only a part of whole work, which I have done. Applying this metric for more consideration an arbitrary object is the next step. The solution for planetary orbit of an ellipsoid planet by using Einstein's field equations also has done. We attempt to solve these equations as the exterior solution for an ellipsoidal planet.

Ellipsoidal shapes in general relativity: general definitions and an application

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.

Space tensors in general relativity III: The structural equations

General Relativity and Gravitation, 1974

A self-consistent theory of spatial differential forms over a pair (M,F) is proposed. The operators a (spatial exterior differentiation), a T (temporal Lie derivative) and ~ (spatial Lie derivative) are defined, and their properties are discussed. These results are then applied to the study of the torsion and curvature tensor fields determined by an arbitrary spatial tensor analysis (?~T) over (M~F). The structural equations of (V~T) and the corresponding spatial Bianchi identities are discussed. The special case (~,VT) = (9',9T*) is examined in detail. The spatial resolution of the Riemann tensor of the manifold M is finally analysed; the resulting structure of Einstein's equations over a pair (Vd~F) is established. An application to the study of the problem of motion in terms of co-moving atlases is proposed.

Space tensors in general relativity I: Spatial tensor algebra and analysis

General Relativity and Gravitation, 1974

A pair (M,F) is defined as a Riemannian manifold M of normal hyperbolic type carrying a distinguished time-like congruence r. The spatial tensor algebra ~ associated with the pair (M,F) is discussed. A general definition of the concept of spatial tensor analysis over (M,r) is then proposed. Basically, this includes a spatial covariant differentiation ~ and a time-derivative 9 T, both acting on ~ and commuting with the process of raising and lowering the tensor indices. The torsion tensor fields of the pair (V,VT) are discussed, as well as the corresponding structural equations. The existence of a distinguished spatial tensor analysis over (M,r) is finally established, and the resulting mathematical structure is examined in detail.