Einstein's Equation of the Gravitational Field

Einstein’s Equation of motion for exterior test particles with spherical mass distribution having varied field, time and radial distance; Golden metric tensors approach.

International Journal of Theoretical & Computational Physics ◽

10.47485/2767-3901.1014 ◽

2021 ◽

Keyword(s):

Gravitational Field ◽

Polar Motion ◽

Equations Of Motion ◽

Radial Distance ◽

Equation Of Motion ◽

Metric Tensor ◽

Test Particles ◽

Einstein's Equation ◽

Spherical Mass ◽

Metric Tensors

Golden metric tensors exterior to hypothetical distribution of mass whose field varies with time and radial distance have been used to construct the coefficient of affine connections that invariably was used to obtained the Einstein’s equations of motion for test particles of non-zero rest masses. The expression for the variation of time on a clock moving in this gravitational field was derived using the time equation of motion. The test particles in this field under the condition of pure polar motion have an inverse square dependence velocity which depends on radial distance. Our result indicates that despite using the golden metric tensor, the inverse square dependence of the velocity on radial distance has not been changed.

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Field Equation of Nonlocal Gravity

10.1093/oso/9780198803805.003.0006 ◽

2017 ◽

Author(s):

Bahram Mashhoon

Keyword(s):

General Relativity ◽

Dark Matter ◽

Gravitational Field ◽

Field Equation ◽

Abelian Group ◽

Gravitational Field Equation ◽

Differential Field ◽

Einstein's Equation ◽

Future Theory ◽

World Function

In extended general relativity (GR), Einstein’s field equation of GR can be expressed in terms of torsion and this leads to the teleparallel equivalent of GR, namely, GR||, which turns out to be the gauge theory of the Abelian group of spacetime translations. The structure of this theory resembles Maxwell’s electrodynamics. We use this analogy and the world function to develop a nonlocal GR|| via the introduction of a causal scalar constitutive kernel. It is possible to express the nonlocal gravitational field equation as modified Einstein’s equation. In this nonlocal gravity (NLG) theory, the gravitational field is local, but satisfies a partial integro-differential field equation. The field equation of NLG can be expressed as Einstein’s field equation with an extra source that has the interpretation of the effective dark matter. It is possible that the kernel of NLG, which is largely undetermined, could be derived from a more general future theory.

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Testing the $f(R)$-theory of Gravity

Communications in Physics ◽

10.15625/0868-3166/29/1/13192 ◽

2019 ◽

Vol 29 (1) ◽

pp. 35 ◽

Cited By ~ 2

Author(s):

Nguyen Anh Ky ◽

Pham Van Ky ◽

Nguyen Thi Hong Van

Keyword(s):

Gravitational Field ◽

Perturbation Method ◽

Theory Of Relativity ◽

Gravitational Systems ◽

Einstein's Equation ◽

Orbital Precession ◽

Theory Of Gravity ◽

Sgr A ◽

Precise Theory ◽

Central Gravitational Field

A procedure of testing the $f(R)$-theory of gravity is discussed. The latter is an extension of the general theory of relativity (GR). In order this extended theory (in some variant) to be really confirmed as a more precise theory it must be tested. To do that we first have to solve an equation generalizing Einstein's equation in the GR. However, solving this generalized Einstein's equation is often very hard, even it is impossible in general to find an exact solution. It is why the perturbation method for solving this equation is used. In a recent work \cite{Ky:2018fer} a perturbation method was applied to the $f(R)$-theory of gravity in a central gravitational field which is a good approximation in many circumstances. There, perturbative solutions were found for a general form and some special forms of $f(R)$. These solutions may allow us to test an $f(R)$-theory of gravity by calculating some quantities which can be verified later by the experiment (observation). In \cite{Ky:2018fer} an illustration was made on the case $f(R)=R+\lambda R^2$. For this case, in the present article, the orbital precession of S2 orbiting around Sgr A* is calculated in a higher-order of approximation. The $f(R)$-theory of gravity should be also tested for other variants of $f(R)$ not considered yet in \cite{Ky:2018fer}. Here, several representative variants are considered and in each case the orbital precession is calculated for the Sun--Mercury- and the Sgr A*--S2 gravitational systems so that it can be compared with the value observed by a (future) experiment. Following the same method of \cite{Ky:2018fer} a light bending angle for an $f(R)$ model in a central gravitational field can be also calculated and it could be a useful exercise.

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Some features of an interaction of electromagnetic radiation quantums with a gravitational field and a problem of graviton

Vestnik of Samara State Technical University. Technical Sciences Series ◽

10.14498/tech.2020.3.6 ◽

2020 ◽

Vol 28 (3) ◽

pp. 90-109

Author(s):

Andrey N. Volobuev ◽

Aleksandr M. Shterenberg ◽

Pavel K. Kuznetsov

Keyword(s):

Gravitational Field ◽

Electromagnetic Radiation ◽

Identical Result ◽

Field Energy ◽

Frequency Change ◽

Radiation Frequency ◽

Double Stars ◽

Einstein's Equation ◽

Field Quantization ◽

Resonant Pumping

The problems connected to propagation of a gravitational field are considered. The law of electromagnetic radiation frequency change in gravitational field is shown. The problem of a gravitational field quantization is investigated. Energy of a graviton is found by two ways. First, on the basis of use quantum gravitational eikonal and Lagrangian of a gravitational field the energy of a solitary graviton is found. It is shown that graviton has the mass proportional to its frequency. Second, due to refusal from symmetric stresses tensor in structure of an energy-impulse tensor the quantum form of the energy-impulse tensor in Einstein's equation is found. It has allowed found the energy of a solitary graviton. Both ways of an energy graviton finding has given the identical result. It is shown that the solution of the Einsteins equation with use of the quantum form of an energy-impulse tensor for the certain direction represents the sum of a gravitational wave and a graviton. It is found out that at approach of a graviton to the massive bodies (double stars) radiating gravitational waves there is a resonant pumping of the gravitational field energy of these bodies to the gravitons with increase in their mass and frequency. It enables registration of the gravitons with the help of the detector located near to massive bodies. The assumption is made that dark energy of a gravitational field is all set of the graviton energies of a space.

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A one-dimensional self-gravitating stellar gas

Symposium - International Astronomical Union ◽

10.1017/s0074180900105261 ◽

1966 ◽

Vol 25 ◽

pp. 46-48 ◽

Cited By ~ 2

Author(s):

M. Lecar

Keyword(s):

Gravitational Field ◽

Phase Space ◽

Motion Picture ◽

Space Distribution ◽

Two Dimensional ◽

One Dimensional ◽

Phase Space Distribution ◽

Dimensional Phase Space ◽

The Mean

“Dynamical mixing”, i.e. relaxation of a stellar phase space distribution through interaction with the mean gravitational field, is numerically investigated for a one-dimensional self-gravitating stellar gas. Qualitative results are presented in the form of a motion picture of the flow of phase points (representing homogeneous slabs of stars) in two-dimensional phase space.

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