The gravitational field of a rotating fluid mass in general relativity

1962 ◽  
Vol 270 (1343) ◽  
pp. 467-492 ◽  
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Residual Energy ◽  
The Body ◽  
Energy Tensor ◽  
Constant Density ◽  
Rotating Fluid ◽  
Fluid Mass

A method recently given by Das, Florides & Synge is now slightly modified and applied to find the gravitational field of a steadily rotating fluid mass, not necessarily of constant density. The result is approximate in the sense that, outside the body, there is a residual energy tensor T ij such that is small of the order (m/a) 3 , where m is the mass of the body and a a typical radius.

The post-Newtonian approximation

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Body Problem ◽  
Equations Of Motion ◽  
Two Body Problem ◽  
Newtonian Approximation ◽  
Actual Motion ◽  
Law Of Attraction ◽  
Universal Law ◽  
Two Bodies

This chapter embarks on a study of the two-body problem in general relativity. In other words, it seeks to describe the motion of two compact, self-gravitating bodies which are far-separated and moving slowly. It limits the discussion to corrections proportional to v2 ~ m/R, the so-called post-Newtonian or 1PN corrections to Newton’s universal law of attraction. The chapter first examines the gravitational field, that is, the metric, created by the two bodies. It then derives the equations of motion, and finally the actual motion, that is, the post-Keplerian trajectories, which generalize the post-Keplerian geodesics obtained earlier in the chapter.

The precise calculations of the constant terms in the equations of motions of planets and photons of general relativity

2021 ◽  
Vol 34 (2) ◽  
pp. 183-192
Author(s):  
Mei Xiaochun
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Deflection Angle ◽  
Constant Term ◽  
Equation Of Motion ◽  
Time Dependent ◽  
Newtonian Gravity ◽  
Newtonian Theory ◽  
The Deflection Angle ◽  
Equations Of Motions

In general relativity, the values of constant terms in the equations of motions of planets and light have not been seriously discussed. Based on the Schwarzschild metric and the geodesic equations of the Riemann geometry, it is proved in this paper that the constant term in the time-dependent equation of motion of planet in general relativity must be equal to zero. Otherwise, when the correction term of general relativity is ignored, the resulting Newtonian gravity formula would change its basic form. Due to the absence of this constant term, the equation of motion cannot describe the elliptical and the hyperbolic orbital motions of celestial bodies in the solar gravitational field. It can only describe the parabolic orbital motion (with minor corrections). Therefore, it becomes meaningless to use general relativity calculating the precession of Mercury's perihelion. It is also proved that the time-dependent orbital equation of light in general relativity is contradictory to the time-independent equation of light. Using the time-independent orbital equation to do calculation, the deflection angle of light in the solar gravitational field is <mml:math display="inline"> <mml:mrow> <mml:mn>1.7</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> . But using the time-dependent equation to do calculation, the deflection angle of light is only a small correction of the prediction value <mml:math display="inline"> <mml:mrow> <mml:mn>0.87</mml:mn> <mml:msup> <mml:mn>5</mml:mn> <mml:mo>″</mml:mo> </mml:msup> </mml:mrow> </mml:math> of the Newtonian gravity theory with a magnitude order of <mml:math display="inline"> <mml:mrow> <mml:msup> <mml:mrow> <mml:mn>10</mml:mn> </mml:mrow> <mml:mrow> <mml:mo>−</mml:mo> <mml:mn>5</mml:mn> </mml:mrow> </mml:msup> </mml:mrow> </mml:math> . The reason causing this inconsistency was the Einstein's assumption that the motion of light satisfied the condition <mml:math display="inline"> <mml:mrow> <mml:mi>d</mml:mi> <mml:mi>s</mml:mi> <mml:mo>=</mml:mo> <mml:mn>0</mml:mn> </mml:mrow> </mml:math> in gravitational field. It leads to the absence of constant term in the time-independent equation of motion of light and destroys the uniqueness of geodesic in curved space-time. Meanwhile, light is subjected to repulsive forces in the gravitational field, rather than attractive forces. The direction of deflection of light is opposite, inconsistent with the predictions of present general relativity and the Newtonian theory of gravity. Observing on the earth surface, the wavelength of light emitted by the sun is violet shifted. This prediction is obviously not true. Practical observation is red shift. Finally, the practical significance of the calculation of the Mercury perihelion's precession and the existing problems of the light's deflection experiments of general relativity are briefly discussed. The conclusion of this paper is that general relativity cannot have consistence with the Newtonian theory of gravity for the descriptions of motions of planets and light in the solar system. The theory itself is not self-consistent too.

THREE DIMENSIONAL INTERPRETATION OF GRAVITATIONAL ANOMALIES

Geophysics ◽  
1952 ◽  
Vol 17 (2) ◽  
pp. 344-364 ◽  
Author(s):  
Fraser S. Grant
Keyword(s):  
Gravitational Field ◽  
Mass Distribution ◽  
Three Dimensional ◽  
Relaxation Method ◽  
The Body ◽  
Multipole Moments ◽  
Size And Shape ◽  
Surface Field ◽  
Derivatives Of

A method is developed for determining the approximate size and shape of the three‐dimensional mass distribution that is required to produce a given gravitational field. The first few reduced multipole moments of the distribution are calculated from the derivatives of the surface field, and the approximative structure is determined from the values of these moments and a knowledge of the density contrast between the body and its surroundings. A system of classification of problems by symmetry is introduced and its practical usage discussed. A relaxation method is described which may be used to adjust the initial solution systematically to give agreement over the whole field. A descriptive discussion is appended.

General Relativity: An Introduction to the Theory of the Gravitational Field

1984 ◽  
Vol 52 (5) ◽  
pp. 478-479
Author(s):  
Hans Stephani ◽  
Jeffrey M. Bowen
Keyword(s):  
General Relativity ◽  
Gravitational Field

scholarly journals Stress Energy Tensor Study in Fluid Mechanics

2019 ◽  
Author(s):  
Roman Baudrimont
Keyword(s):  
General Relativity ◽  
Fluid Mechanics ◽  
Space Time ◽  
Newtonian Fluids ◽  
Third Party ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Perfect Fluids ◽  
Stress Energy

This paper is to summarize the involvement of the stress energy tensor in the study of fluid mechanics. In the first part we will see the implication that carries the stress energy tensor in the framework of general relativity. In the second part, we will study the stress energy tensor under the mechanics of perfect fluids, allowing us to lead third party in the case of Newtonian fluids, and in the last part we will see that it is possible to define space-time as a no-Newtonian fluids.

scholarly journals Foundations of a Theory of Gravity with a Constraint and Its Canonical Quantization

2021 ◽  
Vol 52 (1) ◽  
Author(s):  
Alexander P. Sobolev
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Early Universe ◽  
Canonical Quantization ◽  
Entropy Density ◽  
Maximum Temperature ◽  
Past Century ◽  
Physical Point ◽  
Theory Of Gravity ◽  
The Universe

AbstractThe gravitational equations were derived in general relativity (GR) using the assumption of their covariance relative to arbitrary transformations of coordinates. It has been repeatedly expressed an opinion over the past century that such equality of all coordinate systems may not correspond to reality. Nevertheless, no actual verification of the necessity of this assumption has been made to date. The paper proposes a theory of gravity with a constraint, the degenerate variants of which are general relativity (GR) and the unimodular theory of gravity. This constraint is interpreted from a physical point of view as a sufficient condition for the adiabaticity of the process of the evolution of the space–time metric. The original equations of the theory of gravity with the constraint are formulated. On this basis, a unified model of the evolution of the modern, early, and very early Universe is constructed that is consistent with the observational astronomical data but does not require the hypotheses of the existence of dark energy, dark matter or inflatons. It is claimed that: physical time is anisotropic, the gravitational field is the main source of energy of the Universe, the maximum global energy density in the Universe was 64 orders of magnitude smaller the Planckian one, and the entropy density is 18 orders of magnitude higher the value predicted by GR. The value of the relative density of neutrinos at the present time and the maximum temperature of matter in the early Universe are calculated. The wave equation of the gravitational field is formulated, its solution is found, and the nonstationary wave function of the very early Universe is constructed. It is shown that the birth of the Universe was random.

scholarly journals Simulation of a rotating strong gravity that reverses time

2021 ◽  
pp. 7-16
Author(s):  
Yoshio Matsuki ◽  
Petro Bidyuk
Keyword(s):  
Angular Momentum ◽  
Gravitational Field ◽  
Momentum Density ◽  
Polar Coordinates ◽  
Energy Tensor ◽  
Stress Energy Tensor ◽  
Strong Gravity ◽  
The Past ◽  
Stress Energy ◽  
The Future

In this research we simulated how time can be reversed with a rotating strong gravity. At first, we assumed that the time and the space can be distorted with the presence of a strong gravity, and then we calculated the angular momentum density of the rotating gravitational field. For this simulation we used Einstein’s field equation with spherical polar coordinates and the Euler’s transformation matrix to simulate the rotation. We also assumed that the stress-energy tensor that is placed at the end of the strong gravitational field reflects the intensities of the angular momentum, which is the normal (perpendicular) vector to the rotating axis. The result of the simulation shows that the angular momentum of the rotating strong gravity changes its directions from plus (the future) to minus (the past) and from minus (the past) to plus (the future), depending on the frequency of the rotation.

SPACE–TIME ENERGY–MOMENTUM, THE OBSERVER AND UNCERTAINTY IN GENERAL RELATIVITY

2010 ◽  
Vol 19 (14) ◽  
pp. 2353-2359 ◽  
Author(s):  
F. I. COOPERSTOCK ◽  
M. J. DUPRE
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Uncertainty Principle ◽  
Ricci Tensor ◽  
Space Time ◽  
Energy Integral ◽  
New Approach ◽  
Extended Space ◽  
Energy And Momentum

In this essay, we introduce a new approach to energy–momentum in general relativity. Space–time, as opposed to space, is recognized as the necessary arena for its examination, leading us to define new extended space–time energy and momentum constructs. From local and global considerations, we conclude that the Ricci tensor is the required element for a localized expression of energy–momentum to include the gravitational field. We present and rationalize a fully invariant extended expression for space–time energy, guided by Tolman's well-known energy integral for an arbitrary bounded stationary system. This raises fundamental issues which we discuss. The role of the observer emerges naturally and we are led to an extension of the uncertainty principle to general relativity, of particular relevance to ultra-strong gravity.

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