Matter in curved spacetime

2018 ◽  
Author(s):  
Nathalie Deruelle ◽  
Jean-Philippe Uzan
Keyword(s):  
Angular Momentum ◽  
Gravitational Field ◽  
Opposite Direction ◽  
Equivalence Principle ◽  
Equations Of Motion ◽  
Field Configuration ◽  
Curved Spacetime ◽  
Coordinate Grid ◽  
External Gravitational Field ◽  
Spatio Temporal

This chapter is concerned with the laws of motion of matter—particles, fluids, or fields—in the presence of an external gravitational field. In accordance with the equivalence principle, this motion will be ‘free’. That is, it is constrained only by the geometry of the spacetime whose curvature represents the gravitation. The concepts of energy, momentum, and angular momentum follow from the invariance of the solutions of the equations of motion under spatio-temporal translations or rotations. The chapter shows how the action is transformed, no longer under a modification of the field configuration, but instead under a displacement or, in the ‘passive’ version, under a translation of the coordinate grid in the opposite direction.

Orbital dynamics of the gravitational field of stringy black holes

2014 ◽  
Vol 29 (29) ◽  
pp. 1450144 ◽  
Author(s):  
Yu Zhang ◽  
Jin-Ling Geng ◽  
En-Kun Li
Keyword(s):  
Black Holes ◽  
Angular Momentum ◽  
Gravitational Field ◽  
Phase Plane ◽  
Equations Of Motion ◽  
Orbital Dynamics ◽  
Phase Plane Analysis ◽  
Test Particles ◽  
Stable Circular Orbit ◽  
The Stability

In this paper, we study the orbital dynamics of the gravitational field of stringy black holes by analyzing the effective potential and the phase plane diagram. By solving the equation of Lagrangian, the general relativistic equations of motion in the gravitational field of stringy black holes are given. It is easy to find that the motion of test particles depends on the energy and angular momentum of the test particles. Using the phase plane analysis method and combining the conditions of the stability, we discuss different types of the test particles' orbits in the gravitational field of stringy black holes. We get the innermost stable circular orbit which occurs at r min = 5.47422 and when the angular momentum b ≤ 4.3887 the test particles will fall into the black hole.

scholarly journals Quasi-uniform gravitational field of a disk revisited

2019 ◽  
Vol 34 (33) ◽  
pp. 1950228
Author(s):  
Alexander J. Silenko ◽  
Yury A. Tsalkou
Keyword(s):  
Gravitational Field ◽  
Equivalence Principle ◽  
Equations Of Motion ◽  
Field Approximation ◽  
Weak Field ◽  
Riemann Tensor ◽  
Weak Field Approximation

We calculate the quasi-uniform gravitational field of a disk in the weak-field approximation and demonstrate an inappropriateness of preceding results. The Riemann tensor of this field is determined. The nonexistence of the uniform gravitational field is proven without the use of the weak-field approximation. The previously found difference between equations of motion for the momentum and spin in the accelerated frame and in the quasi-uniform gravitational field also takes place for the disk. However, it does not violate the Einstein equivalence principle because of the nonexistence of the uniform gravitational field.

DELTA GRAVITY AND THE ACCELERATED EXPANSION OF THE UNIVERSE

2011 ◽  
Vol 20 (supp01) ◽  
pp. 65-72
Author(s):  
JORGE ALFARO
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Cosmological Constant ◽  
Equivalence Principle ◽  
Equations Of Motion ◽  
Accelerated Expansion ◽  
Symmetric Tensors ◽  
Test Particles ◽  
Expansion Of The Universe ◽  
The Universe

We study a model of the gravitational field based on two symmetric tensors. The equations of motion of test particles are derived. We explain how the Equivalence principle is recovered. Outside matter, the predictions of the model coincide exactly with General Relativity, so all classical tests are satisfied. In Cosmology, we get accelerated expansion without a cosmological constant.

The effect of the earth’s oblateness on the orbit of a near satellite

1958 ◽  
Vol 247 (1248) ◽  
pp. 49-72 ◽  
Keyword(s):  
Angular Momentum ◽  
Opposite Direction ◽  
Equations Of Motion ◽  
Radial Distance ◽  
Orbital Plane ◽  
Major Axis ◽  
Polar Orbit ◽  
The Mean ◽  
Main Effects ◽  
Earth’S Oblateness

The equations of motion of a satellite in an orbit over an oblate earth in vacuo are solved analytically, by a perturbation method. The solution applies primarily to orbits of eccentricity 0⋅05 or less. The accuracy of the solution for radial distance should then be about 0⋅001%, and the error in angular travel about 0⋅001% per revolution. The earth’s oblateness has four main effects on the motion: (1) The orbital plane, instead of remaining fixed, rotates about the earth’s axis in the opposite direction to the satellite, at a rate of 10⋅00( R / r̄ ) 3.5 cos α deg./day, where α is the inclination of the orbital plane to the equator, R the earth’s equatorial radius and r̄ the satellite’s mean distance from the earth’s centre. (2) The period of revolution of the satellite, from one northward crossing of the equator to the next, is 14⋅5 √( R/r̄ ) sin 2 α sec greater for an inclined orbit than for an equatorial orbit. (3) The radial distance r from the earth’s centre changes. For a given angular momentum the mean r is 14⋅5 R/r̄ nautical miles greater for a polar orbit than an equatorial one. Also, during each revolution r oscillates twice, the amplitude of the oscillation being 0⋅94 ( R/r̄ ) sin 2 α n. miles. (4) The major axis of the orbit rotates in the orbital plane at a rate of 5⋅00( R/r̄ ) 3.5 (5 cos 2 α —1) deg./day. Thus it rotates in the same direction as the satellite if α < 63⋅4°, or in the opposite direction if α > 63⋅4°. A brief comparison is made between theory and observation for Sputniks 1 and 2.

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.

On a non-symmetric theory of the pure gravitational field

1958 ◽  
Vol 54 (1) ◽  
pp. 72-80 ◽  
Author(s):  
D. W. Sciama
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Symmetric Tensor ◽  
Momentum Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Metric Tensor ◽  
Unified Field

ABSTRACTIt is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.

scholarly journals Interesting Examples of Violation of the Classical Equivalence Principle but Not of the Weak One

2018 ◽  
Vol 2018 ◽  
pp. 1-13
Author(s):  
Antonio Accioly ◽  
Wallace Herdy
Keyword(s):  
Wave Function ◽  
Quantum Mechanics ◽  
Gravitational Field ◽  
Experimental Test ◽  
Equivalence Principle ◽  
Free Fall ◽  
Higher Order ◽  
Tree Level ◽  
Quantum Particles ◽  
Bohr Atom

The equivalence principle (EP) and Schiff’s conjecture are discussed en passant, and the connection between the EP and quantum mechanics is then briefly analyzed. Two semiclassical violations of the classical equivalence principle (CEP) but not of the weak one (WEP), i.e., Greenberger gravitational Bohr atom and the tree-level scattering of different quantum particles by an external weak higher-order gravitational field, are thoroughly investigated afterwards. Next, two quantum examples of systems that agree with the WEP but not with the CEP, namely, COW experiment and free fall in a constant gravitational field of a massive object described by its wave-function Ψ, are discussed in detail. Keeping in mind that, among the four examples focused on in this work only COW experiment is based on an experimental test, some important details related to it are presented as well.

scholarly journals Generalizing the coupling between geometry and matter: $$f\left( R,L_m,T\right) $$ gravity

2021 ◽  
Vol 81 (7) ◽  
Author(s):  
Zahra Haghani ◽  
Tiberiu Harko
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Momentum Tensor ◽  
Newtonian Limit ◽  
Gravity Models ◽  
Momentum Balance ◽  
Field Equations ◽  
Gravitational Fields ◽  
Generalized Poisson

AbstractWe generalize and unify the $$f\left( R,T\right) $$ f R , T and $$f\left( R,L_m\right) $$ f R , L m type gravity models by assuming that the gravitational Lagrangian is given by an arbitrary function of the Ricci scalar R, of the trace of the energy–momentum tensor T, and of the matter Lagrangian $$L_m$$ L m , so that $$ L_{grav}=f\left( R,L_m,T\right) $$ L grav = f R , L m , T . We obtain the gravitational field equations in the metric formalism, the equations of motion for test particles, and the energy and momentum balance equations, which follow from the covariant divergence of the energy–momentum tensor. Generally, the motion is non-geodesic, and takes place in the presence of an extra force orthogonal to the four-velocity. The Newtonian limit of the equations of motion is also investigated, and the expression of the extra acceleration is obtained for small velocities and weak gravitational fields. The generalized Poisson equation is also obtained in the Newtonian limit, and the Dolgov–Kawasaki instability is also investigated. The cosmological implications of the theory are investigated for a homogeneous, isotropic and flat Universe for two particular choices of the Lagrangian density $$f\left( R,L_m,T\right) $$ f R , L m , T of the gravitational field, with a multiplicative and additive algebraic structure in the matter couplings, respectively, and for two choices of the matter Lagrangian, by using both analytical and numerical methods.

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.

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