Bondi’s vector, and the Seliger & Whitham Variational Principle for matter fields

1979 ◽  
Vol 368 (1732) ◽  
pp. 59-73
Keyword(s):  
General Relativity ◽  
Variational Principle ◽  
Variational Methods ◽  
Poynting Vector ◽  
Equations Of Motion ◽  
Direct Calculation ◽  
Newtonian Limit ◽  
Conserved Quantities ◽  
Matter Fields

Bondi’s Newtonian Poynting Vector is extended to other ‘Bondi-type’ conserved quantities, which are also derived by variational methods. By means of the technique of Seliger & Whitham it is shown that they are the expressions which arise most naturally in contexts concerned with the equations of motion. The pseudo-tensor derived from the Lagrangian R √(— g) is the corresponding expression in General Relativity; direct calculation verifies that it tends to Bondi’s vector in the Newtonian limit.

ON EQUATIONS OF MOTION IN GENERAL RELATIVITY

1955 ◽  
Vol 33 (12) ◽  
pp. 824-827
Author(s):  
G. E. Tauber
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Charged Particle ◽  
Variational Principle ◽  
Classical Theory ◽  
Equations Of Motion ◽  
Field Equations

It has been shown that both the equations of motion of a charged particle in a gravitational field and the field equations can be obtained from one variational principle by suitably generalizing Dirac's classical theory of electrons.

Conserved quantities of spinning test particles in general relativity. II

1983 ◽  
Vol 385 (1788) ◽  
pp. 229-239 ◽  
Keyword(s):  
General Relativity ◽  
Equations Of Motion ◽  
Conserved Quantities ◽  
Killing Tensor ◽  
Kerr Geometry ◽  
Change Of Variables ◽  
Test Particles ◽  
Suitable Change

In this paper, which completes earlier work on conserved quantities of spinning test particles in general relativity (Rüdiger 1981 a ), quadratic conserved quantities are considered. It is shown that by a suitable change of variables the trivial conserved quantities, which result from a reducible Killing tensor, can essentially be separated from the non-trivial quantities. If the equations of motion are linearized in the spin, it is shown that nontrivial quantities of this type can be constructed for two classes of spacetimes including the Kerr geometry and the Friedman universe.

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 mechanics of general relativity

1959 ◽  
Vol 251 (1264) ◽  
pp. 55-65 ◽  
Keyword(s):  
General Relativity ◽  
General Theory ◽  
Equations Of Motion ◽  
Stress Singularity ◽  
Theory Of Relativity ◽  
General Theory Of Relativity

It is shown how to obtain, within the general theory of relativity, equations of motion for two oscillating masses at the ends of a spring of given law of force. The method of Einstein, Infeld & Hoffmann is used, and the force in the spring is represented by a stress singularity. The detailed calculations are taken to the Newtonian order.

scholarly journals CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

2000 ◽  
Vol 09 (01) ◽  
pp. 13-34 ◽  
Author(s):  
GEN YONEDA ◽  
HISA-AKI SHINKAI
Keyword(s):  
General Relativity ◽  
Hyperbolic System ◽  
Hyperbolic Systems ◽  
Equations Of Motion ◽  
Well Posedness ◽  
Symmetric Hyperbolic System ◽  
The Cauchy Problem ◽  
Long Time ◽  
Vacuum Spacetime ◽  
Gauge Conditions

Hyperbolic formulations of the equations of motion are essential technique for proving the well-posedness of the Cauchy problem of a system, and are also helpful for implementing stable long time evolution in numerical applications. We, here, present three kinds of hyperbolic systems in the Ashtekar formulation of general relativity for Lorentzian vacuum spacetime. We exhibit several (I) weakly hyperbolic, (II) diagonalizable hyperbolic, and (III) symmetric hyperbolic systems, with each their eigenvalues. We demonstrate that Ashtekar's original equations form a weakly hyperbolic system. We discuss how gauge conditions and reality conditions are constrained during each step toward constructing a symmetric hyperbolic system.

scholarly journals Dark matter from non-relativistic embedding gravity

2021 ◽  
pp. 2150101
Author(s):  
S. A. Paston
Keyword(s):  
General Relativity ◽  
Dark Matter ◽  
Explicit Form ◽  
Equations Of Motion ◽  
Additional Contribution ◽  
Relativistic Limit ◽  
The Core ◽  
The Stability ◽  
Dimensional Surface

We study the possibility to explain the mystery of the dark matter (DM) through the transition from General Relativity to embedding gravity. This modification of gravity, which was proposed by Regge and Teitelboim, is based on a simple string-inspired geometrical principle: our spacetime is considered here as a four-dimensional surface in a flat bulk. We show that among the solutions of embedding gravity, there is a class of solutions equivalent to solutions of GR with an additional contribution of non-relativistic embedding matter, which can serve as cold DM. We prove the stability of such type of solutions and obtain an explicit form of the equations of motion of embedding matter in the non-relativistic limit. According to them, embedding matter turns out to have a certain self-interaction, which could be useful in the context of solving the core-cusp problem that appears in the [Formula: see text]CDM model.

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.

NUMERICAL STUDY OF THE INTERACTION BETWEEN POSITIVE AND NEGATIVE MASS OBJECTS IN GENERAL RELATIVITY

2012 ◽  
Vol 21 (07) ◽  
pp. 1250061 ◽  
Author(s):  
ZHOUJIAN CAO
Keyword(s):  
Boundary Condition ◽  
General Relativity ◽  
Numerical Study ◽  
Naked Singularity ◽  
Newtonian Limit ◽  
Computational Domain ◽  
Positive Mass ◽  
Negative Mass ◽  
Mass Particle ◽  
Causal Property

Based on Baumgarte–Shapiro–Shibata–Nakamura formalism and moving puncture method, we demonstrate the first numerical evolutions of the interaction between positive and negative mass objects. Using the causal property of general relativity, we set our computational domain around the positive mass black hole while excluding the region around the naked singularity introduced by the negative mass object. Besides the usual Sommerfeld numerical boundary condition, an approximate boundary condition is proposed for this nonasymptotically-flat computational domain. Careful checks show that either boundary condition introduces smaller error than the numerical truncation errors. This is consistent with the causal property of general relativity. Except for the numerical truncation error and round-off error, our method gives an exact solution to the full Einstein's equation for a portion of spacetime with two objects whose masses have opposite signs. So our method opens the door for numerical explorations with negative mass objects. Based on this method, we investigate the Newtonian limit of spacetime with two objects whose masses have opposite sign. Our result implies that this spacetime does have a Newtonian limit which corresponds to a negative mass particle chasing a positive mass particle. This result sheds some light on an interesting debate about the Newtonian limit of a spacetime with positive and negative point masses.

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