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.

scholarly journals Complete set of quasi-conserved quantities for spinning particles around Kerr

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
Geoffrey Compère ◽  
Adrien Druart
Keyword(s):  
Linear Order ◽  
Conserved Quantities ◽  
Killing Tensor ◽  
Kerr Geometry ◽  
Spinning Particles ◽  
Kerr Spacetime ◽  
Killing Tensors ◽  
Mixed Symmetry ◽  
Constants Of Motion ◽  
Complete Set

We revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles on a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, we demonstrate that besides the two non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, found by Rüdiger, non-trivial quasi-conserved quantities are in one-to-one correspondence with non-trivial mixed-symmetry Killing tensors. We prove that no such stationary and axisymmetric mixed-symmetry Killing tensor exists on the Kerr geometry. We discuss the implications for the motion of spinning particles on Kerr spacetime where the quasi-constants of motion are shown not to be in complete involution.

Spinning test-particles in general relativity. I

1951 ◽  
Vol 209 (1097) ◽  
pp. 248-258 ◽  
Keyword(s):  
General Relativity ◽  
Gravitational Field ◽  
Equations Of Motion ◽  
Covariant Form ◽  
Test Particles

A method for the derivation of the equations of motion of test particles in a given gravitational field is developed. The equations of motion of spinning test particles are derived. The transformation properties are discussed and the equations of motion are written in a covariant form.

Conserved quantities of spinning test particles in general relativity. I

1981 ◽  
Vol 375 (1761) ◽  
pp. 185-193 ◽  
Keyword(s):  
General Relativity ◽  
Vector Field ◽  
Tensor Field ◽  
General Scheme ◽  
Killing Vector ◽  
Conserved Quantity ◽  
General Linear ◽  
Killing Vector Field ◽  
Conserved Quantities ◽  
Test Particles

This is the first of two papers devoted to conserved quantities of spinning test particles in general relativity. In this paper, a general scheme is described according to which these quantities can be investigated. It is shown that the general linear conserved quantity consists of a sum, the first term of which is the well known expression constructed from a Killing vector field, and the second term is of the form U kl S kl , where U* ab is a Killing-Yano tensor field, which is constrained by two additional equations.

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.

scholarly journals Particle paths of general relativity as geodesics of an affine connection

1970 ◽  
Vol 3 (3) ◽  
pp. 325-335 ◽  
Author(s):  
R. Burman
Keyword(s):  
General Relativity ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Affine Connection ◽  
Momentum Tensor ◽  
Field Equations ◽  
Test Particles ◽  
Special Cases ◽  
Particle Paths ◽  
Arbitrary Energy

This paper deals with the motion of incoherent matter, and hence of test particles, in the presence of fields with an arbitrary energy-momentum tensor. The equations of motion are obtained from Einstein's field equations and are written in the form of geodesic equations of an affine connection. The special cases of the electromagnetic field, the Proca field and a scalar field are discussed.

scholarly journals Non-Relativistic Limit of Embedding Gravity as General Relativity with Dark Matter

Universe ◽  
2020 ◽  
Vol 6 (10) ◽  
pp. 163 ◽  
Author(s):  
Sergey Paston
Keyword(s):  
General Relativity ◽  
Dark Matter ◽  
Cold Dark Matter ◽  
Equations Of Motion ◽  
Dynamical Equations ◽  
Relativistic Limit ◽  
Change Of Variables ◽  
Modified Theory ◽  
Mimetic Gravity

Regge-Teitelboim embedding gravity is the modified gravity based on a simple string-inspired geometrical principle—our spacetime is considered here as a 4-dimensional surface in a flat bulk. This theory is similar to the recently popular theory of mimetic gravity—the modification of gravity appears in both theories as a result of the change of variables in the action of General Relativity. Embedding gravity, as well as mimetic gravity, can be used in explaining the dark matter mystery since, in both cases, the modified theory can be presented as General Relativity with additional fictitious matter (embedding matter or mimetic matter). For the general case, we obtain the equations of motion of embedding matter in terms of embedding function as a set of first-order dynamical equations and constraints consistent with them. Then, we construct a non-relativistic limit of these equations, in which the motion of embedding matter turns out to be slow enough so that it can play the role of cold dark matter. The non-relativistic 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 Λ-Cold Dark Matter (ΛCDM) model.

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.

Spinning test-particles in general relativity. II

1951 ◽  
Vol 209 (1097) ◽  
pp. 259-268 ◽  
Keyword(s):  
General Relativity ◽  
Equations Of Motion ◽  
Test Particles

The equations of motion for spinning test particles are discussed for particles characterized by the condition S i 4 ═ 0.

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.

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