Simplicial Palatini action

We consider the piecewise flat spacetime and a simplicial analog of the Palatini form of the general relativity (GR) action where the discrete Christoffel symbols are given on the tetrahedra as variables that are independent of the metric. Excluding these variables with the help of the equations of motion gives exactly the Regge action. This paper continues our previous work. Now, we include the parity violation term and the analog of the Barbero–Immirzi parameter introduced in the orthogonal connection form of GR. We consider the path integral and the functional integration over the connection. The result of the latter (for certain limiting cases of some parameters) is compared with the earlier found result of the functional integration over the connection for the analogous orthogonal connection representation of Regge action. These results, mainly as some measures on the lengths/areas, are discussed for the possibility of the diagram technique where the perturbative diagrams for the Regge action calculated using the measure obtained are finite. This finiteness is due to these measures providing elementary lengths being mostly bounded and separated from zero, just as the finiteness of a theory on a lattice with an analogous probability distribution of spacings.

Download Full-text

The post-Newtonian approximation

10.1093/oso/9780198786399.003.0052 ◽

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.

Download Full-text

PATH INTEGRAL QUANTIZATION OF SUPERSTRING

Modern Physics Letters A ◽

10.1142/s0217732310032287 ◽

2010 ◽

Vol 25 (02) ◽

pp. 135-141

Author(s):

H. A. ELEGLA ◽

N. I. FARAHAT

Keyword(s):

Phase Space ◽

Differential Equations ◽

Path Integral ◽

Equations Of Motion ◽

Singular System ◽

Constrained Systems ◽

Classical Structure ◽

Path Integral Quantization ◽

Total Differential

Motivated by the Hamilton–Jacobi approach of constrained systems, we analyze the classical structure of a four-dimensional superstring. The equations of motion for a singular system are obtained as total differential equations in many variables. The path integral quantization based on Hamilton–Jacobi approach is applied to quantize the system, and the integration is taken over the canonical phase space coordinates.

Download Full-text

The mechanics of general relativity

Proceedings of the Royal Society of London Series A - Mathematical and Physical Sciences ◽

10.1098/rspa.1959.0089 ◽

1959 ◽

Vol 251 (1264) ◽

pp. 55-65 ◽

Cited By ~ 1

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.

Download Full-text

CONSTRUCTING HYPERBOLIC SYSTEMS IN THE ASHTEKAR FORMULATION OF GENERAL RELATIVITY

International Journal of Modern Physics D ◽

10.1142/s0218271800000037 ◽

2000 ◽

Vol 09 (01) ◽

pp. 13-34 ◽

Cited By ~ 6

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.

Download Full-text

Dark matter from non-relativistic embedding gravity

Modern Physics Letters A ◽

10.1142/s0217732321501017 ◽

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.

Download Full-text

Do electromagnetic waves harbour gravitational waves?

Proceedings of The Royal Society A Mathematical Physical and Engineering Sciences ◽

10.1098/rspa.2006.1658 ◽

2006 ◽

Vol 462 (2071) ◽

pp. 1987-2000 ◽

Cited By ~ 6

Author(s):

Brian Bramson

Keyword(s):

General Relativity ◽

Gravitational Waves ◽

Electric Dipole ◽

Electromagnetic Waves ◽

Maxwell Theory ◽

Flat Spacetime

In linearized, Einstein–Maxwell theory on flat spacetime, an oscillating electric dipole is the source of a spin-2 field. Within this approximation to general relativity, it is shown that electromagnetic waves harbour gravitational waves.

Download Full-text

Some Cosmological Solutions of a New Nonlocal Gravity Model

Symmetry ◽

10.3390/sym12060917 ◽

2020 ◽

Vol 12 (6) ◽

pp. 917

Author(s):

Ivan Dimitrijevic ◽

Branko Dragovich ◽

Alexey S. Koshelev ◽

Zoran Rakic ◽

Jelena Stankovic

Keyword(s):

General Relativity ◽

Dark Energy ◽

Analytic Function ◽

Cosmological Constant ◽

Explicit Form ◽

Gravity Model ◽

Necessary Conditions ◽

Equations Of Motion ◽

Nonlocal Operator ◽

Cosmological Solutions

In this paper, we investigate a nonlocal modification of general relativity (GR) with action S = 1 16 π G ∫ [ R − 2 Λ + ( R − 4 Λ ) F ( □ ) ( R − 4 Λ ) ] − g d 4 x , where F ( □ ) = ∑ n = 1 + ∞ f n □ n is an analytic function of the d’Alembertian □. We found a few exact cosmological solutions of the corresponding equations of motion. There are two solutions which are valid only if Λ ≠ 0 , k = 0 , and they have no analogs in Einstein’s gravity with cosmological constant Λ . One of these two solutions is a ( t ) = A t e Λ 4 t 2 , that mimics properties similar to an interference between the radiation and the dark energy. Another solution is a nonsingular bounce one a ( t ) = A e Λ t 2 . For these two solutions, some cosmological aspects are discussed. We also found explicit form of the nonlocal operator F ( □ ) , which satisfies obtained necessary conditions.

Download Full-text