The weak field approximation

2019 ◽  
pp. 66-71
Author(s):  
Steven Carlip
Keyword(s):  
Field Approximation ◽  
Weak Field ◽  
Gravitational Energy ◽  
Newtonian Gravity ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Frame Dragging ◽  
Coupled Partial Differential Equations ◽  
Flat Spacetime ◽  
Weak Fields

The Einstein field equations are a complicated set of coupled partial differential equations, which are usually too complicated to find exact solutions. This chapter introduces a simple approximation for weak fields. It discusses the lowest order solution, which gives back Newtonian gravity, and the next order, which includes “gravitomagnetic” or “frame-dragging” effects. The chapter briefly discusses higher order approximations, expansions around a curved background, and the evidence that gravitational energy itself gravitates. It concludes with a brief description of an alternative derivation of the Einstein field equations, starting from flat spacetime and “bootstrapping” the gravitational self-interaction.

Gravitational waves

2019 ◽  
pp. 72-79
Author(s):  
Steven Carlip
Keyword(s):  
Gravitational Waves ◽  
Gravitational Radiation ◽  
Field Approximation ◽  
Weak Field ◽  
Quadrupole Moments ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Compact Binary ◽  
Speed Of Gravity ◽  
Weak Field Approximation

In the weak field approximation, the Einstein field equations can be solved, and lead to the prediction of gravitational waves. After showing that gravitational radiation depends on changing quadrupole moments, this chapter describes the production, propagation, and detection of gravitational waves. It includes discussions of the speed of gravity, detectors, the “chirp” waveform for a compact binary system, and the nature of astrophysical sources.

The Einstein field equations

2019 ◽  
pp. 52-58
Author(s):  
Steven Carlip
Keyword(s):  
General Relativity ◽  
Conservation Laws ◽  
Newtonian Gravity ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Noether’S Theorem ◽  
Geodesic Deviation ◽  
Fundamental Equations ◽  
Hilbert Action ◽  
Qualitative Discussion

The Einstein field equations are the fundamental equations of general relativity. After a brief qualitative discussion of geodesic deviation and Newtonian gravity, this chapter derives the field equations from the Einstein-Hilbert action. The chapter contains a derivation of Noether’s theorem and the consequent conservation laws, and a brief discussion of generalizations of the Einstein-Hilbert action.

Dynamical system analysis of a nonminimally coupled scalar field

2019 ◽  
Vol 28 (15) ◽  
pp. 1950173 ◽  
Author(s):  
Subhajyoti Pal ◽  
Sudip Mishra ◽  
Subenoy Chakraborty
Keyword(s):  
Scalar Field ◽  
Critical Points ◽  
Center Manifold ◽  
System Analysis ◽  
Nonlinear Differential Equations ◽  
Center Manifold Theory ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Flat Spacetime ◽  
Manifold Theory

This paper deals with a nonminimally coupled scalar field in the background of homogeneous and isotropic Friedmann–Lemaître–Robertson–Walker (FLRW) flat spacetime. As Einstein field equations are coupled second-order nonlinear differential equations, it is very hard to find exact solutions. By suitable choice of variables, we transform Einstein field equations to an autonomous system and critical points are determined. We use center manifold theory to characterize nonhyperbolic critical points and are found to be saddle in nature. We discuss possible bifurcation scenarios, which indicate the existence of the cosmological bouncing model.

scholarly journals The Gravitomagnetism in the Solar System

2017 ◽  
Vol 45 ◽  
pp. 1760052
Author(s):  
Flavia Rocha ◽  
Manuel Malheiro ◽  
Rubens Marinho
Keyword(s):  
Mass Density ◽  
Slow Motion ◽  
Weak Field ◽  
Einstein Equations ◽  
Dipole Potential ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Perihelion Advance ◽  
Motion Approximation

In 1918, Joseph Lense and Hans Thirring discovered the gravitomagnetic (GM) effect of Einstein field equations in weak field and slow motion approximation. They showed that Einstein equations in this approximation can be written as in the same form as Maxwell’s equation for electromagnetism. In these equations the charge and electric current are replaced by the mass density and the mass current. Thus, the gravitomagnetism formalism in astrophysical system is used with the mass assuming the role of the charge. In this work, we present the deduction of gravitoelectromagnetic equations and the analogue of the Lorentz force in the gravitomagnetism. We also discuss the problem of Mercury’s perihelion advance orbit, we propose solutions using GM formalism using a dipole-dipole potential for the Sun-Planet interaction.

scholarly journals Massive Conformal Gravity

2014 ◽  
Vol 2014 ◽  
pp. 1-4 ◽  
Author(s):  
F. F. Faria
Keyword(s):  
Scalar Field ◽  
Field Approximation ◽  
Weak Field ◽  
Vacuum Field ◽  
Conformal Transformations ◽  
Conformal Gravity ◽  
Field Equations ◽  
Metric Tensor ◽  
Weak Field Approximation ◽  
Massive Theory

We construct a massive theory of gravity that is invariant under conformal transformations. The massive action of the theory depends on the metric tensor and a scalar field, which are considered the only field variables. We find the vacuum field equations of the theory and analyze its weak-field approximation and Newtonian limit.

scholarly journals SCHWARZSCHILD SOLUTION OF THE MODIFIED EINSTEIN FIELD EQUATIONS

2017 ◽  
Vol 13 (4) ◽  
pp. 4895-4900
Author(s):  
D.S. Wamalwa ◽  
Carringtone Kinyanjui
Keyword(s):  
Event Horizon ◽  
Interior Solution ◽  
Newtonian Gravity ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Schwarzschild Solution ◽  
Exterior Solution ◽  
Finsler Spacetime ◽  
Energy Approximation ◽  
Three Stages

A reformulation of the Schwarzschild solution of the linearized Einstein field equations in post-Riemannian Finsler spacetime is derived. The solution is constructed in three stages: the exterior solution, the event-horizon solution and the interior solution. It is shown that the exterior solution is asymptotically similar to Newtonian gravity at large distances implying that Newtonian gravity is a low energy approximation of the solution. Application of Eddington-Finklestein coordinates is shown to reproduce the results obtained from standard general relativity at the event horizon. Further application of Kruskal-Szekeres coordinates reveals that the interior solution contains maximally extensible geodesics.

scholarly journals Gravitational Paramagnetism, Diamagnetism and Gravitational Superconductivity

1996 ◽  
Vol 49 (6) ◽  
pp. 1063 ◽  
Author(s):  
M Agop ◽  
C Gh Buzea ◽  
V Griga ◽  
C Ciubotariu ◽  
C Buzea ◽  
...  
Keyword(s):  
Gravitational Field ◽  
Meissner Effect ◽  
Field Approximation ◽  
Weak Field ◽  
Gauge Model ◽  
Closed Geodesics ◽  
Field Equations ◽  
Gravitational Field Equations ◽  
Weak Field Approximation ◽  
Superconducting Parameters

In the weak field approximation to the gravitational field equations, we study gravitational paramagnetism and diamagnetism, the gravitational Meissner effect and gravitational superconductivity. The spontaneous symmetry breaking corresponds to crossing from closed geodesics to open ones, and to the existence of a critical temperature in the frame of a gauge model at finite temperature. In this later case one can obtain expressions giving the dependence of several superconducting parameters on temperature.

scholarly journals HEURISTIC APPROACH TO THE SCHWARZSCHILD GEOMETRY

2005 ◽  
Vol 14 (12) ◽  
pp. 2051-2067 ◽  
Author(s):  
MATT VISSER
Keyword(s):  
Field Approximation ◽  
Weak Field ◽  
Point Particle ◽  
De Sitter ◽  
Field Equations ◽  
Kerr Geometry ◽  
Spacetime Geometry ◽  
Schwarzschild Geometry ◽  
Inertial Frames ◽  
Weak Field Approximation

In this article I present a simple Newtonian heuristic for motivating a weak-field approximation for the spacetime geometry of a point particle. The heuristic is based on Newtonian gravity, the notion of local inertial frames (the Einstein equivalence principle), plus the use of Galilean coordinate transformations to connect the freely falling local inertial frames back to the "fixed stars." Because of the heuristic and quasi-Newtonian manner in which the specific choice of spacetime geometry is motivated, we are at best justified in expecting it to be a weak-field approximation to the true spacetime geometry. However, in the case of a spherically symmetric point mass the result is coincidentally an exact solution of the full vacuum Einstein field equations — it is the Schwarzschild geometry in Painlevé–Gullstrand coordinates. This result is much stronger than the well-known result of Michell and Laplace whereby a Newtonian argument correctly estimates the value of the Schwarzschild radius — using the heuristic presented in this article one obtains the entire Schwarzschild geometry. The heuristic also gives sensible results — a Riemann flat geometry — when applied to a constant gravitational field. Furthermore, a subtle extension of the heuristic correctly reproduces the Reissner–Nordström geometry and even the de Sitter geometry. Unfortunately the heuristic construction is not truly generic. For instance, it is incapable of generating the Kerr geometry or anti-de Sitter space. Despite this limitation, the heuristic does have useful pedagogical value in that it provides a simple and direct plausibility argument (not a derivation) for the Schwarzschild geometry — suitable for classroom use in situations where the full power and technical machinery of general relativity might be inappropriate. The extended heuristic provides more challenging problems — suitable for use at the graduate level.

scholarly journals GRAVITATIONAL FIELD OF A GLOBAL MONOPOLE IN A MODIFIED GRAVITY

2011 ◽  
Vol 03 ◽  
pp. 446-454 ◽  
Author(s):  
T. R. P. CARAMÊS ◽  
E. R. BEZERRA DE MELLO ◽  
M. E. X. GUIMARÃES
Keyword(s):  
Gravitational Field ◽  
Modified Gravity ◽  
Field Approximation ◽  
Weak Field ◽  
Symmetric System ◽  
Radial Coordinate ◽  
Spherically Symmetric ◽  
Global Monopole ◽  
Field Equations ◽  
Metric Tensor

In this paper we analyze the gravitational field of a global monopole in the context of f(R) gravity. More precisely, we show that the field equations obtained are expressed in terms of [Formula: see text]. Since we are dealing with a spherically symmetric system, we assume that F(R) is a function of the radial coordinate only. Moreover, adopting the weak field approximation, we can provide all components of the metric tensor. A comparison with the corresponding results obtained in General Relativity and in the Brans-Dicke theory is also made.

scholarly journals Comparison between the Gravitational Redshift in the Kerr and Schwarzschild Fields

2021 ◽  
Author(s):  
Charles McGruder
Keyword(s):  
Neutron Stars ◽  
White Dwarfs ◽  
Gravitational Redshift ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Frame Dragging ◽  
Gravitational Fields ◽  
Significant Difference ◽  
Spherical Mass ◽  
Kerr Field

Abstract The Schwarzschild and Kerr metrics are solutions of Einstein field equations of general relativity representing the gravitational fields of a non-rotating spherical mass and a rotating black hole respectively. Unlike the Kerr field, the gravitational redshift in the Schwarzschild field is well known. We employ the concept of stationary clocks to derive the gravitational redshift in the Kerr field demonstrating that frame dragging plays no role. We then calculate the Kerr gravitational redshift for the earth, sun, white dwarfs and neutron stars and compare them with the Schwarzschild gravitational redshift, showing that the gravitational redshift on earth and from the sun does not differ from the Schwarzschild gravitational redshift. For extreme cases of rapidly rotating white dwarfs and neutron stars there is a significant difference between the two gravitational redshifts. Unlike the Schwarzschild gravitational redshift, the Kerr gravitational redshift has to date not been put on a firm observational basis. We point out that the gravitational redshift in the Kerr field possess a latitude dependency, which cannot be confirmed through solar or terrestrial observations, but can be on rapidly rotating white dwarfs and neutron stars

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