scholarly journals A scalar geodesic deviation equation and a phase theorem

1983 ◽  
Vol 6 (4) ◽  
pp. 795-802 ◽  
Author(s):  
P. Choudhury ◽  
P. Dolan ◽  
N. S. Swaminarayan
Keyword(s):  
Gravitational Field ◽  
Phase Shift ◽  
Relative Motion ◽  
Plane Waves ◽  
Relative Energy ◽  
Positive Definite ◽  
Geodesic Deviation ◽  
Deviation Equation ◽  
Test Particles ◽  
Geodesic Deviation Equation

A scalar equation is derived forη, the distance between two structureless test particles falling freely in a gravitational field:η¨+(K−Ω2)η=0. An amplitude, frequency and a phase are defined for the relative motion. The phases are classed as elliptic, hyperbolic and parabolic according asK−Ω2>0,<0,=0.In elliptic phases we deduce a positive definite relative energyEand a phase-shift theorem. The relevance of the phase-shift theorem to gravitational plane waves is discussed.

scholarly journals A “constant of the motion” for the geodesic deviation equation

1980 ◽  
Vol 22 (1) ◽  
pp. 28-33 ◽  
Author(s):  
P. Dolan ◽  
P. Choudhury ◽  
J. L. Safko
Keyword(s):  
Relative Motion ◽  
Short Paper ◽  
Linear Superposition ◽  
Geodesic Deviation ◽  
Deviation Equation ◽  
Test Particles ◽  
Geodesic Deviation Equation ◽  
Kinetic And Potential Energies

AbstractIn this short paper, it is shown that the geodesic deviation equation admits a “constant of the motion” and so can be solved exactly. We also derive an expression for the energy E of relative motion between two freely falling test particles. We can infer that, in general, E will not be a linear superposition of kinetic and potential energies.

scholarly journals The relative motion of membranes

Open Physics ◽  
2010 ◽  
Vol 8 (6) ◽  
Author(s):  
Mark Roberts
Keyword(s):  
Relative Motion ◽  
Quantum Fluctuation ◽  
Classical Motion ◽  
Geodesic Deviation ◽  
Deviation Equation ◽  
Geodesic Deviation Equation ◽  
The Universe

AbstractThe relative classical motion of membranes is governed by the equation (w β cα c r βa)a = R δγβα r gb x δa p aγ, where w is the hessian. This is a generalization of the geodesic deviation equation and can be derived from the lagrangian p · ṙ. Quantum mechanically the picture is less clear. Some quantizations of the classical equations are attempted so that the question as to whether the Universe started with a quantum fluctuation can be addressed.

scholarly journals THE STRING DEVIATION EQUATION

1999 ◽  
Vol 14 (25) ◽  
pp. 1739-1751 ◽  
Author(s):  
MARK D. ROBERTS
Keyword(s):  
Relative Motion ◽  
Riemann Tensor ◽  
Geodesic Deviation ◽  
First Variation ◽  
Deviation Equation ◽  
Combined Action ◽  
Combined Actions ◽  
Geodesic Deviation Equation ◽  
The First Variation ◽  
Many Particles

It is well known that the relative motion of many particles can be described by the geodesic deviation equation. Less well known is that the geodesic deviation equation can be derived from the second covariant variation of the point particle's action. Here it is shown that the second covariant variation of the string action leads to a string deviation equation. This equation is a candidate for describing the relative motion of many strings, and can be reduced to the geodesic deviation equation. Like the geodesic deviation equation, the string deviation equation can also be expressed in terms of momenta and projecta. It is also shown that a combined action exists, the first variation of which gives the deviation equations. The combined actions allow the deviation equations to be expressed solely in terms of the Riemann tensor, the coordinates and momenta. In particular geodesic deviation can be expressed as: [Formula: see text] and string deviation can be expressed as: [Formula: see text]

scholarly journals State-dependent graviton noise in the equation of geodesic deviation

2021 ◽  
Vol 81 (1) ◽  
Author(s):  
Z. Haba
Keyword(s):  
Gravitational Field ◽  
Density Matrix ◽  
Transition Probability ◽  
Feynman Integral ◽  
Point Of View ◽  
Geodesic Deviation ◽  
Wave Detection ◽  
State Dependent ◽  
Deviation Equation ◽  
Geodesic Deviation Equation

AbstractWe consider an equation of the geodesic deviation appearing in the problem of gravitational wave detection in an environment of gravitons. We investigate a state-dependent graviton noise (as discussed in a recent paper by Parikh,Wilczek and Zahariade) from the point of view of the Feynman integral and stochastic differential equations. The evolution of the density matrix and the transition probability in an environment of gravitons is obtained. We express the time evolution by a solution of a stochastic geodesic deviation equation with a noise dependent on the quantum state of the gravitational field.

scholarly journals Geodesic motion in a charged 2D stringy black hole spacetime

2014 ◽  
Vol 29 (29) ◽  
pp. 1450157 ◽  
Author(s):  
Rashmi Uniyal ◽  
Hemwati Nandan ◽  
K. D. Purohit
Keyword(s):  
Black Hole ◽  
Tidal Force ◽  
Two Dimensional ◽  
Geodesic Deviation ◽  
Deviation Equation ◽  
Test Particles ◽  
Exactly Solvable ◽  
Black Hole Spacetime ◽  
Geodesic Deviation Equation ◽  
Deviation Vector

We study the time-like geodesics and geodesic deviation for a two-dimensional (2D) stringy black hole (BH) spacetime in Schwarzschild gauge. We have analyzed the properties of effective potential along with the structure of the possible orbits for test particles with different settings of BH parameters. The exactly solvable geodesic deviation equation is used to obtain corresponding deviation vector. The nature of deviation and tidal force is also examined in view of the behavior of corresponding deviation vector. The results are also compared with an another 2D stringy BH spacetime.

scholarly journals k-Jet field approximations to geodesic deviation equations

2018 ◽  
Vol 15 (12) ◽  
pp. 1850199
Author(s):  
Ricardo Gallego Torromé ◽  
Jonathan Gratus
Keyword(s):  
Differential Equations ◽  
Tangent Bundle ◽  
Jacobi Equation ◽  
Smooth Manifold ◽  
Specific Class ◽  
Coordinate Transformations ◽  
Geodesic Deviation ◽  
Deviation Equation ◽  
Geodesic Deviation Equation ◽  
Deviation Functions

Let [Formula: see text] be a smooth manifold and [Formula: see text] a semi-spray defined on a sub-bundle [Formula: see text] of the tangent bundle [Formula: see text]. In this work, it is proved that the only non-trivial [Formula: see text]-jet approximation to the exact geodesic deviation equation of [Formula: see text], linear on the deviation functions and invariant under an specific class of local coordinate transformations, is the Jacobi equation. However, if the linearity property on the dependence in the deviation functions is not imposed, then there are differential equations whose solutions admit [Formula: see text]-jet approximations and are invariant under arbitrary coordinate transformations. As an example of higher-order geodesic deviation equations, we study the first- and second-order geodesic deviation equations for a Finsler spray.

scholarly journals Erratum to: Geodesic deviation equation in f(R) gravity

2015 ◽  
Vol 47 (10) ◽  
Author(s):  
Alejandro Guarnizo ◽  
Leonardo Castañeda ◽  
Juan M. Tejeiro
Keyword(s):  
Geodesic Deviation ◽  
Deviation Equation ◽  
Geodesic Deviation Equation
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