On the Stability of Axisymmetric Systems to Axisymmetric Perturbations in General Relativity. II. a Criterion for the Onset of Instability in Uniformly Rotating Configurations and the Frequency of the Fundamental Mode in Case of Slow Rotation

1972 ◽  
Vol 176 ◽  
pp. 745 ◽  
Author(s):  
S. Chandrasekhar ◽  
John L. Friedman
Keyword(s):  
General Relativity ◽  
Fundamental Mode ◽  
Slow Rotation ◽  
The Stability ◽  
Axisymmetric Perturbations

scholarly journals CAN f(R) GRAVITY MIMIC GENERAL RELATIVITY?

2012 ◽  
Vol 10 ◽  
pp. 63-68 ◽  
Author(s):  
JE-AN GU
Keyword(s):  
General Relativity ◽  
Differential Equations ◽  
Early Time ◽  
Einstein Equations ◽  
Modified Theories Of Gravity ◽  
Cosmological Perturbations ◽  
Long Run ◽  
Theories Of Gravity ◽  
The Stability ◽  
Higher Order Differential Equations

We discuss the stability of the general-relativity (GR) limit in modified theories of gravity, particularly the f(R) theory. The problem of approximating the higher-order differential equations in modified gravity with the Einstein equations (2nd-order differential equations) in GR is elaborated. We demonstrate this problem with a heuristic example involving a simple ordinary differential equation. With this example we further present the iteration method that may serve as a better approximation for solving the equation, meanwhile providing a criterion for assessing the validity of the approximation. We then discuss our previous numerical analyses of the early-time evolution of the cosmological perturbations in f(R) gravity, following the similar ideas demonstrated by the heuristic example. The results of the analyses indicated the possible instability of the GR limit that might make the GR approximation inaccurate in describing the evolution of the cosmological perturbations in the long run.

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 Non-axisymmetric Instabilities in Shocked Accretion Flows

2003 ◽  
Vol 214 ◽  
pp. 95-96
Author(s):  
Wei-Min Gu ◽  
Thierry Foglizzo
Keyword(s):  
Black Hole ◽  
Numerical Integration ◽  
Recent Work ◽  
Growth Rates ◽  
Linearized Equations ◽  
X Ray ◽  
Accretion Flows ◽  
The Stability ◽  
X Ray Binaries ◽  
Axisymmetric Perturbations

We investigate the stability of shocked inviscid isothermal accretion flows onto a black hole. Of the two possible shock positions, the outer one is known to be stable to axisymmetric perturbations, while the inner one is unstable. Our recent work, however, shows that the outer shock is generally linearly unstable to non-axisymmetric perturbations. Eigenmodes and growth rates are obtained by numerical integration of the linearized equations. These results offer new perspectives to interpret the variability of X-ray binaries.

scholarly journals Non-diffusive angular momentum transport in rotating -pinches

2019 ◽  
Vol 85 (6) ◽  
Author(s):  
G. Rüdiger ◽  
M. Schultz
Keyword(s):  
Angular Momentum ◽  
Momentum Flux ◽  
Eddy Viscosity ◽  
Rigid Rotation ◽  
Momentum Transport ◽  
Angular Momentum Transport ◽  
Magnetohydrodynamic Flows ◽  
The Stability ◽  
Background Fields ◽  
Axisymmetric Perturbations

The stability of conducting Taylor–Couette flows under the presence of toroidal magnetic background fields is considered. For strong enough magnetic amplitudes such magnetohydrodynamic flows are unstable against non-axisymmetric perturbations which may also transport angular momentum. In accordance with the often used diffusion approximation, one expects the angular momentum transport to be vanishing for rigid rotation. In the sense of a non-diffusive  $\unicode[STIX]{x1D6EC}$ effect, however, even for rigidly rotating $z$ -pinches, an axisymmetric angular momentum flux appears which is directed outward (inward) for large (small) magnetic Mach numbers. The internal rotation in a magnetized rotating tank can thus never be uniform. Those particular rotation laws are used to estimate the value of the instability-induced eddy viscosity for which the non-diffusive $\unicode[STIX]{x1D6EC}$ effect and the diffusive shear-induced transport compensate each other. The results provide the Shakura & Sunyaev viscosity ansatz leading to numerical values linearly growing with the applied magnetic field.

Conceptual Flutter Analysis of Stepped Labyrinth Seals

2019 ◽  
Author(s):  
Roque Corral ◽  
Almudena Vega ◽  
Michele Greco
Keyword(s):  
Natural Frequency ◽  
Fundamental Mode ◽  
Second Step ◽  
Pressure Side ◽  
Dimensional Model ◽  
Labyrinth Seals ◽  
Acoustic Frequency ◽  
Flutter Analysis ◽  
Dimensional Parameters ◽  
The Stability

Abstract A simple non-dimensional model to describe the flutter onset of two-fin straight labyrinth seals [1] is extended to stepped seals. The effect of the axial displacement of the seal is analyzed first in isolation. It is shown that this fundamental mode is always stable. In a second step, the combination of axial and torsion displacements is used to determine the damping of modes with arbitrary torsion centers. It is concluded that the classical Abbot’s criterion stating that seals supported in the low-pressure side of the seal are stable provided that natural frequency of the mode is greater than the acoustic frequency breaks down under certain conditions. An analytical expression for the non-dimensional work-per-cycle is derived and new non-dimensional parameters controlling the seal stability identified. It is finally concluded the stability of stepped seals can be assimilated to that of a straight through seal if the appropriate distance of the torsion center to the seal is chosen.

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