On the implications of the nonexistence of unstable outgoing normal modes for the stability of spherical stellar models to nonradial perturbations in general relativity

1975 ◽  
Vol 199 ◽  
pp. 220 ◽  
Author(s):  
J. R. Ipser
Keyword(s):  
General Relativity ◽  
Normal Modes ◽  
Stellar Models ◽  
The Stability

An Application of the Poincare´ Map to the Stability of Nonlinear Normal Modes

1980 ◽  
Vol 47 (3) ◽  
pp. 645-651 ◽  
Author(s):  
L. A. Month ◽  
R. H. Rand
Keyword(s):  
Floquet Theory ◽  
Poincaré Map ◽  
Normal Modes ◽  
Nonlinear Normal Modes ◽  
Theory Approach ◽  
Poincare Map ◽  
Linearized Stability ◽  
Nonlinear Normal ◽  
The Stability ◽  
Two Degree Of Freedom

The stability of periodic motions (nonlinear normal modes) in a nonlinear two-degree-of-freedom Hamiltonian system is studied by deriving an approximation for the Poincare´ map via the Birkhoff-Gustavson canonical transofrmation. This method is presented as an alternative to the usual linearized stability analysis based on Floquet theory. An example is given for which the Floquet theory approach fails to predict stability but for which the Poincare´ map approach succeeds.

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.

Stability of the Base Flow to Axisymmetric and Plane-Polar Disturbances in an Electrically Driven Flow Between Infinitely-Long, Concentric Cylinders

2000 ◽  
Vol 123 (1) ◽  
pp. 31-42
Author(s):  
J. Liu ◽  
G. Talmage ◽  
J. S. Walker
Keyword(s):  
Magnetic Field ◽  
Electric Current ◽  
Axial Magnetic Field ◽  
Normal Modes ◽  
Azimuthal Velocity ◽  
Base Flow ◽  
Transition To Turbulence ◽  
Electrically Conducting ◽  
Concentric Cylinders ◽  
The Stability

The method of normal modes is used to examine the stability of an azimuthal base flow to both axisymmetric and plane-polar disturbances for an electrically conducting fluid confined between stationary, concentric, infinitely-long cylinders. An electric potential difference exists between the two cylinder walls and drives a radial electric current. Without a magnetic field, this flow remains stationary. However, if an axial magnetic field is applied, then the interaction between the radial electric current and the magnetic field gives rise to an azimuthal electromagnetic body force which drives an azimuthal velocity. Infinitesimal axisymmetric disturbances lead to an instability in the base flow. Infinitesimal plane-polar disturbances do not appear to destabilize the base flow until shear-flow transition to turbulence.

scholarly journals The stability of massive stars

1994 ◽  
Vol 162 ◽  
pp. 73-74
Author(s):  
W. Glatzel ◽  
M. Kiriakidis ◽  
K.J. Fricke
Keyword(s):  
Massive Stars ◽  
Mass Limit ◽  
Physical Explanation ◽  
Stellar Objects ◽  
Fundamental Interest ◽  
Stability Properties ◽  
Luminous Blue Variables ◽  
Stellar Models ◽  
The Stability

An investigation of the stability properties of stellar models describing massive stars is motivated observationally by the necessity to explain the observed Humphreys - Davidson (HD) limit and the variability of the most massive stars known, i.e. the existence of luminous blue variables (LBVs). Theoretically, a determination of the upper mass limit for stable stellar objects together with its physical explanation and interpretation is of fundamental interest.

scholarly journals The Stability of Relativistic Systems

1974 ◽  
Vol 64 ◽  
pp. 63-81
Author(s):  
S. Chandrasekhar
Keyword(s):  
Azimuthal Angle ◽  
Normal Modes ◽  
Radiation Reaction ◽  
Axisymmetric Deformation ◽  
Alternative Proof ◽  
Newtonian Theory ◽  
Relativistic Systems ◽  
The Stability

The stability of relativistic systems is reviewed against the background of what is known in the corresponding contexts of the Newtonian theory. In particular, the importance of determining whether Dedekind-like points of bifurcation occur along given stationary axisymmetric sequences is emphasized: the occurrence of such points of bifurcation may signal the onset of secular instability induced by radiation-reaction. (At a Dedekind-like point of bifurcation, the system can be subject, quasistationarily, to a non-axisymmetric deformation with an e2iϕ-dependence on the azimuthal angle ϕ.)A formalism is described in terms of which the normal modes of axisymmetric oscillation of axisymmetric systems can be determined. Specialized to neutral modes of oscillation the formalism provides an alternative proof of Carter's theorem and clarifies the minimal requirements for its validity. A parallel formalism is described for ascertaining whether an axisymmetric system can be subject to a quasi-stationary non-axisymmetric deformation. The possibility of applying this latter formalism to determining whether a Dedekind-like point of bifurcation occurs along the Kerr sequence is considered.

Instability of quasi-geostrophic vortices in a two-layer ocean with a thin upper layer

2003 ◽  
Vol 475 ◽  
pp. 303-331 ◽  
Author(s):  
E. S. BENILOV
Keyword(s):  
Boundary Value Problem ◽  
Lower Layer ◽  
Boundary Value ◽  
Stability Boundary ◽  
Normal Modes ◽  
Layer Depth ◽  
Asymptotic Results ◽  
Gaussian Profile ◽  
Swirl Velocity ◽  
The Stability

We examine the stability of a quasi-geostrophic vortex in a two-layer ocean with a thin upper layer on the f-plane. It is assumed that the vortex has a sign-definite swirl velocity and is localized in the upper layer, whereas the disturbance is present in both layers. The stability boundary-value problem admits three types of normal modes: fast (upper-layer-dominated) modes, responsible for equivalent-barotropic instability, and two slow baroclinic types (mixed- and lower-layer-dominated modes). Fast modes exist only for unrealistically small vortices (with a radius smaller than half of the deformation radius), and this paper is mainly focused on the slow modes. They are examined by expanding the stability boundary-value problem in powers of the ratio of the upper-layer depth to the lower-layer depth. It is demonstrated that the instability of slow modes, if any, is associated with critical levels, which are located at the periphery of the vortex. The complete (sufficient and necessary) stability criterion with respect to slow modes is derived: the vortex is stable if and only if the potential-vorticity gradient at the critical level and swirl velocity are of the same sign. Several vortex profiles are examined, and it is shown that vortices with a slowly decaying periphery are more unstable baroclinically and less barotropically than those with a fast-decaying periphery, with the Gaussian profile being the most stable overall. The asymptotic results are verified by numerical integration of the exact boundary-value problem, and interpreted using oceanic observations.

Analytic stellar models in general relativity

1975 ◽  
Vol 198 ◽  
pp. 507 ◽  
Author(s):  
R. C. Adams ◽  
J. M. Cohen
Keyword(s):  
General Relativity ◽  
Stellar Models
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