42 research outputs found
The non-integrability of the Zipoy-Voorhees metric
The low frequency gravitational wave detectors like eLISA/NGO will give us
the opportunity to test whether the supermassive compact objects lying at the
centers of galaxies are indeed Kerr black holes. A way to do such a test is to
compare the gravitational wave signals with templates of perturbed black hole
spacetimes, the so-called bumpy black hole spacetimes. The Zipoy-Voorhees (ZV)
spacetime (known also as the spacetime) can be included in the bumpy
black hole family, because it can be considered as a perturbation of the
Schwarzschild spacetime background. Several authors have suggested that the ZV
metric corresponds to an integrable system. Contrary to this integrability
conjecture, in the present article it is shown by numerical examples that in
general ZV belongs to the family of non-integrable systems.Comment: 10 pages, 13 figure
How to observe a non-Kerr spacetime
We present a generic criterion which can be used in gravitational-wave data
analysis to distinguish an extreme-mass-ratio inspiral into a Kerr background
spacetime from one into a non-Kerr background spacetime. The criterion exploits
the fact that when an integrable system, such as the system that describes
geodesic orbits in a Kerr spacetime, is perturbed, the tori in phase space
which initially corresponded to resonances disintegrate so as to form the so
called Birkhoff chains on a surface of section, according to the
Poincar\'{e}-Birkhoff theorem. The KAM curves of these islands in such a chain
share the same ratio of frequencies, even though the frequencies themselves
vary from one KAM curve to another inside an island. On the other hand, the KAM
curves, which do not lie in a Birkhoff chain, do not share this characteristic
property. Such a temporal constancy of the ratio of frequencies during the
evolution of the gravitational-wave signal will signal a non-Kerr spacetime
which could then be further explored.Comment: 4 pages, 2 figure
Regular and Chaotic Motion in General Relativity: The Case of a Massive Magnetic Dipole
Circular motion of particles, dust grains and fluids in the vicinity of
compact objects has been investigated as a model for accretion of gaseous and
dusty environment. Here we further discuss, within the framework of general
relativity, figures of equilibrium of matter under the influence of combined
gravitational and large-scale magnetic fields, assuming that the accreted
material acquires a small electric charge due to interplay of plasma processes
and photoionization. In particular, we employ an exact solution describing the
massive magnetic dipole and we identify the regions of stable motion. We also
investigate situations when the particle dynamics exhibits the onset of chaos.
In order to characterize the measure of chaoticness we employ techniques of
Poincar\'e surfaces of section and of recurrence plots.Comment: 11 pages, 6 figures, published in the proceedings of the conference
"Relativity and Gravitation: 100 Years after Einstein in Prague" (25. - 29.
6. 2012, Prague
Non-Linear Effects in Non-Kerr spacetimes
There is a chance that the spacetime around massive compact objects which are
expected to be black holes is not described by the Kerr metric, but by a metric
which can be considered as a perturbation of the Kerr metric. These non-Kerr
spacetimes are also known as bumpy black hole spacetimes. We expect that, if
some kind of a bumpy black hole exists, the spacetime around it should possess
some features which will make the divergence from a Kerr spacetime detectable.
One of the differences is that these non-Kerr spacetimes do not posses all the
symmetries needed to make them integrable. We discuss how we can take advantage
of this fact by examining EMRIs into the Manko-Novikov spacetime.Comment: 8 pages, 3 Figures; to appear in the proceedings of the conference
"Relativity and Gravitation: 100 Years after Einstein in Prague" (2012
A Symmetric Integrator for non-integrable Hamiltonian Relativistic Systems
By combining a standard symmetric, symplectic integrator with a new step size
controller, we provide an integration scheme that is symmetric, reversible and
conserves the values of the constants of motion. This new scheme is appropriate
for long term numerical integrations of geodesic orbits in spacetime
backgrounds, whose corresponding Hamiltonian system is non-integrable, and, in
general, for any non-integrable Hamiltonian system whose kinetic part depends
on the position variables. We show by numerical examples that the new
integrator is faster and more accurate i) than the standard symplectic
integration schemes with or without standard adaptive step size controllers and
ii) than an adaptive step Runge-Kutta scheme.Comment: 12 pages, 8 figures, 3 table
Periodic Orbits and Escapes in Dynamical Systems
We study the periodic orbits and the escapes in two different dynamical
systems, namely (1) a classical system of two coupled oscillators, and (2) the
Manko-Novikov metric (1992) which is a perturbation of the Kerr metric (a
general relativistic system). We find their simple periodic orbits, their
characteristics and their stability. Then we find their ordered and chaotic
domains. As the energy goes beyond the escape energy, most chaotic orbits
escape. In the first case we consider escapes to infinity, while in the second
case we emphasize escapes to the central "bumpy" black hole. When the energy
reaches its escape value a particular family of periodic orbits reaches an
infinite period and then the family disappears (the orbit escapes). As this
family approaches termination it undergoes an infinity of equal period and
double period bifurcations at transitions from stability to instability and
vice versa. The bifurcating families continue to exist beyond the escape
energy. We study the forms of the phase space for various energies, and the
statistics of the chaotic and escaping orbits. The proportion of these orbits
increases abruptly as the energy goes beyond the escape energy.Comment: 28 pages, 23 figures, accepted in "Celestial Mechanics and Dynamical
Astronomy
The production of Tsallis entropy in the limit of weak chaos and a new indicator of chaoticity
We study the connection between the appearance of a `metastable' behavior of
weakly chaotic orbits, characterized by a constant rate of increase of the
Tsallis q-entropy (Tsallis 1988), and the solutions of the variational
equations of motion for the same orbits. We demonstrate that the variational
equations yield transient solutions, lasting for long time intervals, during
which the length of deviation vectors of nearby orbits grows in time almost as
a power-law. The associated power exponent can be simply related to the
entropic exponent for which the q-entropy exhibits a constant rate of increase.
This analysis leads to the definition of a new sensitive indicator
distinguishing regular from weakly chaotic orbits, that we call `Average Power
Law Exponent' (APLE). We compare the APLE with other established indicators of
the literature. In particular, we give examples of application of the APLE in
a) a thin separatrix layer of the standard map, b) the stickiness region around
an island of stability in the same map, and c) the web of resonances of a 4D
symplectic map. In all these cases we identify weakly chaotic orbits exhibiting
the `metastable' behavior associated with the Tsallis q-entropy.Comment: 19 pages, 12 figures, accepted for publication by Physica
Stability analysis and quasinormal modes of Reissner Nordstr{\o}m Space-time via Lyapunov exponent
We explicitly derive the proper time principal Lyapunov exponent
() and coordinate time () principal Lyapunov exponent
() for Reissner Nordstr{\o}m (RN) black hole (BH) . We also
compute their ratio. For RN space-time, it is shown that the ratio is
for
time-like circular geodesics and for Schwarzschild BH it is
. We
further show that their ratio may vary from
orbit to orbit. For instance, Schwarzschild BH at innermost stable circular
orbit(ISCO), the ratio is
and at marginally
bound circular orbit (MBCO) the ratio is calculated to be
. Similarly, for extremal RN
BH the ratio at ISCO is
.
We also further analyse the geodesic stability via this exponent. By evaluating
the Lyapunov exponent, it is shown that in the eikonal limit , the real and
imaginary parts of the quasi-normal modes of RN BH is given by the frequency
and instability time scale of the unstable null circular geodesics.Comment: Accepted in Pramana, 07/09/201