58 research outputs found
Quasi-Normal Modes of Stars and Black Holes
Perturbations of stars and black holes have been one of the main topics of
relativistic astrophysics for the last few decades. They are of particular
importance today, because of their relevance to gravitational wave astronomy.
In this review we present the theory of quasi-normal modes of compact objects
from both the mathematical and astrophysical points of view. The discussion
includes perturbations of black holes (Schwarzschild, Reissner-Nordstr\"om,
Kerr and Kerr-Newman) and relativistic stars (non-rotating and
slowly-rotating). The properties of the various families of quasi-normal modes
are described, and numerical techniques for calculating quasi-normal modes
reviewed. The successes, as well as the limits, of perturbation theory are
presented, and its role in the emerging era of numerical relativity and
supercomputers is discussed.Comment: 74 pages, 7 figures, Review article for "Living Reviews in
Relativity
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
Semi-analytic results for quasi-normal frequencies
The last decade has seen considerable interest in the quasi-normal
frequencies [QNFs] of black holes (and even wormholes), both asymptotically
flat and with cosmological horizons. There is wide agreement that the QNFs are
often of the form omega_n = (offset) + i n (gap), though some authors have
encountered situations where this behaviour seems to fail. To get a better
understanding of the general situation we consider a semi-analytic model based
on a piecewise Eckart (Poeschl-Teller) potential, allowing for different
heights and different rates of exponential falloff in the two asymptotic
directions. This model is sufficiently general to capture and display key
features of the black hole QNFs while simultaneously being analytically
tractable, at least for asymptotically large imaginary parts of the QNFs. We
shall derive an appropriate "quantization condition" for the asymptotic QNFs,
and extract as much analytic information as possible. In particular, we shall
explicitly verify that the (offset)+ i n (gap) behaviour is common but not
universal, with this behaviour failing unless the ratio of rates of exponential
falloff on the two sides of the potential is a rational number. (This is
"common but not universal" in the sense that the rational numbers are dense in
the reals.) We argue that this behaviour is likely to persist for black holes
with cosmological horizons.Comment: V1: 28 pages, no figures. V2: 3 references added, no physics changes.
V3: 29 pages, 9 references added, no physics changes; V4: reformatted, now 27
pages. Some clarifications, comparison with results obtained by monodromy
techniques. This version accepted for publication in JHEP. V5: Minor typos
fixed. Compatible with published versio
Quasi-normal frequencies: Key analytic results
The study of exact quasi-normal modes [QNMs], and their associated
quasi-normal frequencies [QNFs], has had a long and convoluted history -
replete with many rediscoveries of previously known results. In this article we
shall collect and survey a number of known analytic results, and develop
several new analytic results - specifically we shall provide several new QNF
results and estimates, in a form amenable for comparison with the extant
literature. Apart from their intrinsic interest, these exact and approximate
results serve as a backdrop and a consistency check on ongoing efforts to find
general model-independent estimates for QNFs, and general model-independent
bounds on transmission probabilities. Our calculations also provide yet another
physics application of the Lambert W function. These ideas have relevance to
fields as diverse as black hole physics, (where they are related to the damped
oscillations of astrophysical black holes, to greybody factors for the Hawking
radiation, and to more speculative state-counting models for the Bekenstein
entropy), to quantum field theory (where they are related to Casimir energies
in unbounded systems), through to condensed matter physics, (where one may
literally be interested in an electron tunelling through a physical barrier).Comment: V1: 29 pages; V2: Reformatted, 31 pages. Title changed to reflect
major additions and revisions. Now describes exact QNFs for the double-delta
potential in terms of the Lambert W function. V3: Minor edits for clarity.
Four references added. No physics changes. Still 31 page
Distributions of charged massive scalars and fermions from evaporating higher-dimensional black holes
A detailed numerical analysis is performed to obtain the Hawking spectrum for
charged, massive brane scalars and fermions on the approximate background of a
brane charged rotating higher-dimensional black hole constructed in
arXiv:0907.5107. We formulate the problem in terms of a "spinor-like" first
order system of differential wave equations not only for fermions, but for
scalars as well and integrate it numerically. Flux spectra are presented for
non-zero mass, charge and rotation, confirming and extending previous results
based on analytic approximations. In particular we describe an inverted charge
splitting at low energies, which is not present in four or five dimensions and
increases with the number of extra dimensions. This provides another signature
of the evaporation of higher-dimensional black holes in TeV scale gravity
scenarios.Comment: 19 pages, 6 figures, minor typos corrected, 1 page added with a
discussion on higher spins, added reference
Gravitational waves from single neutron stars: an advanced detector era survey
With the doors beginning to swing open on the new gravitational wave
astronomy, this review provides an up-to-date survey of the most important
physical mechanisms that could lead to emission of potentially detectable
gravitational radiation from isolated and accreting neutron stars. In
particular we discuss the gravitational wave-driven instability and
asteroseismology formalism of the f- and r-modes, the different ways that a
neutron star could form and sustain a non-axisymmetric quadrupolar "mountain"
deformation, the excitation of oscillations during magnetar flares and the
possible gravitational wave signature of pulsar glitches. We focus on progress
made in the recent years in each topic, make a fresh assessment of the
gravitational wave detectability of each mechanism and, finally, highlight key
problems and desiderata for future work.Comment: 39 pages, 12 figures, 2 tables. Chapter of the book "Physics and
Astrophysics of Neutron Stars", NewCompStar COST Action 1304. Minor
corrections to match published versio
Rotating Stars in Relativity
Rotating relativistic stars have been studied extensively in recent years,
both theoretically and observationally, because of the information one could
obtain about the equation of state of matter at extremely high densities and
because they are considered to be promising sources of gravitational waves. The
latest theoretical understanding of rotating stars in relativity is reviewed in
this updated article. The sections on the equilibrium properties and on the
nonaxisymmetric instabilities in f-modes and r-modes have been updated and
several new sections have been added on analytic solutions for the exterior
spacetime, rotating stars in LMXBs, rotating strange stars, and on rotating
stars in numerical relativity.Comment: 101 pages, 18 figures. The full online-readable version of this
article, including several animations, will be published in Living Reviews in
Relativity at http://www.livingreviews.org
- …