Kilohertz Quasi-Periodic Oscillations (QPOs) have been detected in many
accreting X-ray bi- naries. It has been suggested that the kHz QPO
frequencies observed in the modulation of the X-ray flux reflect a
non-linear resonance between two modes of accreting disk oscillations. A
3:2 resonant ratio of frequencies has been clearly recognized in the
black-hole QPO data supporting the non-linear resonance hypothesis. It
is often discussed whether the same 3:2 ratio appears in case of the
neutron star QPOs as well. For an individual neutron star source, the
relation between the observed lower and upper (νL and νU ) kHz QPO
frequency is well approximated by a linear relation with the slope A and
the shift B, i.e., νU = AνL + B. It was recognized only recently
that for a group of twelve neutron star sources the value of the
individual coefficients A, B is anticorre- lated, obeying the condition
A = 3/2 - 0.0016B. It was also shown that such an anticorrelation is
predicted by the theory of weakly coupled nonlinear oscillators, and one
can argue that it indeed pose the evidence for a 3:2 resonant ratio.
Here, stressing the latest progress, we discuss some of the links
between the QPO observation and the resonance model, and summarize the
appropriate references. Except the slightly updated list of references,
the content of next four pages fully correspond to the A0 poster we
exhibited at the VI Microquasar Workshop. The poster itself is available
at the http://www.physics.cz/research (section Conferences).
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Change of sign of the velocity gradient (mesured with respect to locally
non-rotating frames) has been found for accretion discs orbiting rapidly
rotating Kerr black holes with spin a > 0.9953 for Keplerian discs
[1] and a > 0.99979 for marginally stable thick discs [2]. Such
"humpy" orbital velocity profiles occur close to but above the
marginally stable circular geodesic of the black hole spacetimes. The
maximal positive rate of change of the orbital velocity in terms of the
proper radial distance introduces a locally defined critical frequency
characterizing any processes in the disc capable to excite possible
oscillations connected with the velocity hump. Comparing the "humpy
frequency" related to distant observers with the epicyclic frequencies
we show that in Keplerian discs orbiting extremely rapid Kerr holes (1 -
a < 10-4 ) the ratio of the epicyclic frequencies and the humpy
frequency is nearly constant, i.e., almost independent of a, being
˜ 3 : 2 for the radial epicyclic frequency and ˜ 11 : 2 for
the vertical epicyclic frequency. For black holes with a ˜ 0.996,
i.e., when the resonant phenomena with ratio 3 : 1 between the vertical
and radial epicyclic oscillations occur near the radius of the critical
humpy frequency, the ratio of the radial epicyclic and the humpy
frequency is ˜ 12 : 1, which is close to the ratio between high-
and low- frequency QPO in X-ray systems. For a > 0.996 the resonant
orbit r4:1 (with the ratio 4 : 1 between the vertical and radial
epicyclic oscillations) occurs in the region of the hump. Applying the
model on the nearly extreme black hole candidate GRS 1915+105, we
conclude that for black hole parameters M = 14.8M⊙ and a = 0.9998
the observed high-frequency QPOs could be related to the hump-induced
oscillations in thin accretion disc, as the first two QPOs, 41 Hz and 67
Hz, can be identified with the "humpy frequency" and the radial
epicyclic frequency (at the same orbit). The other observed
QPO-frequencies, 113 Hz and 166 Hz, can be explained as the
combinational ones of the "humpy" and epicyclic frequencies.
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Forces defined in the framework of optical reference geometry are
introduced in the case of stationary and axially symmetric Kerr black
hole and naked-singularity space-times with a repulsive cosmological
constant. Properties of the forces acting on test particles moving along
circular orbits in the equatorial plane are discussed, whereas it is
shown where the gravitational force vanishes and changes its orientation
and where the centrifugal force vanishes and changes its orientation
independently of the velocity of test particles related to the optical
geometry; the Coriolis force does not vanish for the velocity being
nonzero. The space-times are classified according to the number of
circular orbits where the gravitational and centrifugal forces vanish.
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The influence of the observed relict vacuum energy on the fluctuations
of CMBR going through cosmological matter condensations is studied in
the framework of the Einstein-Strauss-de Sitter vakuola model. It is
shown that refraction of light at the matching surface of the vakuola
and the expanding Friedman universe can be very important during
accelerated expansion of the universe, when the velocity of the matching
surface relative to static Schwarzchildian observers becomes
relativistic. Relevance of the refraction effect for the temperature
fluctuations of CMBR is given in terms of the redshift and the angular
extension of the fluctuating region.
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It is hard to imagine curved spacetimes of General Relativity. A simple
but powerful way how to achieve this is visualizing them via embedding
diagrams of both ordinary geometry and optical reference geometry. They
facilitate to gain an intuitive insight into the gravitational field
rendered into a curved spacetime, and to assess the influence of
parameters like electric charge and spin of a black hole, magnetic field
or cosmological constant. Optical reference geometry and related
inertial forces and their relationship to embedding diagrams are
particularly useful for investigation of test particles motion.
Embedding diagrams of static and spherically symmetric, or stationary
and axially symmetric black-hole and naked-singularity spacetimes thus
present a useful concept for intuitive understanding of these
spacetimes' nature. We concentrate on general way of embedding into
3-dimensional Euclidean space, and give a set of illustrative examples.
Read More
Kilohertz Quasi-Periodic Oscillations (QPOs) have been detected in many
accreting X-ray binaries. It has been suggested that the highest QPO
frequencies observed in the modulation of the X-ray flux reflect a
non-linear resonance between two modes of accreting disk oscillation.
This hypothesis implies certain very general predictions, several of
which have been borne out by observations. Some of these follow from
properties of non-linear oscillators, while the others are specific to
oscillations of fluid in strong gravity. A 3:2 resonant ratio of
frequencies can be clearly recognized in the black-hole as well as in
the neutron-star QPO data.
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We show that in the equatorial plane of marginally stable thick discs
(with uniformly distributed specific angular momentum l(r, θ) =
const) the orbital velocity relative to the LNRF has a positive radial
gradient in the vicinity of black holes with a > 0.99979. The change
of sign of the velocity gradient occurs just above the center of the
thick toroidal discs, in the region where stable circular geodesics of
the Kerr spacetime are allowed. The global character of the phenomenon
is given in terms of topology changes of the von Zeipel surfaces
(equivalent to equivelocity surfaces in the tori with l(r, θ) =
const). Toroidal von Zeipel surfaces exist around the circle
corresponding to the minimum of the equatorial LNRF velocity profile,
indicating a possibility of development of some vertical instabilities
in those parts of marginally stable tori with positive gradient of the
LNRF velocity. Eventual oscillatory frequencies connected with the
phenomenon are given in a coordinate-independent form.
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Perfect fluid tori with uniform distribution of the specific angular
momentum orbiting the Kerr-de Sitter black holes or naked singularities
are studied. Closed equipotential surfaces corresponding to stationary
toroidal discs are allowed only in the spacetimes admitting stable
circular geodesics. The last closed surface crosses itself in the
cusp(s) enabling outflow(s) of matter from the torus due to the
violation of hydrostatic equilibrium. The repulsive cosmological
constant, Λ > 0, implies the existence of the outer cusp (with
a stabilizing effect on the tori because of excretion, i.e., outflow of
matter from the torus into the outer space) and the strong collimation
of open equipotential surfaces along the rotational axis. Both the
effects take place nearby so-called static radius where the
gravitational attraction is just balanced by the cosmic repulsion. The
plus-family discs (which are always corotating in the black-hole
backgrounds but can be counterrotating, even with negative energy of the
fluid elements, in some naked-singularity backgrounds) are thicker and
more extended than the minus-family ones (which are always
counterrotating in all backgrounds). If parameters of the
naked-singularity spacetimes are very close to the parameters of extreme
black-hole spacetimes, the family of possible disc-like configurations
includes members with two isolated discs where the inner one is always a
counterrotating accretion disc. Mass estimates for tori with
nonrelativistic adiabatic equation of state give limits on their central
mass-density, for which the approximation of test fluid is adequate.
Read More
The influence of the observed relict vacuum energy on the fluctuations of CMBR going through cosmological matter condensations is studied in the framework of the Einstein-Strauss-de Sitter vakuola model. It is shown that refraction of light at the matching surface of the vakuola and the expanding Friedman universe can be very important during accelerated expansion of the universe, when the velocity of the matching surface relative to static Schwarzchildian observers becomes relativistic. Relevance of the refraction effect for the temperature fluctuations of CMBR is given in terms of the redshift and the angular extension of the fluctuating region.
Read More
It is hard to imagine curved spacetimes of General Relativity. A simple but powerful way how to achieve this is visualizing them via embedding diagrams of both ordinary geometry and optical reference geometry. They facilitate to gain an intuitive insight into the gravitational field rendered into a curved spacetime, and to assess the influence of parameters like electric charge and spin of a black hole, magnetic field or cosmological constant. Optical reference geometry and related inertial forces and their relationship to embedding diagrams are particularly useful for investigation of test particles motion. Embedding diagrams of static and spherically symmetric, or stationary and axially symmetric black-hole and naked-singularity spacetimes thus present a useful concept for intuitive understanding of these spacetimes' nature. We concentrate on general way of embedding into 3-dimensional Euclidean space, and give a set of illustrative examples.
Read More
