21 research outputs found
On aberration in gravitational lensing
It is known that a relative translational motion between the deflector and
the observer affects gravitational lensing. In this paper, a lens equation is
obtained to describe such effects on actual lensing observables. Results can be
easily interpreted in terms of aberration of light-rays. Both radial and
transverse motions with relativistic velocities are considered. The lens
equation is derived by first considering geodesic motion of photons in the
rest-frame Schwarzschild spacetime of the lens, and, then, light-ray detection
in the moving observer's frame. Due to the transverse motion images are
displaced and distorted in the observer's celestial sphere, whereas the radial
velocity along the line of sight causes an effective re-scaling of the lens
mass. The Einstein ring is distorted to an ellipse whereas the caustics in the
source plane are still point-like. Either for null transverse motion or up to
linear order in velocities, the critical curve is still a circle with its
radius corrected by a factor (1+z_d) with respect to the static case, z_d being
the relativistic Doppler shift of the deflector. From the observational point
of view, the orbital motion of the Earth can cause potentially observable
corrections of the order of the microarcsec in lensing towards the
super-massive black hole at the Galactic center. On a cosmological scale,
tangential peculiar velocities of cluster of galaxies bring about a typical
flexion in images of background galaxies in the weak lensing regime but future
measurements seem to be too much challenging.Comment: 8 pages, 2 figures, in press on PR
Strong deflection limit of black hole gravitational lensing with arbitrary source distances
The gravitational field of supermassive black holes is able to strongly bend
light rays emitted by nearby sources. When the deflection angle exceeds ,
gravitational lensing can be analytically approximated by the so-called strong
deflection limit. In this paper we remove the conventional assumption of
sources very far from the black hole, considering the distance of the source as
an additional parameter in the lensing problem to be treated exactly. We find
expressions for critical curves, caustics and all lensing observables valid for
any position of the source up to the horizon. After analyzing the spherically
symmetric case we focus on the Kerr black hole, for which we present an
analytical 3-dimensional description of the higher order caustic tubes.Comment: 20 pages, 8 figures, appendix added. In press on Physical Review
Kerr black hole lensing for generic observers in the strong deflection limit
We generalize our previous work on gravitational lensing by a Kerr black hole
in the strong deflection limit, removing the restriction to observers on the
equatorial plane. Starting from the Schwarzschild solution and adding
corrections up to the second order in the black hole spin, we perform a
complete analytical study of the lens equation for relativistic images created
by photons passing very close to a Kerr black hole. We find out that, to the
lowest order, all observables (including shape and shift of the black hole
shadow, caustic drift and size, images position and magnification) depend on
the projection of the spin on a plane orthogonal to the line of sight. In order
to break the degeneracy between the black hole spin and its inclination
relative to the observer, it is necessary to push the expansion to higher
orders. In terms of future VLBI observations, this implies that very accurate
measures are needed to determine these two parameters separately.Comment: 17 pages, 4 figures, one section added, to appear on Physical Review
Viewing the Shadow of the Black Hole at the Galactic Center
In recent years, the evidence for the existence of an ultra-compact
concentration of dark mass associated with the radio source Sgr A* in the
Galactic Center has become very strong. However, an unambiguous proof that this
object is indeed a black hole is still lacking. A defining characteristic of a
black hole is the event horizon. To a distant observer, the event horizon casts
a relatively large ``shadow'' with an apparent diameter of ~10 gravitational
radii due to bending of light by the black hole, nearly independent of the
black hole spin or orientation. The predicted size (~30 micro-arcseconds) of
this shadow for Sgr A* approaches the resolution of current
radio-interferometers. If the black hole is maximally spinning and viewed
edge-on, then the shadow will be offset by ~8 micro-arcseconds from the center
of mass, and will be slightly flattened on one side. Taking into account
scatter-broadening of the image in the interstellar medium and the finite
achievable telescope resolution, we show that the shadow of Sgr A* may be
observable with very long-baseline interferometry at sub-millimeter
wavelengths, assuming that the accretion flow is optically thin in this region
of the spectrum. Hence, there exists a realistic expectation of imaging the
event horizon of a black hole within the next few years.Comment: 5 pages, 1 figure (color), (AAS)Tex, to appear in The Astrophysical
Journal Letters, Vol. 528, L13 (Jan 1, 2000 issue); also available at
http://www.mpifr-bonn.mpg.de/staff/hfalcke/publications.html#bhimag
Kerr-Sen dilaton-axion black hole lensing in the strong deflection limit
In the present work we study numerically quasi-equatorial lensing by the
charged, stationary, axially-symmetric Kerr-Sen dilaton-axion black hole in the
strong deflection limit. In this approximation we compute the magnification and
the positions of the relativistic images. The most outstanding effect is that
the Kerr-Sen black hole caustics drift away from the optical axis and shift in
clockwise direction with respect to the Kerr caustics. The intersections of the
critical curves on the equatorial plane as a function of the black hole angular
momentum are found, and it is shown that they decrease with the increase of the
parameter . All of the lensing quantities are compared to particular
cases as Schwarzschild, Kerr and Gibbons-Maeda black holes.Comment: 31 pages, 17 figures; V2 references added, some typos corrected, V3
references added, language corrections, V4 table added, minor technical
correction
Strong field limit of black hole gravitational lensing
We give the formulation of the gravitational lensing theory in the strong
field limit for a Schwarzschild black hole as a counterpart to the weak field
approach. It is possible to expand the full black hole lens equation to work a
simple analytical theory that describes at a high accuracy degree the physics
in the strong field limit. In this way, we derive compact and reliable
mathematical formulae for the position of additional critical curves,
relativistic images and their magnification, arising in this limit.Comment: 11 pages, 3 figure
Analytical Kerr black hole lensing in the weak deflection limit
We present an analytical treatment of gravitational lensing by a Kerr black
hole in the weak deflection limit. Lightlike geodesics are expanded as a Taylor
series up to and including third-order terms in m/b and a/b, where m is the
black hole mass, a the angular momentum and b the impact parameter of the light
ray. Positions and magnifications of individual images are computed with a
perturbative analysis. At this order, the degeneracy with the translated
Schwarzschild lens is broken. The critical curve is still a circle displaced
from the black hole position in the equatorial direction and the corresponding
caustic is point-like. The degeneracy between the black hole spin and its
inclination relative to the observer is broken through the angular coordinates
of the perturbed images.Comment: 12 pages; minor changes to match the version in press on Phys. Rev.
Optical caustics of Kerr spacetime: the full structure
We present an exhaustive numerical investigation of the optical caustics in
gravitational lensing by a spinning black hole for an observer at infinity.
Besides the primary caustic, we examine higher order caustics, formed by
photons performing one or several loops around the black hole. Our
investigation covers the whole parameter space, including the black hole spin,
its inclination with respect to the line of sight, the source distance, and the
caustic order. By comparing our results with the available analytical
approximations, we find perfect agreement in their respective domains of
validity. We then prove that all caustics maintain their shape (a tube with
astroidal cross-section) in the entire parameter space without suffering any
transitions to different caustic shapes. For nearly extremal spin, however,
higher order caustics grow so large that their cross-sections at fixed radii
wind several times around the black hole. As a consequence, for each caustic
order, the number of images ranges from 2 to 2(n+1), where n is the number of
loops spanned by the caustic. As for the critical curves, we note that for high
values of the spin they develop a small dip on the side corresponding to
prograde orbits.Comment: 20 pages, 21 figure
Gravitational lensing in the strong field limit
We provide an analytic method to discriminate among different types of black
holes on the ground of their strong field gravitational lensing properties. We
expand the deflection angle of the photon in the neighbourhood of complete
capture, defining a strong field limit, in opposition to the standard weak
field limit. This expansion is worked out for a completely generic spherically
symmetric spacetime, without any reference to the field equations and just
assuming that the light ray follows the geodesics equation. We prove that the
deflection angle always diverges logarithmically when the minimum impact
parameter is reached. We apply this general formalism to Schwarzschild,
Reissner-Nordstrom and Janis-Newman-Winicour black holes. We then compare the
coefficients characterizing these metrics and find that different collapsed
objects are characterized by different strong field limits. The strong field
limit coefficients are directly connected to the observables, such as the
position and the magnification of the relativistic images. As a concrete
example, we consider the black hole at the centre of our galaxy and estimate
the optical resolution needed to investigate its strong field behaviour through
its relativistic images.Comment: 10 pages, 5 figures, in press on Physical Review
An extended scheme for fitting X-ray data with accretion disk spectra in the strong gravity regime
Accreting black holes are believed to emit X-rays which then mediate
information about strong gravity in the vicinity of the emission region. We
report on a set of new routines for the Xspec package for analysing X-ray
spectra of black-hole accretion disks. The new computational tool significantly
extends the capabilities of the currently available fitting procedures that
include the effects of strong gravity, and allows one to systematically explore
the constraints on more model parameters than previously possible (for example
black-hole angular momentum). Moreover, axial symmetry of the disk intrinsic
emissivity is not assumed, although it can be imposed to speed up the
computations. The new routines can be used also as a stand-alone and flexible
code with the capability of handling time-resolved spectra in the regime of
strong gravity. We have used the new code to analyse the mean X-ray spectrum
from the long XMM--Newton 2001 campaign of the Seyfert 1 galaxy MCG--6-30-15.
Consistent with previous findings, we obtained a good fit to the broad Fe K
line profile for a radial line intrinsic emissivity law in the disk which is
not a simple power law, and for near maximal value of black hole angular
momentum. However, equally good fits can be obtained also for small values of
the black hole angular momentum. The code has been developed with the aim of
allowing precise modelling of relativistic effects. Although we find that
current data cannot constrain the parameters of black-hole/accretion disk
system well, the approach allows, for a given source or situation, detailed
investigations of what features of the data future studies should be focused on
in order to achieve the goal of uniquely isolating the parameters of such
systems.Comment: Accepted for publication in ApJ S