155 research outputs found
Magnification relations for Kerr lensing and testing Cosmic Censorship
A Kerr black hole with mass parameter m and angular momentum parameter a
acting as a gravitational lens gives rise to two images in the weak field
limit. We study the corresponding magnification relations, namely the signed
and absolute magnification sums and the centroid up to post-Newtonian order. We
show that there are post-Newtonian corrections to the total absolute
magnification and centroid proportional to a/m, which is in contrast to the
spherically symmetric case where such corrections vanish. Hence we also propose
a new set of lensing observables for the two images involving these
corrections, which should allow measuring a/m with gravitational lensing. In
fact, the resolution capabilities needed to observe this for the Galactic black
hole should in principle be accessible to current and near-future
instrumentation. Since a/m >1 indicates a naked singularity, a most interesting
application would be a test of the Cosmic Censorship conjecture. The technique
used to derive the image properties is based on the degeneracy of the Kerr lens
and a suitably displaced Schwarzschild lens at post-Newtonian order. A simple
physical explanation for this degeneracy is also given.Comment: 13 pages, version 2: references added, minor changes. To appear in
Phys. Rev.
Center of Light Curves for Whitney Fold and Cusp
The generic, qualitative, local behavior of center-of-light curves near folds
and cusps are studied. The results apply to any finite number of lens planes.Comment: 2 pages, 1 figure, to appear in the ``Proceedings of the Ninth Marcel
Grossmann Meeting on General Relativity,'' eds. V. Gurzadyan, R. Jantzen, &
R. Ruffini, World Scientific (Singapore
On Relativistic Corrections to Microlensing Effects: Applications to the Galactic Black Hole
The standard treatment of gravitational lensing by a point mass lens M is
based on a weak-field deflection angle a = 2/x, where x = (r c^2)/(2 G M) with
r the distance of closest approach to the mass of a lensed light ray. It was
shown that for a point mass lens, the total magnification and image centroid
shift of a point source remain unchanged by relativistic corrections of second
order in 1/x. This paper considers these issues analytically taking into
account the relativistic images, under three standard lensing configuration
assumptions, for a Schwarzschild black hole lens with background point and
extended sources having arbitrary surface brightness profiles. We apply our
results to the Galactic black hole for lensing scenarios where our assumptions
hold. We show that a single factor characterizes the full relativistic
correction to the weak-field image centroid and magnification. As the
lens-source distance increases, the relativistic correction factor strictly
decreases. In particular, we find that for point and extended sources about 10
pc behind the black hole (which is a distance significantly outside the tidal
disruption radius of a sun-like source), the relativistic correction factor is
miniscule, of order 10^{-14}. Therefore, for standard lensing configurations,
any detectable relativistic corrections to microlensing by the Galactic black
hole will most likely have to come from sources significantly closer to the
black hole.Comment: To appear in MNRAS, 8 pages, 4 figure
A Universal Magnification Theorem III. Caustics Beyond Codimension Five
In the final paper of this series, we extend our results on magnification
invariants to the infinite family of A, D, E caustic singularities. We prove
that for families of general mappings between planes exhibiting any caustic
singularity of the A, D, E family, and for a point in the target space lying
anywhere in the region giving rise to the maximum number of lensed images (real
pre-images), the total signed magnification of the lensed images will always
sum to zero. The proof is algebraic in nature and relies on the Euler trace
formula.Comment: 8 page
A Mathematical Theory of Stochastic Microlensing II. Random Images, Shear, and the Kac-Rice Formula
Continuing our development of a mathematical theory of stochastic
microlensing, we study the random shear and expected number of random lensed
images of different types. In particular, we characterize the first three
leading terms in the asymptotic expression of the joint probability density
function (p.d.f.) of the random shear tensor at a general point in the lens
plane due to point masses in the limit of an infinite number of stars. Up to
this order, the p.d.f. depends on the magnitude of the shear tensor, the
optical depth, and the mean number of stars through a combination of radial
position and the stars' masses. As a consequence, the p.d.f.s of the shear
components are seen to converge, in the limit of an infinite number of stars,
to shifted Cauchy distributions, which shows that the shear components have
heavy tails in that limit. The asymptotic p.d.f. of the shear magnitude in the
limit of an infinite number of stars is also presented. Extending to general
random distributions of the lenses, we employ the Kac-Rice formula and Morse
theory to deduce general formulas for the expected total number of images and
the expected number of saddle images. We further generalize these results by
considering random sources defined on a countable compact covering of the light
source plane. This is done to introduce the notion of {\it global} expected
number of positive parity images due to a general lensing map. Applying the
result to microlensing, we calculate the asymptotic global expected number of
minimum images in the limit of an infinite number of stars, where the stars are
uniformly distributed. This global expectation is bounded, while the global
expected number of images and the global expected number of saddle images
diverge as the order of the number of stars.Comment: To appear in JM
A comparison of approximate gravitational lens equations and a proposal for an improved new one
Keeping the exact general relativistic treatment of light bending as a
reference, we compare the accuracy of commonly used approximate lens equations.
We conclude that the best approximate lens equation is the Ohanian lens
equation, for which we present a new expression in terms of distances between
observer, lens and source planes. We also examine a realistic gravitational
lensing case, showing that the precision of the Ohanian lens equation might be
required for a reliable treatment of gravitational lensing and a correct
extraction of the full information about gravitational physics.Comment: 11 pages, 6 figures, to appear on Physical Review
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