In a recent paper [P. V. P. Cunha, C. A. R. Herdeiro, E. Radu, and H. F. Runarsson, Phys. Rev. Lett. 115, 211102 (2015).], it was shown that the lensing of light around rotating boson stars and Kerr black holes with scalar hair can exhibit chaotic patterns. Since no separation of variables is known (or expected) for geodesic motion on these backgrounds, we examine the 2D effective potentials for photon trajectories, to obtain a deeper understanding of this phenomenon. We find that the emergence of stable light rings on the background spacetimes allows the formation of "pockets" in one of the effective potentials, for open sets of impact parameters, leading to an effective trapping of some trajectories, dubbed "quasibound orbits." We conclude that pocket formation induces chaotic scattering, although not all chaotic orbits are associated to pockets. These and other features are illustrated in a gallery of examples, obtained with a new ray-tracing code, PYHOLE, which includes tools for a simple, simultaneous visualization of the effective potential, together with the spacetime trajectory, for any given point in a lensing image. An analysis of photon orbits allows us to further establish a positive correlation between photon orbits in chaotic regions and those with more than one turning point in the radial direction; we recall that the latter is not possible around Kerr black holes. Moreover, we observe that the existence of several light rings around a horizon (several fundamental orbits, including a stable one), is a central ingredient for the existence of multiple shadows of a single hairy black hole. We also exhibit the lensing and shadows by Kerr black holes with scalar hair, observed away from the equatorial plane, obtained with PYHOLE.
We establish the multiparameter quantum Cramér-Rao bound for simultaneously estimating the centroid, the separation, and the relative intensities of two incoherent optical point sources using a linear imaging system. For equally bright sources, the Cramér-Rao bound is independent of the source separation, which confirms that the Rayleigh resolution limit is just an artifact of the conventional direct imaging and can be overcome with an adequate strategy. For the general case of unequally bright sources, the amount of information one can gain about the separation falls to zero, but we show that there is always a quadratic improvement in an optimal detection in comparison with the intensity measurements. This advantage can be of utmost important in realistic scenarios, such as observational astronomy.The time-honored Rayleigh criterion [1] specifies the minimum separation between two incoherent optical sources using a linear imaging system. As a matter of fact, it is the size of the point spread function [2] that determines the resolution: two points closer than the PSF width will be difficult to resolve due to the substantial overlap of their images.Thus far, this Rayleigh criterion has been considered as a fundamental limit. Resolution can only be improved either by reducing the wavelength or by building higher numericalaperture optics, thereby making the PSF narrower. Nonetheless, outstanding methods have been developed lately that can break the Rayleigh limit under special circumstances [3][4][5][6][7][8][9][10][11][12]. Though promising, these techniques are involved and require careful control of the source, which is not always possible, especially in astronomical applications.Despite being very intuitive, the common derivation of the Rayleigh limit is heuristic and it is deeply rooted in classical optical technology [13]. Recently, inspired by ideas of quantum information, Tsang and coworkers [14][15][16][17] have revisited this problem using the Fisher information and the associated Cramér-Rao lower bound (CRLB) to quantify how well the separation between two point sources can be estimated. When only the intensity at the image is measured (the basis of all the conventional techniques), the Fisher information falls to zero as the separation between the sources decreases and the CRLB diverges accordingly; this is known as the Rayleigh curse [14]. However, when the Fisher information of the complete field is calculated, it stays constant and so does the CRLB, revealing that the Rayleigh limit is not essential to the problem.These remarkable predictions prompted a series of experimental implementations [18][19][20] and further generalizations [21][22][23][24][25], including the related question of source localization [26][27][28]. All this previous work has focused on the estimation of the separation, taking for granted a highly symmetric configuration with identical sources. In this Letter, we approach the issue in a more realistic scenario, where both sources may have unequal intensities. This involves the si...
Utilizing concepts from dynamical systems theory, we demonstrate how the existence of light rings, or fixed points, in a spacetime will give rise to families of periodic orbits and invariant manifolds in phase space. It is shown that these structures define the shape of the black hole shadow as well as a number of salient features of the spacetime lensing. We illustrate this through the analysis of lensing by a hairy black hole.
Abstract:We prove that the near-horizon geometries of minimal gauged five-dimensional supergravity preserve at least half of the supersymmetry. If the near-horizon geometries preserve a larger fraction, then they are locally isometric to AdS 5 . Our proof is based on Lichnerowicz type theorems for two horizon Dirac operators constructed from the supercovariant connection restricted to the horizon sections, and on an application of the index theorem. An application is that all half-supersymmetric five-dimensional horizons admit an sl(2, R) symmetry subalgebra.
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