EDITORIAL: “Six Sigma - Its Application, Practice, and Utility”

Editorial for Special Issue on Six Sigm

Quasi-Equatorial Gravitational Lensing by Spinning Black Holes in the Strong Field Limit

Spherically symmetric black holes produce, by strong field lensing, two infinite series of relativistic images, formed by light rays winding around the black hole at distances comparable to the gravitational radius. In this paper, we address the relevance of the black hole spin for the strong field lensing phenomenology, focusing on trajectories close to the equatorial plane for simplicity. In this approximation, we derive a two-dimensional lens equation and formulae for the position and the magnification of the relativistic images in the strong field limit. The most outstanding effect is the generation of a non trivial caustic structure. Caustics drift away from the optical axis and acquire finite extension. For a high enough black hole spin, depending on the source extension, we can practically observe only one image rather than two infinite series of relativistic images. In this regime, additional non equatorial images may play an important role in the phenomenology.Comment: 13 pages, 9 figures. Improved version with detailed physical discussio

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

Stability analysis and quasinormal modes of Reissner Nordstr{\o}m Space-time via Lyapunov exponent

We explicitly derive the proper time $(\tau)$ principal Lyapunov exponent ($\lambda_{p}$) and coordinate time ($t$) principal Lyapunov exponent ($\lambda_{c}$) 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 $\frac{\lambda_{p}}{\lambda_{c}}=\frac{r_{0}}{\sqrt{r_{0}^2-3Mr_{0}+2Q^2}}$ for time-like circular geodesics and for Schwarzschild BH it is $\frac{\lambda_{p}}{\lambda_{c}}=\frac{\sqrt{r_{0}}}{\sqrt{r_{0}-3M}}$. We further show that their ratio $\frac{\lambda_{p}}{\lambda_{c}}$ may vary from orbit to orbit. For instance, Schwarzschild BH at innermost stable circular orbit(ISCO), the ratio is $\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=6M}=\sqrt{2}$ and at marginally bound circular orbit (MBCO) the ratio is calculated to be $\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{mb}=4M}=2$. Similarly, for extremal RN BH the ratio at ISCO is $\frac{\lambda_{p}}{\lambda_{c}}\mid_{r_{ISCO}=4M}=\frac{2\sqrt{2}}{\sqrt{3}}$. 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

Non-Commutative Correction to Thin Shell Collapse in Reissner Nordstro¨\ddot{o}m Geometry

This paper investigates the polytropic matter shell collapse in the non-commutative Reissner-Nordstr$\ddot{o}$m geometry. Using the Israel criteria, equation of motion for the polytropic matter shell is derived. In order to explore the physical aspects of this equation, the most general equation of state, $p=k{\rho}^{({1+\frac{1}{n}})}$, has been used for finite and infinite values of $n$. The effective potentials corresponding to the equation of motion have been used to explain different states of the matter shell collapse. The numerical solution of the equation of motion predicts collapse as well as expansion depending on the choice of initial data. Further, in order to include the non-commutative correction, we modify the matter components and re-formulate the equation of motion as well as the corresponding effective potentials by including non-commutative factor and charge parameter. It is concluded that charge reduces the velocity of the expanding or collapsing matter shell but does not bring the shell to static position. While the non-commutative factor with generic matter favors the formation of black hole.Comment: 18 pages,17 figure

Bile Acid Sequestrants for Lipid and Glucose Control

Bile acids are generated in the liver and are traditionally recognized for their regulatory role in multiple metabolic processes including bile acid homeostasis, nutrient absorption, and cholesterol homeostasis. Recently, bile acids emerged as signaling molecules that, as ligands for the bile acid receptors farnesoid X receptor (FXR) and TGR5, activate and integrate multiple complex signaling pathways involved in lipid and glucose metabolism. Bile acid sequestrants are pharmacologic molecules that bind to bile acids in the intestine resulting in the interruption of bile acid homeostasis and, consequently, reduction in low-density lipoprotein cholesterol levels in hypercholesterolemia. Bile acid sequestrants also reduce glucose levels and improve glycemic control in persons with type 2 diabetes mellitus (T2DM). This article examines the mechanisms by which bile acid–mediated activation of FXR and TGR5 signaling pathways regulate lipid and glucose metabolism and the potential implications for bile acid sequestrant–mediated regulation of lipid and glucose levels in T2DM

Creating a proof-of-concept climate service to assess future renewable energy mixes in Europe: an overview of the C3S ECEM project

The EU Copernicus Climate Change Service (C3S) European Climatic Energy Mixes (ECEM) has produced, in close collaboration with prospective users, a proof-of-concept climate service, or Demonstrator, designed to enable the energy industry and policy makers assess how well different energy supply mixes in Europe will meet demand, over different time horizons (from seasonal to long-term decadal planning), focusing on the role climate has on the mixes. The concept of C3S ECEM, its methodology and some results are presented here. The first part focuses on the construction of reference data sets for climate variables based on the ERA-Interim reanalysis. Subsequently, energy variables were created by transforming the bias-adjusted climate variables using a combination of statistical and physically-based models. A comprehensive set of measured energy supply and demand data was also collected, in order to assess the robustness of the conversion to energy variables. Climate and energy data have been produced both for the historical period (1979–2016) and for future projections (from 1981 to 2100, to also include a past reference period, but focusing on the 30 year period 2035–2065). The skill of current seasonal forecast systems for climate and energy variables has also been assessed. The C3S ECEM project was designed to provide ample opportunities for stakeholders to convey their needs and expectations, and assist in the development of a suitable Demonstrator. This is the tool that collects the output produced by C3S ECEM and presents it in a user-friendly and interactive format, and it therefore constitutes the essence of the C3S ECEM proof-of-concept climate service

From Geometry to Numerics: interdisciplinary aspects in mathematical and numerical relativity

This article reviews some aspects in the current relationship between mathematical and numerical General Relativity. Focus is placed on the description of isolated systems, with a particular emphasis on recent developments in the study of black holes. Ideas concerning asymptotic flatness, the initial value problem, the constraint equations, evolution formalisms, geometric inequalities and quasi-local black hole horizons are discussed on the light of the interaction between numerical and mathematical relativists.Comment: Topical review commissioned by Classical and Quantum Gravity. Discussion inspired by the workshop "From Geometry to Numerics" (Paris, 20-24 November, 2006), part of the "General Relativity Trimester" at the Institut Henri Poincare (Fall 2006). Comments and references added. Typos corrected. Submitted to Classical and Quantum Gravit

Gravitational Lensing by Black Holes

We review the theoretical aspects of gravitational lensing by black holes, and discuss the perspectives for realistic observations. We will first treat lensing by spherically symmetric black holes, in which the formation of infinite sequences of higher order images emerges in the clearest way. We will then consider the effects of the spin of the black hole, with the formation of giant higher order caustics and multiple images. Finally, we will consider the perspectives for observations of black hole lensing, from the detection of secondary images of stellar sources and spots on the accretion disk to the interpretation of iron K-lines and direct imaging of the shadow of the black hole.Comment: Invited article for the GRG special issue on lensing (P. Jetzer, Y. Mellier and V. Perlick Eds.). 31 pages, 12 figure

Theorems on existence and global dynamics for the Einstein equations

This article is a guide to theorems on existence and global dynamics of solutions of the Einstein equations. It draws attention to open questions in the field. The local-in-time Cauchy problem, which is relatively well understood, is surveyed. Global results for solutions with various types of symmetry are discussed. A selection of results from Newtonian theory and special relativity that offer useful comparisons is presented. Treatments of global results in the case of small data and results on constructing spacetimes with prescribed singularity structure or late-time asymptotics are given. A conjectural picture of the asymptotic behaviour of general cosmological solutions of the Einstein equations is built up. Some miscellaneous topics connected with the main theme are collected in a separate section.Comment: Submitted to Living Reviews in Relativity, major update of Living Rev. Rel. 5 (2002)