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

Semi-analytic results for quasi-normal frequencies

The last decade has seen considerable interest in the quasi-normal frequencies [QNFs] of black holes (and even wormholes), both asymptotically flat and with cosmological horizons. There is wide agreement that the QNFs are often of the form omega_n = (offset) + i n (gap), though some authors have encountered situations where this behaviour seems to fail. To get a better understanding of the general situation we consider a semi-analytic model based on a piecewise Eckart (Poeschl-Teller) potential, allowing for different heights and different rates of exponential falloff in the two asymptotic directions. This model is sufficiently general to capture and display key features of the black hole QNFs while simultaneously being analytically tractable, at least for asymptotically large imaginary parts of the QNFs. We shall derive an appropriate "quantization condition" for the asymptotic QNFs, and extract as much analytic information as possible. In particular, we shall explicitly verify that the (offset)+ i n (gap) behaviour is common but not universal, with this behaviour failing unless the ratio of rates of exponential falloff on the two sides of the potential is a rational number. (This is "common but not universal" in the sense that the rational numbers are dense in the reals.) We argue that this behaviour is likely to persist for black holes with cosmological horizons.Comment: V1: 28 pages, no figures. V2: 3 references added, no physics changes. V3: 29 pages, 9 references added, no physics changes; V4: reformatted, now 27 pages. Some clarifications, comparison with results obtained by monodromy techniques. This version accepted for publication in JHEP. V5: Minor typos fixed. Compatible with published versio

Quasi-normal frequencies: Key analytic results

The study of exact quasi-normal modes [QNMs], and their associated quasi-normal frequencies [QNFs], has had a long and convoluted history - replete with many rediscoveries of previously known results. In this article we shall collect and survey a number of known analytic results, and develop several new analytic results - specifically we shall provide several new QNF results and estimates, in a form amenable for comparison with the extant literature. Apart from their intrinsic interest, these exact and approximate results serve as a backdrop and a consistency check on ongoing efforts to find general model-independent estimates for QNFs, and general model-independent bounds on transmission probabilities. Our calculations also provide yet another physics application of the Lambert W function. These ideas have relevance to fields as diverse as black hole physics, (where they are related to the damped oscillations of astrophysical black holes, to greybody factors for the Hawking radiation, and to more speculative state-counting models for the Bekenstein entropy), to quantum field theory (where they are related to Casimir energies in unbounded systems), through to condensed matter physics, (where one may literally be interested in an electron tunelling through a physical barrier).Comment: V1: 29 pages; V2: Reformatted, 31 pages. Title changed to reflect major additions and revisions. Now describes exact QNFs for the double-delta potential in terms of the Lambert W function. V3: Minor edits for clarity. Four references added. No physics changes. Still 31 page

Repaired tetralogy of Fallot: the roles of cardiovascular magnetic resonance in evaluating pathophysiology and for pulmonary valve replacement decision support

Surgical management of tetralogy of Fallot (TOF) results in anatomic and functional abnormalities in the majority of patients. Although right ventricular volume load due to severe pulmonary regurgitation can be tolerated for many years, there is now evidence that the compensatory mechanisms of the right ventricular myocardium ultimately fail and that if the volume load is not eliminated or reduced by pulmonary valve replacement the dysfunction might be irreversible. Cardiovascular magnetic resonance (CMR) has evolved during the last 2 decades as the reference standard imaging modality to assess the anatomic and functional sequelae in patients with repaired TOF. This article reviews the pathophysiology of chronic right ventricular volume load after TOF repair and the risks and benefits of pulmonary valve replacement. The CMR techniques used to comprehensively evaluate the patient with repaired TOF are reviewed and the role of CMR in supporting clinical decisions regarding pulmonary valve replacement is discussed

Exploring new physics frontiers through numerical relativity

The demand to obtain answers to highly complex problems within strong-field gravity has been met with significant progress in the numerical solution of Einstein's equations - along with some spectacular results - in various setups. We review techniques for solving Einstein's equations in generic spacetimes, focusing on fully nonlinear evolutions but also on how to benchmark those results with perturbative approaches. The results address problems in high-energy physics, holography, mathematical physics, fundamental physics, astrophysics and cosmology

Modelling Survival and Mortality Risk to 15 Years of Age for a National Cohort of Children with Serious Congenital Heart Defects Diagnosed in Infancy

Congenital heart defects (CHDs) are a significant cause of death in infancy. Although contemporary management ensures that 80% of affected children reach adulthood, post-infant mortality and factors associated with death during childhood are not well-characterised. Using data from a UK-wide multicentre birth cohort of children with serious CHDs, we observed survival and investigated independent predictors of mortality up to age 15 years. Methods Data were extracted retrospectively from hospital records and death certificates of 3,897 children (57% boys) in a prospectively identified cohort, born 1992–1995 with CHDs requiring intervention or resulting in death before age one year. A discrete-time survival model accounted for time-varying predictors; hazards ratios were estimated for mortality. Incomplete data were addressed through multilevel multiple imputation. Findings By age 15 years, 932 children had died; 144 died without any procedure. Survival to one year was 79.8% (95% confidence intervals [CI] 78.5, 81.1%) and to 15 years was 71.7% (63.9, 73.4%), with variation by cardiac diagnosis. Importantly, 20% of cohort deaths occurred after age one year. Models using imputed data (including all children from birth) demonstrated higher mortality risk as independently associated with cardiac diagnosis, female sex, preterm birth, having additional cardiac defects or non-cardiac malformations. In models excluding children who had no procedure, additional predictors of higher mortality were younger age at first procedure, lower weight or height, longer cardiopulmonary bypass or circulatory arrest duration, and peri-procedural complications; non-cardiac malformations were no longer significant. Interpretation We confirm the high mortality risk associated with CHDs in the first year of life and demonstrate an important persisting risk of death throughout childhood. Late mortality may be underestimated by procedure-based audit focusing on shorter-term surgical outcomes. National monitoring systems should emphasise the importance of routinely capturing longer-term survival and exploring the mechanismsThis work was supported by a British Heart Foundation project grant (reference PG/02/065/13934). RLK was awarded an MRC Special Training Fellowship in Health of the Public and Health Services Research (reference G106/1083). HG and the Centre for Paediatric Epidemiology and Biostatistics benefited from Medical Research Council funding support to the MRC Centre of Epidemiology for Child Health (reference G04005546). Great Ormond St Hospital for Children NHS Trust and the UCL Institute of Child Health receives a proportion of funding from the Department of Health's NIHR Biomedical Research Centres schem