Extremal black holes, gravitational entropy and nonstationary metric fields

We show that extremal black holes have zero entropy by pointing out a simple fact: they are time-independent throughout the spacetime and correspond to a single classical microstate. We show that non-extremal black holes, including the Schwarzschild black hole, contain a region hidden behind the event horizon where all their Killing vectors are spacelike. This region is nonstationary and the time $t$ labels a continuous set of classical microstates, the phase space $[\,h_{ab}(t), P^{ab}(t)\,]$, where $h_{ab}$ is a three-metric induced on a spacelike hypersurface $\Sigma_t$ and $P^{ab}$ is its momentum conjugate. We determine explicitly the phase space in the interior region of the Schwarzschild black hole. We identify its entropy as a measure of an outside observer's ignorance of the classical microstates in the interior since the parameter $t$ which labels the states lies anywhere between 0 and 2M. We provide numerical evidence from recent simulations of gravitational collapse in isotropic coordinates that the entropy of the Schwarzschild black hole stems from the region inside and near the event horizon where the metric fields are nonstationary; the rest of the spacetime, which is static, makes no contribution. Extremal black holes have an event horizon but in contrast to non-extremal black holes, their extended spacetimes do not possess a bifurcate Killing horizon. This is consistent with the fact that extremal black holes are time-independent and therefore have no distinct time-reverse.Comment: 12 pages, 2 figures. To appear in Class. and Quant. Gravity. Based on an essay selected for honorable mention in the 2010 gravity research foundation essay competitio

General C–H Arylation Strategy for the Synthesis of Tunable Visible Light-Emitting Benzo[<i>a</i>]imidazo[2,1,5‑<i>c</i>,<i>d</i>]indolizine Fluorophores

Herein we report the discovery of the benzo­[<i>a</i>]­imidazo­[2,1,5-<i>c</i>,<i>d</i>]­indolizine motif displaying tunable emission covering most of the visible spectrum. The polycyclic core is obtained from readily available amides via a chemoselective process involving Tf₂O-mediated amide cyclodehydration, followed by intramolecular C–H arylation. Additionally, these fluorescent heterocycles are easily functionalized using electrophilic reagents, enabling divergent access to varied substitution. The effects of said substitution on the compounds’ photophysical properties were rationalized by density functional theory calculations. For some compounds, emission wavelengths are directly correlated to the substituent’s Hammett constants. Easily introduced nonconjugated reactive functional groups allow the labeling of biomolecules without modification of emissive properties. This work provides a straightforward platform for the synthesis of new moderately bright fluorescent dyes remarkable for their chemical stability, predictability, and unusually high excitation–emission differential

The determinants of company response to environmental regulation

The impact of governmental regulatory action on the economic performance of mining companies has been the focus of a great deal of attention by industry commentators and academic scholarship alike. The influence of environmental regulation on the strategic objectives of mining firms is less well understood, however. This article investigates what influences the way mining companies react to environmental approvals regulation. It presents the results of a recent cross-national survey of Australian and Canadian mining companies into the effect of a range of possible determinants on company response to environmental approvals regulation. Possible influences included both external pressures (stakeholder pressure, jurisdictional culture and market characteristics) and internal pressures (organizational culture, organizational learning, the influence of individuals within firms and company size). While cross-national comparison revealed some differences with respect to the influence of particular pressures, on balance the results suggest that for mining companies in both countries, internal pressures exert the greatest influence on company response. These results contradict a prevailing view in the literature, which suggests that external factors, particularly stakeholder groups, exert the most influence on the environmental responses of firms. The article concludes that the existing emphasis on external pressures to explain corporate environmental behaviour should be supplemented by a focus on the internal dynamics of firms