Fluctuations of an Evaporating Black Hole from Back Reaction of Its Hawking Radiation: Questioning a Premise in Earlier Work

This paper delineates the first steps in a systematic quantitative study of the spacetime fluctuations induced by quantum fields in an evaporating black hole. We explain how the stochastic gravity formalism can be a useful tool for that purpose within a low-energy effective field theory approach to quantum gravity. As an explicit example we apply it to the study of the spherically-symmetric sector of metric perturbations around an evaporating black hole background geometry. For macroscopic black holes we find that those fluctuations grow and eventually become important when considering sufficiently long periods of time (of the order of the evaporation time), but well before the Planckian regime is reached. In addition, the assumption of a simple correlation between the fluctuations of the energy flux crossing the horizon and far from it, which was made in earlier work on spherically-symmetric induced fluctuations, is carefully analyzed and found to be invalid. Our analysis suggests the existence of an infinite amplitude for the fluctuations of the horizon as a three-dimensional hypersurface. We emphasize the need for understanding and designing operational ways of probing quantum metric fluctuations near the horizon and extracting physically meaningful information. Dedicated to Rafael Sorkin on the occasion of his 60th birthday.     

The three space problem in topological groups

We study compact, countably compact, pseudocompact, and functionally bounded sets in extensions of topological groups. A property P is said to be a three space property if, for every topological group G and a closed invariant subgroup N of G, the fact that both groups N and G/N have P implies that G also has P. It is shown that if all compact (countably compact) subsets of the groups N and G/N are metrizable, then G has the same property. However, the result cannot be extended to pseudocompact subsets, a counterexample exists under p=c. Another example shows that extensions of groups do not preserve the classes of realcompact, Dieudonné complete and μ-spaces: one can find a pseudocompact, non-compact Abelian topological group G and an infinite, closed, realcompact subgroup N of G such that G/N is compact and all functionally bounded subsets of N are finite. Several examples given in the article destroy a number of tempting conjectures about extensions of topological groups.     

Bounded sets in topological groups and embeddings

We show that the existence of a non-metrizable compact subspace of a topological group G often implies that G contains an uncountable supersequence (a copy of the one-point compactification of an uncountable discrete space). The existence of uncountable supersequences in a topological group has a strong impact on bounded subsets of the group. For example, if a topological group G contains an uncountable supersequence and K is a closed bounded subset of G which does not contain uncountable supersequences, then any subset A of K is bounded in G?(K?A). We also show that every precompact Abelian topological group H can be embedded as a closed subgroup into a precompact Abelian topological group G such that H is bounded in G and all bounded subsets of the quotient group G/H are finite. This complements Ursul's result on closed embeddings of precompact groups to pseudocompact groups.     

On the effect of montmorillonite in the curing reaction of epoxy nanocomposites

The procedure for the fabrication of epoxy-based polymer layered silicate nanocomposites is important in respect of the nanostructure that is developed. To further our understanding of this, the influence of an organically modified clay (montmorillonite, MMT) on the curing kinetics of an epoxy resin has been studied by differential scanning calorimetry. Clay loadings of 10 and 20 mass% are used, and isothermal as well as dynamic cures have been investigated. For both cure schedules the effect of the MMT is to advance the reaction. Kinetic analysis yields values for the activation energy, but shows that the reaction cannot be described simply by the usual autocatalytic equation. The glass transition of the cured nanocomposites is lower than that for the cured neat resin, a result that is attributed to homopolymerisation taking place in addition to the epoxy–amine reaction.     

Multivariate resolution of NMR labile signals by means of hard- and soft-modelling methods

One of the difficulties frequently encountered when studying acid–base equilibria with NMR spectroscopy is the labile behaviour of the measured signal, which hinders the application of bilinear multivariate data analysis methods. In this work, a mathematical transformation is proposed for the conversion of NMR labile signals to inert signals, which make possible the application of multivariate data analysis methods, based on bilinear data models. The procedure has been applied to the analysis of NMR data corresponding to the acid–base equilibria of nucleotides dCMP and dGMP. Both hard-modelling (EQUISPEC) and soft-modelling (MCR-ALS) approaches have been applied for the analysis and resolution of transformed bilinear NMR data matrices.     

Syntheses and Structural Aspects of Cyclodextrin/Dialkylamine Inclusion Compounds

We report the syntheses and structural aspects of cyclodextrin host–guest inclusion compounds containing linear secondary alkylamines (dipropyl, dibutyl, dipentyl, dihexyl, and dioctyl) at 25 °C. Elemental analysis, ¹³C CP-MAS NMR spectroscopy, and powder X-ray diffraction analysis confirm the inclusion process. The basic host structure of the products is similar to that of typical cyclodextrin inclusion systems. ¹³C MAS NMR experiments show a different resonance pattern for the confined guest molecules with respect to the amine in the liquid phase. The presence of different resonance signals for the homologous carbon atoms of both dialkylamine branches is evidence for the non-symmetric location of the amine in the cyclodextrin channels.     

Enzymatic desaturation of fatty acids: delta11 desaturase activity on cyclopropane acid probes

The formation of methylenecyclopropanes by enzymatic desaturation of 11-cyclopropylundecanoic acid (1) and its disubstituted derivatives cis- and trans-3-5 has been investigated using the Delta(11) desaturase of Spodoptera littoralis as model enzyme. Gas chromatography coupled to mass spectrometry analyses of methanolyzed lipidic extracts from tissues incubated with each probe revealed that all the cyclopropyl fatty acids were transformed into the corresponding 11-cyclopropylidene acids, except for compound trans-5 (5b), which was not desaturated at C11. The formation of methylenecyclopropane 9 as the only reaction product from 1 indicates that a potential radical intermediate is too short-lived to allow rearrangement reactions. Information on the Delta(11) desaturase substrate binding domain is provided considering the cyclopropyl probes 3-5 as conformationally restricted analogues of the straight-chain substrates.     

Crystal structure oftrans-ethyl(1,5,6-trimethylbenzimidazole)-bis(dimethylglyoximato)cobalt(III). Relationships between structural and spectroscopic properties in compounds of the general formulae [Co(dmgH)2(R)(1,5,6-Me3Bzm)]

Summary The synthesis and x-ray crystal structure oftrans-[Co(dmgH)₂(Et)(1,5,6-Me₃Bzm)] where dmgH=dimethylglyoximate(–1), and 1,5,6-Me₃Bzm=1,5,6-trimethylbenzimidazole, is reported. The compound C₁₉H₂₆N₆O₄Co is monoclinic, space group P2₁/n;a=11.700(4);b=24.205(6);c=8.500(3) Å and =101.63(3)°. D(calcd) 1.299 g cm^–3; Z=4 and R=0.066 for 2359 independent reflections. Comparison of Co-N(axial ligand) bond lengths for compounds of general formulaetrans-[Co(dmgH)₂(R)(L)], with L=pyridine or 1,5,6-trimethylbenzimidazole and R=CH(CN)Cl, CH₂NO₂, Me, Et,i-Pr, cyclo-hexyl or adamantyl is made. The Co–N(1,5,6-Me₃Bzm) bond lengths of the trimethylbenzimidazole derivatives show a fairly linear relationship with the electronic parameter of the axial R group, derived from the¹³C-n.m.r. spectra of their pyridine analogues. The influence of steric effects on the properties of these Co^III compounds is discussed.     

Synthesis,Spectroscopic and EPR Results and Crystal Structure of Copper(II) Formato Derivatives with Pyrazine and Dimethylpyrazines

The complexes of copper formato with pyrazine and dimethylpyrazines: Cu(HCOO)₂(pyrz) ( 1 ), Cu₂(HCOO)₄(pyrz) ( 2 ), Cu₂(HCOO)₄(2,3‐Me₂pyrz) ( 3 ), Cu₂(HCOO)₄(2,5‐Me₂pyrz) ( 4 ), and Cu₂(HCOO)₄(2,6‐Me₂pyrz)₂ ( 5 ) have been synthesized and characterized by chemical analysis and IR and electronic spectro scopies. The three‐dimensional structure of compound 1 consists of –Cu–pyrz–Cu–pyrz– chains joined by bridging formate groups coordinated, in configuration syn‐anti, to two copper atoms. The coordination around the copper atom is orthorhombic with two Cu–O distances of 2.374(1) Å, two Cu–O of 1.952 Å and two Cu–N of 2.080 Å. Compound 5 is formed by molecular dimers with the [Cu₂(μ‐HCOO)₄] unit, two copper atoms and four syn‐syn fomate groups, and two ligands coordinated to the copper atoms in the axial positions. In compounds 2 and 3 chains of [Cu₂(μ‐HCOO)₄] dimers with pyrz or 2,3‐Me₂pyrz as bridging ligands are formed. The EPR signal of 1 is orthorhombic (g = 2.23, 2.20 and 2.06). In the EPR spectra of 2 – 5 compounds the triplet (S = 1) signals are observed. The g_?, g_∥ and D values are been calculated.     

Synthesis and Characterization of Heterodinuclear IrCo, RuCo, IrNi, and RuNi Complexes Containing Pyrazolate and Pyrazolylborate Ligands

Treatment of the metallo ligands [ML(pz)(2)(Hpz)] (pz = pyrazolate; L = C(5)Me(5), M = Ir (1); L = mesitylene, M = Ru (3)) with [M'Cl{HB(3-i-Pr-4-Br-pz)(3)}] (M' = Co (4), Ni (5)) yields heterodinuclear complexes of formula [LM(&mgr;-pz)(2)(&mgr;-Cl)M'{HB(3-i-Pr-4-Br-pz)(3)}] (L = C(5)Me(5); M = Ir; M' = Co (6), Ni (7). L = mesitylene; M = Ru; M' = Co (8)). The related complex [Ru(eta(6)-p-cymene)(pz)(2)(Hpz)] (2) reacts with equimolar amounts of 4 or 5 to give mixtures of the corresponding bis(&mgr;-pyrazolato) &mgr;-chloro complexes [(eta(6)-p-cymene)Ru(&mgr;-pz)(2)(&mgr;-Cl)M'{HB(3-i-Pr-4-Br-pz)(3)}] (M' = Co (9), Ni (10)) and the triply pyrazolato-bridged complexes [(eta(6)-p-cymene)Ru(&mgr;-pz)(3)M'{HB(3-i-Pr-4-Br-pz)(3)}] (M' = Co (11), Ni (12)). Complex 1 reacts with 5 in the presence of KOH to give the IrNi complex [(eta(5)-C(5)Me(5))Ir(&mgr;-pz)(3)Ni{HB(3-i-Pr-4-Br-pz)(3)}] (13) whereas its reaction with 4 and KOH rendered the bis(&mgr;-pyrazolato) &mgr;-hydroxo complex [(eta(5)-C(5)Me(5))Ir(&mgr;-pz)(2)(&mgr;-OH)Co{HB(3-i-Pr-4-Br-pz)(3)}] (14). The molecular structure of the heterobridged IrCo complex (6) and those of the homobridged RuNi (12) and IrNi (13) complexes have been determined by X-ray analyses. Compound 6 crystallizes in the monoclinic space group P2(1)/n, with a = 10.146(5) ?, b = 18.435(4) ?, c = 22.187(13) ?, beta = 97.28(4) degrees, and Z = 4. Complex 12 is monoclinic, space group P2(1), with a = 10.1169(7) ?, b = 21.692(2) ?, c = 11.419(1) ?, beta = 112.179(7) degrees, and Z = 2. Compound 13 crystallizes in the monoclinic space group Cc, with a = 13.695(2) ?, b = 27.929(6) ?, c = 13.329(2) ?, beta = 94.11(4) degrees, and Z = 4. All the neutral complexes 6, 12, and 13 consist of linear M.M'.B backbones with two (6) or three (12, 13) pyrazolate ligands bridging the dimetallic M.M' units and three substituted 3-i-Pr-4-Br-pz groups joining M' to the boron atoms. The presence in the proximity of the first-row metal M' of the three space-demanding isopropyl substituents of the pyrazolate groups induces a significant trigonal distortion of the octahedral symmetry, yielding clearly different M'-N bond distances on both sides of the ideal octahedral coordination sphere of these metals.