Embedding Spacetime via a Geodesically Equivalent Metric of Euclidean Signature

Starting from the equations of motion in a 1 + 1 static, diagonal, Lorentzian spacetime, such as the Schwarzschild radial line element, I find another metric, but with Euclidean signature, which produces the same geodesics x(t). This geodesically equivalent, or dual, metric can be embedded in ordinary Euclidean space. On the embedded surface freely falling particles move on the shortest path. Thus one can visualize how acceleration in a gravitational field is explained by particles moving freely in a curved spacetime. Freedom in the dual metric allows us to display, with substantial curvature, even the weak gravity of our earth. This may provide a nice pedagogical tool for elementary lectures on general relativity. I also study extensions of the dual metric scheme to higher dimensions.     

Stability of a microwave heated fluid Layer

When a time harmonic electromagnetic wave impinges on a slaba certain portion of the wave creates heat within the slab throughdipolar and ohmic heating. The electrical and thermal propertiesof the material dictate the dynamical nature of the heatingprocess, as well as the steady-state temperature profile. Thematerial considered here is a slab of fluid. We consider thecase where the fluid is bounded by thin rigid layers of transparentmaterial. The steady-state heating profile governs the typesof convective motions that can occur and also affects the stabilitycharacteristics of temperature, pressure and velocity perturbationsintroduced in the slab. The main objective here is to examinesuch stability characteristics, initially in the linear regime.Both rigid-rigid and rigid-free configurations are considered.     

Idempotent Modules in the Stable Category

Let G be a finite group and k be an algebraically closed fieldof prime characteristic. Corresponding to each closed homogeneoussubvariety W of the maximal ideal spectrum of H*(G, k) we construct(usually infinite-dimensional) kG-modules E(W) and F(W) whichare idempotent in the sense that E(W) and F(W) are isomorphic(up to projective summands) to E(W) E(W) and F(W) F(W) respectively.We study the properties of these modules, and as an applicationwe use them to describe natural direct sum decompositions ofmodules in quotient categories.     

Compounds of nickel(II) with 5SR,7RS,12SR,14SR-5,12-dimethyl-7,14-diphenyl-1,4,8,11-tetraazacyclotetradecane, rmL: The structures of [Ni(rmL)](ClO4)2 · 0.5H2O and cis-[Ni(rmL)(acac)]ClO4

The imine functions of [Ni(mL¹)](ClO₄)₂ (mL¹ = meso-7RS,14SR-5,12-dimethyl-7,14-diphenyl-1,4,8,11-tetraazacyclotetradeca-4,11-diene) are reduced by using NaBH₄ in acetonitrile/methanol to form the meso–meso and rac–meso isomeric cyclic tetramine complex cations [Ni(mmL²)]²⁺ and [Ni(rmL²)]²⁺ (mml² = 5RS,7RS,12SR,14SR- and rmL² = 5SR,7RS,12SR,14SR-5,12-dimethyl-7,14-diphenyl-1,4,8,11-tetraazacyclotetradecane) in ca. 8:1 proportions. [Ni(rmL²)]²⁺ is also prepared from rmL², formed in <1% yield by the reduction of mL¹ by NaBH₄ in ethanol. Square planar singlet ground state (S = 1) salts [Ni(rmL²)](ClO₄)₂ and [Ni(rmL²)][ZnCl₄] and triplet ground state (S = 3) trans-di-ligand octahedral compounds trans-[Ni(rmL²)X₂] ,μ-Y-trans-[Ni(rmL²)Y] and folded macrocycle compounds cis-[Ni(rmL²)(acac)]CIO₄ (acac⁻ = pentane-2,4-dionato), cis-[{Ni(rmL²)}₂(C₂O₄)](ClO₄)₂, cis-[Ni(rmL²)(H₂O)₂](ClO₄)₂ and cis-[Ni(rmL²)X₂], X⁻ = Cl⁻, Br⁻, are described. The S = 1 salt 1SR,4SR,5SR,7RS,8RS,11RS,12SR,14SR-[Ni(rmL²)](ClO₄)₂ · 0.5H₂O has a disordered structure with Ni(II) in square planar coordination by the nitrogen atoms of the macrocycle, in N-configuration III, with Ni–N_mean = 1.96(2) Å. The six-membered chelate rings both have chair conformations, with the phenyl substituents equatorially oriented and with the methyl substituents disordered over axial and equatorial orientations. The S = 3 compound cis-1SR,4SR,5SR,7RS,8SR,11SR,12SR,14SR-[Ni(rmL²)(acac)]ClO₄ has N-configuration V. The macrocycle is folded along N¹–Ni–N⁸, adjacent to the phenyl substituents {N1–Ni–N8 = 176.45(6), N4–Ni–N11 = 98.16(6)°}, with mean Ni–N = 2.09(2) Å and mean Ni–O = 2.121(5) Å. Both six-membered chelate rings have chair conformations with the methyl substituents equatorially oriented, while one has the phenyl substituent equatorially and the other has it axially oriented. The structures of the isomeric [M(rmL²)(acac)]ClO₄, [M(rrL²)(acac)]CIO₄ and [M(mmL²)(acac)]ClO₄ compounds are compared.     

Four isomers from the oxidative addition of Me3SnH to Os(CO)2(PPh3)3 and the crystal structure of Os(SnMeI2)I(CO)2(PPh3)2, in which the pairs of CO and PPh3 ligands are mutually trans

Reaction between Os(CO)₂(PPh₃)₃ and Me₃SnH produces Os(SnMe₃)H(CO)₂(PPh₃)₂ (1). Multinuclear NMR studies of solutions of 1 reveal the presence of four geometrical isomers, the major one being that with mutually cis triphenylphosphine ligands and mutually trans CO ligands. Os(SnMe₃)H(CO)₂(PPh₃)₂ undergoes a redistribution reaction, at the trimethylstannyl ligand, when treated with Me₂SnCl₂ giving Os(SnMe₂Cl)H(CO)₂(PPh₃)₂ (2). Solutions of 2 again show the presence of four isomers but now the major isomer is that with mutually trans triphenylphosphine ligands and mutually cis CO ligands. The redistribution reaction of 1 with SnI₄ produces Os(SnMeI₂)H(CO)₂(PPh₃)₂ (3) which exists in solution as only one isomer, that with mutually trans triphenylphosphine ligands and mutually trans CO ligands. Treatment of 3 with I₂ cleaves the Os-H bond with retention of geometry giving Os(SnMeI₂)I(CO)₂(PPh₃)₂ (4). The crystal structure of 4 has been determined. No isomerization of the trans dicarbonyl complex 4 occurs when 4 is heated, instead there is a formal loss of “MeSnI” and formation of OsI₂(CO)₂(PPh₃)₂ (5).