Multipole moments in general relativity and dynamical perturbations of black-hole magnetospheres

This thesis consists of two parts. In Part I, we develop a multipole moment formalism in general relativity and use it to analyze the motion and precession of compact bodies. More specifically, the generic, vacuum, dynamical gravitational field of the exterior universe in the vicinity of a freely moving body is expanded in positive powers of the distance r away from the body's spatial origin (i.e., in the distance r from its timelike-geodesic world line). The expansion coefficients, called "external multipole moments,'' are defined covariantly in terms of the Riemann curvature tensor and its spatial derivatives evaluated on the body's central world line. In a carefully chosen class of de Donder coordinates, the expansion of the external field involves only integral powers of r ; no logarithmic terms occur. The expansion is used to derive higher-order corrections to previously known laws of motion and precession for black holes and other bodies. The resulting laws of motion and precession are expressed in terms of couplings of the time derivatives of the body's quadrupole and octopole moments to the external moments, i.e., to the external curvature and its gradient.

In part II, we study the interaction of magnetohydrodynamic (MHD) waves in a black-hole magnetosphere with the "dragging of inertial frames" effect of the hole's rotation - i.e., with the hole's "gravitomagnetic field." More specifically: we first rewrite the laws of perfect general relativistic magnetohydrodynamics (GRMHD) in 3+1 language in a general spacetime, in terms of quantities (magnetic field, flow velocity, ...) that would be measured by the ''fiducial observers” whose world lines are orthogonal to (arbitrarily chosen) hypersurfaces of constant time. We then specialize to a stationary spacetime and MHD flow with one arbitrary spatial symmetry (e.g., the stationary magnetosphere of a Kerr black hole); and for this spacetime we reduce the GRMHD equations to a set of algebraic equations. The general features of the resulting stationary, symmetric GRMHD magnetospheric solutions are discussed, including the Blandford-Znajek effect in which the gravitomagnetic field interacts with the magnetosphere to produce an outflowing jet. Then in a specific model spacetime with two spatial symmetries, which captures the key features of the Kerr geometry, we derive the GRMHD equations which govern weak, linealized perturbations of a stationary magnetosphere with outflowing jet. These perturbation equations are then Fourier analyzed in time t and in the symmetry coordinate x, and subsequently solved numerically. The numerical solutions describe the interaction of MHD waves with the gravitomagnetic field. It is found that, among other features, when an oscillatory external force is applied to the region of the magnetosphere where plasma (e⁺e^-) is being created, the magnetosphere responds especially strongly at a particular, resonant, driving frequency. The resonant frequency is that for which the perturbations appear to be stationary (time independent) in the common rest frame of the freshly created plasma and the rotating magnetic field lines. The magnetosphere of a rotating black hole, when buffeted by nonaxisymmetric magnetic fields anchored in a surrounding accretion disk, might exhibit an analogous resonance. If so then the hole's outflowing jet might be modulated at resonant frequencies ω=(m/2) Ω_H where m is an integer and Ω_H is the hole's angular velocity.

Electron transfer in chemically and genetically modified myoglobins

The temperature dependences of the reduction potentials (E^o') of wildtype human myoglobin (Mb) and three site-directed mutants have been measured by using thin-layer spectroelectrochemistry. Residue Val68, which is in van der Waals contact with the heme in Mb, has been replaced by Glu, Asp, and Asn. At pH 7.0, reduction of the heme iron (III) in the former two proteins is accompanied by uptake of a proton by the protein. The changes in E^o', and the standard entropy (ΔS^o') and enthalpy (ΔH^o') of reduction in the mutant proteins were determined relative to values for wild-type; the change in E^o' at 25°C was about -200 millivolts for the Glu and Asp mutants, and about -80 millivolts for the Asn mutant. Reduction of Fe(III) to Fe(II) in the Glu and Asp mutants is accompanied by uptake of a proton. These studies demonstrate that Mb can tolerate substitution of a buried hydrophobic group by potentially charged and polar residues, and that such amino acid replacements can lead to substantial changes in the redox thermodynamics of the protein.

Through analysis of the temperature dependence and shapes of NMR dispersion signals, it is determined that a water molecule is bound to the sixth coordination site of the ferric heme in the Val68Asp and in the Val68Asn recombinant proteins while the carboxyl group of the sidechain of Glu68 occupies this position in Val68Glu. The relative rhombic distortions in the ESR spectra of these mutant proteins combined with H₂¹⁷O and spin interconversion experiments performed on them confirm the conclusions of the NMRD study.

The rates of intramolecular electron transfer (ET) of (NH₃)₅Ru-His48 (Val68Asp, His81GIn, Cys110AIa)Mb and (NH₃)₅Ru-His48 (Val68GIu,His81GIn,Cys110Ala)Mb were measured to be .85(3)s^-1 and .30(2)s^-1, respectively. This data supports the hypothesis that entropy of 111 reduction and reorganization energy of ET are inversely related. The rates of forward and reverse ET for (NH₃)₅ Ru-His48 (Val68GIu, His81 GIn, Cys110AIa)ZnMb -7.2(5)•10⁴s^-1and 1.4(2)•10⁵s^-1, respectively- demonstrate that the placement of a highly polar residue nearby does not significantly change the reorganization energy of the photoactive Zn porphyrin.

The distal histidine imidazoles of (NH₃)₄isnRu-His48 SWMb and (NH₃)₅Ru-His48 SWMb were cyanated with BrCN. The intramolecular ET rates of these BrCN-modified Mb derivatives are 5.5(6)s^-1 and 3.2(5)s^-1, respectively. These respective rates are 20 and 10 times faster than those of their noncyanated counterparts after the differences in ET rate from driving force are scaled according to the Marcus equation. This increase in ET rate of the cyanated Mb derivatives is attributed to lower reorganization energy since the cyanated Mb heme is pentacoordinate in both oxidation states; whereas, the native Mb heme loses a water molecule upon reduction so that it changes from six to five coordinate. The reorganization energy from Fe-OH₂ dissociation is estimated to be .2eV. This conclusion is used to reconcile data from previous experiments in our lab. ET in photoactive porphyrin-substituted myoglobins proceed faster than predicted by Marcus Theory when it is assumed that the only difference in ET parameters between photoactive porphyrins and native heme systems is driving force. However, the data can be consistently fit to Marcus Theory if one corrects for the smaller reorganization in the photoactive porphyrin systems since they do not undergo a coordination change upon ET.

Finally, the intramolecular ET rate of (NH₃)₄isnRu-His48 SWMb was measured to be 3.0(4)s^-1. This rate is within experimental error of that for (NH₃)₄pyrRu-His48 SWMb even though the former has 80mV more driving force. One likely possibility for this observation is that the tetraamminepyridineruthenium group undergoes less reorganization upon ET than the tetraammineisonicotinamideruthenium group. Moreover, analysis of the (NH₃)₄isnRu-His48 SWMb experimental system gives a likely explanation of why ET was not observed previously in (NH₃)₄isnRu-Cytochrome C.

Essays on Correlated Equilibrium and Voter Turnout

This thesis consists of three essays in the areas of political economy and game theory, unified by their focus on the effects of pre-play communication on equilibrium outcomes.

Communication is fundamental to elections. Chapter 2 extends canonical voter turnout models, where citizens, divided into two competing parties, choose between costly voting and abstaining, to include any form of communication, and characterizes the resulting set of Aumann's correlated equilibria. In contrast to previous research, high-turnout equilibria exist in large electorates and uncertain environments. This difference arises because communication can coordinate behavior in such a way that citizens find it incentive compatible to follow their correlated signals to vote more. The equilibria have expected turnout of at least twice the size of the minority for a wide range of positive voting costs.

In Chapter 3 I introduce a new equilibrium concept, called subcorrelated equilibrium, which fills the gap between Nash and correlated equilibrium, extending the latter to multiple mediators. Subcommunication equilibrium similarly extends communication equilibrium for incomplete information games. I explore the properties of these solutions and establish an equivalence between a subset of subcommunication equilibria and Myerson's quasi-principals' equilibria. I characterize an upper bound on expected turnout supported by subcorrelated equilibrium in the turnout game.

Chapter 4, co-authored with Thomas Palfrey, reports a new study of the effect of communication on voter turnout using a laboratory experiment. Before voting occurs, subjects may engage in various kinds of pre-play communication through computers. We study three communication treatments: No Communication, a control; Public Communication, where voters exchange public messages with all other voters, and Party Communication, where messages are exchanged only within one's own party. Our results point to a strong interaction effect between the form of communication and the voting cost. With a low voting cost, party communication increases turnout, while public communication decreases turnout. The data are consistent with correlated equilibrium play. With a high voting cost, public communication increases turnout. With communication, we find essentially no support for the standard Nash equilibrium turnout predictions.

Characterization and modeling of premixed turbulent n-heptane flames in the thin reaction zone regime

n-heptane/air premixed turbulent flames in the high-Karlovitz portion of the thin reaction zone regime are characterized and modeled in this thesis using Direct Numerical Simulations (DNS) with detailed chemistry. In order to perform these simulations, a time-integration scheme that can efficiently handle the stiffness of the equations solved is developed first. A first simulation with unity Lewis number is considered in order to assess the effect of turbulence on the flame in the absence of differential diffusion. A second simulation with non-unity Lewis numbers is considered to study how turbulence affects differential diffusion. In the absence of differential diffusion, minimal departure from the 1D unstretched flame structure (species vs. temperature profiles) is observed. In the non-unity Lewis number case, the flame structure lies between that of 1D unstretched flames with "laminar" non-unity Lewis numbers and unity Lewis number. This is attributed to effective Lewis numbers resulting from intense turbulent mixing and a first model is proposed. The reaction zone is shown to be thin for both flames, yet large chemical source term fluctuations are observed. The fuel consumption rate is found to be only weakly correlated with stretch, although local extinctions in the non-unity Lewis number case are well correlated with high curvature. These results explain the apparent turbulent flame speeds. Other variables that better correlate with this fuel burning rate are identified through a coordinate transformation. It is shown that the unity Lewis number turbulent flames can be accurately described by a set of 1D (in progress variable space) flamelet equations parameterized by the dissipation rate of the progress variable. In the non-unity Lewis number flames, the flamelet equations suggest a dependence on a second parameter, the diffusion of the progress variable. A new tabulation approach is proposed for the simulation of such flames with these dimensionally-reduced manifolds.

Conserved quantities and the formation of black holes in the Brans-Dicke theory of gravitation

In Part I, we construct a symmetric stress-energy-momentum pseudo-tensor for the gravitational fields of Brans-Dicke theory, and use this to establish rigorously conserved integral expressions for energy-momentum Pⁱ and angular momentum J^ik. Application of the two-dimensional surface integrals to the exact static spherical vacuum solution of Brans leads to an identification of our conserved mass with the active gravitational mass. Application to the distant fields of an arbitrary stationary source reveals that Pⁱ and J^ik have the same physical interpretation as in general relativity. For gravitational waves whose wavelength is small on the scale of the background radius of curvature, averaging over several wavelengths in the Brill-Hartle-Isaacson manner produces a stress-energy-momentum tensor for gravitational radiation which may be used to calculate the changes in Pⁱ and J^ik of their source.

In Part II, we develop strong evidence in favor of a conjecture by Penrose--that, in the Brans-Dicke theory, relativistic gravitational collapse in three dimensions produce black holes identical to those of general relativity. After pointing out that any black hole solution of general relativity also satisfies Brans-Dicke theory, we establish the Schwarzschild and Kerr geometries as the only possible spherical and axially symmetric black hole exteriors, respectively. Also, we show that a Schwarzschild geometry is necessarily formed in the collapse of an uncharged sphere.

Appendices discuss relationships among relativistic gravity theories and an example of a theory in which black holes do not exist.

Effects of density differences on lateral mixing in open-channel flows

This study investigates lateral mixing of tracer fluids in turbulent open-channel flows when the tracer and ambient fluids have different densities. Longitudinal dispersion in flows with longitudinal density gradients is investigated also.

Lateral mixing was studied in a laboratory flume by introducing fluid tracers at the ambient flow velocity continuously and uniformly across a fraction of the flume width and over the entire depth of the ambient flow. Fluid samples were taken to obtain concentration distributions in cross-sections at various distances, x, downstream from the tracer source. The data were used to calculate variances of the lateral distributions of the depth-averaged concentration. When there was a difference in density between the tracer and the ambient fluids, lateral mixing close to the source was enhanced by density-induced secondary flows; however, far downstream where the density gradients were small, lateral mixing rates were independent of the initial density difference. A dimensional analysis of the problem and the data show that the normalized variance is a function of only three dimensionless numbers, which represent: (1) the x-coordinate, (2) the source width, and (3) the buoyancy flux from the source.

A simplified set of equations of motion for a fluid with a horizontal density gradient was integrated to give an expression for the density-induced velocity distribution. The dispersion coefficient due to this velocity distribution was also obtained. Using this dispersion coefficient in an analysis for predicting lateral mixing rates in the experiments of this investigation gave only qualitative agreement with the data. However, predicted longitudinal salinity distributions in an idealized laboratory estuary agree well with published data.

Theoretical investigation of turbulent boundary layer over a wavy surface

The important features of the two-dimensional incompressible turbulent flow over a wavy surface of wavelength comparable with the boundary layer thickness are analyzed.

A turbulent field method using model equation for turbulent shear stress similar to the scheme of Bradshaw, Ferriss and Atwell (1967) is employed with suitable modification to cover the viscous sublayer. The governing differential equations are linearized based on the small but finite amplitude to wavelength ratio. An orthogonal wavy coordinate system, accurate to the second order in the amplitude ratio, is adopted to avoid the severe restriction to the validity of linearization due to the large mean velocity gradient near the wall. Analytic solution up to the second order is obtained by using the method of matched-asymptotic-expansion based on the large Reynolds number and hence the small skin friction coefficient.

In the outer part of the layer, the perturbed flow is practically "inviscid." Solutions for the velocity, Reynolds stress and also the wall pressure distributions agree well with the experimental measurement. In the wall region where the perturbed Reynolds stress plays an important role in the process of momentum transport, only a qualitative agreement is obtained. The results also show that the nonlinear second-order effect is negligible for amplitude ratio of 0.03. The discrepancies in the detailed structure of the velocity, shear stress, and skin friction distributions near the wall suggest modifications to the model are required to describe the present problem.

A generalized Hausdorff dimension for functions and sets

Let E be a compact subset of the n-dimensional unit cube, 1_n, and let C be a collection of convex bodies, all of positive n-dimensional Lebesgue measure, such that C contains bodies with arbitrarily small measure. The dimension of E with respect to the covering class C is defined to be the number

d_C(E) = sup(β:H_{β, C}(E) > 0),

where H_{β, C} is the outer measure

inf(Ʃm(C_i)^β:UC_i Ↄ E, C_i ϵ C) .

Only the one and two-dimensional cases are studied. Moreover, the covering classes considered are those consisting of intervals and rectangles, parallel to the coordinate axes, and those closed under translations. A covering class is identified with a set of points in the left-open portion, 1’_n, of 1_n, whose closure intersects 1_n - 1’_n. For n = 2, the outer measure H_{β, C} is adopted in place of the usual:

Inf(Ʃ(diam. (C_i))^β: UC_i Ↄ E, C_i ϵ C),

for the purpose of studying the influence of the shape of the covering sets on the dimension d_C(E).

If E is a closed set in 1₁, let M(E) be the class of all non-decreasing functions μ(x), supported on E with μ(x) = 0, x ≤ 0 and μ(x) = 1, x ≥ 1. Define for each μ ϵ M(E),

d_C(μ) = lim/c → inf/0 log ∆μ(c)/log c , (c ϵ C)

where ∆μ(c) = v/x (μ(x+c) – μ(x)). It is shown that

d_C(E) = sup (d_C(μ):μ ϵ M(E)).

This notion of dimension is extended to a certain class Ӻ of sub-additive functions, and the problem of studying the behavior of d_C(E) as a function of the covering class C is reduced to the study of d_C(f) where f ϵ Ӻ. Specifically, the set of points in 1₁,

(*) {d_B(F), d_C(f)): f ϵ Ӻ}

is characterized by a comparison of the relative positions of the points of B and C. A region of the form (*) is always closed and doubly-starred with respect to the points (0, 0) and (1, 1). Conversely, given any closed region in 1₂, doubly-starred with respect to (0, 0) and (1, 1), there are covering classes B and C such that (*) is exactly that region. All of the results are shown to apply to the dimension of closed sets E. Similar results can be obtained when a finite number of covering classes are considered.

In two dimensions, the notion of dimension is extended to the class M, of functions f(x, y), non-decreasing in x and y, supported on 1₂ with f(x, y) = 0 for x · y = 0 and f(1, 1) = 1, by the formula

d_C(f) = lim/s · t → inf/0 log ∆f(s, t)/log s · t , (s, t) ϵ C

where

∆f(s, t) = V/x, y (f(x+s, y+t) – f(x+s, y) – f(x, y+t) + f(x, t)).

A characterization of the equivalence d_C1(f) = d_C2(f) for all f ϵ M, is given by comparison of the gaps in the sets of products s · t and quotients s/t, (s, t) ϵ C_i (I = 1, 2).

Proton-Coupled Reduction of N2 Facilitated by Molecular Fe Complexes

The activation of Fe-coordinated N₂ via the formal addition of hydrogen atom equivalents is explored in this thesis. These reactions may occur in nitrogenase enzymes during the biological conversion of N₂ to NH₃. To understand these reactions, the N₂ reactivity of a series of molecular Fe(N₂) platforms is investigated. A trigonal pyramidal, carbon-ligated Fe^I complex was prepared that displays a similar geometry to that of the resting state 'belt' Fe atoms of nitrogenase. Upon reduction, this species was shown to coordinate N₂, concomitant with significant weakening of the C-Fe interaction. This hemilability of the axial ligand may play a critical role in mediating the interconversion of Fe(N_xH_y) species during N₂ conversion to NH₃. In fact, a trigonal pyramidal borane-ligated Fe complex was shown to catalyze this transformation, generating up to 8.49 equivalents of NH₃. To shed light on the mechanistic details of this reaction, protonation of a borane-ligated Fe(N₂) complex was investigated and found to give rise to a mixture of species that contains an iron hydrazido(2-) [Fe(NNH₂)] complex. The identification of this species is suggestive of an early N-N bond cleavage event en route to NH₃ production, but the highly-reactive nature of this complex frustrated direct attempts to probe this possibility. A structurally-analogous silyl-ligated Fe(N₂) complex was found to react productively with hydrogen atom equivalents, giving rise to an isolable Fe(NNH₂) species. Spectroscopic and crystallographic studies benefited from the enhanced stability of this complex relative to the borane analogue. One-electron reduction of this species initiates a spontaneous disproportionation reaction with an iron hydrazine [Fe(NH₂NH₂)] complex as the predominant reaction product. This transformation provides support for an Fe-mediated N₂ activation mechanism that proceeds via a late N-N bond cleavage. In hopes of gaining more fundamental insight into these reactions, a series of Fe(CN) complexes were prepared and reacted with hydrogen-atom equivalents. Significant quantities of CH₄ and NH₃ are generated in these reactions as a result of complete C-N bond activation. A series of Fe(CNH_x) were found to be exceptionally stable and may be intermediates in these reactions. The stability of these compounds permitted collection of thermodynamic parameters pertinent to the unique N-H bonds. This data is comparatively discussed with the theoretically-predicted data of the N₂-derived Fe(NNH_x) species. Exceptionally-weak N-H bond enthalpies are found for many of these compounds, and sheds light on their short-lived nature and tendency to evolve H₂. As a whole, these works both establish and provide a means to understand Fe-mediated N₂ activation via the addition of hydrogen atom equivalents.

A scaling technique for the design of idealized electromagnetic lenses

A technique is developed for the design of lenses for transitioning TEM waves between conical and/or cylindrical transmission lines, ideally with no reflection or distortion of the waves. These lenses utilize isotropic but inhomogeneous media and are based on a solution of Maxwell's equations instead of just geometrical optics. The technique employs the expression of the constitutive parameters, ɛ and μ, plus Maxwell's equations, in a general orthogonal curvilinear coordinate system in tensor form, giving what we term as formal quantities. Solving the problem for certain types of formal constitutive parameters, these are transformed to give ɛ and μ as functions of position. Several examples of such lenses are considered in detail.

Design, Synthesis, and Study of Novel Platforms for Iron-N2 Chemistry and Photoinduced, Copper-mediated C-N Bond Formation

Several new ligand platforms designed to support iron dinitrogen chemistry have been developed. First, we report Fe complexes of a tris(phosphino)alkyl (CP^iPr₃) ligand featuring an axial carbon donor intended to conceptually model the interstitial carbide atom of the nitrogenase iron-molybdenum cofactor (FeMoco). It is established that in this scaffold, the iron center binds dinitrogen trans to the C_alkyl anchor in three structurally characterized oxidation states. Fe-C_alkyl lengthening is observed upon reduction, reflective of significant ionic character in the Fe-C_alkyl interaction. The anionic (CP^iPr₃)FeN₂^- species can be functionalized by a silyl electrophile to generate (CP^iPr₃)Fe-N₂SiR₃. This species also functions as a modest catalyst for the reduction of N₂ to NH₃. Next, we introduce a new binucleating ligand scaffold that supports an Fe(μ-SAr)Fe diiron subunit that coordinates dinitrogen (N₂-Fe(μ-SAr)Fe-N₂) across at least three oxidation states (Fe^IIFe^II, Fe^IIFe^I, and Fe^IFe^I). Despite the sulfur-rich coordination environment of iron in FeMoco, synthetic examples of transition metal model complexes that bind N₂ and also feature sulfur donor ligands remain scarce; these complexes thus represent an unusual series of low-valent diiron complexes featuring thiolate and dinitrogen ligands. The (N₂-Fe(μ-SAr)Fe-N₂) system undergoes reduction of the bound N₂ to produce NH₃ (~50% yield) and can efficiently catalyze the disproportionation of N₂H₄ to NH₃ and N₂. The present scaffold also supports dinitrogen binding concomitant with hydride as a co-ligand. Next, inspired by the importance of secondary-sphere interactions in many metalloenzymes, we present complexes of iron in two new ligand scaffolds ([SiP^NMe₃] and [SiP^iPr₂P^NMe]) that incorporate hydrogen-bond acceptors (tertiary amines) which engage in interactions with nitrogenous substrates bound to the iron center (NH₃ and N₂H₄). Cation binding is also facilitated in anionic Fe(0)-N₂ complexes. While Fe-N₂ complexes of a related ligand ([SiP^iPr₃]) lacking hydrogen-bond acceptors produce a substantial amount of ammonia when treated with acid and reductant, the presence of the pendant amines instead facilitates the formation of metal hydride species.

Additionally, we present the development and mechanistic study of copper-mediated and copper-catalyzed photoinduced C-N bond forming reactions. Irradiation of a copper-amido complex, ((m-tol)₃P)₂Cu(carbazolide), in the presence of aryl halides furnishes N-phenylcarbazole under mild conditions. The mechanism likely proceeds via single-electron transfer from an excited state of the copper complex to the aryl halide, generating an aryl radical. An array of experimental data are consistent with a radical intermediate, including a cyclization/stereochemical investigation and a reactivity study, providing the first substantial experimental support for the viability of a radical pathway for Ullmann C-N bond formation. The copper complex can also be used as a precatalyst for Ullmann C-N couplings. We also disclose further study of catalytic C_alkyl-N couplings using a CuI precatalyst, and discuss the likely role of [Cu(carbazolide)₂]^- and [Cu(carbazolide)₃]^- species as intermediates in these reactions.

Finally, we report a series of four-coordinate, pseudotetrahedral P₃Fe^II-X complexes supported by tris(phosphine)borate ([PhBP₃Fe^R]^-) and phosphiniminato X-type ligands (-N=PR'₃) that in combination tune the spin-crossover behavior of the system. Low-coordinate transition metal complexes such as these that undergo reversible spin-crossover remain rare, and the spin equilibria of these systems have been studied in detail by a suite of spectroscopic techniques.