Delta-function Source Research Articles

On the absence of supergravity solutions for localized, intersecting sources

For decades intersecting D-branes and O-planes have been playing a very important role in string phenomenology in the context of particle physics model building and in the context of flux compactifications. The corresponding supergravity equations are hard to solve so generically solutions only exist in a so-called smeared limit where the delta function sources are replaced by constants. We are showing here that supergravity solutions for two perpendicularly intersecting localized sources in flat space do not exist for a generic diagonal metric Ansatz. We show this for two intersecting sources with p = 1, 2, 3, 4, 5, 6 spatial dimensions that preserve 8 supercharges, and we allow for fully generic fluxes.

Holography for bulk states in 3D quantum gravity

In this work we discuss the holographic description of states in the Hilbert space of (2+1)-dimensional quantum gravity, living on a time slice in the bulk. We focus on pure gravity coupled to pointlike sources for heavy spinning particles. We develop a formulation where the equations for the backreacted metric reduce to two decoupled Liouville equations with delta-function sources under pseudosphere boundary conditions. We show that both the semiclassical wavefunction and the gravity solution are determined by a universal object, namely a classical Virasoro vacuum block on the sphere. In doing so we derive a version of Polyakov’s conjecture, as well as an existence criterion, for classical Liouville theory on the pseudosphere. We also discuss how some of these results are modified when considering closed universes with compact spatial slices.

From sources to initial data and back again: on bulk singularities in Euclidean AdS/CFT

A common method to prepare states in AdS/CFT is to perform the Euclidean path integral with sources turned on for single-trace operators. These states can be interpreted as coherent states of the bulk quantum theory associated to Lorentzian initial data on a Cauchy slice. In this paper, we discuss the extent to which arbitrary initial data can be obtained in this way. We show that the initial data must be analytic and define the subset of it that can be prepared by imposing bulk regularity. Turning this around, we show that for generic analytic initial data the corresponding Euclidean section contains singularities coming from delta function sources in the bulk. We propose an interpretation of these singularities as non-perturbative objects in the microscopic theory.

T-branes and defects

We study T-branes on compact Kähler surfaces, in the presence of fields localised at curves. If such fields are treated as defects, their vevs induce delta-function sources for the 7-brane background, possibly leading to profiles with poles. We find that the presence of defect sources relaxes the constraints on globally well-defined T-brane configurations, avoiding the obstruction to building them on surfaces of positive curvature. Profiles with poles can be understood, from a 4d viewpoint, as non-trivial vevs for massive modes induced by the defects, and come with their own set of constraints. In the special case of fields localised on a self-intersection curve, we show how to retrieve the Hitchin system with defects from an ordinary global one with enhanced symmetry.

Multi-centered higher spin solutions from {mathcal{W}}_N conformal blocks

Motivated by the question of bulk localization in holography, we study the problem of constructing multi-centered solutions in higher spin gravity which describe point particles in the interior of AdS3. In the Chern-Simons formulation these take into account the backreaction after adding Wilson line sources. We focus on chiral solutions where only the left-moving sector is excited. In that case it is possible to choose a gauge where the dynamical variables are a set of Toda fields living in the bulk. The problem then reduces to solving the {{mathcal{A}}_N}_{-1} Toda equations with delta function sources, which in turn requires solving an associated monodromy problem. We show that this monodromy problem is equivalent to the monodromy problem for a particular {mathcal{W}}_N vacuum conformal block at large central charge. Therefore, knowledge of the {mathcal{W}}_N vacuum block determines the multi-centered solution. Our calculations go beyond the heavy-light approximation by including the backreaction of all higher spin particles.

Some notes about scanning probe microscopy, nanoengineering and methods of quantum mechanics

In the presented paper, a research program for new vision of nanoengineering has been suggested. In the framework of this program, growth of a solid state surface under the influence of external source of sputtering particles is considered as the problem of optimal control by distributed parameter system, the source being a control. As one of cornerstones in this approach, general theory of the one-dimensional KPZ-equation in approximation of small angles under the action of the spatially-inhomogeneous source has been developed. The complete mathematical model of this process with the Dirac delta function source of external particles illustrates this theory, Green’s function for this system having been calculated exactly. Scheme of physical realization of delta-functional source by means of series of carbon nanotubes has been discussed. Opportunity of combined characterization of solid state surface by both atomic force microscopy and scattering data for electromagnetic waves has been demonstrated. The height of the sample has been constructed with help of the Cole-Hopf substitution from single eigenfunction of discrete spectrum being an example of the surface shape under investigation.

Generation-recombination voltage noise spectrum in uniformly doped majority-carrier semiconductor samples

The formula for the voltage noise spectrum due to generation-recombination (G-R) rate fluctuations is derived for uniformly doped majority-carrier n-type semiconductor samples. In the derivation of the formula, the ohmic boundary condition is used to obtain physically sound results, correcting and extending the previously published formulas. It is shown that the G-R voltage noise spectrum becomes saturated in the high electric field region, which is the signature feature of the G-R noise. Furthermore, the electron density fluctuation is developed and calculated due to the delta-function population source. It is shown that as the electric field increases, the profile of the electron density fluctuation becomes quite asymmetric due to the strong drift velocity, and the peak value of the electron density fluctuation at the delta-function source location decreases, resulting in the saturation of the G-R noise voltage spectrum at high electric fields.

A relook at radiation by a point charge. I

Efforts to suggest a classical model for the hydrogen atom are discouraged by a conclusion, based on the principles of electrodynamics, that an accelerating charged particle necessarily radiates. In this paper, we re-examine the steps leading to this conclusion. We start with the relativistic expressions for energy and momentum of a particle and establish the relationship between special relativity and electrodynamics. The standard field expression and its relativistic transformations are then studied for a point charge source, represented by a delta function. In conventional Poynting’s theorem analysis, the rate of change of work done on a system of charges is written as addition of two terms, rate of change of stored energy, and surface integral of Poynting vector. For a delta function source, the first two terms of this equation are either non-integrable or difficult to evaluate. Only the third surface integration term can be evaluated, which is said to give radiation by the point charge. Thus, the statement that an accelerated charge radiates is a conclusion based on this Poynting vector analysis. We examine it and realize that this statement, namely, that a point charge radiates continuously just because it is accelerating, does not have adequate theoretical justification.

Photoacoustic effect generated by moving optical sources: Motion in one dimension

Although the photoacoustic effect is typically generated by pulsed or amplitude modulated optical beams, it is clear from examination of the wave equation for pressure that motion of an optical source in space will result in the production of sound as well. Here, the properties of the photoacoustic effect generated by moving sources in one dimension are investigated. The cases of a moving Gaussian beam, an oscillating delta function source, and an accelerating Gaussian optical sources are reported. The salient feature of one-dimensional sources in the linear acoustic limit is that the amplitude of the beam increases in time without bound.

Conduction Invariance in Similarity Solutions for Compressible Flow Code Verification

In 1991, Coggeshall published a series of 22 closed-form solutions of the Euler compressible flow equations with a heat conduction term included. A remarkable feature of some of these solutions is invariance with respect to conduction; this phenomenon follows from subtle ancillary constraints wherein a heat flux term is assumed to be either identically zero or nontrivially divergence-free. However, the solutions featuring the nontrivial divergence-free heat flux constraint can be shown to be incomplete, using a well-known result most commonly encountered in elementary electrostatic theory. With this result, the application of the divergence operator to the heat flux distributions exhibited by many of the solutions yields a delta function source term instead of identically zero. In theory, the relevant solutions will be conduction invariant only if the appropriate source term is included. This result has important implications for the use of the Coggeshall similarity solutions as code verification test problems for simulation codes featuring coupled compressible fluid flow and heat conduction processes. Computational reproduction of the conduction invariance property represents a conceptually simple check for verifying the robustness of a multiphysics algorithm. In this work, it is demonstrated in the context of various computational instantiations of Coggeshall solution #8 (Cog8) that to maintain any semblance of conduction invariance, a heat source term must be included even with a simple nonlinear heat conduction process. The efficacy of the heat source term is shown to depend not only on values of the various free parameters included in the Coggeshall mathematical model but also the representation of heat sources in multiphysics simulation codes of interest.

Novel Remarks on Point Mass Sources, Firewalls, Null Singularities and Gravitational Entropy

A continuous family of static spherically symmetric solutions of Einstein’s vacuum field equations with a spatial singularity at the origin \( r = 0 \) is found. These solutions are parametrized by a real valued parameter \( \lambda \) (ranging from 0 to 1) and such that the radial horizon’s location is displaced continuously towards the singularity (\( r = 0 \)) as \( \lambda \) increases. In the extreme limit \( \lambda = 1\), the location of the singularity and horizon merges leading to a null singularity. In this extreme case, any infalling observer hits the null singularity at the very moment he/she crosses the horizon. This fact may have important consequences for the resolution of the fire wall problem and the complementarity controversy in black holes. An heuristic argument is provided how one might avoid the Hawking particle emission process in this extreme case when the singularity and horizon merges. The field equations due to a delta-function point-mass source at \( r = 0 \) are solved and the Euclidean gravitational action corresponding to those solutions is evaluated explicitly. It is found that the Euclidean action is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions \( D \ge 3 \).

Strings on AdS wormholes and nonsingular black holes

Certain AdS black holes in the STU model can be conformally scaled to wormhole and black hole backgrounds which have two asymptotically AdS regions and are completely free of curvature singularities. While there is a delta-function source for the dilaton, classical string probes are not sensitive to this singularity. According to the AdS/CFT correspondence, the dual field theory lives on the union of the disjoint boundaries. For the wormhole background, causal contact exists between the two boundaries and the structure of certain correlation functions is indicative of an interacting phase for which there is a coupling between the degrees of freedom living at each boundary. The nonsingular black hole describes an entangled state in two non-interacting identical conformal field theories. By studying the behavior of open strings on these backgrounds, we extract a number of features of the ‘quarks’ and ‘anti-quarks’ that live in the field theories. In the interacting phase, we find that there is a maximum speed with which the quarks can move without losing energy, beyond which energy is transferred from a quark in one field theory to a quark in the other. We also compute the rate at which moving quarks within entangled states lose energy to the two surrounding plasmas. While a quark–antiquark pair within a single field theory exhibits Coulomb interaction for small separation, a quark in one field theory exhibits spring-like confinement with an anti-quark in the other field theory. For the entangled states, we study how the quark–antiquark screening length depends on temperature and chemical potential.

Erratum and Addendum: "A confining quark model and new gauge symmetry"

In the calculations of the static gauge fields in (14) and (22)–(24), the normalization constant of the color singlet charmonium state ( 1/ √ 3 ) δij |cicj〉 should have been taken into account in the delta function sources related to three color quarks. As a result, the factor g s involved in the results (17)–(21) and (25)–(27) should be replaced by g s/3. Moreover, the potential energies VXst, VY st and VZst associated with color carrying confions, X , Y and Z, should also be included in the total potential energy because these contributions are consistent with color singlet state of the charmonium. Thus, the strong confining potential energy Vst of the charmonium should be

Vector BPS baby Skyrme model

We investigate the relation between the BPS baby Skyrme model and its vector meson formulation, where the baby Skyrme term is replaced by a coupling between the topological current $B_\mu$ and the vector meson field $\omega_\mu$. The vector model still possesses infinitely many symmetries leading to infinitely many conserved currents which stand behind its solvability. It turns out that the similarities and differences of the two models depend strongly on the specific form of the potential. We find, for instance, that compactons (which exist in the BPS baby Skyrme model) disappear from the spectrum of solutions of the vector counterpart. Specifically, for the vector model with the old baby Skyrme potential we find that it has compacton solutions only provided that a delta function source term effectively screening the topological charge is inserted at the compacton boundary. For the old baby Skyrme potential squared we find that the vector model supports exponentially localized solitons, like the BPS baby Skyrme model. These solitons, however, saturate a BPS bound which is a nonlinear function of the topological charge and, as a consequence, higher solitons are unstable w.r.t. decay into smaller ones, which is at variance with the more conventional situation (a linear BPS bound and stable solitons) in the BPS baby Skyrme model.

Points of general relativistic shock wave interaction are ‘regularity singularities’ where space–time is not locally flat

We show that the regularity of the gravitational metric tensor in spherically symmetric space–times cannot be lifted from C 0,1 to C 1,1 within the class of C 1,1 coordinate transformations in a neighbourhood of a point of shock wave interaction in General Relativity, without forcing the determinant of the metric tensor to vanish at the point of interaction. This is in contrast to Israel's theorem, which states that such coordinate transformations always exist in a neighbourhood of a point on a smooth single shock surface. The results thus imply that points of shock wave interaction represent a new kind of regularity singularity for perfect fluids evolving in space–time, singularities that make perfectly good sense physically, that can form from the evolution of smooth initial data, but at which the space–time is not locally Minkowskian under any coordinate transformation. In particular, at regularity singularities, delta function sources in the second derivatives of the metric exist in all coordinate systems of the C 1,1 -atlas, but due to cancellation, the full Riemann curvature tensor remains supnorm bounded .

Pickup ion distributions from three-dimensional neutral exospheres

Pickup ions formed from ionized neutral exospheres in flowing plasmas have phase space distributions that reflect their source's spatial distributions. Phase space distributions of the ions are derived from the Vlasov equation with a delta function source using three.dimensional neutral exospheres. The ExB drift produced by plasma motion picks up the ions while the effects of magnetic field draping, mass loading, wave particle scattering, and Coulomb collisions near a planetary body are ignored. Previously, one.dimensional exospheres were treated, resulting in closed form pickup ion distributions that explicitly depend on the ratio rg/H, where rg is the ion gyroradius and H is the neutral scale height at the exobase. In general, the pickup ion distributions, based on three.dimensional neutral exospheres, cannot be written in closed form, but can be computed numerically. They continue to reflect their source's spatial distributions in an implicit way. These ion distributions and their moments are applied to several bodies, including He(+) and Na(+) at the Moon, H(+2) and CH(+4) at Titan, and H+ at Venus. The best places to use these distributions are upstream of the Moon's surface, the ionopause of Titan, and the bow shock of Venus.

Orientifolded locally AdS3 geometries

Continuing the analysis of [Loran F and Sheikh-Jabbari M M 2010 Phys. Lett. B 693 184–7], we classify all locally AdS3 stationary axi-symmetric unorientable solutions to AdS3 Einstein gravity and show that they are obtained by applying certain orientifold projection on AdS3, BTZ or AdS3 self-dual orbifold, respectively, O-AdS3, O-BTZ and O-SDO geometries. Depending on the orientifold fixed surface, the O-surface, which is either a space-like 2D plane or a cylinder, or a light-like 2D plane or a cylinder, one can distinguish four distinct cases. For the space-like orientifold plane or cylinder cases, these geometries solve AdS3 Einstein equations and are hence locally AdS3 everywhere except at the O-surface, where there is a delta-function source. For the light-like cases, the geometry is a solution to Einstein equations even at the O-surface. We discuss the causal structure for static, extremal and general rotating O-BTZ and O-SDO cases as well as the geodesic motion on these geometries. We also discuss orientifolding Poincaré patch AdS3 and AdS2 geometries as a way to geodesic completion of these spaces and comment on the 2D CFT dual to the O-geometries.

Regularization of fields for self-force problems in curved spacetime: Foundations and a time-domain application

We propose an approach for the calculation of self-forces, energy fluxes and waveforms arising from moving point charges in curved spacetimes. As opposed to mode-sum schemes that regularize the self-force derived from the singular retarded field, this approach regularizes the retarded field itself. The singular part of the retarded field is first analytically identified and removed, yielding a finite, differentiable remainder from which the self-force is easily calculated. This regular remainder solves a wave equation which enjoys the benefit of having a non-singular source. Solving this wave equation for the remainder completely avoids the calculation of the singular retarded field along with the attendant difficulties associated with numerically modeling a delta function source. From this differentiable remainder one may compute the self-force, the energy flux, and also a waveform which reflects the effects of the self-force. As a test of principle, we implement this method using a 4th-order (1+1) code, and calculate the self-force for the simple case of a scalar charge moving in a circular orbit around a Schwarzschild black hole. We achieve agreement with frequency-domain results to ~ 0.1% or better.

The Euclidean gravitational action as black hole entropy, singularities, and spacetime voids

We argue why the static spherically symmetric vacuum solutions of Einstein’s equations described by the textbook Hilbert metric gμν(r) is not diffeomorphic to the metric gμν(∣r∣) corresponding to the gravitational field of a point mass delta function source at r=0. By choosing a judicious radial function R(r)=r+2G∣M∣Θ(r) involving the Heaviside step function, one has the correct boundary condition R(r=0)=0, while displacing the horizon from r=2G∣M∣ to a location arbitrarily close to r=0 as one desires, rh→0, where stringy geometry and quantum gravitational effects begin to take place. We solve the field equations due to a delta function point mass source at r=0, and show that the Euclidean gravitational action (in ℏ units) is precisely equal to the black hole entropy (in Planck area units). This result holds in any dimensions D⩾3. In the Reissner–Nordstrom (massive charged) and Kerr–Newman black hole case (massive rotating charged) we show that the Euclidean action in a bulk domain bounded by the inner and outer horizons is the same as the black hole entropy. When one smears out the point-mass and point-charge delta function distributions by a Gaussian distribution, the area-entropy relation is modified. We postulate why these modifications should furnish the logarithmic corrections (and higher inverse powers of the area) to the entropy of these smeared black holes. To finalize, we analyze the Bars–Witten stringy black hole in 1+1 dimension and its relation to the maximal acceleration principle in phase spaces and Finsler geometries.

Problems and Techniques

With major elections in many countries during 2004, what could be more timely than the two papers in this SIAM Review issue about preferential voting---the Survey and Review paper by Roberto Serrano and our first Problems and Techniques article, "Single Transferable Votes with Tax Cuts," by Eivind Stensholt? In the Problems and Techniques paper, the author begins by explaining the philosophy and strategy of different ways to tally preferential votes, giving sometimes surprising illustrations of their strengths and weaknesses. Even using the "single transferable vote" (STV) systems described in the paper, which are designed to avoid certain defects of other systems, there will always be collections of individual preference relations that allow election manipulation---for example, under some conditions a voter can actually hurt candidate x by giving x the top rank. A further wrinkle is learning how a "tax cut" (not income tax...) can be included to address the "free ride" problem, in which some voters end up with more influence than others. The paper includes data from a real case of preference voting, analyzing (among several interesting insights) the situation of an "anti-establishment" voter. Moving from voting to fundamental numerical algorithms in scientific and engineering applications, our second paper, by Leslie Greengard and June-Yub Lee, concerns acceleration of the nonuniform fast Fourier transform (FFT). For applications with an irregular sampling of data in the frequency domain---for example, image reconstruction or radio astronomy---the standard FFT is inappropriate because it requires uniform sampling. Greengard and Lee stress that three separate issues arise in reconstructing functions from nonuniform data in the Fourier domain: acquisition of data, choice of a quadrature scheme, and existence of a fast algorithm for computing the discrete transform. Given that a quadrature approach has been chosen, the authors describe techniques for rapid and accurate computation of the needed discrete sums---in effect, combining an interpolation scheme with the standard FFT by replacing a Dirac delta function source with a sharply peaked Gaussian function, thereby "smearing" the source strength onto a regular grid. An especially interesting part of the paper is its characterization of fast Gaussian gridding, which relies on the fact that the values to be evaluated are negligible except at nearby grid points. Numerical examples are included that verify the speed and accuracy of the proposed method, illustrate the benefits of fast gridding, and display results on the ubiquitous Shepp--Logan phantom. The third paper, "Symplectic Rotational Geometry in Human Biomechanics," by V. Ivancevic, begins with informative background on the study of human motion, characterized by complex, multidimensional, highly nonlinear, and hierarchical dynamics---i.e., a very difficult problem! Classical biomechanics relies on vector geometry for the more than six hundred skeletal muscles, each generating forces and torques. With the widely used technique of inverse dynamics, segmental movements are measured using motion capture, often combined with other sensor data. For the more unusual forward dynamics approach, in which translational and rotational motion are simulated based on joint forces and torques, the author proposes the formalism of rotational symplectic geometry. A review of symplectic geometry in manifolds leads to an analysis of how different features of human motion (such as humanoid joint angles, "hinge" joints, and "ball-and-socket" joints) can be modeled in terms of constrained Lie groups, followed by a mathematical tour of (among other topics) segment dynamics, full body dynamics, joint dynamics, and muscular dynamics. We thank the authors of all three papers for proving by example the rich variety of the Problems and Techniques section.