Scalar Green's Function Research Articles

Approximations to the 3-D Free-Space Scalar Green's Function in Terms of the Taylor Product Theorem

In electromagnetics and acoustics, difficulties are encountered in evaluations of integrals containing the free-space scalar Green's Function <inline-formula xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink"><tex-math notation="LaTeX">$G(\mathbf{r},{{\bf r^{\prime}}})$</tex-math></inline-formula> . So-called far-field approximations have long been used for obtaining approximate solutions. We have developed new approximations by using the Taylor Product Theorem that are equally simple but far more accurate as demonstrated by numerical examples. Additionally, we have developed error estimates.

Boundary-integral representation sans waterline integral for flows around ships steadily advancing in calm water

The boundary-integral flow-representation associated with the boundary-value problem, commonly called Neumann–Kelvin (NK) problem, that corresponds to linear potential flow around a ship steadily advancing in calm water involves an integral around the mean waterline of the ship. This ‘waterline integral’ is a notorious source of numerical difficulties and has been extensively studied. The waterline integral in the NK theory is largely – but not fully – eliminated in the modification, called Neumann–Michell (NM) theory, of the NK theory. Specifically, the NM theory includes a residual waterline-distribution of weak Rankine singularities, ignored in practical applications. A crucial element of the NM theory is a mathematical transformation that is based on a vector Green function, which is associated with the common scalar Green function used in the NK theory. This transformation is revisited in the present study. A rigorous analysis yields an answer to a fifty-year old puzzle: an exact boundary-integral flow representation that does not include a waterline integral. A remarkable feature of this new flow representation, which is a modification of the NM flow representation given previously, is that it explicitly determines the flow potential and the flow velocity at a ship hull surface in terms of the flow velocity at the hull surface, rather than in terms of the hull-surface potential as in usual boundary-integral flow representations obtained via Green’s classical identity.

Regularized calculation of the retarded Green function in a Schwarzschild spacetime

The retarded Green function for linear field perturbations of black hole spacetimes is notoriously difficult to calculate. One of the difficulties is due to a Dirac-$\delta$ divergence that the Green function possesses when the two spacetime points are connected by a "direct" null geodesic. We present a procedure which notably aids its calculation in the case of Schwarzschild spacetime by separating this direct $\delta$-divergence from the remainder of the retarded Green function. More precisely, the method consists of calculating the multipolar $\ell$-modes of the direct $\delta$-divergence and subtracting them from the corresponding modes of the retarded Green function. We illustrate the usefulness of the method with some specific calculations in the case of the scalar Green function and self-field for a point scalar charge in Schwarzschild spacetime.

Vectorial model for guided-mode resonance gratings

We propose a self-consistent vectorial method, based on a Green's function technique, to describe the Fano resonances that appear in guided mode resonance gratings. The model provides intuitive expressions of the reflectivity and transmittivity matrices of the structure, involving coupling integrals between the modes of a planar reference structure and radiative modes. These expressions are used to derive a physical analysis in configurations where the effect of the incident polarization is not trivial. We provide numerical validations of our model. On a technical point of view, we show how the Green's tensor of our planar reference structure can be expressed as two scalar Green's functions, and how to deal with the singularity of the Green's tensor.

Green’s functions and Hadamard parametrices for vector and tensor fields in general linear covariant gauges

We determine the retarded and advanced Green's functions and Hadamard parametrices in curved spacetimes for linearized massive and massless gauge bosons and linearized Einstein gravity with a cosmological constant in general linear covariant gauges. These vector and tensor parametrices contain additional singular terms compared with their Feynman/de Donder-gauge counterpart. We also give explicit recursion relations for the Hadamard coefficients, and indicate their generalization to $n$ dimensions. Furthermore, we express the divergence and trace of the vector and tensor Green's functions in terms of derivatives of scalar and vector Green's functions, and show how these relations appear as Ward identities in the free quantum theory.

Scalar field Green functions on causal sets

We examine the validity and scope of Johnston’s models for scalar field retarded Green functions on causal sets in 2 and 4 dimensions. As in the continuum, the massive Green function can be obtained from the massless one, and hence the key task in causal set theory is to first identify the massless Green function. We propose that the 2d model provides a Green function for the massive scalar field on causal sets approximated by any topologically trivial 2-dimensional spacetime. We explicitly demonstrate that this is indeed the case in a Riemann normal neighbourhood. In 4d the model can again be used to provide a Green function for the massive scalar field in a Riemann normal neighbourhood which we compare to Bunch and Parker’s continuum Green function. We find that the same prescription can also be used for de Sitter spacetime and the conformally flat patch of anti-de Sitter spacetime. Our analysis then allows us to suggest a generalisation of Johnston’s model for the Green function for a causal set approximated by 3-dimensional flat spacetime.

Simplifying the one-loop five graviton amplitude in type IIB string theory

We consider the [Formula: see text] and [Formula: see text] terms in the low momentum expansion of the five graviton amplitude in type IIB string theory at one loop. They involve integrals of various modular graph functions over the fundamental domain of [Formula: see text]. Unlike the graphs which arise in the four graviton amplitude or at lower orders in the momentum expansion of the five graviton amplitude where the links are given by scalar Green functions, there are several graphs for the [Formula: see text] and [Formula: see text] terms where each of these two links are given by a derivative of the Green function. Starting with appropriate auxiliary diagrams, we show that these graphs can be expressed in terms of those which do not involve any derivatives. This results in considerable simplification of the amplitude.

Vacuum polarization and classical self-action near higher-dimensional defects

We analyze the gravity-induced effects associated with a massless scalar field in a higher-dimensional spacetime being the tensor product of (d-n)-dimensional Minkowski space and n-dimensional spherically/cylindrically symmetric space with a solid/planar angle deficit. These spacetimes are considered as simple models for a multidimensional global monopole (if ngeqslant 3) or cosmic string (if n=2) with (d-n-1) flat extra dimensions. Thus, we refer to them as conical backgrounds. In terms of the angular-deficit value, we derive the perturbative expression for the scalar Green function, valid for any dgeqslant 3 and 2leqslant nleqslant d-1, and compute it to the leading order. With the use of this Green function we compute the renormalized vacuum expectation value of the field square {langle phi ^{{}2}(x)rangle }_{mathrm{ren}} and the renormalized vacuum averaged of the scalar-field energy-momentum tensor {langle T_{M N}(x)rangle }_{mathrm{ren}} for arbitrary d and n from the interval mentioned above and arbitrary coupling constant to the curvature xi . In particular, we revisit the computation of the vacuum polarization effects for a non-minimally coupled massless scalar field in the spacetime of a straight cosmic string. The same Green function enables to consider the old purely classical problem of the gravity-induced self-action of a classical point-like scalar or electric charge, placed at rest at some fixed point of the space under consideration. To deal with divergences, which appear in consideration of the two problems, we apply the dimensional-regularization technique, widely used in quantum field theory. The explicit dependence of the results upon the dimensionalities of both the bulk and conical submanifold is discussed.

Poisson equation for the three-loop ladder diagram in string theory at genus one

The three-loop ladder diagram is a graph with six links and four cubic vertices that contributes to the [Formula: see text] amplitude at genus one in type II string theory. The vertices represent the insertion points of vertex operators on the toroidal worldsheet and the links represent scalar Green functions connecting them. By using the properties of the Green function and manipulating the various expressions, we obtain a modular invariant Poisson equation satisfied by this diagram, with source terms involving one-, two- and three-loop diagrams. Unlike the source terms in the Poisson equations for diagrams at lower orders in the momentum expansion or the Mercedes diagram, a particular source term involves a five-point function containing a holomorphic and a antiholomorphic worldsheet derivative acting on different Green functions. We also obtain simple equalities between topologically distinct diagrams, and consider some elementary examples.

Proving relations between modular graph functions

We consider modular graph functions that arise in the low energy expansion of the four graviton amplitude in type II string theory. The vertices of these graphs are the positions of insertions of vertex operators on the toroidal worldsheet, while the links are the scalar Green functions connecting the vertices. Graphs with four and five links satisfy several non-trivial relations, which have been proved recently. We prove these relations by using elementary properties of Green functions and the details of the graphs. We also prove a relation between modular graph functions with six links.

SuperDARN scalar radar equations

The quadratic scalar radar equations are obtained for SuperDARN radars that are suitable for the analysis and interpretation of experimental data. The paper is based on a unified approach to the obtaining radar equations for the monostatic and bistatic sounding with use of hamiltonian optics and ray representation of scalar Green's function and without taking into account the polarization effects. The radar equation obtained is the sum of several terms corresponding to the propagation and scattering over the different kinds of trajectories, depending on their smoothness and the possibility of reflection from the ionosphere. It is shown that the monostatic sounding in the media with significant refraction, unlike the case of refraction-free media, should be analyzed as a combination of monostatic and bistatic scattering. This leads to strong dependence of scattering amplitude on background ionospheric density due to focusing mechanism and appearance of new (bistatic) areas of effective scattering with significant distortion of the scattered signal spectrum. Selective properties of the scattering also have been demonstrated.

Scalar Green function bounds for instantaneous shock location and one-dimensional stability of viscous shock waves

In this paper, we investigate and prove the nonlinear stability of viscous shock wave solutions of a scalar viscous conservation law, using the methods developed for general systems of conservation laws by Howard, Mascia, Zumbrun and others, based on instantaneous tracking of the location of the perturbed viscous shock wave. In some sense, this paper extends the treatment in a previous expository work of Zumbrun [Instantaneous shock location ...] on Burgers equation to the general case, giving an exposition of these methods in the simplest setting of scalar equations. In particular we give by a rescaling argument a simple treatment of nonlinear stability in the small-amplitude case.

Bessel beam CARS of axially structured samples.

We report about a Bessel beam CARS approach for axial profiling of multi-layer structures. This study presents an experimental implementation for the generation of CARS by Bessel beam excitation using only passive optical elements. Furthermore, an analytical expression is provided describing the generated anti-Stokes field by a homogeneous sample. Based on the concept of coherent transfer functions, the underling resolving power of axially structured geometries is investigated. It is found that through the non-linearity of the CARS process in combination with the folded illumination geometry continuous phase-matching is achieved starting from homogeneous samples up to spatial sample frequencies at twice of the pumping electric field wave. The experimental and analytical findings are modeled by the implementation of the Debye Integral and scalar Green function approach. Finally, the goal of reconstructing an axially layered sample is demonstrated on the basis of the numerically simulated modulus and phase of the anti-Stokes far-field radiation pattern.

Towards obtaining Green functions for a Casimir cavity in de Sitter spacetime

Recent work in the literature has studied rigid Casimir cavities in a weak gravitational field, or in de Sitter spacetime, or yet other spacetime models. The present review paper studies the difficult problem of direct evaluation of scalar Green functions for a Casimir-type apparatus in de Sitter spacetime. Working to first order in the small parameter of the problem, i.e. twice the gravity acceleration times the plates’ separation divided by the speed of light in vacuum, suitable coordinates are considered for which the differential equations obeyed by the zeroth- and first-order Green functions can be solved in terms of special functions. This result can be used, in turn, to obtain, via the point-split method, the regularized and renormalized energy–momentum tensor both in the scalar case and in the physically more relevant electromagnetic case.

Publisher’s Note: Biconformal symmetry and static Green functions in the higher-dimensional Reissner-Nordström spacetimes [Phys. Rev. D91, 064037 (2015)

We study a static scalar massless field created by a source located near an electrically charged higher dimensional spherically symmetric black hole. We demonstrated that there exist bi-conformal transformations relating static field solutions in the metric with different parameters of the mass M and charge Q. Using this symmetry we obtain the static scalar Green function in the higher dimensional Reissner-Nordstrom spacetimes.

Biconformal symmetry and static Green functions in the higher-dimensional Reissner-Nordström spacetimes

We study a static scalar massless field created by a source located near an electrically charged higher dimensional spherically symmetric black hole. We demonstrated that there exist bi-conformal transformations relating static field solutions in the metric with different parameters of the mass M and charge Q. Using this symmetry we obtain the static scalar Green function in the higher dimensional Reissner-Nordstrom spacetimes.

A SEMI-ANALYTICAL METHOD TO CALCULATE THE ENTRIES OF THE METHOD OF MOMENTS MATRIX FOR THE MIXED POTENTIAL INTEGRAL EQUATION OF A SOURCE RECONSTRUCTION PROBLEM

In this article, the mixed potential integral equation is discretized using the Rao-Wilton- Glisson basis functions in order to obtain a method of moments matrix equation for a source reconstruction problem. The weighting functions used in the setup of the moments equation are Dirac delta functions. The entries of the moments matrix are computed using a semi-analytical method which is applicable to any method of moments problem with point matching. The analytical calculation is made possible by employing a differentiation property of the scalar Green function and the properties of the mesh elements of the source plane. The semi-analytical method makes it easier to increase the accuracy of the moments matrix elements. The accuracy of the method is shown by comparing the results obtained using the semi-analytical method to those obtained by a fully numerical procedure. The source reconstruction for a specific electromagnetic field distribution is the procedure of either setting up equivalent electromagnetic problems in order to determine equivalent sources or determining the actual sources by solving the related integral equations. The source reconstruction method is used in several areas which can be mentioned as near-field to far-field (NF-FF) transformation, microwave imaging, antenna design, antenna characterization, antenna diagnostics, electromagnetic interference estimation in electronic circuits, corrections in antenna measurements, radome defect localization. The most important aspect of the source reconstruction method is that it takes the electromagnetic problem into consideration just as it is, with all of the real world factors included through the electromagnetic field measurement results. The literature on the source reconstruction method can be divided into two parts with respect to the type of the integral equation formulation for the problem. In the first group, a single integral equation formulation is done whereas in the second group, a dual-integral equation formulation is preferred. In the single equation formulation, the equivalent sources obtained using the equivalence principle are related to the field measurement results using the radiation integrals. In the dual equation formulation, an extra equation is set up using the Love's equivalence principle at the measurement surface. Some of the articles in the single equation formulation group are to be cited now. In (7-30), several NF-FF transformation formulations using the source reconstruction method and the related applications are studied. In (7), equivalent magnetic currents are constructed on a plane including the aperture of an antenna using the near electric field data on a plane. These currents are then used to calculate the far-field. In (8), equivalent magnetic currents are built on a plane including the aperture of an antenna again. However, the near electric field data is not necessarily to be obtained on a plane and can be gotten from a surface with an arbitrary geometry. In (9), an equivalent electric current is set up on a

A line source in Minkowski for the de Sitter spacetime scalar Green's function: Massless minimally coupled case

Motivated by the desire to understand the causal structure of physical signals produced in curved spacetimes – particularly around black holes – we show how, for certain classes of geometries, one might obtain its retarded or advanced minimally coupled massless scalar Green's function by using the corresponding Green's functions in the higher dimensional Minkowski spacetime where it is embedded. Analogous statements hold for certain classes of curved Riemannian spaces, with positive definite metrics, which may be embedded in higher dimensional Euclidean spaces. The general formula is applied to (d ≥ 2)-dimensional de Sitter spacetime, and the scalar Green's function is demonstrated to be sourced by a line emanating infinitesimally close to the origin of the ambient (d + 1)-dimensional Minkowski spacetime and piercing orthogonally through the de Sitter hyperboloids of all finite sizes. This method does not require solving the de Sitter wave equation directly. Only the zero mode solution to an ordinary differential equation, the “wave equation” perpendicular to the hyperboloid – followed by a one-dimensional integral – needs to be evaluated. A topological obstruction to the general construction is also discussed by utilizing it to derive a generalized Green's function of the Laplacian on the (d ≥ 2)-dimensional sphere.

Modelling of Scalar Wave Propagation Problems in Heterogeneous Media by the Explicit Green's Approach Method

The Explicit Green's approach is an effective method for solving initial-boundary value problems in any kind of medium and geometry. This is due to the fact that numerical Green's functions are adopted instead of analytical ones, rendering a more general approach. When the Explicit Green's approach is applied to the scalar wave equation and Green's functions are computed by the finite element method (FEM) in conjunction with the central difference time integration scheme, unstable results are observed. To circumvent this drawback, the present paper discusses the application of a time substep procedure to the central difference time integration for the Green's functions computation. The substep procedure has the advantage of stabilizing the solution as well as increasing the accuracy order in the time domain from two to four. Numerical examples that demonstrate the accuracy and effectiveness of the proposed methodology are provided.

Scalar Green function of the Kerr spacetime

In this paper we study the scalar Green function in the Kerr spacetime using Wentzel-Kramers-Brillouin (WKB) methods. The Green function can be expressed by Fourier-transforming to its frequency-domain counterpart, and with the help of complex analysis it can be divided into parts: 1) the ``direct part,'' which propagates on the light cone and dominates at very early times; 2) the ``quasinormal-mode part,'' which represents the waves traveling around the photon sphere and is important at early and intermediate times; and 3) the ``tail part,'' which is due to scattering by the Coulomb-type potential and becomes more important at later times. We focus on the ``quasinormal-mode part'' of the Green function and derive an approximate analytical formula for it using WKB techniques. This approximate Green function diverges at points that are connected by null geodesics, and it recovers the fourfold singular structure of Green functions that are seen in Schwarzschild and other spacetimes. It also carries unique signatures of the Kerr spacetime such as frame dragging. Along the way, we also derive approximate quasinormal-mode wave functions and expressions for the black hole excitation factors in the Kerr spacetime. We expect this work to benefit the understanding of both wave propagation and the problem of self-force in the Kerr spacetime.