Use Of Conformal Transformations Research Articles

A New 2D Formulation of Modified General Relativity

AbstractIt is well known that the Einstein–Hilbert action in two dimensions is topological and yields an identically vanishing Einstein tensor. Consequently one is faced with difficulties when formulating a non‐trivial gravity model. The authors present a new, intrinsically two‐dimensional, approach to this problem based on the Einstein action. This yields a well defined variational approach which results in new field equations that break diffeomorphism invariance. This proposed approach does not require the introduction of additional scalar fields, nor the use of conformal transformations. However, how including conformal counter terms leads to equivalent results is shown. In doing so, an explanation for why previous approaches worked can be provided. Solutions to the field equations are briefly discussed.

General Formulas of Effective Earth Radius and Modified Refractive Index

The effective earth radius can transform the curved ray path into a straight line, and the modified refractive index can transform the spherical ground into a plane. Using them can simplify the propagation modeling of electromagnetic waves in the atmosphere, so they are widely used in the calculation of long-range propagation of electromagnetic wave, and adopted by the ITU Recommendations and the IEEE Standards. In this paper, upon employing simple geometric methods, we provide general formulas for the effective earth radius and the modified refractive index, pointing out that they both originate from the use of conformal transformations. In turn, we prove that for both quantities the formulas contained in the ITU Recommendations and the IEEE Standards are special cases of the general formulas given in this paper. Our results provide a solid mathematical background to the concepts of effective earth radius and modified refractive index, and pave the way for their use beyond the propagation of electromagnetic waves in the earth atmosphere, e.g. to the propagation of other types of waves in generic media.

Reconstructing exact scalar-tensor cosmologies via conformal transformations

We propose a new reconstruction method for scalar-tensor gravity based on the use of conformal transformations. The new method allows the derivation of a set of interesting exact cosmological solutions in Brans-Dicke gravity as well as other extensions of general relativity.

Zeta functions in brane world cosmology

We present a calculation of the zeta function and of the functional determinant for a Laplace-type differential operator, corresponding to a scalar field in a higher-dimensional de Sitter brane background, which consists of a higher-dimensional anti de Sitter bulk spacetime bounded by a de Sitter section, representing a brane. Contrary to the existing examples, which all make use of conformal transformations, we evaluate the zeta function working directly with the higher-dimensional wave operator. We also consider a generic mass term and coupling to curvature, generalizing previous results. The massless, conformally coupled case is obtained as a limit of the general result and compared with known calculations. In the limit of large anti de Sitter radius, the zeta determinant for the ball is recovered in perfect agreement with known expressions, providing an interesting check of our result and an alternative way of obtaining the ball determinant.

Casimir effect between anti–de Sitter braneworlds

We calculate the one-loop effective action of a scalar field with general mass and coupling to the curvature in the detuned Randall-Sundrum brane world scenario, where the four-dimensional branes are anti-de Sitter. We make use of conformal transformations to map the problem to one on the direct product of the hyperbolic space H^4 and the interval. We also include the cocycle function for this transformation. This Casimir potential is shown to give a sizable correction to the classical radion potential for small values of brane separation.

On the use of conformal transformations to calculate the eigenvalues of the Helmholtz equation for cylinders with polygonal cross section

Solutions of the Helmholtz equation, as it arises in reactor criticality calculations, have been obtained in cylinders of polygonal cross section. The method used employs the Schwarz–Christoffel transformation which maps the vertices of the polygon onto the circumference of the unit circle. Thus the polygonal geometry is converted to r– θ geometry thereby allowing a solution to be constructed which satisfies the symmetry and boundary conditions of zero flux exactly. Three cases have been studied numerically; the equilateral triangle, the square and the hexagon. The square is particularly useful because an exact solution can be obtained by separation of variables in x– y geometry and hence allows the accuracy of the conformal mapping procedure to be assessed. Numerical results are obtained in the r– θ domain using a variational method and appropriate trial functions. The accuracy of the eigenvalues is very high and by comparison with the exact value in the square it is shown that only one term in the trial function leads to an accuracy in the eigenvalue of 0.1%. The lowest order approximation to the current at the square centre edge gives an accuracy of 4%. Details of convergence for all cases are given, as well as some useful approximate analytical formulae for eigenvalues and edge current. ©

Relativistic static thin discs with radial stress support

New solutions for static non-rotating thin discs of finite radius with non-zero radial stress are studied. A method to introduce either radial pressure or radial tension is presented. The method is based on the use of conformal transformations.

Theories of inflation and conformal transformation

We show that seberal different theories of inflation including R 2, Brans-Dicke, and induced-gravity inflation are all related to generalized or power-law inflation by means of conformal transformations. These theories all involve non-standard gravity, and the use of conformal transformations allows one to obtain standard inflationary predictions such as the expansion time-scale, reheating and density perturbations in each case very simply. We also discuss the possibilities of this method to be applied to string theory.

A transmission line model of cable to antenna coupling inside a screened room

A coupled-transmission-line model that evaluates the coupling of emissions from a cable to an antenna when both are located within a screened room is presented. The cable is positioned over a conducting bench in the room. The model developed is valid over the frequency range 10 kHz to 30 MHz. Its parameters can be found quickly and can be easily adjusted to account for changes in height, length, location across the bench, and radius of the cable. A method for evaluating the coupling parameters of the model is given. It involves the use of conformal transformations and image techniques.< <ETX xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:xlink="http://www.w3.org/1999/xlink">&gt;</ETX>

Comments on "Capacitance between Perpendicular Conducting Planes Separated by a Gap"

A recent paper [1] attempted to solve the problem of the capacitance between a semi-infinite ground plane and a symmetrically placed strip perpendicular to its edge, by use of conformal transformations and an approximate variational Green's-function formulation. This note points out that the results given in the paper are in error by more than a factor of 2, and that no approximation is necessary to solve the stated problem. The exact solution is given.

The Use of Conformal Transformations for the Numerical Solution of Elliptic Boundary Value Problems with Boundary Singularities

The Use of Conformal Transformations for the Numerical Solution of Elliptic Boundary Value Problems with Boundary Singularities

On the use of conformal transformations for the numerical solution of harmonic boundary value problems

Abstract Certain aspects concerned with the application of a conformal transformation method for the numerical solution of harmonic boundary value problems are considered. In particular, a procedure is decribed for removing the effects of singularities which are sometimes introduced by the transformations in the solution of the transformed problem. The transformed problem is then solved by a collocation technique.

The two-dimensional magnetic or electric field of a single isolated pole-piece

The two-dimensional magnetic or electric field of a single isolated pole-piece consisting of a thick parallel plate terminated by a concave semi-circular cylinder is investigated theoretically. The method involves the use of conformal transformations and elliptic functions. The variation of the field strength along the edge of the pole-piece and along the external axis of symmetry is calculated and shown graphically.

The two-dimensional magnetic or electric field above and below an infinite corrugated sheet

The field is investigated above and below the surface of an infinite corrugated sheet. The sheet may be regarded as a magnetized ferromagnetic body, or a charged electrical conductor. The corrugations are parallel, semi-circular cylinders. The method involves the use of conformal transformations and elliptic integrals. Tables and graphs of the field strength at important points are given.