Reference frames in harmonic coordinates: geometric and physical aspects (original) (raw)

Invariant vector harmonics. The ellipsoidal case

Journal of Mathematical Analysis and Applications, 2013

We introduce a complete set of vector harmonic functions in an invariant form, that is, in a form that is independent of any coordinate system. In fact, we define three vector differential operators of the first order which, when they act on a scalar harmonic function they generate three independent vector harmonic functions. Then, we prove the relative independence properties and we investigate the characterization of every harmonic as an irrotational or solenoidal field. We also prove that this set of functions forms a complete set of vector harmonics. Finally, we use these invariant expressions to recover the vector spherical harmonics of Hansen and to introduce vector ellipsoidal harmonics in R 3. Our method can be applied to any other coordinate system to produce the corresponding vector harmonics.

The coordinate-free approach to spherical harmonics

2008

We present in a unified and self-contained manner the coordinate-free approach to spherical harmonics initiated in the mid 19th century by James Clerk Maxwell, William Thomson and Peter Guthrie Tait. We stress the pedagogical advantages of this approach which leads in a natural way to many physically relevant results that students find often difficult to work out using spherical coordinates and associated Legendre functions. It is shown how most physically relevant results of the theory of spherical harmonics - such as recursion relations, Legendre's addition theorem,surface harmonics expansions, the method of images, multipolar charge distributions, partial wave expansions, Hobson's integral theorem, rotation matrix and Gaunt's integrals - can be efficiently derived in a coordinate free fashion from a few basic elements of the theory of solid and surface harmonics discussed in the paper.

Axisymmetric stationary solutions as harmonic maps

General Relativity and Gravitation, 1994

We present a method for generating exact solutions of Einstein equations in vacuum using harmonic mal~, when the spacetime possesses two commuting K.iHing vectors. This method consists of writin K the axisymmetrlc stationary Einstein equations in vacuum as a harmonic map which belongs to the group SL(2t~), and decomposing it into its harmonic "suhmaps". This method provides a natural classification of the solutions in classes (Weyl's class, Lewis' class etc.).

On geometrical and physical consequences of using certain special classes of comoving physical frames of reference

Journal of Interdisciplinary Mathematics, 2009

This paper derrelops the studv of the geometrical characteristics of the class of frames of reference associated rvith the tr.r'o u'ell known sets of coorclinates: Levi-Civita's curvature coclrdinates and ganssian polar coordinates. In particular, thr:ough the separate invariant forrnulation of the rrritial clata problem ancl oi the restricted errolntion problem, the analysis is applierl to a sphericallr, syrnmetric perfect flr-rid, r,r'hose stream lines coincide r,r'ith those of the above rnentiorrecl class of frames.

A simple approach to the transformation of spherical harmonic models under coordinate system rotation

Geophysical Journal International, 1996

The transformation of a set of spherical harmonic coefficients characterizing a model of the geomagnetic field, or a general function defined on a sphere, subject to a rotation of the coordinate system, is given by the direct relations between the coefficients and then by using a numerical approach. The parameters for a pair of such rotations (from one set to another, and vice versa) are given, along with a few examples of their application. The method is particularly useful for the comparison of geophysical characteristics derived from models developed under different coordinate systems. It offers a practical solution to the problem, which can be implemented without difficulty.

Generalized harmonic spatial coordinates and hyperbolic shift conditions

Physical Review D, 2005

We propose a generalization of the condition for harmonic spatial coordinates analogous to the generalization of the harmonic time slices introduced by Bona et al., and closely related to dynamic shift conditions recently proposed by Lindblom and Scheel, and Bona and Palenzuela. These generalized harmonic spatial coordinates imply a condition for the shift vector that has the form of an evolution equation for the shift components. We find that in order to decouple the slicing condition from the evolution equation for the shift it is necessary to use a rescaled shift vector. The initial form of the generalized harmonic shift condition is not spatially covariant, but we propose a simple way to make it fully covariant so that it can be used in coordinate systems other than Cartesian. We also analyze the effect of the shift condition proposed here on the hyperbolicity of the evolution equations of general relativity in 1+1 dimensions and 3+1 spherical symmetry, and study the possible development of blow-ups. Finally, we perform a series of numerical experiments to illustrate the behavior of this shift condition.

Orthogonality of harmonic potentials and fields in spheroidal and ellipsoidal coordinates: application to geomagnetism and geodesy

Geophysical Journal International, 2012

We investigate the orthogonality of the potential distributions that are the basis solutions of Laplace's equation appropriate to 3-D ellipsoidal (including spheroidal) coordinate systems, and also the orthogonality of the corresponding vector gradient fields, both over the surface of the ellipsoid, and for integration over the volume of the annular shell between two confocal ellipsoids. The only situation for which there is orthogonality is for the vector gradients when integrated over the annular shell. In the other three cases (potential over surface or annulus, and field over surface) orthogonality can be restored by using an appropriate geometrical weighting factor applied to the integrand; it is therefore still possible to perform the equivalent of a classical spherical harmonic analysis. In the special case of the sphere, there is real orthogonality in all four cases; in effect the weighting factors are all unity. In geodesy, spheroidal harmonic analysis is done using a method that relies on a particular result valid only for potential; it cannot be extended to the corresponding vector field, or to ellipsoidal geometry. The lack of orthogonality over the surface means that care must be taken when interpreting conventional geomagnetic 'power spectra', and geodetic 'degree variance', as these no longer correspond exactly to the mean-square values over the actual ellipsoidal surface. We illustrate some of the problems by comparing different versions of the power spectrum for a spheroidal analysis of the global lithospheric magnetic field. We use only simple vector algebra, and do not need to know the details of the actual basis solutions, only that they are the product of three functions, one for each coordinate and involving only that coordinate, and that they satisfy Laplace's equation. Similarly, our results do not depend on the normalization used in the basis functions.

Spherical Harmonic on a Four Sphere

Applied Mathematics and Physics, 2014

In this paper, we will analyse the scalar harmonics on a four sphere using a associated Legendre function. Then, we will use these modes to construct two types of vector harmonics on a four sphere. Finally, we will also construct three types of tensor harmonics on a four sphere. As there is a relation between de Sitter spacetime and four sphere, these modes are related to the modes on de Sitter spacetime.