Gravity

3-D Spinors, Spin-Weighted Functions and their Applications by Gerardo F. Torres del Castillo

By Gerardo F. Torres del Castillo

This booklet at the thought of three-d spinors and their functions fills a massive hole within the literature. It offers an introductory remedy of spinors.

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"Gathers a lot of what might be performed with 3-D spinors in an easy-to-read, self-contained shape designed for purposes that would complement many on hand spinor remedies. The book…should be attractive to graduate scholars and researchers in relativity and mathematical physics." -―MATHEMATICAL REVIEWS

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Extra info for 3-D Spinors, Spin-Weighted Functions and their Applications (Progress in Mathematical Physics)

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Stillwell 1992) or the Poincare half-plane, which models the Lobachevsky geometry. 1 Spherical harmonics The spherical harmonics can be defined in various ways; they are eigenfunctions of the Laplace-Beltrami operator of the sphere and they are the angular part of the separable solutions in spherical coordinates of the Laplace equation in Euclidean space. Another useful characterization is given by the following result. Proposition. k are symmetric in their I indices (i, j, ... = 1, ... m = O.

30) contains an additional factor e-isx . 40) + s"f)1+s ar((1 + s"f)-Sl1) (cf. Eastwood and Tod 1982, Penrose and Rindler 1984, Stewart 1990). 17) (which correspond to X = 0). 41) which reduces to the associated Legendre equation when m or s is equal to O. 42) or, equivalently, [ 2 d2 d (m + s)2 (m - s)2 .. ] (1-x )dx 2 -2x dx - 2(I-x) - 2(1+x) +J{]+I) s'iljm =0, 2. m-s. (a + fJ) = j - max{lml, lsI}. fJ)(x) is a Jacobi polynomial (the subscript n is equal to the degree of the polynomial). Similarly, letting s1:ljm(x) = (1- x)(m+s)/2(1 + x)j-(m+s)/2g(~+D, we find that g obeys the hypergeometric equation d 2g(x) x(l-x)(l;2 + [m +s + 1- (1 +m +s - dg(x) .

CDE ... F ~PVoE; •. CDE ... F a(oAoB ••• oC)OVoE •.. CDE ... F( -croB .. 0C - oA(jB ... x OVoE •. ·oF oc - ... - oAoB •.. CDE ... F~p A B -'::C~ ~ = - (J' +s 0 O. , apart from a constant factor, the effect of or on a spin-weighted spherical harmonic is to replace a factor () by a factor 0 or vice versa. Thus, if j =1= ±s, after applying and in any order, to a spin-weighted spherical harmonic, the result is a multiple of the same spin-weighted spherical harmonic. 22) showing that the spin-weighted spherical harmonics are eigenfunctions of aa and ofaa.

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