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Edth-a differential operator on the sphere

  • Michael Eastwood (a1) and Paul Tod (a2)

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Introduction. In (9) Newman and Penrose introduced a differential operator which they denoted ð, the phonetic symbol edth. This operator acts on spin weighted, or spin and conformally weighted functions on the two-sphere. It turns out to be very useful in the theory of relativity via the isomorphism of the conformal group of the sphere and the proper inhomogeneous Lorentz group (11, 4). In particular, it can be viewed (2) as an angular momentum lowering operator for a suitable representation of SO(3) and can be used to investigate the representations of the Lorentz group (4). More recently, edth has appeared in the good cut equation describing Newman's ℋ-space for an asymptotically flat space-time (10). This development is closely related to Penrose's theory of twistors and, in particular, to asymptotic twistors (14).

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(1)Curtis, W. D. and Lerner, D. E.Complex line bundles in relativity. J. Math. Phys. 19 (1978), 874877.
(2)Goldberg, J. N., MacFarlane, A. J., Newman, E. T., Rohrlich, F. and Sudarshan, E. C. G. Spin-s spherical harmonics and ð. J. Math. Phys. 8 (1967), 21552161.
(3)Hansen, R. O., Newman, E. T., Penrose, R. and Tod, K. P.The metric and curvature propeities of ℋ-space. Proc. Roy. Soc. London, Ser. A 363 (1978), 445468.
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(5)Higgins, J. R.Completeness and basis properties of sets of special functions (Cambridge University Press, 1977).
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(9)Newman, E. T. and Penrose, R.A note on the Bondi-Metzner-Sachs group. J. Math.Phys. 7 (1966), 863870.
(10)Newman, E. T. Heaven and its properties. Gen. Rel. Grav. 7 (1976), 107111.
(11)Penrose, R.The apparent shape of a relativistically moving sphere. Proc. Cambridge Philos. Soc. 55 (1959), 137139.
(12)Penrose, R. The structure of space-time. In Battelle rencontres, ed. DeWitt, C. M. and Wheeler, J. A., pp. 121235 (Benjamin, New York, 1968), pp. 121235.
(13)Penrose, R.Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7 (1976), 3152.
(14)Penrose, R. and Ward, R. S. Twistors for flat and curved space-time. In Einstein centennial volume, ed. Bergman, P. G., Goldberg, J. N. and Held, A. P.. (To appear.)
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(16)Wells, R. O. Jr,. Differential analysis on complex manifolds (Springer, Berlin–Heidelberg–New York, 1980).

Edth-a differential operator on the sphere

  • Michael Eastwood (a1) and Paul Tod (a2)

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