Hostname: page-component-cd9895bd7-p9bg8 Total loading time: 0 Render date: 2024-12-27T19:49:33.877Z Has data issue: false hasContentIssue false
  • English
  • Français

Edth-a differential operator on the sphere

Published online by Cambridge University Press:  24 October 2008

Michael Eastwood  and
Paul Tod
Michael Eastwood
Affiliation:
Institute for Advanced Study, Princeton
Paul Tod
Affiliation:
Mathematical Institute, Oxford University
Get access Rights & Permissions [Opens in a new window]

Extract

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).

Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 92 , Issue 2 , September 1982 , pp. 317 - 330
Copyright
Copyright © Cambridge Philosophical Society 1982

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

REFERENCES

(1)Curtis, W. D. and Lerner, D. E.Complex line bundles in relativity. J. Math. Phys. 19 (1978), 874–877.CrossRefGoogle Scholar
(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), 2155–2161.CrossRefGoogle Scholar
(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), 445–468.Google Scholar
(4)Held, A., Newman, E. T. and Posadas, R.The Lorentz group and the sphere. J. Math. Phys. 11 (1970), 3145–3154.CrossRefGoogle Scholar
(5)Higgins, J. R.Completeness and basis properties of sets of special functions (Cambridge University Press, 1977).CrossRefGoogle Scholar
(6)Ko, M., Ludvigsen, M., Newman, E. T. and Tod, K. P.The theory of ℋ-space. Phys. Rep. 71 (1981), 51–139.CrossRefGoogle Scholar
(7)Milnor, J. and Stasheff, J.Characteristic classes (Princeton University Press, 1974).CrossRefGoogle Scholar
(8)Morrow, J. and Kodaira, K.Complex manifolds (Holt, Rinehart and Winston, New York–Montreal–London, 1971).Google Scholar
(9)Newman, E. T. and Penrose, R.A note on the Bondi-Metzner-Sachs group. J. Math.Phys. 7 (1966), 863–870.CrossRefGoogle Scholar
(10)Newman, E. T. Heaven and its properties. Gen. Rel. Grav. 7 (1976), 107–111.CrossRefGoogle Scholar
(11)Penrose, R.The apparent shape of a relativistically moving sphere. Proc. Cambridge Philos. Soc. 55 (1959), 137–139.CrossRefGoogle Scholar
(12)Penrose, R. The structure of space-time. In Battelle rencontres, ed. DeWitt, C. M. and Wheeler, J. A., pp. 121–235 (Benjamin, New York, 1968), pp. 121–235.Google Scholar
(13)Penrose, R.Nonlinear gravitons and curved twistor theory. Gen. Rel. Grav. 7 (1976), 31–52.CrossRefGoogle Scholar
(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.)Google Scholar
(15)Serre, J.-P.Un théorème de dualité. Comment. Math. Helv. 29 (1955), 9–26.CrossRefGoogle Scholar
(16)Wells, R. O. Jr,. Differential analysis on complex manifolds (Springer, Berlin–Heidelberg–New York, 1980).CrossRefGoogle Scholar