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An Approach to Gravitational Radiation by a Method of Spin Coefficients
1.A. Z. Petrov, Sci. Not. Kazan State Univ. 114, 55 (1954).
2.F. A. E. Pirani, Acta Phys. Polon. 15, 389 (1956);
2.F. A. E. Pirani, Phys. Rev. 105, 1089 (1957);
2.F. A. E. Pirani, Bull. Acad. Polon. Sci. 5, 143 (1957a).
3.E. Newman, J. Math. Phys. 2, 324 (1961).
4.J. Goldberg and R. Kerr, J. Math. Phys. 2, 327 (1961).
5.R. Sachs, Infeld Volume and preprints.
6.H. Bondi and R. Sachs (private communication).
7.L. Witten, Phys. Rev. 113, 357 (1959).
8.R. Penrose, Ann. Phys. (New York) 10, 171 (1960).
9.J. Ehlers, Hamburg Lectures.
10.It is possible, if one has no familiarity with spinors to omit Sec. III, with but a small loss of continuity.
11.J. Goldberg and R. Sachs (to be published).
12.R. Sachs (to be published).
13.I. Robinson and A. Trautman, Phys. Rev. Letters 4, 431 (1960).
14.Greek indices (values 1, 2, 3, 4) are tensor indices, bold face (values 1, 2, 3, 4) are tetrad indices, capital latin (values 0, 1) are spinor indices and small lightface latin (values 0, 1) are spinor “dyad” indices.
15.L. P. Eisenhart, Riemannian Geometry (Princeton University Press, Princeton, New Jersey, 1960).
16.See, for example, W. L. Bade and H. Jehle, Revs. Modern Phys. 25, 714 (1953).
17.We use primed rather than dotted indices for typographical reasons.
18.Many authors omit the bar over the complex conjugate.
19.The quantities (3.9) can be defined directly in terms of derivatives of the as follows: where or .
20.These definitions of differ by a factor 2 from those given in reference 8. Also, the Riemann tensor used here is the negative of that used in reference 8.
21.For completeness, though it is never used in this paper, we give in the Appendix the formulas for the Bianchi identities in the presence of a Maxwell field as well as the Maxwell equations using the notation of this section.
22.I. Robinson, J. Math. Phys. 2, 290 (1961).
23.Though we have not seen all the details of the Goldberg‐Sachs proof, we believe our proof to be essentially equivalent, but, due to the conciseness of our notation, much shorter.
24.An affine parameter is a parameter along the geodesic, such that the equation for the geodesic takes the standard form. See, for example, E. M. Schrödinger, Expanding Universes (Cambridge University Press, New York, 1956).
25.The meaning of the order symbols used here is that means for some function F independent of r and for all large r, and means .
26.These assumptions, though stated in terms of a particular coordinate system appear to have a considerable amount of coordinate independence. For example, given a null geodesic with affine parameter r and tangent vector if the r parameter of the original coordinate system can be so adjusted that , then (7.4) implies that also, where is the complex Riemann tensor component associated with However, additional global assumptions appear to be necessary to ensure that r can always be so chosen.
27.This may be a fairly strong restriction. It is, of course, stronger than just local analyticity in r since, for example, ln r cannot be expanded in negative powers of r.
28.More properly, the quantity may be the most significant one to specify on the hypersurface.
29.R. Penrose (to be published).
30.The necessity of (7.3) for the deduction of (7.4) and of ruling out the “asymptotically plane” case can be illustrated by considerations of certain plane waves. Plane waves can also be used to show that, for example, a local assumption merely of or even is quite inadequate for obtaining (7.4).
31.E. Coddington and N. Levinson, Theory of Ordinary Differential Equations (McGraw‐Hill Book Publishers Inc., New York, 1955), p. 103.
32.It is permissible to integrate order symbols formally but not to differentiate them.
33.N. Levinson, Am. J. Math. 68, 1 (1946).
34.It may be seen from the proof to the lemma that conditions (7.16) are in fact, rather stronger than is necessary. They may be weakened to B, where This enables condition (7.2) to be weakened to and (7.4) can still be obtained. Conditions (7.3) [and even (7.1)] can also be correspondingly weakened.
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