A new approach to general relativity by means of a tetrad or spinor formalism is presented. The essential feature of this approach is the consistent use of certain complex linear combinations of Ricci rotation coefficients which give, in effect, the spinor affine connection. It is applied to two problems in radiation theory; a concise proof of a theorem of Goldberg and Sachs and a description of the asymptotic behavior of the Riemann tensor and metric tensor, for outgoing gravitational radiation.

## REFERENCES

1.

2.

3.

4.

5.

R. Sachs, Infeld Volume and preprints.

6.

H. Bondi and R. Sachs (private communication).

7.

8.

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.

14.

Greek indices (values 1, 2, 3, 4) are tensor indices, bold face $a,b\cdots $ (values 1, 2, 3, 4) are tetrad indices, capital latin $A,B\cdots $ (values 0, 1) are spinor indices and small lightface latin $a,b\cdots $ (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 $\sigma \mu ab\u2032,$ as follows: where or .

$\Gamma abcd\u2032\u2009=\u200912\epsilon p\u2032q\u2032{\sigma cd\u2032ap\u2032bq\u2032\u2212\sigma cd\u2032bq\u2032ap\u2032\u2212\sigma ap\u2032bq\u2032cd\u2032}$

$\sigma ab\u2032cd\u2032ef\u2032\u2009=\u2009\sigma cd\u2032[\mu ab\u2032\sigma \nu ]\sigma \mu ef\u2032,\nu $

$\Gamma abcd\u2032\u2009=\u200912\epsilon p\u2032a\u2032\sigma ag\u2032\mu \sigma cd\u2032\nu \sigma \mu bp\u2032;\nu $

20.

These definitions of $\Psi ABCD,$ $\Phi ABC\u2032D\u2032$ 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.

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 $f(r,u,xi)\u2009=\u2009O[g(r)]$ means $|f(r,u,xi)|<g(r)F(u,xi)$ for some function .

*F*independent of*r*and for all large*r*, and $f(r,u,xi)\u2009=\u2009o[g(r)]$ means$limr\u2192\u221ef(r,u,xi)g(r)\u2009=\u20090\u2009for\u2009each\u2009u,xi$

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 , then (7.4) implies that $\Psi \u03030\u2009=\u2009O(r\u22125)$ also, where $\Psi \u03030$ is the complex Riemann tensor component associated with $l\u0304\mu .$ However, additional global assumptions appear to be necessary to ensure that

*r*and tangent vector $l\mu ,$ if the*r*parameter of the original coordinate system can be so adjusted that$r\u0303\u2009=\u2009r+o(r),l\u0304\mu \u2009=\u2009l\mu +O(r\u22121)$

*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, $r\u2212n$ ln*r*cannot be expanded in negative powers of*r*.28.

More properly, the quantity $D\Psi 0\u22125\rho \Psi 0$ 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 $\Psi 4\u2009=\u2009O(r\u22121)$ or even $R\mu \nu \rho \sigma \u2009=\u2009O(r\u22121)$ 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.

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*, $b\u2009=\u2009O[f(r)]$ where $\u222bfdr\u2009=\u20090(1),$ $f>0.$ This enables condition (7.2) to be weakened to $D\Psi 0\u2009=\u2009O(r\u22124f(r))$ and (7.4) can still be obtained. Conditions (7.3) [and even (7.1)] can also be correspondingly weakened.
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© 1962 American Institute of Physics.

1962

American Institute of Physics

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