We describe how, in spite of differing conventions regarding spacetime signature, sign of the Riemann tensor, and definition of the Ricci tensor, we were able to construct a precise dictionary relating various notations which are being employed in the null tetrad formulation of general relativity. In addition, we give in appendices the forms assumed by the Newman–Penrose equations and the corresponding abstract structural equations when nontraditional assumptions are made with respect to the three sign conventions.
REFERENCES
1.
All the languages with which we are concerned regard as skew‐symmetric in α and β as well as in γ and δ.
2.
It is convenient to select in order to preserve the form of
3.
E.
Newman
and R.
Penrose
, 4
, 998
(1963
). In determining we took note of Sec. II and Eqs. (2.7) and (2.8)., J. Math. Phys.
4.
L. P. Eisenhart, Riemannian Geometry (Princeton U.P., Princeton, N.J., 1926).
5.
I.
Hauser
and R. J.
Malhiot
, 16
, 150
, 1625
(1975
). Our identifications of and may be based upon Sec. 1 of the last paper, Footnote 10 of the second paper, and Eq. (39) of the first paper., J. Math. Phys.
6.
R.
Debever
, Cah. Phys.
168–169
, 303
(1964
). In determining and we took note of Sec. I.1, Eq. (5.6), and Sec. I.6, Debever’s notation is currently in a state of flux!7.
J. F. Plebański, Spinors, Tetrads and Forms, a proto‐book representing lecture notes from a course on advanced relativity given at the Centro de Investigación y de Estudios Avanzados del IPN (México), 1974. In determining and we took note of Sec. I.1 and Eqs. (V.2.21), (V.2.12), (II.1.11), and (II.1.12). The reader should note that at an earlier time Plebański employed signature but currently uses signature
8.
9.
I thank R. Isaacson for suggesting that I look into the relationship between the IIT language and the Newman—Penrose language when he and I were colleagues at IIT.
10.
At IIT we generally suppress the symbol ∧ between differential forms.
11.
We would continue to use for the corresponding bivector components of the Weyl conform tensor if we were to switch signature.
12.
Our symbols generally have subscripts associated with spin weight. However, the symbols are used in order to avoid having two different types of subscripts on spincoefficients.
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© 1978 American Institute of Physics.
1978
American Institute of Physics
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