# Talk:Spin-coefficient formalism

## Contents |

## Reviewer A

The article is clearly written, up-to-date and provides a clear panorama on the use of the Newman-Penrose formalism. My only suggestion is to correct the paper at some places, mainly in the equations.

The file with reviewer A corrections and the author's answer are full of formulas and too lenghty to be uploaded here. If you need them please contact the editorial board.

## Reviewer B

I have gone through the article making many cosmetic but absolutely minor changes. I would like to make some comments, which the authors might want to consider

- I think it would be more appropriate to talk about principal null
**directions**and not vectors because this is what they are. - In order to keep some consistency with published work it might be a useful to use w instead of t for the boost-weight.
- Should the indices in eqn. 34-37 not be concrete i.e., bold indices?
- I have rewritten the second sentence in the section "Null coordinates and associated tetrad conditions" for clarity. Please check!
- I have changed the indices of the metric in the conformal transformation in the Remark in this section. Please check!
- The spin-coefficient equations as given in the article are not the general ones but only those for vacuum space-times, is this intended?
- I have changed the \(\Delta\lambda\) equation (1) because I think it was wrong. Please check!
- The same goes for the simplified \(\Delta\lambda\) equation and the \(\delta U\) equation.
- In the section "The results" the index A is specialised to be \(\zeta\) or \(\bar\zeta\). While it is clear what it means, it is not consistent with A being 3 or 4.

## Reviewer C:

Overall, this is, as one would expect, an authoritative, clear, and well-thought out article. I think the selection of topics included is reasonable, although it results in a piece rather longer than Scholarpaedia aspires to, and the enormous literature is dealt with, by the random selection, about as well as it could be. I do have a number of less major points to make, though.

In the opening paragraph, other techniques with some related features are mentioned. It would be useful to know which of these will have articles of their own. If they will not, perhaps additional references are needed, e.g. for the orthonormal tetrad technique one should mention Ellis's work, or the exposition of it given in MacCallum's Cargese lectures. One technique not mentioned but which seems to be of importance is the single null vector method of Edgar, Ramos and Vickers.

In the next paragraph, point 1, 'linear' means linear in the derivatives only, i.e. of degree 1. They are not linear equations (the right sides being quadratic in general). This should perhaps be clarified.

A previous reviewer has queried the restriction to vacuum. I feel more strongly than him/her. In an encyclopaedia article the basic equations (the ones in the paragraph "The spin-coefficient equations") should give the most general form, although I am very happy with the decision elsewhere to focus on the vacuum case. In particular, it is quite hard to find a misprint/mistake-free complete set anywhere in the literature (I think the second edition of the well-known exact solutions book, or Stewart's book, may have one) so a correct online version would be very useful.

Relatedly, when the Goldberg-Sachs theorem is discussed, 'algebraically special' is defined as referring only to vacuum. It should be made clear that it can apply regardless of matter content.

On the integration and the use of the various sets of equations, the authors might cite the work of Edgar on the integrability of the NP set and their redundancies. (Incidentally, I appreciate the authors' modesty, but it would help readers to mention that the NP abbreviation for Newman-Penrose is widely used, and, for people reading other papers and using this as a guide, also to give the set of equations coming from the Ricci identity the commonly-used subnumbering by letter, as in the original article.)

On "Notation" I'm sure the authors' statement about the origin of the combinations must be true (who could possibly know better!?) but it could be worth saying they also occur naturally in a differential forms notation based on the tetrad (e.g. equation (7.9) of the exact solutions book). That might also prompt a self-deprecating footnote that with hindsight the names $\pi$ and $\kappa$ should have been swapped.

It may be worth noting that the 'metric equations' are often applied, as differential operator equations, to other spin coefficients or curvature components.

There is no totally satisfactory set of sign and ordering conventions, and I do not wish to suggest changes, but variations do make it harder to check correctness. I did not spot any problems in those equations I was able to check without undue further delay to this review, but I was far from checking all of them.

On a very minor point, the word 'geodicity' is not in my dictionary. Maybe 'geodesic character' or something similar is safer.

Another addition at the end which might be useful for pedagogical purposes would be a short list of texts where the formalism is described or used at greater length, in addition to the lists of sample applications and the citations of original articles. The Penrose-Rindler books would appear there, but I would add Stewart's short book and a few others.

## Authors' answer to reviewer C

We sincerely thank this reviewer and the earlier ones for their careful reading and comments.

1. *Overall, this is, as one would expect, an authoritative, clear, and well-thought out article. I think the selection of topics included is reasonable, although it results in a piece rather longer than Scholarpaedia aspires to, and the enormous literature is dealt with, by the random selection, about as well as it could be. I do have a number of less major points to make, though.*

- Thanks

2. *In the opening paragraph, other techniques with some related features are mentioned. It would be useful to know which of these will have articles of their own. If they will not, perhaps additional references are needed, e.g. for the orthonormal tetrad technique one should mention Ellis's work, or the exposition of it given in MacCallum's Cargese lectures. One technique not mentioned but which seems to be of importance is the single null vector method of Edgar, Ramos and Vickers.*

- Authors were unfamiliar with the work of Edger et al, but the reference has been added.

3. *In the next paragraph, point 1, 'linear' means linear in the derivatives only, i.e. of degree 1. They are not linear equations (the right sides being quadratic in general). This should perhaps be clarified.*

- There is here a misunderstanding on the reviewers part. There are many places and uses, where the equations are truly linear. This occurs in several places. By a simple transformation the optical equations for sigma and rho are made linear and then using the 'known' rho and sigma many other equations do become truly linear. The suspect sentence in the Intro "and often they can be partially grouped together into sets of linear equations (Newman and Penrose, 1962; Newman and Unti, 1962)", is thus true in that sense.

4. *A previous reviewer has queried the restriction to vacuum. I feel more strongly than him/her. In an encyclopaedia article the basic equations (the ones in the paragraph "The spin-coefficient equations") should give the most general form, although I am very happy with the decision elsewhere to focus on the vacuum case. In particular, it is quite hard to find a misprint/mistake-free complete set anywhere in the literature (I think the second edition of the well-known exact solutions book, or Stewart's book, may have one) so a correct online version would be very useful.*

- We have compromised here with the reviewer. We very strongly felt that the non-vacuum case does not belong in the text - it would make it far too long and take it far afield. But we have included the non-vacuum equations in an appendix.

5. *Relatedly, when the Goldberg-Sachs theorem is discussed, 'algebraically special' is defined as referring only to vacuum. It should be made clear that it can apply regardless of matter content.*

- We have added a sentence explaining that the classification is for the Weyl tensor and thus extends to the non-vacuum case.

6. *On the integration and the use of the various sets of equations, the authors might cite the work of Edgar on the integrability of the NP set and their redundancies. (Incidentally, I appreciate the authors' modesty, but it would help readers to mention that the NP abbreviation for Newman-Penrose is widely used, and, for people reading other papers and using this as a guide, also to give the set of equations coming from the Ricci identity the commonly-used subnumbering by letter, as in the original article.)*

- The Edger paper is referenced and the Editor has taken care of the SC vs the NP. Our modesty is thus preserved.

7. *On "Notation" I'm sure the authors' statement about the origin of the combinations must be true (who could possibly know better!?) but it could be worth saying they also occur naturally in a differential forms notation based on the tetrad (e.g. equation (7.9) of the exact solutions book). That might also prompt a self-deprecating footnote that with hindsight the names $\pi$ and $\kappa$ should have been swapped.*

- Please allow us not to get involved with that. It is not clear at this point how that would make the paper any clearer and it certainly would make it longer. Also note that 'kappa' does refer to a 'curvature

8. *It may be worth noting that the 'metric equations' are often applied, as differential operator equations, to other spin coefficients or curvature components.*

- Again, please allow us to avoid further issues. We (I) are not even certain what the reviewer is asking for or would want here.

9. *There is no totally satisfactory set of sign and ordering conventions, and I do not wish to suggest changes, but variations do make it harder to check correctness. I did not spot any problems in those equations I was able to check without undue further delay to this review, but I was far from checking all of them.*

- It is not clear if there is any requested change here. We just mention that the accuracy of the SC equations is a very serous concern. The reviewer is probably well aware of how an earlier error (misprint) has impacted on us. We have tried to be as careful as possible - and even checked our equations with the probably most accurate form - namely those in the Stewart book. That however is not conclusive evidence that we have it correct. They are truly a mess and an error or misprint is easy to make.

10. *On a very minor point, the word 'geodicity' is not in my dictionary. Maybe 'geodesic character' or something similar is safer.*

- The word 'geodicity' is quite commonly used. One finds it by simply going to google and typing' geodicity'. A rather humorous example is that the Reviewers comments have themselves now appeared on google using the word 'geodicity'. An example of Russell's self-referencial statements!!!! Fortunately, there are many other examples of its use.

Another addition at the end which might be useful for pedagogical purposes would be a short list of texts where the formalism is described or used at greater length, in addition to the lists of sample applications and the citations of original articles. The Penrose-Rindler books would appear there, but I would add Stewart's short book and a few others.

- Agreed. We have put in several others

## Reviewer C's further comments

Here I use the same numbering but only the ones where I have further comments are mentioned. I am glad that in general my comments were felt to be helpful.

2. I could not see the mention in the version currently on the web page. The most relevant papers are probably\\ M.P.M. Ramos and J.A.G. Vickers, Proc. Roy. Soc. A 1940, 693 (1995)\\ M.P.M. Ramos and J.A.G. Vickers, A spacetime calculus based on a single null direction, Class. Quant. Grav. 13, 1579 (1996)\\ though there have been applications by Edgar, Ramos, Vickers and others\\ e.g. Class.Quant.Grav. 22 (2005) 791-802\\

What is there is a mention of the Edgar stuff on integration in SC. (my comment 6)

3. That certainly explains the point but I wonder if some rephrasing is still needed to avoid confusing readers in the same way as I was unable to guess what was meant (despite doing just that sort of calculation more than once).

8. I had in mind that one can apply the commutator relations for the tetrad vectors, treated as equations in differential operators, to (for example) a spin coefficient, as yet another route in solving the whole system. In my experience this is rather useful.

9. Indeed I did not propose a change. I was only telling the authors I had not checked their equations. I have now checked the equations in the appendix against the set in Stephani et al and note the following differences which should be checked (some are obvious misprints but others may need calculations)\\

Eq (6) beta -> 2 beta\\ (8) the subscript 1 is missing in the second term\\ The \Phi_11 terms is missing a 2, and there is an extra 2 in the \Phi_02 term \\ (9) the sign of the 4th term on the left is different\\ (12) the \Phi_11 term on the right is missing a 2\\ (14) the indices on the second tem should print as subscripts

## Author's answer to "Reviewer C's further comments"

We have agreed to everything except item #8 - but with a compromise:

- Item #2. The suggested reference have been added in the application section

- Item #3.We have put the following sentence into the text since it does not belong in the Introduction - "By a simple transformation (see the discussion after Eq.(54) the optical equations for sigma and rho are made linear and then using the 'known' rho and sigma many other equations become linear."

- Item #8 - Re "equations in differential operators" . In the reviewers first set of Comments he wanted the remark to go with the 'metric equations' . Both I and Roger had no idea what he was referring to. In the second set of comments he wants it to apply to the 'commutator equations'. Now I understand what he is referring to. But the commutator equations are never once mentioned here. They were used originally to derive the 'metric equations' - but play no role in this paper. I do not believe that the remark belongs here without considerable addition material being included. I suggest that the reviewer give us a reference that we could add to the application references.

- Item #9. I thank the reviewer for catching all my typing misprints in the appendix.

## Reviewer C on #8

A couple of papers that definitely use commutators explicitly are

- Kinnersley, W., ``Type D vacuum metrics
*, {\em J. Math. Phys.}, {\bf 10}, 1195 (1969).*

- Siklos, S.T.C., ``Some Einstein spaces and their global properties
*, {\em J. Phys. A}, {\bf 14}, 395 (1981).*

I hope one of those would plug the gap OK