Talk:Bondi-Sachs Formalism

Comments on the article:

The article is a very well presented account of the Bondi-Sachs formalism in GR. In it the authors discuss several issues pertinent to this important step in the understanding of gravitational waves.

I want to make a couple of comments that may help to improve the article.

1. In the introduction after (1) the authors talk about the retarded and advanced time being characteristic hypersurfaces. This is clearly not correct. It might be better to say that they define such hypersurfaces, or, even better, that their level sets are characteristic hypersurfaces.
2. The fact that the conditions in (3) hold is not obvious even though it is just a rewriting of the earlier geometric assumptions. But since the article should be accessible to graduate students I think it would be useful to add a few explanatory remarks.
3. In (13) suddenly the symbol scri pops up. It has not been defined or otherwise been introduced. It is discussed later in the paper but at this stage nothing is known about it. It might be helpful to say a couple of words about its meaning already at this stage. In fact, the only information needed here is that it symbolizes \(\lim_{r\to\infty} \).
4. After (22) the discussion of the PDE abruptly changes into the presentation of a numerical evolution scheme. It might be helpful to discuss why.
5. Before (24) it might be better to say divergence-free condition instead of local conservation condition for the energy-momentum tensor, since the equation is not a conservation law.
6. After (31) the authors mention the conformal 2-metric \(h_{AB}\). For someone in the know this sentence is clear, but thinking again of someone who reads these things for the first time I am doubtful. This metric was introduced without making explicit reference to the conformal structure of the 2-sphere. It might be helpful to mention the fact that \(h_{AB}\) was introduced as a special representative of the conformal class of the 2-sphere metric \(g_{AB}\).
7. The first item (35) in this list is not an asymptotic datum in contrast to what is said in the sentence before the list.
8. In the discussion of the news tensor before (43) the word gauge-invariant is used. In what sense is it meant here? I would suspect it means that the news tensor is invariant under the choice of basis on the 2-sphere. It is certainly not meant to be invariant under diffeomorphisms.
9. The derivation of (49) is very brief. I have not been able to get the third line. I get an additional term proportional to \(q^A q^B d_{AB}\) containing \(\bar{q}^E\eth_E q^B\). The conclusion is still true. Even if this term is not there it is not obvious to the untrained eye why \(\bar{q}^E\eth_E (q^A q^B d_{AB}) = 0\) implies \(q^A q^Bd_{AB} = 0\). A remark about the spin-weight of that quantity or something to that effect might be useful.
10. I do not understand the meaning of the sentence before (62). What is done here? Is the vanishing of the conformal Einstein tensor used in order to get the form of the asymptotic expansion, or are these equations an ansatz? What are the conformal space Einstein equations?
11. I corrected (59) with a conformal factor on the left hand side. Please check!
12. I corrected several other more obvious typos.

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Authors answer to Referee

The article is a very well presented account of the Bondi-Sachs formalism in GR. In it the authors discuss several issues pertinent to this important step in the understanding of gravitational waves.

We thank the referee for his positive and constructive comments which helped a lot to amend our review article. Among other small changes to the pre-refereed version, please, find below our specific answers to your comments:

1. In the introduction after (1) the authors talk about the retarded and advanced time being characteristic hypersurfaces. This is clearly not correct. It might be better to say that they define such hypersurfaces, or, even better, that their level sets are characteristic hypersurfaces.

We changed the statement after Eq. 2 to "define characteristic hypersurfaces".

2. The fact that the conditions in (3) hold is not obvious even though it is just a rewriting of the earlier geometric assumptions. But since the article should be accessible to graduate students I think it would be useful to add a few explanatory remarks.

We added these explanatory details to the construction of the Bondi-Sachs coordinate system in the text before Eq. 2.

3. In (13) suddenly the symbol scri pops up. It has not been defined or otherwise been introduced. It is discussed later in the paper but at this stage nothing is known about it. It might be helpful to say a couple of words about its meaning already at this stage. In fact, the only information needed here is that it symbolizes \(\lim_{r\to\infty} \).

We substituted the symbol scri for the $\lim_{r\rightarrow\infty}$ in Eq. 12 (previously Eq.13) and everywhere before 'scri', $\mathcal{I}^+$, is defined.

4. After (22) the discussion of the PDE abruptly changes into the presentation of a numerical evolution scheme. It might be helpful to discuss why.

This evolution scheme is a strategy to solve the main equations which are the hierarchical set of ordinary differential equations along the null rays. We added a clarification (after Eq. 21) before discussing the algorithm.

5. Before (24) it might be better to say divergence-free condition instead of local conservation condition for the energy-momentum tensor, since the equation is not a conservation law.

We changed the statement to 'divergence-free condition'.

6. After (31) the authors mention the conformal 2-metric \(h_{AB}\). For someone in the know this sentence is clear, but thinking again of someone who reads these things for the first time I am doubtful. This metric was introduced without making explicit reference to the conformal structure of the 2-sphere. It might be helpful to mention the fact that \(h_{AB}\) was introduced as a special representative of the conformal class of the 2-sphere metric \(g_{AB}\).

The conformal 2-metric $h_{AB}$ is now introduced when the Bondi-Sachs metric is first explained (see point 2).

7. The first item (35) in this list is not an asymptotic datum in contrast to what is said in the sentence before the list.

It is true the left hand side of Eq. (34) (previously Eq. 35) is not an asymptotic datum because it depends on $r$. We corrected the misleading sentence before Eq. (34).

8. In the discussion of the news tensor before (43) the word gauge-invariant is used. In what sense is it meant here? I would suspect it means that the news tensor is invariant under the choice of basis on the 2-sphere. It is certainly not meant to be invariant under diffeomorphisms.

We changed the phrase gauge invariant news tensor to a tensor field that is independent of the $u-$foliation at scri and refer to a later discussion we have added in connection with Eq. (70).

9. The derivation of (49) is very brief. I have not been able to get the third line. I get an additional term proportional to \(q^A q^B d_{AB}\) containing \(\bar{q}^E\eth_E q^B\). The conclusion is still true. Even if this term is not there it is not obvious to the untrained eye why \(\bar{q}^E\eth_E (q^A q^B d_{AB}) = 0\) implies \(q^A q^Bd_{AB} = 0\). A remark about the spin-weight of that quantity or something to that effect might be useful.

This original derivation was indeed misleading and abstruse. We added a more transparent derivation of the property that $d_{AB}$ is pure trace between Eq's. (47) and (56).

10. I do not understand the meaning of the sentence before (62). What is done here? Is the vanishing of the conformal Einstein tensor used in order to get the form of the asymptotic expansion, or are these equations an ansatz? What are the conformal space Einstein equations?

We specified that the equations which are now Eq. (63)-(66) are subject to the physical space Einstein equations.

11. I corrected (59) with a conformal factor on the left hand side. Please check!'

We thank the referee to include the missing conformal factor on the left hand side of Eq. 59.

12. I corrected several other more obvious typos.

Thank you.