Twisting asymptotically-flat spacetimes

1. The paper, while long and technical, is explained clearly, as a whole.
2. It covers and brings together several different topics on asymptotically flat spacetimes.
3. Several avenues of future research, which build on the results in this work, were clearly described.

1. There were a few places where further discussion or clarifications would be beneficial.
2. The discussion of 3D (2+1) spacetimes was brief compared with the main results of the paper in 4D (3+1).

1. Sec. 1: I think it would help to summarize the conventions used for different index types and for the curvature tensors in one place, for example, at the end of the introduction. This could be text or a table.

2. Sec. 2.2: I think it would be helpful to describe the order in which the NP radial equations were integrated to obtain the results in (2.9)-(2.11).

3. Sec. 3.2: I found the discussion of gauge freedom somewhat confusing in this section, as I tend to distinguish between tetrad (local Lorentz) versus coordinate gauge freedom. For reference, the tetrad freedom in the NP formalism is the type I, II, and III transformations, as discussed in the text below Eq. (2.4). Coordinate gauge freedom refers to the standard general coordinate transformations of the form $y^{\mu'}(x^\nu)$, where the metric would transform as $g_{\mu\nu} =(\partial y^{\alpha'}/\partial x^\mu)(\partial y^{\beta'}/\partial x^\nu) g_{\alpha'\beta'}$. These two freedoms are independent in the sense that the NP variables are scalars under general coordinate transformations, but transform under tetrad transformations. Tensors (aside from the tetrad, for example) are scalars under tetrad transformations. This includes the spacetime metric. However, at the start of Sec. 3.2, there is discussion of replacing a coordinate gauge condition in Bondi gauge $g_{ra}=0$ with a tetrad gauge condition $\kappa = 0$. Is this because the coordinates are being defined in terms of the particular tetrad gauge-fixed vectors from Sec. 2? Further clarifications on this point would be helpful.

4. Sec. 3.5 and 3.6: As described in Ref. [65], for example, the hierarchy in Bondi gauge applies independent of the asymptotic boundary conditions or other assumptions on the solutions (e.g., polyhomogeneous versus $1/r$). The hierarchy here was shown for a $1/r$ type expansion with standard Peeling behavior. Does the hierarchy hold more generally without these assumptions once twist is permitted?

5. p. 25: Can you elaborate more on what the "mismatch of charges" means? Are these inequivalent notions of charges, or would one expect them to agree?

6. Finally, below are a number of small suggestions:

7. Abstract and several places later: "enables" should be "enables us"
8. Footnote 1: Add citations for unimodular, double null, and Scherk-Schwarz gauges.
9. p. 2: The phrase "contravariant gauge condition" seems non-standard. Perhaps "gauge condition on the components of the inverse metric" is clearer.
10. p. 3: I believe Kerr's original coordinates and what you call "Eddington-Finkelstein coordinates" (see, e.g., arXiv:0706.0622) are equivalent. I think it would be beneficial to cite Kerr's original paper in this context.
11. Eq. (2.1): I think it would be clearer to add indices to $\eta$ (i.e., $\eta^{ij}$) so that the index-free notation in (2.2) and elsewhere is reserved for spacetime vectors.
12. p. 11: "appart" should be "apart".
13. p. 13: I recommend deleting "downstairs" or replacing it with "covariant".
14. p. 16: "futur" should be "future" and "formes" should be "forms".
15. p. 19: Define the $\stackrel{B}{=}$ notation.
16. p. 32: Is $\Psi_{3,4}$ equivalent to $\Psi_3$, $\Psi_4$?
17. Eq. (6.1): Is there a reason that the spatial vector is normalized to $1/2$ rather than $1$ here? See also the comment about Eq. (2.1).
18. Eq. (A5): Define the $F$, $F_a$ and $F_{ab}$ in this equation.
19. Eq. (A.6m): I did not see where $\mathcal D_a$ was defined. Is it the same as the $D_a$ in Eq. (3.17), which I assume is the covariant derivative compatible with $q_{ab}$?

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• Complete description of an enlarged Bondi-Sachs coordinate systems that allows for arbitrary twist of the null congruence $\partial_r$ (and therefore arbitrary Ehresmann connection on the boundary).
• Clarity of the presentation heavily compensating the technicality of the paper.
• Exhaustive discussion of the solution space in two complementary viewpoints (including the very practical Newman-Penrose formalism).
• Application to the important class of algebraically special solutions.
• Evident mastery of advanced covariant phase space methods in gravity.

• Although coming from a slightly different perspective, the results present some overlaps with the covariant Newman-Unti gauge worked out by Petropoulos et al.
• Discussion of the charges and asymptotic symmetries a bit short.

Please refer to the attached PDF report.

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