This short note concerns gauge choices for metric perturbations, which are important tools in black hole perturbation theory (including for the self-force program). The note proves that a commonly employed gauge — known as radiation gauge — can leave the position of the event horizon untouched to first order in the metric perturbation. This is, to my knowledge, a new result.

The theorem’s proof is sound and clearly presented. I therefore recommend the paper for publication in CQG, provided the author addresses the following question.

Regarding the assumption that the Ricci tensor be second order in the perturbation (or its weaker version in Remark 2): When is this assumption satisfied? Is it sufficient for the background to be Ricci flat? It would be good to comment on this point, also in relation to the example.

I thank the author for addressing my questions. This clarified the relevance of the theorem for an important problem in gravitational physics, i.e., the modelling of extreme mass ratio inspirals (EMRIs). I recommend publication.