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Carroll Expansion of General Relativity
by Dennis Hansen, Niels A. Obers, Gerben Oling, Benjamin T. Søgaard
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Submission summary
| Authors (as registered SciPost users): | Gerben Oling |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2112.12684v3 (pdf) |
| Date submitted: | 2022-07-04 11:32 |
| Submitted by: | Oling, Gerben |
| Submitted to: | SciPost Physics |
| Ontological classification | |
|---|---|
| Academic field: | Physics |
| Specialties: |
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| Approach: | Theoretical |
Abstract
We study the small speed of light expansion of general relativity, utilizing the modern perspective on non-Lorentzian geometry. This is an expansion around the ultra-local Carroll limit, in which light cones close up. To this end, we first rewrite the Einstein-Hilbert action in pre-ultra-local variables, which is closely related to the 3+1 decomposition of general relativity. At leading order in the expansion, these pre-ultra-local variables yield Carroll geometry and the resulting action describes the electric Carroll limit of general relativity. We also obtain the next-to-leading order action in terms of Carroll geometry and next-to-leading order geometric fields. The leading order theory yields constraint and evolution equations, and we can solve the evolution analytically. We furthermore construct a Carroll version of Bowen-York initial data, which has associated conserved boundary linear and angular momentum charges. The notion of mass is not present at leading order and only enters at next-to-leading order. This is illustrated by considering a particular truncation of the next-to-leading order action, corresponding to the magnetic Carroll limit, where we find a solution that describes the Carroll limit of a Schwarzschild black hole. Finally, we comment on how a cosmological constant can be incorporated in our analysis.
Author comments upon resubmission
List of changes
First, we have amended our Introduction with a paragraph detailing our precise physical and technical motivations for studying the Carroll expansion of general relativity. These points, which were previously mentioned only later on in the introduction, are now collected in the fourth and fifth paragraph of our Introduction. Additionally, in the ninth paragraph of the Introduction, we have clarified the relation of our expansion to the electric and magnetic limits, emphasizing that the latter are only special cases of our more general result. Furthermore, we have added a roadmap of our paper at the end of the Introduction, as requested in Report 1.
To address the third general comment in Report 1, we have expanded a previous footnote into a paragraph below Equation (2.4), explaining the assumptions underlying that expansion. We have expanded the first, second, third and last paragraph of Section 2.5 to address the specific comments 8 and 9 of Report 1. Additionally, we have added a paragraph below (3.6), addressing both the specific comments 11 and 12 from Report 1 and comment 7 from Report 2. Finally, we have fixed several other typos (including the one pointed out in the specific comment 2 of Report 1).
Current status:
Reports on this Submission
Report 2 by Luca Ciambelli on 2022-7-18 (Invited Report)
Report
The authors carefully addressed all my points in a systematic way.
There is still a minor detail (that I will not need to review) that I would like to point at, in the amended version. The discussion below eq. (2.30) is unclear. First, I do not understand the sentence "The local Carroll boost transformations (2.9) act by shifting bi → bi + λi, corresponding to the choice of Ehresmann connection." Perhaps the authors meant "corresponding to a different the choice of Ehresmann connection." ? In the sentence after, there is a closing round parenthesis ")" that I believe is a typo. In general, this paragraph could be amended and explained better.
I am glad to recommend this paper for publication in SciPost.