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Carroll Expansion of General Relativity
by Dennis Hansen, Niels A. Obers, Gerben Oling, Benjamin T. Søgaard
This is not the latest submitted version.
Submission summary
| Authors (as registered SciPost users): | Niels Obers · Gerben Oling |
| Submission information | |
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| Preprint Link: | https://arxiv.org/abs/2112.12684v3 (pdf) |
| Date submitted: | July 4, 2022, 11:32 a.m. |
| Submitted by: | Gerben Oling |
| Submitted to: | SciPost Physics |
| Ontological classification | |
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| 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
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 (Referee 1) on 2022-7-18 (Invited Report)
Report
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.