https://doi.org/10.1140/epjc/s10052-024-13355-8
Regular Article - Theoretical Physics
A note on the Cotton flow and the Ricci flow for three-manifolds, and the Hořava–Lifshitz gravity
1
Instituto de Física, Universidad Autónoma de Puebla, Apartado Postal J-48, 72570, Puebla Pue, Mexico
2
Escuela de Ciencias, Universidad Autónoma Benito Juárez de Oaxaca, Apartado Postal, 68120, Oaxaca de Juárez, Oaxaca, Mexico
Received:
6
June
2024
Accepted:
10
September
2024
Published online:
27
September
2024
We consider the more general geometrical flow in the space of metrics for three-manifolds that consists of a combination of two flows, the Cotton flow and the Ricci flow; by playing a fundamental role in the detailed balance principle of the four dimensional Hořava–Lifshitz gravity, this generalized flow reveals another difficulty with this theory that attempts to be a candidate for an UV completion of Einstein general relativity, namely, the supposed emergency of the speed of light, the Newton constant, and the cosmological constant, from a deeply nonrelativistic theory of gravity. Respecting that principle, the generalized flow shows the proliferation of different limits of the theory with an unwanted behavior at both the IR and UV regimes.
© The Author(s) 2024
Open Access This article is licensed under a Creative Commons Attribution 4.0 International License, which permits use, sharing, adaptation, distribution and reproduction in any medium or format, as long as you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons licence, and indicate if changes were made. The images or other third party material in this article are included in the article’s Creative Commons licence, unless indicated otherwise in a credit line to the material. If material is not included in the article’s Creative Commons licence and your intended use is not permitted by statutory regulation or exceeds the permitted use, you will need to obtain permission directly from the copyright holder. To view a copy of this licence, visit http://creativecommons.org/licenses/by/4.0/.
Funded by SCOAP3.