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Evolution of Lifshitz metric anisotropies in Einstein–Proca theory under the Ricci–DeTurck flow

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Abstract

By starting from a Perelman entropy functional and considering the Ricci–DeTurck flow equations, we analyze the behavior of Einstein–Hilbert and Einstein–Proca theories with Lifshitz geometry as functions of a flow parameter. In the former case, we found one consistent fixed point that represents flat space-time as the flow parameter tends to infinity. Massive vector fields in the latter theory enrich the system under study and have the same fixed point achieved at the same rate as in the former case. The geometric flow is parametrized by the metric coefficients and represents a change in anisotropy of the geometry toward an isotropic flat space-time as the flow parameter evolves. Indeed, the flow of the Proca fields depends on certain coefficients that vanish when the flow parameter increases, rendering these fields constant. We have been able to write down the evolving Lifshitz metric solution with positive, but otherwise arbitrary, critical exponents relevant to geometries with spatially anisotropic holographic duals. We show that both the scalar curvature and matter contributions to the Ricci–DeTurck flow vanish under the flow at a fixed point consistent with flat space-time geometry. Thus, the behavior of the scalar curvature always increases, homogenizing the geometry along the flow. Moreover, the theory under study keeps positive-definite, but decreasing, the entropy functional along the Ricci–DeTurck flow.

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Data Availability Statement

Data sharing is not applicable to this article as no datasets were generated or analyzed during the current study.

Notes

  1. And in particular, Ricci flows.

  2. Like Massive vector field, supergravity and string theory [21].

  3. See for example [23] for some applications of Ricci flows in General Relativity.

  4. We mentioned in the introduction section that, within the holographic framework, the so-called critical exponents \(z_{i}\) of a Lifshitz geometry are to be related with order parameters in quantum critical points of a dual field theory. So far these order parameters are found to be positive in real field theories. Therefore, we considered the corresponding \(z_{i}>0\). However, some or all of these critical exponents can be negative, leading to a scaling that can be compensated by the dilatation of the corresponding \(x_{i}\) coordinates. Therefore, under the evolution of the Ricci flow parameter \(\lambda\), the metric tends to the same fixed point, independently of the sign of the \(\gamma _{i}\) coefficients.

  5. In principle, one can integrate Eq. (4.7a) and get linear in r expression supplemented by an exponential term for the DeTurck vector field. However, there is no way to get rid of this term in Eq. (4.7c) without trivializing the vector field. Hence, this case reduces to the one treated in Sect. 3.

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Acknowledgements

All authors have been benefited from the CONACYT Grant No. A1-S-38041, while RCF and AHA were supported by the VIEP-BUAP Grants Nos. 113 and 122. MCL thanks the financial assistance provided by a CONACYT postdoctoral Grant No. 30563. JAHM acknowledges support from CONACYT through a PhD Grant No. 750974. DFHB is also grateful to CONACYT for a Postdoc por México Grant No. 372516.

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Authors and Affiliations

  1. Instituto de Física, Benemérita Universidad Autónoma de Puebla, Apartado Postal J-48, 72570, Puebla, Mexico

    Roberto Cartas-Fuentevilla, Manuel de la Cruz, Alfredo Herrera-Aguilar, Jhony A. Herrera-Mendoza & Daniel F. Higuita-Borja

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  1. Roberto Cartas-Fuentevilla
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  2. Manuel de la Cruz
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  3. Alfredo Herrera-Aguilar
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Correspondence to Alfredo Herrera-Aguilar.

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Cartas-Fuentevilla, R., de la Cruz, M., Herrera-Aguilar, A. et al. Evolution of Lifshitz metric anisotropies in Einstein–Proca theory under the Ricci–DeTurck flow. Eur. Phys. J. Plus 137, 1379 (2022). https://doi.org/10.1140/epjp/s13360-022-03606-6

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