The well-known shear viscosity to entropy density ratio ($\eta /s$) cannot be computed when the black hole space-time has zero thermodynamic entropy. This is the case, for example, when general relativity in four dimensions is complemented with critical gravity, or in particular scenarios within the four-dimensional scalar Gauss-Bonnet theories. Recently, it has been shown that the zero-entropy situation can be overcome in these examples by introducing suitable matter fields. With this at hand, in this paper we analyze each case and the impact of these extra sources in the ratio, in terms of their new parameters. We find that while the $\eta /s$ ratio remains constant and insensitive in the former, this is not the case for the latter. To perform the calculations, we construct a Noether charge using a spacelike Killing vector. The accuracy of the aforementioned findings is supported by the Kubo formalism.

• Received 21 March 2023
• Accepted 27 April 2023

DOI:https://doi.org/10.1103/PhysRevD.107.104032

Published by the American Physical Society under the terms of the Creative Commons Attribution 4.0 International license. Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Funded by SCOAP³.

Published by the American Physical Society