Redundant and Physical Black Hole Parameters: Is there an independent physical dilaton charge?

TL;DR

This paper establishes criteria to distinguish physical black hole parameters from redundant ones, showing that dilaton moduli are redundant parameters that do not contribute to the first law of black hole thermodynamics.

Contribution

It introduces necessary conditions for parameters to be physical and demonstrates that dilaton moduli are redundant, contrasting with previous claims of scalar charges affecting the first law.

Findings

Dilaton moduli are redundant parameters for certain black holes.

Redundant parameters do not appear in the first law of thermodynamics.

Variation of redundant parameters leads to non-integrable thermodynamic quantities.

Abstract

Black holes as solutions to gravity theories, are generically identified by a set of parameters. Some of these parameters are associated with black hole physical conserved charges, like ADM charges. There can also be some "redundant parameters." We propose necessary conditions for a parameter to be physical. The conditions are essentially integrability and non-triviality of the charge variations arising from "parametric variations," variation of the solution with respect to the chosen parameters. In addition, we prove that variation of the redundant parameters which do not meet our criteria do not appear in the first law of thermodynamics. As an interesting application, we show that dilaton moduli are redundant parameters for black hole solutions to Einstein-Maxwell-(Axion)-Dilaton theories, because variations in dilaton moduli would render entropy, mass, electric charges or angular momenta non-integrable. Our results are in contrast with modification of the first law due to scalar charges suggested in Gibbons-Kallosh-Kol paper arXiv:hep-th/9607108 and its follow-ups. We also briefly discuss implications of our results for the attractor behavior of extremal black holes.