Black hole solutions and Euler equation in Rastall and generalized Rastall theories of gravity

TL;DR

This paper investigates black hole solutions in generalized Rastall gravity, examining their thermodynamic properties and the validity of the Euler equation at horizons, revealing that thermodynamic pressure may differ from gravitational pressure components.

Contribution

It introduces new classes of black hole solutions in generalized Rastall theory and analyzes their thermodynamic behavior, highlighting differences between thermodynamic and gravitational pressures.

Findings

Thermodynamic pressure often differs from gravitational pressure components.

Solutions with equal thermodynamic and Hawking temperatures are identified.

The validity of the Euler equation depends on the energy definition used.

Abstract

Focusing on the special case of generalized Rastall theory, as a subclass of the non-minimal curvature-matter coupling theories in which the field equations are mathematically similar to the Einstein field equations in the presence of cosmological constant, we find two classes of black hole (BH) solutions including $i$) conformally flat solutions and $ii$) non-singular BHs. Accepting the mass function definition and by using the entropy contents of the solutions along with thermodynamic definitions of temperature and pressure, we study the validity of Euler equation on the corresponding horizons. Our results show that the thermodynamic pressure, meeting the Euler equation, is not always equal to the pressure components appeared in the gravitational field equations and satisfies the first law of thermodynamics, a result which in fact depends on the presumed energy definition. The requirements of having solutions with equal thermodynamic and Hawking temperatures are also studied. Additionally, we study the conformally flat BHs in the Rastall framework. The consequences of employing generalized Misner-Sharp mass in studying the validity of the Euler equation are also addressed.