Generalized Euler Equation from Effective Action: Implications for the Smarr Formula in AdS Black Holes

TL;DR

The authors derive a generalized Euler equation for perfect fluids by extending the effective action with a geometric scale variable y, yielding ε+p = sT+μq+y ∂p/∂y. When applied to holographic fluids dual to AdS black holes, this framework reproduces the Smarr formula, providing a local, field-theoretic reinterpretation of black hole thermodynamics that does not rely on AdS/CFT as a necessity. The work further argues that treating the cosmological constant Λ as a thermodynamic variable is not physically justified within this formalism and discusses implications for extended black hole thermodynamics and central-charge interpretations. It also outlines several avenues for future work, including extending to higher-derivative terms, rotating spacetimes, and Carrollian fluid descriptions, as well as exploring connections to horizon symmetries and emergent gauge structures.

Abstract

We derive a generalized Euler equation, $ε+p=sT+μq+y\frac{\partial p}{\partial y}$, using the effective field theory formulation of perfect fluids. This generalization was achieved by introducing a new variable $y$ into the effective action, which encodes a geometrical scale of the spacetime where the fluid is on. Notably, the generalized Euler equation is independent of the AdS/CFT correspondence. However, when applied to a holographic perfect fluid, this equation naturally recovers the Smarr formula for AdS black holes, thus situating the physical interpretation of the Smarr formula within the framework of well-established physics. Finally, our findings raise important questions regarding the validity of treating the cosmological constant $Λ$ as a thermodynamic variable, as proposed in certain frameworks within the literature.