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

We derive a generalized Euler equation, $\epsilon+p=sT+\mu q+y\frac{\partial p}{\partial y}$, 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 $\Lambda$ as a thermodynamic variable, as proposed in certain frameworks within the literature.