Generalized Euler equation from effective action: implications for the smarr formula in AdS black holes

A bstract We derive a generalized Euler equation, ϵ + p = sT + μq + $$ y\frac{\partial p}{\partial y} $$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:mi>y</mml:mi> <mml:mfrac> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>p</mml:mi> </mml:mrow> <mml:mrow> <mml:mi>∂</mml:mi> <mml:mi>y</mml:mi> </mml:mrow> </mml:mfrac> </mml:math> , 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.

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