An explicit covariant formula for thermodynamic volume is derived that universally decomposes into explicit Lagrangian coupling dependence plus dynamical field response contributions.
In this case, the sym- plectic currentw[δϕ,L ξϕ]≡δC[L ξϕ]− LξC[δϕ] must be added to Eq.(10)[27, 45, 46]
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Explicit and covariant formula for thermodynamic volume in extended black hole thermodynamics
An explicit covariant formula for thermodynamic volume is derived that universally decomposes into explicit Lagrangian coupling dependence plus dynamical field response contributions.