Reply to ‘‘Comment on ‘Length and curvature in the geometry of thermodynamics’ ’’

Previous authors have introduced a Riemannian surface, [(S,V),$U_{\alpha \beta }$(S,V)], and claimed that this surface could be identified with the surface U=U(S,V) for a substance in thermodynamic equilibrium. No proof has ever been offered of this supposed equivalence. The paper commented on showed that such a proof is not possible unless a new class of thermodynamic inequalities were to exist (they do not). The two preceding Comments also fail to provide a proof that these two surfaces can be identified. In the present Comment we prove once again, citing fundamental theorems of differential geometry, that this identification is not possible. We show specifically that any Riemannian metric chosen to measure distances in the equilibrium surface must lead, through the curvature tensor, to a Gaussian sectional curvature which is everywhere positive semidefinite if the convexity condition of the second law of thermodynamics is not to be violated. Two previous choices of metric do not possess this property.