Gibbs–Duhem equation for Kerr–Newman black holes

Abstract We show that the intensive variables associated with the outer and inner horizons of the Kerr–Newman black hole: absolute temperatures, angular velocities and electromagnetic potentials, obey a Gibbs–Duhem type relation, which is closely related, but however distinct, to the Smarr equation. Thus, analogously as in ordinary thermodynamics, $$T_{\pm }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>T</mml:mi> <mml:mo>±</mml:mo> </mml:msub> </mml:math> , $$\Omega _{\pm }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Ω</mml:mi> <mml:mo>±</mml:mo> </mml:msub> </mml:math> and $$\Phi _{\pm }$$ <mml:math xmlns:mml="http://www.w3.org/1998/Math/MathML"> <mml:msub> <mml:mi>Φ</mml:mi> <mml:mo>±</mml:mo> </mml:msub> </mml:math> are not independent quantities. The relation also holds in the Kerr and Reissner–Nordström cases, while in the Schwarzschild case it is trivially fulfilled due to the zeroth law of black hole mechanics.