Black hole interior mass formula

We argue by explicit computations that, although the area product, horizon radii product, entropy product, and irreducible mass product of the event horizon and Cauchy horizon are universal, the surface gravity product, the surface temperature product and the Komar energy product of the said horizons do not seem to be universal for Kerr–Newman black hole spacetimes. We show the black hole mass formula on the Cauchy horizon following the seminal work by Smarr [Phys Rev Lett 30:71 (1973), Phys Rev D 7:289 (1973)] for the outer horizon. We also prescribe the four laws of black hole mechanics for the inner horizon. A new definition of the extremal limit of a black hole is discussed.