Euler's Theorem and the Smarr Relation

The area of a charged rotating (Kerr) black hole is given by

$$\label{eqn:kerr-area-relation}\nA = 4\pi \left[ 2 M^2 + 2 (M^4 - L^2 - M^2 Q^2)^{1/2} - Q^2 \right] \\$This relation can be inverted to express the mass$M$as a function of charge$Q$, area$A$and angular momentum$L$:$\label{eqn:kerr-mass-relation}\nM = \left[ \frac{A}{16\pi} + \frac{4\pi L^2}{A} + \frac{Q^2}{2} + \frac{\pi Q^4}{A} \right]^{1/2} \\$As can be seen this is a homogenous function of the variables$(A,L,Q^2)$of degree$n = 1/2$.

From the first law of black hole thermodynamics¹, we know that the change in the mass of the black hole can be written as:$\label{eqn:kerr-first-law}\ndM = T dA + \Omega dL + \Phi dQ \\$where$T, \Omega, \Phi$are the temperature ², angular velocity and electric potential of the black hole respectively.

Now, Euler’s homogenous function theorem³ tells us that any function homogenous in its variables of degree$n$, satisfies:$\label{eqn:euler-homogenous}\nn f(\vec{x}) = \sum_{i=1}^N x_i \frac{\partial f}{\partial x_i} \\$where$\vec{x} = {x_1,x_2,\ldots,x_N}$. $\ref{eqn:euler-homogenous}$ along with $\ref{eqn:kerr-mass-relation}$, implies that: CODEXMATHTOKEN0END where we have used $\ref{eqn:kerr-first-law}$ to determine the differentials of$M$with respect to the other variables. In the third term on the r.h.s, we have used:$\nQ^2 \frac{\partial M}{\partial Q^2} = Q^2 \frac{\partial M}{\partial Q} \frac{\partial Q}{\partial Q^2} = \frac{Q}{2} \frac{\partial M}{\partial Q} \\$This gives us the Smarr formula relating the mass of a charged, rotating black hole to its charge, angular momentum and area:$\label{eqn:smarr-relation}\nM = 2AT + 2L\Omega + Q\Phi \\$$

Bardeen, Carter, Hawking; Four laws of black hole thermodynamics; CMP; 1973
Or as Smarr calls it, the “effective surface tension” of the black hole horizon.
Euler’s Homogenous Function Theorem