Research

Euler’s Homogenous Function Theorem

Homogenous Functions A homogenous function of order $n$ satisfies: $$ f(\lambda x_1, \lambda x_2, \ldots, \lambda x_m) =\lambda^n f(x_1, x_2, \ldots, x_m) $$ For e.g. $$ f(x,y) = \sqrt{x} y^2 $$ is homogenous of order $n = 3/2$ . However: $$ f(x,y) = \sqrt x y^2 + x^2 y^2 $$ is not a homogenous function… Continue reading Euler’s Homogenous Function Theorem

Research

Euler’s Theorem and the Smarr Relation

The area of a charged rotating (Kerr) black hole is given by \begin{equation} \label{eqn:kerr-area-relation} A = 4\pi \left[ 2 M^2 + 2 (M^4 – L^2 – M^2 Q^2)^{1/2} – Q^2 \right] \end{equation} This relation can be inverted to express the mass $M$ as a function of charge $Q$, area $A$ and angular momentum $L$: \begin{equation}… Continue reading Euler’s Theorem and the Smarr Relation