Research

The Planck Scale May Be Closer Than It Appears

Objects in the rearview mirror may be closer than they seem. This statutory warning is familiar to anyone who has ever ridden in a passenger vehicle. Its intent is to warn the driver and the passengers to beĀ careful while backing up in case the car collides with something which is closer than it actually seemsĀ in… Continue reading The Planck Scale May Be Closer Than It Appears

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