Universality of tunnelling particles in Hawking radiation

The complex path (or Hamilton-Jacobi) approach to Hawking radiation corresponds to the intuitive picture of particles tunnelling through the horizon and forming a thermal radiation. This method computes the tunnelling rate of a given particle from its equation of motion and equates it to the Boltzmann distribution of the radiation from which the Hawking temperature is identified. In agreement with the original derivation by Hawking and the other approaches, it has been checked case by case that the temperature is indeed universal for a number of backgrounds and the tunnelling of particles from spins $0$ to $1$ mostly, spins $3/2$ and $2$ in some instances. In this letter, we give a general proof that the temperature is indeed equal for all (massless and massive) particles with spins from $0$ to $2$ on an arbitrary background (limited to be Einstein for spin greater than $1$) in any number of dimensions. Moreover, we propose a general argument to extend this result to any spin greater than $2$.