Thermodynamics and Luminosities of Rainbow Black Holes

Doubly special relativity (DSR) is an effective model for encoding quantum gravity in flat spacetime. As a result of the nonlinearity of the Lorentz transformation, the energy-momentum dispersion relation is modified. One simple way to import DSR to curved spacetime is \textquotedblleft Gravity's rainbow", where the spacetime background felt by a test particle would depend on its energy. Focusing on the \textquotedblleft Amelino-Camelia dispersion relation" which is $E^{2}=m^{2}+p^{2}\left[ 1-\eta\left( E/m_{p}\right) ^{n}\right]$ with $n>0$, we investigate the thermodynamical properties of a Schwarzschild black hole and a static uncharged black string for all possible values of $\eta$ and $n$ in the framework of rainbow gravity. It shows that there are non-vanishing minimum masses for these two black holes in the cases with $\eta<0$ and $n\geq2$. Considering effects of rainbow gravity on both the Hawking temperature and radius of the event horizon, we use the geometric optics approximation to compute luminosities of a 2D black hole, a Schwarzschild one and a static uncharged black string. It is found that the luminosities can be significantly suppressed or boosted depending on the values of $\eta$ and $n$.