A generalization of the model of Hawking radiatio with modified high frequency dispersion relation

The existence of Hawking radiation is closely related to the fact that a particle which marginally escapes from collapsing into a black hole is suffered from infinite redshift. In other words, the particle observed in the future infinity had a very high frequency when it was near the event horizon. Motivated by the possibility that the new physics which may exist beyond some critical scale may alter the property of Hawking radiation, Unruh proposed a model with higher order spatial derivative terms. In his model, the effects of unknown physics are modeled so as to be suppressed for the waves with a wavelength much longer than the critical scale, $k_0^{-1}$. Surprisingly, it was shown that the thermal spectrum is recovered for such modified models. To introduce such higher order spatial derivative terms, the Lorentz invariance must be violated because one special spatial direction needs to be chosen. In previous works, the rest frame of freely-falling observers was employed as this special reference frame. Here we give an extension by allowing a more general choice of the reference frame. Developing the method taken by Corley, we show that the resulting spectrum of created particles again becomes the thermal one at the Hawking temperature even if the choice of the reference frame is generalized. Using the technique of the matched asymptotic expansion, we also show that the correction to the thermal radiation stays of order $k_0^{-2}$ or smaller when the spectrum of radiated particle around its peak is concerned.