Accelerated Particle Detectors with Modified Dispersion Relations

There is increasing interest in discrete or "pixelated" spacetime models as a foundation for a satisfactory theory of quantum gravity. If spacetime possesses a cellular structure, there should be observable consequences: for example, the vacuum becomes a dispersive medium. Of obvious interest are the implications for the thermodynamic properties of quantum black holes. As a first step to investigating that topic, we present here a calculation of the response of a uniformly accelerating particle detector in the (modified) quantum vacuum of a background pixelated spacetime, which is well known to mimic some features of the Hawking effect. To investigate the detector response we use the standard DeWitt treatment, with a two-point function modified to incorporate the dispersion. We use dispersion relations taken from the so-called doubly special relativity (DSR) and Ho\v{r}ava-Lifshitz gravity. We find that the correction terms retain the Planckian nature of particle detection, but only for propagation faster than the speed of light, a possibility that arises in this treatment because the dispersion relations violate Lorentz invariance. A fully Lorentz-invariant theory requires additional features; however, we believe the thermal response will be preserved in the more elaborate treatment.