Gravitational tunneling in Lorentz violating gravity

Black holes in Lorentz violating gravity, such as Einstein-Aether or Hořava-Lifshitz gravity, are drastically different from their general relativistic siblings. Although they allow for superluminal motion in their vicinity, they still exhibit an absolute causal boundary in the form of a universal horizon. By working in the tunneling picture for a gravitating scalar field, we show that universal horizons emit Hawking radiation in a manner akin to standard results in general relativity, with a temperature controlled by the high-energy behavior of the dispersion relation of the gravitating field, and in agreement with alternative derivations in the literature. Our results substantiate the link between the universal horizon and thermodynamics in Lorentz violating theories.

Figure 1

The Killing horizon is displayed as a null line that separates the normal (unshaded) from the trapped region (gray shaded). The tunneling path shows a positive energy particle that starts in the interior on a past-directed outgoing null path, crosses the horizon on a complex path (dashed), and scatters into a future-directed outgoing null path, once it has crossed the horizon. This is equivalent to the negative energy particle tunneling inwards.

Figure 2

Structure of the modes. Close to the UH and outside of it we have four modes—ingoing ${\varphi }_{\mathrm{in}}$, outgoing ${\varphi }_{\mathrm{out}}$, and the two tails of the trapped one ${\varphi }_{\mathrm{tr}}$. However, the latter is indistinguishable from the former in the region $r\sim r_{\mathrm{UH}}$. Note that the mode ${\varphi }_{\mathrm{p}}$ is the Hawking partner to the red mode. The UH is denoted by the purple line, while the Killing horizon is shown as a dotted line. The zigzag line on the left depicts the singularity.