Generalized Uncertainty Principle effects in the Ho\v{r}ava-Lifshitz quantum theory of gravity

TL;DR

This paper investigates the effects of a generalized uncertainty principle on the Wheeler-DeWitt equation within Hořava-Lifshitz gravity, providing analytic solutions in specific limits and exploring the implications for quantum cosmology.

Contribution

It introduces a novel approach to incorporate the generalized uncertainty principle into Hořava-Lifshitz quantum gravity and derives approximate solutions in different regimes.

Findings

Analytic solutions in infrared and ultraviolet limits

Oscillatory behavior possible under certain conditions

Modified Wheeler-DeWitt equation incorporating GUP effects

Abstract

The Wheeler-DeWitt equation for a Kantowski-Sachs metric in Ho\v{r}ava-Lifshitz gravity with a set of coordinates in minisuperspace that obey a generalized uncertainty principle is studied. We first study the equation coming from a set of coordinates that obey the usual uncertainty principle and find analytic solutions in the infrared as well as a particular ultraviolet limit that allows us to find the solution found in Ho\v{r}ava-Lifshitz gravity with projectability and with detailed balance but now as an approximation of the theory without detailed balance. We then consider the coordinates that obey the generalized uncertainty principle by modifying the previous equation using the relations between both sets of coordinates. We describe two possible ways to obtain the Wheeler-DeWitt equation. One of them is useful to present the general equation but it is found to be very difficult to solve. Then we use the other proposal to study the limiting cases considered before, that is, the infrared limit that can be compared to the equation obtained by using general relativity and the particular ultraviolet limit. For the second limit we use a ultraviolet approximation and then solve analytically the resulting equation. We find that oscillatory behaviour is possible to be found but it is not a general feature for any values of the parameters involved.