Schr\"odinger-like Dilaton Gravity

TL;DR

This paper explores a Schr"odinger-like gravity model with a dynamical critical exponent of 2, incorporating a dilaton field to enable non-trivial dynamics, potentially serving as a foundation for quantum dilaton gravity and membrane quantization.

Contribution

It introduces a novel $z=2$ gravity model with a dilaton field, extending Horava gravity to include dynamical features suitable for quantum gravity applications.

Findings

Constructed a $z=2$ dispersion relation with a dilaton field.

Demonstrated the model's potential for quantum dilaton gravity.

Provided a classical framework for membrane quantization.

Abstract

We investigate possibilities for a Schr\"odinger-like gravity with the dynamical critical exponent $z=2$, where the action only contains the first-order time derivative. The Horava gravity always admits such a relevant deformation because the full $(d+1)$ dimensional diffeomorphism of the Einstein gravity is replaced by the foliation preserving diffeomorphism. The dynamics is locally trivial or topological in the pure gravity case, but we can construct a dynamical field theory with a $z=2$ dispersion relation by introducing a dilaton degree of freedom. Our model provides a classical starting point for the possible quantum dilaton gravity which may be applied to a membrane quantization.