# $AdS_2$ dilaton gravity from reductions of some nonrelativistic theories

## TL;DR

This paper derives $AdS_2$ dilaton-gravity backgrounds from reductions of higher-dimensional nonrelativistic theories, analyzing fluctuations and linking Schwarzian action coefficients to black brane entropy.

## Contribution

It demonstrates how $AdS_2$ backgrounds emerge from nonrelativistic higher-dimensional theories and relates fluctuations to black brane entropy.

## Key findings

- $AdS_2$ backgrounds obtained from nonrelativistic reductions
- Fluctuation analysis links Schwarzian coefficient to entropy
- Relates relativistic and nonrelativistic black brane theories

## Abstract

We study dilaton-gravity theories in 2-dimensions obtained by dimensional reduction of higher dimensional nonrelativistic theories. Focussing on certain families of extremal charged hyperscaling violating Lifshitz black branes in Einstein-Maxwell-scalar theories with an extra gauge field in 4-dimensions, we obtain $AdS_2$ backgrounds in the near horizon throats. We argue that these backgrounds can be obtained in equivalent theories of 2-dim dilaton-gravity with an extra scalar, descending from the higher dimensional scalar, and an interaction potential with the dilaton. A simple subcase here is the relativistic black brane in Einstein-Maxwell theory. We then study linearized fluctuations of the metric, dilaton and the extra scalar about these $AdS_2$ backgrounds. The coefficient of the leading Schwarzian derivative term is proportional to the entropy of the (compactified) extremal black branes.