# Time-dependent $NAdS_2$ holography with applications

## TL;DR

This paper develops a method for exact time-dependent solutions in $NAdS_2$ holography, explores holographic quenches, and introduces a semi-holographic impurity model with energy transfer dynamics and black hole mass evolution.

## Contribution

It presents a novel exact solution approach for time-dependent $NAdS_2$ holography and proposes a semi-holographic impurity model with conserved energy and dynamic black hole mass behavior.

## Key findings

- Black hole mass increases during holographic quenches.
- The impurity's energy is conserved and transferred during evolution.
- Black hole mass can change sign depending on initial velocity.

## Abstract

We develop a method for obtaining exact time-dependent solutions in Jackiw-Teitelboim gravity coupled to non-conformal matter and study consequences for $NAdS_2$ holography. We study holographic quenches in which we find that the black hole mass increases. A semi-holographic model composed of an infrared $NAdS_2$ holographic sector representing the mutual strong interactions of trapped impurities confined at a spatial point is proposed. The holographic sector couples to the position of a displaced impurity acting as a self-consistent boundary source. This effective $0+1-$dimensional description has a total conserved energy. Irrespective of the initial velocity of the particle, the black hole mass initially increases, but after the horizon runs away to infinity in the physical patch, the mass vanishes in the long run. The total energy is completely transferred to the kinetic energy or the self-consistent confining potential energy of the impurity. For initial velocities below a critical value determined by the mutual coupling, the black hole mass changes sign in finite time. Above this critical velocity, the initial condition of the particle can be retrieved from the $SL(2,R)$ invariant exponent that governs the exponential growth of the bulk gravitational $SL(2,R)$ charges at late time.