Generation and purification of excited spacetimes using Schwarzian derivative

In this article, we use the expression of the Schwarzian derivative to set up differential equations to find answers to three fundamental questions in the context of QFT in curved spacetime, specifically in two dimensions.

One of the ways in which one can derive the Unruh effect in two dimensions is to use the anomalous transformation law of the energy-momentum tensor for a CFT that involves a Schwarzian derivative (Virasoro Anomaly).

We answer the following three questions.

The first question is as follows: If we have a spacetime with a massless scalar field in vacuum, what are all the subsets of spacetime such that the subset has a thermal distribution of particles for the left-moving and/or right-moving sectors?

We obtain a general solution to this question by setting up and solving a third-order nonlinear differential equation based on the expression of Schwarzian.

Based on the general solution, we can generate various subsets of the given spacetime that have a thermal flux/density of particles, of which the Rindler spacetime is one.

The second question is an inverse question in which we suppose we are given a spacetime with a thermal distribution of particles; what are the possible purifying spacetimes (the ``parent'' spacetimes with the field in vacuum state whose reduced state in the given spacetime yields the observed particle content)?

We similarly obtain a general class of solutions by setting up and solving a second differential equation.

In this context, we also define ``partial purification'' where we obtain a spacetime that purifies only the left-moving or right-moving sector.

The third question concerns locating spacetimes with the same particle content starting from the same ``parent'' spacetime.

These sibling spacetimes are generated again by obtaining the general solution of a third differential equation based on the expression of Schwarzian.