# Inflying perspectives of reduced phase space

## TL;DR

This paper explores how coordinate choices affect the quantization of a self-gravitating shell model, introducing 'inflying' coordinates that interpolate between known coordinate systems, and discusses implications for quantum gravity observer roles.

## Contribution

It extends previous analysis by developing a reduced phase space with residual coordinate freedom using inflying coordinates, enabling study of coordinate effects in quantum gravity.

## Key findings

- Coordinate freedom can be retained in reduced phase space.
- Introduction of 'inflying' coordinates allows arbitrary boosting between coordinate systems.
- Discussion on the utility of reduced systems for understanding observer roles in quantum gravity.

## Abstract

There is widespread disagreement about how the general covariance of a theory affects its quantization. Without a complete quantum theory of gravity, one can examine quantum consequences of coordinate choices only in highly idealized `toy' models. In this work, we extend our previous analysis of a self-gravitating shell model [1], and demonstrate that coordinate freedom can be retained in a reduced phase space description of the system. We first consider a family of coordinate systems discussed by Martel and Poisson [2], which have time coordinates that coincide with the proper times of ingoing and outgoing geodesics (for concreteness, we only consider the former). Included in this family are Painlev\'e-Gullstrand coordinates, related to a network of infalling observers that are asymptotically at rest, and Eddington-Finkelstein coordinates, related to a network of infalling observers that travel at the speed of light. We then introduce "inflying" coordinates - a hybrid coordinate system that allows the infalling observers to be arbitrarily boosted from one member of the aforementioned family to another. We perform a phase space reduction using inflying coordinates with an unspecified boosting function, resulting in a reduced theory with residual coordinate freedom. Finally, we discuss quantization, and comment on the utility of the reduced system for the study of coordinate effects and the role of observers in quantum gravity.