Schr\"odinger Symmetry in Spherically-symmetric Static Mini-superspaces with Matter Fields

TL;DR

This paper demonstrates the emergence of Schr"odinger symmetry in classical mini-superspace models with matter fields, revealing its robustness and potential implications for quantum gravity and matter dynamics.

Contribution

It develops a canonical transformation method to show Schr"odinger symmetry in spherically-symmetric static mini-superspaces with matter, and interprets the symmetry under Hamiltonian constraints.

Findings

Schr"odinger symmetry emerges in 3D and (2+n)D models with matter fields.

Solutions include (anti-) de Sitter Reissner-Nordstr"om and Janis-Newman-Winicour spacetimes.

Symmetry persists in matter decoupling limit, indicating covariance.

Abstract

Schr\"odinger symmetry emerged in a ``fluid limit" from a full superspace to several mini-superspace models. We consider two spherically-symmetric static mini-superspace models with matter fields and verify the robustness of this emergent symmetry at the classical level: (i) Maxwell field with cosmological constant and (ii) $n$ massless scalar fields. We develop a method based on canonical transformations and show that: for model (i), 3D Schr\"odinger symmetry emerges, and the solution is the (anti-) de Sitter Reissner-Nordstr\"om spacetime, and for model (ii), $(2+n)$D Schr\"odinger symmetry appears, and the solution is a generalized Janis-Newman-Winicour spacetime and its ``interior", a Kantowski-Sachs type closed universe. In the matter decoupling limit, both cases lead to 2D Schr\"odinger symmetry in different lapse functions and mini-superspace coordinates, which implies the covariance of Schr\"odinger symmetry. Finally, we propose a physical interpretation of the symmetry under Hamiltonian constraints $H$ and explain it with examples: Symmetry generators commuting with $H$ map a solution to another one, while those non-commuting with $H$ generate a new theory with the Schr\"odinger symmetry and the transformed configuration is a solution to the new theory. These support the robustness of the emergence of Schr\"odinger symmetry and open new possibilities for exploring quantum dynamics of matter and gravity based on the symmetry.