Schrödinger Symmetry in Spherically-symmetric Static Mini-superspaces with Matter Fields

TL;DR

This work investigates the robustness of Schrödinger symmetry in gravity by focusing on two spherically-symmetric static mini-superspace models with matter: a Maxwell field with a cosmological constant and a system of $n$ massless scalars. A canonical-transformation method is developed to map the mini-superspace Hamiltonian to the free-particle form, revealing a 3D Schrödinger symmetry for the Maxwell case and a $(2+n)$-dimensional Schrödinger symmetry for the scalar case, with corresponding spacetime solutions given by AdS–RN and a generalized JNW metric (and its Kantowski–Sachs interior) respectively. In the matter decoupling limit, both systems reduce to 2D Schrödinger symmetry in different lapse choices, indicating covariance of the 2D symmetry in the mini-superspace and supporting the symmetry’s robustness. The authors further interpret Schrödinger generators under Hamiltonian constraints, showing that commuting generators map physical solutions within the same theory, while non-commuting generators can generate a new theory whose solutions are governed by an updated Hamiltonian constraint $H^{\rm New}$. These results underscore the potential universality of Schrödinger symmetry in quantum gravity’s fluid-like regimes and open avenues for exploring quantum dynamics of gravity coupled to matter via symmetry-guided constructions.

Abstract

Schrödinger 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 Schrödinger symmetry emerges, and the solution is the (anti-) de Sitter Reissner-Nordström spacetime, and for model (ii), $(2+n)$D Schrödinger 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 Schrödinger symmetry in different lapse functions and mini-superspace coordinates, which implies the covariance of Schrödinger 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 Schrödinger symmetry and the transformed configuration is a solution to the new theory. These support the robustness of the emergence of Schrödinger symmetry and open new possibilities for exploring quantum dynamics of matter and gravity based on the symmetry.

Figures (2)

• Figure 1: For the case of $\epsilon=-1$, the areal radius $r(R)^2$ (Left) and energy density $-T^t{}_t(R)$ (Right). $\mu=1.4$ and $r_0=100$.
• Figure 2: For the case of $\epsilon=+1$, the areal radius $r(R)^2$ (Left) and energy density $-T^R{}_R(R)$ (Right). $\mu=1.4$ and $r_0=100$. Here, $R$ plays a role of the time coordinate.