Schrödinger Symmetry Achieves 3D Emergence in Static Mini-superspace Models with Matter

The fundamental nature of symmetry plays a crucial role in understanding the universe, and recent research explores the emergence of Schrödinger symmetry within simplified cosmological models. Taishi Sano from Waseda University and Yuki Yokokura from the Theory Center at KEK investigate this symmetry in two distinct scenarios, one involving Maxwell fields with a cosmological constant, and another with massless scalar fields, to determine its robustness at a classical level. Their work demonstrates that both models exhibit Schrödinger symmetry, yielding solutions corresponding to well-known spacetimes, including anti-de Sitter Reissner-Nordström and generalized Janis-Newman-Winicour universes. By establishing a connection between this symmetry and Hamiltonian constraints, the researchers reveal that it not only links existing solutions but also generates entirely new theoretical frameworks, potentially offering a powerful tool for exploring the dynamics of matter and the universe itself.

They developed a method based on canonical transformations and demonstrate that, for the first model, a three-dimensional Schrödinger symmetry emerges, yielding the (anti-)de Sitter Reissner-Nordström spacetime as a solution. For the second model, a (2 + n)-dimensional Schrödinger symmetry appears, resulting in a generalized Janis, Newman, Winicour spacetime and, within it, a Kantowski-Sachs type closed universe representing its “interior”. Importantly, in the matter decoupling limit, both cases converge to 2D Schrödinger symmetry, albeit with different lapse functions and mini-superspace coordinates.

Black Holes and Cosmology Share Hidden Symmetry

This research demonstrates the emergence of Schrödinger symmetry within simplified models of gravity, revealing a deep connection between black hole and cosmological spacetimes. Scientists developed a method using canonical transformations to explore this symmetry in two distinct models: one featuring a Maxwell field with a cosmological constant and another with massless scalar fields. Results show that the Maxwell model exhibits a three-dimensional Schrödinger symmetry, yielding the anti-de Sitter Reissner-Nordström spacetime, a well-known black hole geometry. The scalar field model, conversely, produces a solution corresponding to a generalized Janis-Newman-Winicour spacetime and its interior, resembling a Kantowski-Sachs closed universe, a type of cosmological model.

Schrödinger Symmetry in Black Hole Mechanics

This research reveals a surprising link between black holes and the universe, demonstrating that both systems share a hidden symmetry known as Schrödinger symmetry. Scientists explored this symmetry using simplified models of gravity, employing a mathematical technique called canonical transformations. Their work focused on two distinct scenarios: one involving a Maxwell field with a cosmological constant and another with multiple massless scalar fields. The results show that the Maxwell model exhibits a three-dimensional Schrödinger symmetry, leading to the anti-de Sitter Reissner-Nordström spacetime, a solution describing a black hole. Conversely, the scalar field model produces a solution corresponding to a generalized Janis-Newman-Winicour spacetime and its interior, resembling a Kantowski-Sachs closed universe, a type of cosmological model.

Emergent Schrödinger Symmetry in Simplified Gravity Models

This research demonstrates the emergence of Schrödinger symmetry within simplified models of gravity, known as mini-superspace models. By applying canonical transformations to these models, specifically those incorporating a Maxwell field with a cosmological constant and multiple massless scalar fields, the team found that a three-dimensional Schrödinger symmetry arises in the first case, corresponding to the (anti-)de Sitter Reissner-Nordström spacetime, and a higher-dimensional symmetry appears in the second, relating to a generalized Janis-Newman-Winicour spacetime and a closed universe. Importantly, this symmetry appears consistently even when considering different coordinate systems and approaches to simplification, suggesting its robustness. The findings reveal that this emergent Schrödinger symmetry is not merely a mathematical artifact, but has a physical interpretation linked to the Hamiltonian constraint, a fundamental equation governing the dynamics of gravity. The researchers show that symmetry generators either transform solutions into other solutions, or generate entirely new theories possessing the Schrödinger symmetry, thereby expanding the possibilities for exploring gravitational dynamics.

The team discovered that the interior of the black hole horizon, as described by the Schwarzschild metric, directly corresponds to a Kantowski-Sachs cosmological spacetime when coordinates are appropriately transformed, suggesting a fundamental link between these seemingly disparate solutions. Furthermore, the research establishes that the scalar field model unifies the Janis-Newman-Winicour solution and the Kantowski-Sachs spacetime into a single metric, with the latter representing the interior of the former. Measurements confirm that the canonical transformation method simplifies the system’s motion to uniform linear motion on the mini-superspace, governed by a Hamiltonian constraint. This linearity allows for the superposition of states with different charges at the quantum level.

Scientists also demonstrated that the Komar mass formula, used to calculate the ADM energy of asymptotically-flat spacetimes, aligns with the mass definitions derived within the study for both Λ = 0 and ε = −1. Similarly, calculations of electric charge using the standard formula and Maxwell’s equations are consistent with the derived results. These findings open new avenues for exploring the dynamics of matter and gravity based on Schrödinger symmetry and provide a promising framework for connecting quantum black hole and quantum universe models.