Researchers Resolve Schwarzschild Singularity Using Unimodular Time and Discover Theories Allowing Only One Mass Sign

The enigmatic singularities at the heart of black holes represent a fundamental challenge to our understanding of gravity, signalling the breakdown of classical theory, but new research offers a potential resolution. Sofie Ried from the University of Sheffield, and colleagues, demonstrate a method for resolving the singularity within Schwarzschild-(Anti-)de Sitter black holes by enforcing the principles of quantum mechanics with respect to a specific type of time measurement. The team’s approach, utilising a particular mathematical framework, transforms the complex equations governing black holes into a more manageable form resembling the familiar Schrödinger equation, allowing them to explore scenarios where the singularity disappears. This work reveals a family of theoretical models where black holes transition into white holes, governed by a new length scale, and suggests that these exotic objects may only exist with a single, defined mass, offering a significant step towards reconciling quantum mechanics and general relativity.

Canonical quantization of unimodular gravity, performed on a symmetry-reduced black hole model, yields a Wheeler-DeWitt equation that effectively behaves as a Schrödinger equation in unimodular time. Imposing unitarity reveals a family of quantum theories capable of resolving the classical singularity, each permitting only semiclassical states corresponding to a single mass sign, either positive, negative, or zero. Furthermore, the research derives an analytical expression for the quantum-corrected Schwarzschild metric, which is modified by a new length scale, rmin, governing the black hole’s transition to a white hole.

Unimodular Gravity and Quantum-Corrected Trajectories

In black holes, classical general relativity breaks down as the predictability of infalling objects is lost, and quantum gravitational effects might resolve singularities by ensuring unitary time evolution. This work explores how imposing unitarity can extend trajectories beyond classical singularities, focusing on relational dynamics where a degree of freedom serves as a clock for evolution while maintaining gauge invariance. The theoretical framework begins with unimodular gravity, a formulation classically equivalent to general relativity, by promoting the cosmological constant to a field and adding a Lagrange multiplier, introducing new degrees of freedom related to the determinant of the metric. This links the cosmological constant to the conjugate momentum of a field, identified with unimodular time, defined as the integral of a field over a hypersurface.

To simplify the quantisation process, the research utilises symmetries of the black hole spacetime, reducing the degrees of freedom and allowing for the calculation of the Hamiltonian, which is then canonically quantised. This leads to the Wheeler-DeWitt equation, formulated as a Schrödinger equation in unimodular time, with expectation values of operators computed at a fixed unimodular time, yielding gauge-invariant expressions. The analysis employs semiclassical states to explore the quantum system, characterised by expectation values and variances of relevant operators, and the self-adjoint extension parameter arises when seeking self-adjoint extensions of the Hamiltonian, introducing a one-parameter family of quantum theories. Remarkably, the singularity is resolved in all quantum theories, as the expectation value of the scale factor remains finite where the classical solution diverges.

A key result is the link between the semiclassical black hole mass and the self-adjoint extension parameter, ensuring all semiclassical states share the same mass sign, avoiding vacuum instability. By replacing metric components with their expectation values, a quantum-corrected Schwarzschild-(Anti-)de Sitter metric is derived, exhibiting corrections to the classical Schwarzschild metric, governed by the new length scale, rmin, representing the transition from a black hole to a white hole geometry. The research provides analytic expressions for expectation values, yielding a Schwarzschild metric with explicit quantum corrections, and deviations from the classical metric are controlled by the minimal length scale rmin.

Singularity Resolution via Unimodular Time Evolution

Researchers have successfully resolved the singularity at the center of black holes by imposing the principle of unitary evolution with respect to unimodular time. Employing a specific mathematical formulation, the team performed a quantum analysis on a simplified black hole model, effectively transforming the governing equations into a form resembling the Schrödinger equation. This allowed them to discover a family of theories where the classical singularity disappears, yielding finite and physically meaningful results, and these theories constrain all possible states to possess either positive, negative, or zero mass, avoiding instabilities. The analysis yields an analytical expression for a quantum-corrected Schwarzschild metric, describing the spacetime around the black hole, modified by a new length scale, rmin, governing the transition between a black hole and a white hole.

The team demonstrated that, regardless of the specific quantum parameters chosen, the singularity is consistently resolved, with the scale factor remaining finite where the classical solution diverges. Furthermore, the research establishes a direct link between the mass of the semiclassical black hole and a parameter defining the quantum theory, ensuring all states share the same mass sign. The quantum-corrected Schwarzschild metric reveals deviations from the classical theory, controlled by the minimum radius rmin. For large distances, the metric’s corrections can be approximated, demonstrating the influence of quantum effects on the spacetime geometry. This minimum radius represents the scale at which the black hole transitions to a white hole, offering a potential pathway to understanding the ultimate fate of these enigmatic objects.

Quantum Gravity Resolves Black Hole Singularities

This research demonstrates a resolution to the singularity at the centre of black holes by applying the principles of unitary evolution within the framework of unimodular time. By performing a canonical quantization on a simplified black hole model, the team found that imposing unitarity leads to theories where the classical singularity disappears, restricting semiclassical states to having a single mass sign, positive, negative, or zero. The resulting metric incorporates quantum corrections and introduces a new length scale that governs the transition from a black hole to a white hole geometry. While the study utilizes a symmetry-reduced model, it provides a pathway towards understanding how quantum effects might resolve the problematic singularities predicted by classical general relativity.