Stochastic Quantum Mechanics and Black Hole Geometry: A New Approach to Painlevé–Gullstrand Coordinates

Stochastic Quantum Mechanics and Black Hole Geometry: A New Approach to Painlevé–Gullstrand Coordinates

Introduction:
​Black holes have long been a source of mystery and fascination in both physics and philosophy. Their dense gravitational fields warp spacetime to such extremes that the rules of classical physics often fail to provide a clear picture. The event horizon, where the nature of space and time becomes deeply distorted, presents a particularly difficult challenge for understanding both the classical and quantum worlds. In this article, we explore the intersection of quantum mechanics, stochastic processes, and black hole geometry through the lens of Painlevé–Gullstrand coordinates and Jacob Barandes' recent work on unistochastic processes. By bridging these concepts, we aim to uncover new ways of thinking about black hole singularities, quantum gravity, and the fundamental structure of reality.
Painlevé–Gullstrand Coordinates: A Classical PerspectivePainlevé–Gullstrand coordinates offer a significant advance in the study of black holes. Unlike traditional Schwarzschild coordinates, which become singular at the event horizon, GP coordinates describe a smooth and well-behaved passage through the event horizon. These coordinates are particularly useful for understanding the free-falling observer or "raindrop," an object that falls into the black hole, starting from rest at infinity.
In these coordinates, the spacetime around a black hole can be described without the singularities encountered in other coordinate systems. For instance, the event horizon, the point beyond which not even light can escape, appears as a regular coordinate point in the GP system, allowing us to study the internal structure of a black hole without encountering undefined values.
Quantum Mechanics and the GP Coordinates: Discontinuities and Quantum GravityAs we introduce quantum mechanics into the black hole scenario, things get more complex. Classical theories like general relativity describe the smooth geometry of a black hole using coordinates such as the Painlevé–Gullstrand system. However, when quantum effects are included, they introduce a set of discontinuities into the otherwise smooth flow of spacetime.
Recent studies by researchers such as Fazzini, Rovelli, and Soltani have suggested that quantum gravity may cause discontinuities in the evolution of the GP coordinates as we approach the event horizon. These discontinuities do not necessarily represent a physical singularity but may instead reflect the limitations of the classical coordinate system when quantum effects come into play. In quantum gravity, the spacetime geometry near the event horizon could be governed by more complex, stochastic dynamics than classical models can account for.
Jacob Barandes and the Unistochastic ApproachJacob Barandes, a physicist at Harvard University, has proposed a unistochastic reformulation of quantum mechanics, which offers a fresh perspective on how to understand quantum systems, including those near extreme conditions like black holes. The term unistochastic refers to a form of randomness in quantum evolution that proceeds in a single, directed probabilistic step rather than a complex web of conflicting possibilities.
Barandes' unistochastic processes aim to simplify quantum dynamics by introducing directed randomness—where the quantum system evolves probabilistically but in a manner that is more transparent and consistent with classical understanding. This approach seeks to provide a realist, local, and probabilistic interpretation of quantum systems without invoking the often controversial concepts like wave-function collapse or non-locality.
Stochastic Processes in Black Holes: Bridging Classical and Quantum RealitiesBy incorporating unistochastic processes into black hole physics, we can begin to reframe the understanding of black holes from both a quantum and classical perspective. Painlevé–Gullstrand coordinates describe the black hole in a smooth, continuous manner for classical objects. However, when quantum mechanics is considered, stochastic fluctuations could cause deviations from this smooth behavior, potentially explaining the observed discontinuities near the event horizon.
In this framework:

• Stochastic quantum effects near the event horizon may introduce uncertainty in the smoothness of spacetime, leading to a more nuanced understanding of black hole dynamics.
• Unistochastic processes could help describe how quantum objects (including particles and light) behave as they interact with the intense gravitational field of the black hole, accounting for both the probabilistic nature of quantum states and the smooth geometry of the black hole's classical structure.

What Does This Mean for Our Understanding of Black Holes?Integrating stochastic processes into the study of black holes allows us to consider both probabilistic and deterministic elements of black hole physics. If the evolution of quantum systems inside the black hole is stochastic, as Barandes proposes, this could provide a pathway to resolving some of the paradoxes and discontinuities that arise in the classical models. For instance, the smooth flow through the event horizon in GP coordinates could be understood as the result of quantum fluctuations that are random, yet directed by the system’s underlying dynamics.
This insight could provide a better framework for understanding quantum gravity, potentially leading to a deeper understanding of the connection between the classical geometry of black holes and the quantum processes that might govern their interior.

Conclusion: Toward a New Theory of Black Hole Quantum GravityThe integration of quantum mechanics, stochastic processes, and black hole physics offers a promising new avenue of exploration in understanding the fundamental structure of the universe. Barandes’ unistochastic approach, combined with the smooth geometry provided by Painlevé–Gullstrand coordinates, could offer a unified framework that bridges the gap between classical general relativity and the uncertainty inherent in quantum mechanics.
As we continue to explore the nature of black holes and quantum gravity, the interplay between these classical and quantum perspectives will undoubtedly lead to new insights. By adopting a stochastic view of quantum systems in extreme environments like black holes, we may move closer to resolving the age-old mysteries of the event horizon, singularities, and the nature of spacetime itself.