Stochastic Quantum Gravity

TL;DR

This work develops a covariant stochastic quantum-mechanics framework in curved spacetime and applies it to the Schwarzschild background. By linking a stochastic 4-velocity formalism with the Klein–Gordon equation through Madelung and Cole–Hopf transformations, it derives forward/backward stochastic differential equations and identifies two velocities that govern quantum trajectories. In Schwarzschild spacetime, massless scalar fields reduce to a radial equation of confluent Heun form, with solutions expressed via confluent Heun functions, which feed into the stochastic dynamics. Numerical simulations show that stochastic fluctuations influence trajectories near the horizon, with higher particle frequency yielding more deterministic paths and short-time behavior preserving near-geodesic dynamics, suggesting a viable route to explore quantum effects in curved spacetime and motivating extensions to more general geometries.

Abstract

This work explores the possibility of applying stochastic quantum mechanics to curved spacetimes, with an emphasis on the Schwarzschild black hole. After reviewing the fundamental concepts of this approach, the quantum stochastic equations are extended to curved spacetime using a fully covariant treatment. Subsequently, the Klein-Gordon equation is solved for scalar perturbations, and the resulting stochastic trajectories are analyzed by varying parameters such as angular momentum, particle frequency, and computational integration time. In conclusion, we find that the trajectories are influenced by gravitational fluctuations in spacetime and that, depending on the variation of the fundamental parameters, different types of stochastic trajectories are obtained.

Figures (3)

• Figure 1: Multipanel visualization of the stochastic trajectories corresponding to the angular momentum values $l=1,2,3$. Top row: Radial trajectories $r(\tau)$ (solid red) and time trajectories $v(\tau)$ (dashed blue) as a function of the affine parameter $\tau$, along with the event horizon $r=1$ (black horizontal line) and the first horizon crossing marker (green dot). Middle row: Parametric plots $r(\tau)$ versus $v(\tau)$, showing the coupled stochastic evolution of both variables for each value of $l$. Bottom row: Radial coordinate $r$ as a function of time $t$, illustrating how all trajectories asymptotically approach the event horizon, depending on the angular momentum. In the middle and bottom rows, the trajectories are represented with different colors, and the event horizon $r=1$ (black horizontal line) is included again. Each graph contains seven realizations of the stochastic trajectories.
• Figure 2: Multipanel visualization of the stochastic trajectories corresponding to the frequency values $\omega_{0}=1000, 35, 1$. Top row: Radial trajectories $r(\tau)$ (solid red) and time trajectories $v(\tau)$ (dashed blue) as a function of the affine parameter $\tau$, along with the event horizon $r=1$ (black horizontal line) and the first horizon crossing marker (green dot). Middle row: Parametric plots $r(\tau)$ versus $v(\tau)$, showing the coupled stochastic evolution of both variables for each value of $\omega_{0}$. Bottom row: Radial coordinate $r$ as a function of time $t$, illustrating how all trajectories asymptotically approach the event horizon, depending on the angular momentum. In the middle and bottom rows, the trajectories are represented with different colors, and the event horizon $r=1$ (black horizontal line) is included again. Each graph contains seven realizations of the stochastic trajectories.
• Figure 3: Visualization of coordinates, $r(\tau)$ and $v(\tau),$ evolution for short times, the corresponding parameters are $\omega_0=35$, the initial point is $r(0)=1.05r_s$. The event horizon is located at $r=1,$ the times goes from $0$ to $0.03$.