Gravitational Wave Generation via the Einstein-Langevin Equation

TL;DR

This work probes whether graviton fluctuations confined within a contracting hollow mass-shell can generate gravitational waves in a stochastic, Brownian manner. By applying the Einstein-Langevin equation to the interior and discretizing with a forward-Euler scheme, the authors derive a quanta-dissipation kernel that scales as $K_3\propto V^{-1}$ and construct an effective damping framework accompanied by Gaussian noise, yielding a Wiener-process-like evolution for graviton fluctuations. Analytical steps—including renormalization of divergences and a Kerr/Laplace-Beltrami-inspired energy functional—lead to a tractable equation for the conformal-time dynamics of the system and a final Langevin-like equation for the metric perturbation, $\dot a$, whose behavior mirrors the increasing GW amplitude toward merger. Numerical results in a Mathematica implementation show GW-like chirp features emerging from stochastic interior dynamics, while the discussion situates the approach within gravitational EFTs and outlines avenues for refining the noise model and cross-section analyses. Overall, the paper provides a computational framework linking microscopic graviton fluctuations to macroscopic GW signals, offering a fresh stochastic perspective on GW generation and its connections to quantum gravity concepts.

Abstract

Detections of gravitational waves (GWs) since GW150914 has gained a contemporary interest in a potential quantum-classical correspondence between GWs and hypothetical gravitons. One such correspondence theory is stochastic gravity, whereby graviton fluctuations are treated as the stochastic noise embedded in globally-flat manifolds and local gravitational interactions. Utilizing the Einstein-Langevin equation that describes graviton fluctuations, in attempt to form a correlation with GW generation, we utilize the hollow mass-shell model of coalescing compact binaries. This is to explore the second Newtonian postulate of neutralized internal gravitational fields, i.e. the stochastic noise of an enclosed, internal Minkowski manifold. This stochatic picture of GW formation implies the treatment of the enclosed gravitons as a Brownian bath. From the Einstein-Langevin equation, we establish a scaling relation where quanta dissipation depends inversely with the contracting volume (i.e., emission increases during coalescence). Using an Euler iteration scheme, we simulate the graviton fluctuations from inspiral to merger as a Wiener process, revealing a signal that qualitatively resembles macroscopic GW waveforms. While inherently heuristic and phenomenological, this approach provides a computational framework for exploring graviton-scale perturbations in GW formation. We discuss furthermore analytical waveform matching with the iteration scheme, as well as the justification of a Brownian analogy amidst current and state-of-the-art effective field theory frameworks.

Figures (1)

• Figure 1: A graviton fluctuation simulation under the Gaussian distribution. Fluctuations in the "strain" (i.e. the unitless $x(\tau)/\gamma^*$ in blue) are shown under two plotting scales: panel (a) plots the iterations in the log-linear scale, and panel (b) plots the same iterations in the default linear-linear scale.