Anomalous contribution to galactic rotation curves due to stochastic spacetime (2402.19459v3)

Abstract: We consider a proposed alternative to quantum gravity, in which the spacetime metric is treated as classical, even while matter fields remain quantum. Consistency of the theory necessarily requires that the metric evolve stochastically. Here, we show that this stochastic behaviour leads to a modification of general relativity at low accelerations. In the low acceleration regime, the variance in the acceleration produced by the gravitational field is high in comparison to that produced by the Newtonian potential, and can act as an entropic force, causing a deviation from Einstein's theory of general relativity. We show that in this "diffusion regime", the entropic force acts from a gravitational point of view, as if it were a contribution to the matter distribution. We compute modifications to the expectation value of the metric via the path integral formalism, and find an anomalous contribution which corresponds to a cosmological constant, anti-correlated with a contribution which has been used to fit galactic rotation curves without dark matter. We caution that a greater understanding of this effect is needed before conclusions can be drawn, most likely through numerical simulations, and provide a template for computing the deviation from general relativity which serves as an experimental signature of the Brownian motion of spacetime.

• The paper proposes that stochastic fluctuations in the spacetime metric yield entropic forces that alter gravity, potentially explaining galactic rotation curves without dark matter.
• Using path integral methods, the authors compute effective contributions that mimic a small cosmological constant and produce MOND-like behavior at low accelerations.
• The study provides a testable framework that reconciles gravitational discrepancies and suggests new experimental targets for astrophysical observations.

Implications of Stochastic Spacetime on Galactic Rotation Curves

The paper by Oppenheim and Russo challenges conventional understanding of gravity by proposing that spacetime may inherently possess stochastic, rather than deterministic, dynamics. Their hypothesis stems from an alternative approach to quantum gravity where the spacetime metric is treated as classical, but evolving stochastically, while matter fields remain quantum. This conceptual shift leads to significant modifications in the classical theory of general relativity, particularly in regimes of low acceleration.

Theoretical Foundation

The authors present a framework where stochastic behaviors in the spacetime metric can induce an entropic force significant enough to alter traditional gravitational predictions at low acceleration scales. In this framework, the variance in acceleration due to gravitational fields can manifest as an effective matter distribution. Through the path integral formalism, they compute expectation values of the metric and identify stochastic contributions akin to a cosmological constant. Interestingly, these contributions show an inverse correlation with a term historically proposed as a fit for galactic rotation curves without invoking dark matter.

Numerical Insights

Key numerical insights from the paper reveal that even when a bare cosmological constant is set to zero, stochastic fluctuations lead to the emergence of a small cosmological constant term. The paper demonstrates that these fluctuations can induce a gravitational force in line with Modified Newtonian Dynamics (MOND) behaviors. The findings correlate with observational coincidences such as Milgrom's observation relating the MOND acceleration scale to the cosmological constant, albeit the theory does not precisely predict these coincidences.

Practical & Theoretical Implications

The implications of this work are manifold:

1. Reinterpretation of Dark Matter: By offering an explanation for galactic rotation curves without relying on dark matter, the paper suggests paths for reconciling galactic dynamics through modifications of gravity rather than introducing unseen matter constituents.
2. Alternative Cosmological Constant: It proposes a mechanism for explaining the small, non-zero cosmological constant through stochastic spacetime dynamics rather than a fundamental constant of nature, potentially resolving the long-standing cosmological constant problem.
3. Astrophysical and Cosmological Tests: The paper provides a quantitative framework that can be tested against astronomical observations. The characteristic stochastic contributions serve as experimental signatures potentially distinguishable from dark matter effects.
4. Acceleration Scale in Gravity: The paper identifies a diffusion regime where traditional gravitational laws modify, providing a natural acceleration scale that aligns with MOND phenomenology — a critical area of examination for contemporary astrophysics.
5. Path Integral Approach to Stochastic Gravity: The reliance on a path integral formulation underscores an advanced methodological shift, allowing for stochastic effects which could be pivotal in unifying gravity with quantum mechanics.

Speculation on Future Developments

Future explorations could involve numerical simulations to elucidate the full implications of the theory. The authors suggest that understanding the effect of stochastic spacetime on large scale cosmic structures and galaxy formation is essential. Moreover, mapping out the exact contribution of stochastic forces relative to established dark matter models remains a significant frontier.

Conclusively, this paper offers profound insights into gravitational theory by suggesting that inherent stochasticity in spacetime might play a crucial role in celestial dynamics, challenging established paradigms of dark energy and dark matter. The theoretical architecture provided by this paper opens new avenues for scientific inquiry and experimental verification in the quest to comprehend the universe's gravitational underpinnings.