Unified Framework Models Geodesic Dynamics with Conservative, Dissipative Effects and GUP Corrections

The motion of objects through spacetime, traditionally described by geodesic equations, faces ongoing refinement as physicists explore the influence of external forces and quantum gravity effects. Gaurav Bhandari, S. D. Pathak, and Harjit Ghotra from Lovely Professional University present a unified framework that incorporates conservative forces, energy dissipation, and corrections stemming from the Generalized Uncertainty Principle, a theory suggesting a minimal length scale in the universe. This work establishes generalized geodesic equations capable of modelling these combined effects, revealing that external potentials introduce velocity-dependent corrections to particle trajectories and potentially challenge the equivalence principle, a cornerstone of general relativity. By applying this framework to cosmological models, including universes dominated by various forms of matter and dark energy, the team provides a powerful new tool for investigating the interplay between fundamental forces, spacetime curvature, and the elusive realm of Planck-scale physics.

Minimum Length and Modified Uncertainty Relations

This extensive research details investigations into the implications of the Generalized Uncertainty Principle (GUP) across diverse areas of physics, from cosmology and quantum gravity to classical mechanics and atomic spectroscopy. This modification introduces a minimum length scale, profoundly impacting the structure of spacetime and the behavior of particles at extremely small scales. Researchers explored the GUP’s influence on several key areas, including cosmology, where it can affect the equation of state of dark energy and potentially revise cosmological models.

In quantum gravity, the GUP appears in theories like string theory or loop quantum gravity, impacting black hole thermodynamics and potentially resolving singularities predicted by classical general relativity. The team also investigated how GUP-inspired modifications can be incorporated into classical mechanics, providing descriptions of dissipative systems. A significant focus lies in spectroscopic effects, specifically the Zeeman and Stark effects, where GUP corrections can lead to measurable shifts in spectral lines, offering a potential avenue for experimentally testing the GUP in a laboratory setting. A central goal of this work is to find ways to experimentally verify the GUP, with predicted shifts in atomic spectra offering a promising route.

The research consistently presents the GUP as a potential consequence of a more fundamental theory of quantum gravity, demonstrating that it leads to deviations from standard physics in various contexts. Researchers developed and applied mathematical tools to incorporate GUP corrections into physical models, showing that these corrections lead to predictable shifts in the energy levels of atoms subjected to magnetic and electric fields, and alter the dispersion relations and wave functions of quantum fields. This research contributes to the ongoing effort to develop a consistent theory of quantum gravity and to understand the fundamental nature of spacetime. The emphasis on experimental verification is particularly important, as it provides a pathway for testing theoretical predictions and potentially uncovering new physics, highlighting the potential for using precision measurements in atomic physics to probe the realm of quantum gravity.

Geodesic Equations with Dissipation and Minimal Lengths

Scientists developed a novel theoretical framework for geodesic equations, extending standard models to incorporate conservative forces, dissipative effects, and corrections arising from a minimal length scale predicted by some quantum gravity theories. The work modifies the equations governing particle motion in curved spacetime, introducing external vector potentials to represent conservative interactions and employing an exponential rescaling of the Lagrangian to model dissipation. Crucially, the team incorporated the Generalized Uncertainty Principle (GUP) through deformed Poisson brackets within the Hamiltonian formulation, allowing them to explore the implications of minimal length scales on geodesic trajectories. The introduction of external potentials induces velocity-dependent corrections, suggesting potential violations of the equivalence principle. To explore these effects, the team analyzed modified trajectories within Friedmann-Lemaitre-Robertson-Walker (FLRW) universes, considering cosmological scenarios dominated by dust, radiation, stiff matter, and dark energy. This study pioneered a unified approach to conservative, dissipative, and GUP-corrected geodesics, enabling a comprehensive investigation of the interplay between external forces, spacetime curvature, and Planck-scale physics.

Geodesic Corrections From Dissipation and Quantum Scales

Scientists have derived generalized geodesic equations that account for conservative forces, dissipative effects, and corrections motivated by minimal-length scales predicted by some quantum gravity theories. The work establishes a unified approach to understanding particle motion in curved spacetime, incorporating influences beyond gravity itself. Researchers employed a variational approach to derive these equations, initially focusing on conservative external forces before extending the analysis to include dissipative dynamics arising from an exponential rescaling of the particle Lagrangian. The introduction of external potentials induces velocity-dependent corrections, suggesting potential violations of the equivalence principle. To explore these effects in a cosmological context, the team analyzed modified trajectories within Friedmann-Lemaitre-Robertson-Walker (FLRW) universes, considering universes dominated by dust, radiation, stiff matter, and dark energy. Researchers derived modified geodesic equations demonstrating that, in the absence of external interactions, free-particle motion and the equivalence principle remain valid, even with GUP corrections. However, the presence of external potentials introduces velocity-dependent corrections to these equations, indicating a potential violation of the equivalence principle. The team investigated these modified trajectories within various cosmological scenarios.