Unified Framework for Geodesic Dynamics with Conservative, Dissipative, and GUP Effects

TL;DR

Problem: how non-gravitational forces, dissipation, and quantum-gravity–motivated minimal length affect geodesic motion in curved spacetime. Approach: develop a unified framework combining variational external forces, exponential Lagrangian dissipation, and GUP-deformed Poisson brackets in a curved-space Hamiltonian setting. Contributions: explicit modified geodesic equations for conservative forces, dissipation, and GUP; leading-order result that free geodesics stay intact, but potentials induce velocity-dependent corrections; application to FLRW cosmologies demonstrates qualitative effects. Significance: provides a prototype to probe Planck-scale physics in cosmological and astrophysical contexts and clarifies the interplay between forces, curvature, and quantum corrections.

Abstract

We derive generalized geodesic equations in curved spacetime that include conservative forces, dissipative effects, and quantum-gravity-motivated minimal-length corrections. Conservative interactions are incorporated through external vector potentials, while dissipative dynamics arise from an exponential rescaling of the particle Lagrangian. Phenomenological study of Quantum-gravity effects is introduced via Generalized Uncertainty Principle (GUP) deformed Poisson brackets in the Hamiltonian framework. We show that free-particle geodesics remain unaffected at leading order, but external potentials induce velocity-dependent corrections, implying possible violations of the equivalence principle. As an application, we analyze modified trajectories in Friedmann-Lemaitre-Robertson-Walker (FLRW) universes dominated by dust, radiation, stiff matter, and dark energy. Our results establish a unified approach to conservative, dissipative, and GUP-corrected geodesics, providing a framework to probe the interplay between external forces, spacetime curvature, and Planck-scale physics.

Figures (3)

• Figure 1: Representative geodesic curves for different curvature parameter $k$ and different values of $\alpha$. Top row: Dust-dominated universe, $a(t) \propto t^{2/3}$. Second row: Radiation-dominated universe, $a(t) \propto t^{1/2}$. Third row: Stiff-fluid universe, $a(t) \propto t^{1/3}$. Bottom row: Dark energy-dominant universe. In each row, plots correspond to $k=-1$ (left), $k=0$ (center), and $k=1$ (right).
• Figure 2: The plot Represent geodesic curves for different curvature parameter $k$ with different dissipative term values $Q$. Top row: Dust-dominated universe, $a(t) \propto t^{2/3}$. Second row: Radiation-dominated universe, $a(t) \propto t^{1/2}$. Third row: Stiff-fluid universe, $a(t) \propto t^{1/3}$. Bottom row: Dark energy-Dominated universe. In each row, plots correspond to $k=-1$ (left), $k=0$ (center), and $k=1$ (right).
• Figure 3: Geodesic trajectories in the presence of the GUP parameter $\beta$ for different spatial curvature values $k = -1, 0, 1$ respectively. Top row: Dust-dominated universe with scale factor $a \propto t^{2/3}$. Increasing $\beta$ leads to a greater deviation of the timelike geodesics from the pure geodesic, while the orange shaded region denotes the causal region. Second row: Radiation-dominated universe with $a \propto t^{1/2}$. Third row: Stiff-fluid-dominated universe with $a\propto t^{1/3}$. Bottom row: Dark energy-dominated universe with $a \propto e^{H t}$.