Geodesics in Quantum Gravity

TL;DR

This work introduces q-desics, quantum-corrected analogs of classical geodesics, defined by the expectation value of the affine connection operator rather than the classical metric alone. It provides both a Lagrangian and a Hamiltonian derivation, showing that particle trajectories satisfy $\frac{d^2 x^\beta}{d\lambda^2} + \left\langle \hat{\Gamma}^\beta_{\mu\nu}\right\rangle \frac{dx^\mu}{d\lambda} \frac{dx^\nu}{d\lambda} = 0$, thereby encoding quantum-geometric effects. The framework is applied to static, spherically symmetric quantum backgrounds (SER), where a set of operator expectation values and integration constants $C_{i,j}$ capture quantum-state information; this yields horizon shifts and modified circular velocities, depending on the quantum state $|\Psi\rangle$. The results show that quantum fluctuations of spacetime can impart observable corrections to motion across short and long distances, offering a concrete link between quantum gravity and astrophysical tests, while connecting to broader discussions on effective actions and non-geodesic motion. Overall, q-desics provide a principled, testable route to probe quantum spacetime effects in semiclassical gravity and beyond.

Abstract

We investigate the motion of test particles in quantum-gravitational backgrounds by introducing the concept of q--desics, quantum-corrected analogs of classical geodesics. Unlike standard approaches that rely solely on the expectation value of the spacetime metric, our formulation is based on the expectation value of quantum operators, such as the the affine connection-operator. This allows us to capture richer geometric information. We derive the q--desic equation using both Lagrangian and Hamiltonian methods and apply it to spherically symmetric static backgrounds obtained from canonical quantum gravity. Exemplary results include, light-like radial motion and circular motion with quantum gravitational corrections far above the Planck scale. This framework provides a refined description of motion in quantum spacetimes and opens new directions for probing the interface between quantum gravity and classical general relativity.

Figures (2)

• Figure 1: Conceptual path to geodesic equation(s). Hatted quantities denote quantum operators. Along the conventional red path, all vacuum expectation values (VEVs) are taken before variation and extremization, yielding the standard geodesic equation. In contrast, the blue path defers the VEV of the metric degrees of freedom until the final step. This leads to a quantum-modified equation of motion, which we refer to as the q--desic equation. Solving this equation requires input from quantum gravity in the form of operator averages such as $\left\langle \hat{\Gamma}^\beta_{\mu \nu} \right\rangle$.
• Figure 2: Orbital velocity as a function of the radius. For all curves we used in natural untits $G=1,\; M=0.1,$ and $\Lambda=-0.05$. The black curve corresponds to the classical relation (\ref{['eq_v2cl']}). In terms of (\ref{['eq_v2circ']}) this classical velocity-distance relation corresponds to $\epsilon_{0,0}=\epsilon_{1,0}=\epsilon_{0,1}=\epsilon_{1,2}=\epsilon_{2,2}=0$. The colored curves are also based on (\ref{['eq_v2circ']}), but allowing one single parameter to be different from zero. For the blue curves we chose $\epsilon_{2,2}=\{0.3,\; 0.4,\; 0.5\}$, while for the red curves we we used $\epsilon_{0,0}=\{-0.2,\; -0.4,\; -0.6\}$.