Newton's Equation on the kappa space-time and the Kepler problem

TL;DR

This paper investigates how Newton's laws are altered in $kappa$-space-time, deriving a modified Hamiltonian and equations of motion, and uses observational data to constrain the deformation parameter and test the equivalence principle.

Contribution

It derives the first-order deformation of Newton's laws in $kappa$-space-time and analyzes its physical implications, including force modifications and bounds on the deformation parameter.

Findings

Identifies an additional radial force in the modified equations.

Uses Pioneer anomaly data to constrain the deformation parameter $a$.

Shows violation of the equivalence principle in the $kappa$-space-time framework.

Abstract

We study the modification of Newton's second law, upto first order in the deformation parameter $a$, in the $\kappa$-space-time. We derive the deformed Hamiltonian, expressed in terms of the commutative phase space variables, describing the particle moving in a central potential in the $\kappa$-space-time. Using this, we find the modified equations of motion and show that there is an additional force along the radial direction. Using Pioneer anomaly data, we set a bond as well as fix the sign of $a$. We also analyse the violation of equivalence principle predicted by the modified Newton's equation, valid up to first order in $a$ and use this also to set an upper bound on $a$.