Maximal acceleration in a Lorentz invariant non-commutative space-time

TL;DR

This paper explores how non-commutative geometry affects the maximal acceleration in DFR space-time, deriving corrections, bounds, and modified relations, thus advancing understanding of quantum gravity effects in a Lorentz-invariant framework.

Contribution

It introduces non-commutative corrections to maximal acceleration, derives bounds, and formulates modified uncertainty and commutation relations in DFR space-time.

Findings

Non-commutativity decreases maximal acceleration in the commutative limit.

Upper bound on acceleration along non-commutative coordinates established.

Modified uncertainty and commutation relations derived.

Abstract

In this paper, we derive the non-commutative corrections to the maximal acceleration in the Doplicher-Fredenhagen-Roberts (DFR) space-time and show that the effect of the non-commutativity is to decrease the magnitude of the value of the maximal acceleration in the commutative limit. We also obtain an upper bound on the acceleration along the non-commutative coordinates using the positivity condition on the magnitude of the maximal acceleration in the commutative space-time. From the Newtonian limit of the geodesic equation and Einstein's equation for linearised gravity, we derive the explicit form of Newton's potential in DFR space-time. By expressing the non-commutative correction term of the maximal acceleration in terms of Newton's potential and applying the positivity condition, we obtain a lower bound on the radial distance between two particles under the gravitational attraction in DFR space-time. We also derive modified uncertainty relation and commutation relation between coordinates and its conjugate, due to the existence of maximal acceleration.