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

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.