must be associated to interaction processes, so it is sufficient and logically convenient to define them only in interaction regions, as it was done in [@SH], whereas the asymptotic variables are the physical ones. The inclusion of interacting objects opens a door to the introduction of gravity as a further interaction, leading to a small perturbation, and not as a quantum property of space.

The only further ingredients, used beside Newtonian gravity in the next sections, are the mass-energy relation and the de Broglie wavelength of particles. In detail we will use the following approximations:

-   Newtonian gravity, understood as lowest-order approximation of general relativity, in other words, as a simplified substitute for a curved background.

-   Quantum field theory in first-order perturbative approximation.

-   A general lowest-order ansatz for DSR-like corrections of SR.

We are going to compare only unspecified interactions in the absence and in the presence of classical gravity, so the considerations are independent of specific high-energy quantum effects, like varying coupling constants. In the next two sections, $c$ and $\hbar$ are set equal to one, they will have to be restored in section 4.

Gravity in two-particle interactions
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Central collision
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We consider the scattering 