Wavefunction collapse induced by gravity in a relativistic Schr{\''o}dinger-Newton model

Abstract: A relativistic version of the Schr{\"o}dinger-Newton equation is analyzed within the recently proposed Grave de Peralta approach [L. Grave de Peralta, {\em Results Phys.} {\bf 18} (2020) 103318], which include relativistic effects by a parametrization of the non-relativistic hamiltonian, so as to impose that the average kinetic energy of the system coincide with its relativistic kinetic energy. The reliability of this method is tested for the particle in a box. By applying this method to the Schr{\"o}dinger-Newton equation we shows that the characteristic length of the model [L. Di{\'o}si, {\em Phys. Lett}. {\bf 105A} (1984) 199] goes to zero for a mass of the order of the Planck mass, suggesting a collapse of the wavefuncton, induced by gravity.