Particle production induced by a Lorentzian non-commutative spacetime

In this paper, we examine particle production, evaporation, and greybody factors for a Lorentzian non-commutative black hole. We begin by analyzing particle creation for bosons, considering scalar perturbations to compute the Bogoliubov coefficients, which enable the determination of the Hawking temperature $T_{\Theta}$. Subsequently, we describe Hawking radiation as a tunneling process using the Painlev\'e-Gullstrand metric representation, allowing the evaluation of divergent integrals via the residue method. This approach yields the particle creation density for bosonic modes. Next, we extend the analysis to fermions, obtaining the corresponding particle creation density. The black hole evaporation is then examined through the Stefan-Boltzmann law, leading to an estimate of the black hole's lifetime. In this context, we identify the presence of a remnant mass when the black hole reaches the final stage of its evaporation. Furthermore, we compute greybody factors for bosons, taking into account scalar, vector, and tensorial perturbations. Finally, we determine the greybody factors for fermions as well. Overall, compared to the Schwarzschild case ($\Theta = 0$), the presence of the non-commutative parameter $\Theta$ lowers the Hawking temperature and reduces the particle creation density for both bosons and fermions, causing the evaporation process to proceed more slowly. Additionally, $\Theta$ decreases the magnitude of the greybody factors for bosons and fermions across all perturbations considered in this analysis.