Quasinormal modes of scalar field coupled to Einstein's tensor in the non-commutative geometry inspired black hole

Abstract

We investigate the quasinormal modes (QNMs) of the scalar field coupled to the Einstein's tensor in the non-commutative geometry inspired black hole spacetime. It is found that the lapse function of the non-commutative black hole metric can be represented by a Kummer's confluent hypergeometric function, which can effectively solve the problem that the numerical results of the QNMs are sensitive to the model parameters and make the QNMs values more reliable. We make a careful analysis of the scalar QNM frequencies by using several numerical methods, and find that the numerical results obtained by the new WKB method (the Padé approximants) and the Mashhoon method (P$\ddot{\text{o}}$schl-Teller potential method) are quite different from those obtained by the asymptotic iterative method (AIM) and time-domain integration method when the non-commutative parameter $\theta$ and coupling parameter $\eta$ are large. The most obvious difference is that the numerical results obtained by the AIM and the time-domain integration method appear a critical value ${\eta }_c$ with an increase of $\eta$, which leads to the dynamical instability. After carefully analyzing the numeral results, we conclude that the numerical results obtained by the AIM and the time-domain integration method are closer to the theoretical values than those obtained by the WKB method and the Mashhoon method, when the $\theta$ and $\eta$ are large. Moreover, through a numerical fitting, we obtain that the functional relationship between the threshold ${\eta }_c$ and the non-commutative parameter $\theta$ satisfies ${\eta }_c=a{\theta }^b+c$ for a fixed $l$ approximately. We find that the stability of dynamics can be ensured in the $\eta <{\eta }_c(\theta ,l)$ region.

Comments: 28 pages, 7 figures. Owing to the first author's nice insight, the Kummer's confluent hypergeometric function is first found as the representation of the lapse function of the non-commutative black hole metric