Noncommutative Regge-Wheeler potential: some nonperturbative results

TL;DR

This work develops a nonperturbative, theta-exact framework for axial gravitational perturbations of a Schwarzschild black hole in spacetimes with radial noncommutativity, defined by $[t\stackrel{\star}{,} r] = i a \alpha A(r)$ and $[\varphi\stackrel{\star}{,} r] = i a \beta A(r)$. By employing a semipseudo-Killing twist and a Bopp-shift approach, star-products between background functions and perturbations reduce to translations, yielding a master formula for the Regge-Wheeler potential $V$ that is exact to all orders in the NC parameter $a$, with $r_\pm = r(\hat{r} \pm \lambda a/2)$ and $A_\pm = A(r_\pm)$. In the commutative limit $a\to 0$, the classical Regge-Wheeler potential is recovered, while explicit NC potentials are provided for Moyal and $\kappa$-Minkowski choices of $A(r)$, including horizon behaviors and Planck-scale regimes. The results illuminate the role of noncommutativity in black-hole perturbations, stability, and possible gravitational-wave phenomenology, and pave the way for nonperturbative analyses of NC effects in black hole physics.

Abstract

We study the gravitational perturbation theory of black holes in noncommutative spacetimes with noncommutativity of the type $[t\stackrel{\star}{,} r] = i a αA(r)$ and $[\varphi \stackrel{\star}{,} r] = i a βA(r)$ for arbitrary $A(r)$, which includes several Moyal-type spaces and also the $κ$-Minkowski space. The main result of this paper is an analytical expression for the effective potential of the axial perturbation modes, valid to all orders in the noncommutativity parameter. This is achieved by evaluating the $\star$-products using translations in the radial direction, i.e., Bopp shift. We comment on various regimes, such as Planck-scale black holes, where the noncommutativity length scale is of the same order of magnitude as the black hole horizon.

Figures (6)

• Figure 1: Noncommutative potentials for the $\ell = 2, M = 1$ case and $A = 1$. First-order in $a$, given in \ref{['pot0']}, is on the left and up to all orders, given in \ref{['pot1']}, is on the right.
• Figure 2: Noncommutative potentials with $A(r) = 1-R/r$ for the $\ell = 2, M = 1$ case.
• Figure 3: Potentials for $A(r) = r/R, \alpha = 0, \beta = 1.$
• Figure 4: Potentials for large values of $am$. Positive values are on the left figure and negative on the right.
• Figure 5: Potentials $|am| < 50$. Positive values are on the left figure and negative on the right.