Kerr-CFT and gravitational perturbations

Abstract

Motivated by the Kerr-CFT conjecture, we investigate perturbations of the near-horizon extreme Kerr spacetime. The Teukolsky equation for a massless field of arbitrary spin is solved. Solutions fall into two classes: normal modes and traveling waves. Imposing suitable (outgoing) boundary conditions, we find that there are no unstable modes. The explicit form of metric perturbations is obtained using the Hertz potential formalism, and compared with the Kerr-CFT boundary conditions. The energy and angular momentum associated with scalar field and gravitational normal modes are calculated. The energy is positive in all cases. The behaviour of second order perturbations is discussed.

Figures (4)

• Figure 1: Values of $\eta^2$, defined in \ref{['eqn:etadef']}, for $|s|=2$, and a) $l=2$ and b) $l=3$.
• Figure 2: Values of $\eta^2$, defined in \ref{['eqn:etadef']}, for $|s|=2$, and a) $l=4$ and b) $l=16$.
• Figure 3: Ratio of the conserved charges $\mathcal{J}/\mathcal{E}$ for spin $|s|=2$ perturbations as a function of the azimuthal angular number $m$ for a) $l=2$ and b) $l=3$. We only consider normal modes, i.e., values of $m$ that yield $\eta^2>0$ as defined in \ref{['eqn:etadef']}. Data points corresponding to $n=0,1,2$ are plotted, with the solid line representing $n=2$ and the dashed line $n=0$.
• Figure 4: Ratio of the conserved charges $\mathcal{J}/\mathcal{E}$ for spin $|s|=2$ perturbations as a function of the azimuthal angular number $m$ for a) $l=4$ and b) $l=6$.