Strings and Quantum Gravity








RESEARCH > Quasi-normal modes > Black holes in asymptotically flat space

The study of quasi-normal modes in asymptotically flat space-times has attracted a lot of attention recently [15,16,17,18,19] because the asymptotic form of quasi-normal frequencies was shown to be related to the Barbero-Immirzi parameter [20] of Loop Quantum Gravity (see [21] and references therein). The asymptotic form of high overtones is

\frac{\omega_n}{T_H} = (2n+1)\pi i + \ln 3 \end{displaymath} (3)

This has been derived numerically [22,23] and subsequently confirmed analytically [15]. The imaginary part is large making the numerical analysis cumbersome, but is easy to understand by noting that the spacing of the frequencies $2\pi i T_H$ coincides with the spacing of the poles of a thermal Green function on the Schwarzschild black hole background. The analytical value of the real part was first conjectured by Hod [24] based on the general form of the area spectrum of black holes proposed by Mukhanov and Bekenstein [25] ( $\delta A =4G \ln k$, $k = 2,3,\dots$, where $\delta A$ is the spacing of eigenvalues and $G$ is Newton's constant in units such that $\hbar = c = 1$) which is related to the number of microstates of the black hole through the entropy-area relation $S = \frac{1}{4G}\ A$ [26]. It has an intriguing value from the loop quantum gravity point of view suggesting that the gauge group should be SO(3) rather than SU(2) (since we have k = 3 instead of k = 2). Thus the study of quasi-normal modes may lead to a deeper understanding of black holes and quantum gravity.

Motl and Neitzke [15] showed that (3) may be extended to arbitrary spin j and analytically derived the generalized expression

\frac{\omega_n}{T_H} = (2n+1)\pi i + \ln (1+2\cos(\pi j) ) + o(1/\sqrt n) \end{displaymath} (4)

Their derivation offered a new surprise as it heavily relied on the black hole singularity. It is intriguing that the ``unphysical'' region beyond the horizon influences the behavior of physical quantities. Dr. Siopsis and Dr. Musiri extended the result of ref. [15] by calculating the first-order correction to the asymptotic formula (4). They solved the wave equation perturbatively for arbitrary spin of the wave and obtained an explicit expression for the quasi-normal frequencies [27],
\frac{\omega_n}{T_H} = (2n+1)\pi i + \ln (1+2\cos(\pi j) ) + \frac{\hat\Omega_1}{\sqrt{n+1/2}} + o(1/n) \end{displaymath} (5)

\begin{displaymath}\hat\Omega_1 = (1-i) \ e^{\pi ij/2}\
\frac{3\ell(\ell +1) +1-...
...pi j/2)}
\Gamma^2 (1/4)\ \Gamma(1/4 + j/2)
\ \Gamma (1/4 - j/2)\end{displaymath} (6)

which includes the $o(1/\sqrt n)$ correction to the $o(1)$ asymptotic formula (4). This result is in agreement with numerical results [23,18] in the case of gravitational and scalar waves as well as results from a WKB analysis [19] in the case of gravitational waves.

Extending the above calculation to rotating (Kerr) black holes appears to be far from straightforward. One needs to solve a set of coupled differential equations, because the angular and radial equations do not neatly separate as in the Schwarzschild case [28]. Based on Bohr's correspondence principle, Hod has argued [29] that the real part of the quasi-normal frequencies of gravitational waves ought to be given by the asymptotic expression (cf. eq. (3))

\Re\omega_n = T_H \ln 3 + m\Omega \end{displaymath} (7)

where $m$ is the azimuthal eigenvalue of the wave and $\Omega$ is the angular velocity of the horizon. On the other hand, numerical results have been obtained [17] which appear to contradict the above assertion, suggesting instead an asymptotic expression independent of the temperature,
\Re\omega_n = m\Omega \end{displaymath} (8)

To shed light on this issue, Dr. Siopsis and Dr. Musiri obtained an expression for the asymptotic form of quasi-normal modes of a Kerr black hole [30] by solving the Teukolsky wave equation [28] and applying a monodromy argument similar to the Schwarzschild case [15]. The equation reduced to Whittaker's equation and the final expression was in agreement with Hod's formula (7). However, this result is only valid for asymptotic values of $\omega_n$ which are bounded from above by $1/a$, where $a$ is the angular momentum per unit mass of the Kerr black hole. Thus, its validity is limited to small values of the parameter $a$. This includes the Schwarzschild limit in which the range of frequencies extends to infinity. It appears that Bohr's correspondence principle is not applicable to the true asymptotic regime $1/a \lesssim \Re\omega_n$ in general and the numerical result (8) may well be valid. Understanding it analytically is a challenge.

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