Quasinormal modes and horizon area quantisation in Loop Quantum Gravity

The connection between quasinormal modes and quantisation of the horizon area has drawn some attention in the recent past after a pioneering work by Hod Hod . On the basis of his approach, the Barbero-Immirzi parameter of Loop Quantum Gravity (LQG) could be fixed dreyer , but it was soon shown in disagreement with the value obtained from the entropy of large black holes. In addition, the Hod’s main assumption was later criticised by Maggiore Maggiore , namely the use of the real part $\omega_{R}$ of the QNM frequencies in the area quantum estimation. Maggiore considered, instead, a corresponding damped oscillator and used the difference between the natural frequencies of subsequent modes, $\delta\omega_{0}$. In the high overtones limit of a Schwarzschild black hole of mass $M$, it is possible to show that $\omega_{0}\approx\omega_{I}$ (where $\omega_{I}$ is the imaginary part of QNM frequencies) and there is a constant shift between consecutive modes, given by $\delta\omega_{I}=1/(4M)$. This led Maggiore to $\delta A=8\pi l_{P}^{2}$, the quantum of area early suggested by Bekenstein on the basis of a different reasoning.¹¹1$l_{P}^{2}=1$ is the Planck length in natural units $\hbar=c=G=1$. In the present paper we argue that the correspondence between the horizon area levels and LQG area eigenvalues should only be valid for strictly isolated horizons, for which the Hawking radiation is exactly null. In particular, for the case of a Kerr extremal horizon, Berti et al. Kerr numerically showed that $\delta\omega_{I}=1/(4M)$ is still correct within $98\%$ precision. We will show that this allows us to obtain $\delta A=4\pi l_{P}^{2}$, half of the Bekenstein’s value.

On the other hand, in Loop Quantum Gravity Ashtekar ; Rovelli ; Thiemann ; Gambini the area created when a surface is pierced by a spin network line in the fundamental representation $j=1/2$ is given by $4\pi\sqrt{3}\gamma l_{P}^{2}$ Smolin , where $\gamma$ is the Barbero-Immirzi parameter, the only free parameter in the theory. Furthermore, isolated horizons are characterised by the so called projection constraint, which implies that lines with $j=1/2$ must pierce the horizon at least twice, which leads again to $\delta A=4\pi l_{P}^{2}$, provided that $\gamma=\sqrt{3}/6$. This is in $95\%$ agreement to the approximate value derived from the entropy of large horizons Meissner . It also provides, through an exact computation of micro-states of Planck scale horizons, a precise fit to the Bekenstein-Hawking entropy at leading order Pigozzo .

As we have seen, for an extremal rotating black hole the value of $\delta\omega_{I}$ given by (14) is $2.6\%$ above $1/(4M)$. Therefore, within the numerical precision of (14), the value of $\delta{A}$ is higher than (15) by the same amount and, when comparing with (16), we obtain a Barbero-Immirzi parameter $2.6\%$ higher than $\sqrt{3}/6$. As the latter gives a very good fit of the Bekenstein-Hawking entropy Pigozzo , we should be concerned about this small difference. We have then redone the analysis of Pigozzo , comparing the results derived with three different values of $\gamma$, namely $\gamma=0.274$ (the approximate value obtained from the entropy of large horizons), $\gamma=\sqrt{3}/6$ and $\gamma=0.296$ (the precise value got from (14)-(16)). Since, in the limit of large areas, the slope of the entropy $\times$ area relation is inversely proportional to $\gamma$ Meissner , in the latter case we expect a leading order linear relation with slope $\approx 2.6\%$ smaller than $1/4$.

Our computational analysis consists of counting the number of spin network configurations that satisfies the area operator eigenvalues (for details, see Pigozzo and references therein). The results depend on the bin used in the variation of $A$, and they are shown in Figs. 1 and 2 for a maximum area of $250\,l_{P}^{2}$. In Fig. 1 we display the entropy $\times$ area relation for the three above values of $\gamma$, without imposing the projection constraint, for a bin $\delta{\cal A}=0.5$. We see that the oscillations characteristic of small areas are attenuated in the thermodynamic limit of larger areas, as expected barbero2 . We also see that the slope of the leading linear fit decreases with $\gamma$. In Fig. 2 we show the slope as a function of the are bin, with and without the projection constraint. For large bins the Hawking slope $1/4$ is obtained with $\gamma=0.274$, as in the limit of large areas Meissner . In the limit of small bins, however, the best fit is given by $\gamma=\sqrt{3}/6$. For $\gamma=0.296$ we have a slope slightly below $1/4$, as expected. The understanding of this small discrepancy between the Barbero-Immirzi parameter derived from quasinormal modes and that obtained from the Bekenstein-Hawking entropy is something deserving further investigation.

The presence of only one free parameter in the theory turns Loop Quantum Gravity falsifiable if we can find two independent ways of fixing this parameter. Quasinormal modes and horizon thermodynamics are not really independent as both come from the same classical theory. However, the precise coefficient in the linear relation between entropy and area of large horizons has a quantum origin related to black hole evaporation. In this respect, it seems intriguing that LQG can fix this coefficient with (almost) the same Barbero-Immirzi parameter obtained when we derive the fundamental quantum of area from the Correspondence Principle applied to quasinormal modes. Note that with QNM we only test the “area gap” of LQG, while the horizon entropy is a test for the whole area spectrum. None of them can be considered an empirical test, as QNM and the Bekenstein-Hawking entropy are pure theoretical results. Together, they constitute rather a consistency test for the theory, that in this way has no free parameter anymore and can, in principle, be falsified in a single experiment. Another possible signature for the value of $\gamma$ has been recently suggested in Pigozzo . There, assuming that quantum corrections to the horizon area are negligible for Planck size horizons⁵⁵5This assumption is corroborated by effective models of quantum black holes, as e.g. that proposed in olmedo . See Pigozzo for details., it is shown that the horizon area of a Planck mass extremal rotating black hole is in the spectrum of the LQG area operator, provided that $\gamma\approx\sqrt{3}/6$. Incidentally, such stable black holes could be produced at the end of inflation at a rate large enough to be interpreted as composing the presently observed dark matter PLB .

We are thankful to Alberto Saa for helpful discussions and comments. SC is partially support by CNPq (Brazil) with grants # 311584/2020-9 and 420641/2018-1.