aa r X i v : . [ g r- q c ] M a y On the Kerr Quantum Area Spectrum
A.J.M. MedvedPhysics DepartmentUniversity of SeoulSeoul 130-743KoreaE-Mail(1): [email protected](2): joey [email protected] 25, 2018
Abstract
Suppose that there is a quantum operator that describes the horizon area of ablack hole. Then what would be the form of the ensuing quantum spectrum? Inthis regard, it has been conjectured that the characteristic frequencies of the blackhole oscillations can be used to calibrate the spacing between the spectral levels.The current article begins with a brief review of this conjecture and some of itssubsequent developments. We then suggest a simple but vital modification to arecent treatment on the Kerr (or rotating black hole) spectrum. As a consequenceof this refinement, we are able to rectify a prior inconsistency (as was found betweentwo distinct calculations) and to establish, unambiguously, a universal form for theKerr and Schwarzschild spectra. A ) of ablack hole was shown to be an adiabatic invariant [2] and, as such, should logically obtaina discrete spectrum upon a suitable process of quantization. With a few other pertinentinputs, Bekenstein went on to propose his now-famous evenly spaced spectrum: A n = ǫ ~ n where n = 0 , , , ... . (1)Here, n is the associated quantum number, all fundamental constants besides Planck’sconstant have been set to one and ǫ should be regarded as a numerical coefficient of theorder of unity.[Various schools of thought have questioned the viability of an evenly spaced areaspectrum; most notably, many proponents of loop quantum gravity [3]. It should, however,be pointed out that the rest of the discussion focuses on the semi-classical or large n limit.Hence, one may — if so preferred — regard Eq.(1) as an asymptotic form of a possiblymore convoluted spectrum.]Along with the form of the spectrum itself, the value of ǫ has been somewhat contro-versial. On this point, Hod made an interesting suggestion; namely, ǫ can be fixed byutilizing the quasi-normal mode frequencies of an oscillating black hole [4]. The premiseof this so-called Hod Conjecture goes as follows: A perturbed black hole will tend to equi-librate by radiating away energy in the form of gravitational waves. This radiation will,predominantly, consist of damped oscillations that can be characterized by a discrete setof complex frequencies. Then, “in the spirit of the
Bohr Correspondence Principle ”, itbecomes natural to associate the classical limit of these frequencies (say ω c ) with the large n limit of the aforementioned (quantum) area spectrum. Inasmuch as a given frequencyrepresents a transition between spectral levels, this association can be made rather precise: A n +1 − A n = ∂A ( M ) ∂M ~ ω c = 32 πM ~ ω c as n → ∞ , (2)where M is the black hole mass and the well-known Schwarzschild relation A = 16 πM has been invoked. [We limit considerations, for the time being, to the Schwarzschildcase of a non-rotating (and uncharged) black hole. The rotating or Kerr scenario willbe addressed in due course.] Note that this difference in levels is, in Planck units, quitesimply the numerical coefficient ǫ .Hod went on to propose that, from the quasi-normal-mode side of things, the relevantclassical limit should be that of infinite damping (or vanishing decay time); with thelogic being that a truly classical black hole can not conceivably emit any radiation. Morespecifically, he suggested that ω c should be regarded as the real part of the quasi-normalfrequency after the imaginary part (which determines the damping rate) has been sentoff to infinity. Using the numerical evidence of the time [5] (which was later verified byanalytic means [6]), Hod identified the asymptotic form for the mode frequencies as ~ ω ( k ) ≈ i πT k + T ln 3 as k → ∞ , (3)where k is the quasi-normal “discreteness index” and T = ~ / πM is the Hawking2Schwarzschild) temperature [7]. Then, by way of the above line of reasoning, ~ ω c = T ln 3 = ~ ln 38 πM . (4)Finally, combining Eqs.(2) and (4), one can reproduce Hod’s important realization: ǫ = 32 πM · ln 38 πM = 4 ln 3 . (5)Kunstatter furthered this tapestry of ideas with the following insightful observations [8]:A natural adiabatic invariant for a system with energy E and vibrational frequency ω ( E )is the ratio E/ω ( E ). So, by way of the Bohr–Sommerfeld quantization condition (whichessentially relates an adiabatic invariant to a quantum number n in the semi-classical orlarge n limit), it follows that Z dEω ( E ) ≈ n ~ as n → ∞ . (6)Casting a Schwarzschild black hole into this framework, Kunstatter replaced E with M and identified ω c as the most appropriate choice for the frequency. Making the recom-mended substitutions and then integrating, one readily obtains4 πM ln 3 ≈ n ~ . (7)This obviously implies the spectral form A n = 4 n ~ ln 3 (8)or ǫ = 4 ln 3 in agreement with Hod.Recently, Maggiore refined the Hod treatment by arguing that — insofar as a blackhole is to be viewed as a damped harmonic oscillator — the physically relevant frequencywould actually be [9] ω p = q ω R + | ω I | , (9)where ω R and ω I are, respectively, the real and imaginary part of the quasi-normal modefrequency. In the large-damping limit, this “physical frequency” necessarily translatesinto [ cf , Eq.(3)] ~ ω p ≈ ~ | ω I | ≈ πT k = ~ k M as k → ∞ . (10)To effectively regulate this large k divergence, Maggiore sensibly proposed that the char-acteristic classical frequency (our ω c ) should now be identified with a transition between(adjacent) quasi-normal frequency levels. That is, ω c = ω p ( k + 1) − ω p ( k ) as k → ∞≈ M . (11)It should not be difficult to convince oneself that — given this revised state of affairs —the spectral-spacing coefficient now becomes ǫ = 8 π . (12)3his outcome can easily be verified by either Hod or Kunstatter’s choice of methodology.[It may be of interest that this particular value of ǫ occurs frequently in the literaturefor a variety of analytic techniques (see [10] and references therein). Many other “sellingpoints” for Maggiore’s revision are discussed in his already-cited work. Conversely, themain attribute of the original Hod calculation is that it complies with ǫ = 4 ln m (for somepositive integer m > S = A/ ~ [7, 12]) along with Boltzmann–Einstein statistics. On the other hand, thereare still viable reasons [9, 10] why this last point might reasonably be overlooked.]Even more recently, Vagenas applied the above machinery to the Kerr or rotating(but still uncharged) black hole case [13]. It is worthwhile to first recall the relevantthermodynamic quantities for this scenario: A = 8 π (cid:16) M + √ M − J (cid:17) , (13) T = ~ √ M − J πM (cid:0) M + √ M − J (cid:1) , (14)Ω = J M (cid:0) M + √ M − J (cid:1) . (15)Here, J is the angular momentum of the black hole and Ω is the rotational velocity at itshorizon. (Both of which will be regarded as positive, without any loss of generality.)Also of relevance to the current discussion, there has been some recent progress inunderstanding the highly damped quasi-normal frequencies for the Kerr solution. Infact, an analytic result has now been obtained [17] and, reassuringly, this computationcomplies with some older numerical work [18]. Most pertinently, the imaginary part ofthe frequency ascends asymptotically as ~ | ω I | ≈ πT k as k → ∞ , (16)whereas the real part remains finite. The parameter T can be viewed as an effective“temperature” and is defined by T ≡ T ( J = 0) = ~ / πM .Because of the simplicity of Eq.(16), it is quite evident that the Kerr analogue ofMaggiore’s revised (Hod) calculation goes through unfettered; so that, once again, ǫ = 8 π . (17)Indeed, it is this very universality that was the main observation made by Vagenas.Unfortunately, the same type of consistency failed to transpire for the Kerr analogue ofthe Kunstatter calculation.Given the serious nature of the last claim, let us elaborate much further on this issue.With cognizance of the first law of black hole thermodynamics [19], Vagenas perceptivelydeduced the Kerr analogue of Kunstatter’s adiabatic-invariant integral as being Z dM − Ω dJω c . (18) Other (not necessarily related) derivations of the Kerr quantum area spectrum include [14–16].
Z (cid:20) M dM − J dJM + √ M − J (cid:21) ≈ n ~ as n → ∞ . (19)Some straightforward integration then reveals that M + 2 √ M − J − M ln (cid:20) M + √ M − J ~ (cid:21) ≈ n ~ ; (20)which — by way of Eq.(13) — can be reinterpreted as the following quantum spectrumfor the area: A n − πM ln (cid:20) A n π ~ (cid:21) = 4 π ~ n . (21)The reader has most probably noticed the conspicuous presence of a logarithmic termin the spectrum. Certainly, it is true that logarithmic corrections to the horizon areahave had a long and storied tradition in the literature [20]. But the key word here is“correction”. In Eq.(21), the logarithmic term becomes the dominant one for any blackhole larger than (at most) a few hundred Planck areas. Meanwhile, by invoking a semi-classical quantization condition, we are — by implication — talking about black holeswith A >> ~ . Hence, for the obligatory regime of validity, the spectrum is far fromevenly spaced (although still discrete). To further complicate matters, A >> ~ ensuresus that the logarithm is positive, and so the (typically dominant) logarithmic term isnegative. Meanwhile, the left-hand side of the equation is, in reality, inherently positive![This follows directly from the first and second laws of thermodynamics; inasmuch as theintegrand of Eq.(18) is essentially the (differential) change in entropy of a rotating blackhole, albeit with an unorthodox choice for the temperature.] Finally, even if there could besome conceivable means of dismissing away the logarithmic term, one would then deducea spacing coefficient of ǫ = 4 π — disturbingly off by a factor of two from the previousKerr result [ cf , Eq.(17)].So what exactly went wrong? The problem, as we see it, can be traced to the evaluationof Eq.(19), where the integration variables ( M and J ) have been placed on an equalfooting. But such a democracy of variables can not actually be justified. To see this, letus first reconsider the application of the Bohr–Sommerfeld quantization condition. Thisimmediately implies a semi-classical regime, which requires n to be a very large number,as the area (in Planck units) must be as well. Although not so immediately obvious, thedomain of semi-classicality indicates yet another constraint that must inevitably be dealtwith. In short, the black hole must be far away from extremality . (Keep in mind thatthe Kerr extremal limit is J = M . For J > M , the black hole is replaced by a nakedsingularity, in a grievous violation of cosmic censorship.)To elucidate, it has been demonstrated that, at least in the context of Bekenstein-inspired spectroscopy, a near-extremal black hole (whether charged [21] and/or rotating[22] / [15]) is a highly quantum object. This is because, as clarified in the cited articles, thequantum number n is actually a measure of the areal deviation from extremality, ratherthan the area itself. Meaning that a near-extremal black hole is necessarily associated We obtain an ~ in the denominator of the logarithm by cutting the integration off at the Planckscale; this being a very natural choice given our ignorance of sub-Planckian physics. Nothing that is saidbelow will, however, depend on the precise value of this ultraviolet regulator. J ; that is, J << M . It should be clear that no such restriction has yet been imposed. A simple but reasonable way to constrain the angular momentum is to treat the in-tegrand of Eq.(19) as a perturbative expansion in
J/M . On this basis, one would beinclined to recast the expression as follows: Z (cid:20) M dM − J + O ( J ) M dJ (cid:21) ≈ n ~ . (22)The integration now leads to 2 M − J M + O ( J ) ≈ n ~ (23)or, equivalently, M + √ M − J + O ( J ) ≈ n ~ . (24)Comparing with Eq.(13), we can now extrapolate an area spectral form of A n + O ( J n ) = 8 π ~ n . (25)So that (for the semi-classical domain of both large n and small J ) the Kerr spectral-spacing coefficient is reconfirmed as taking on the universal value of ǫ = 8 π . Moreover, theonce-dominant logarithmic term has safely been eradicated and both sides of the equationare now manifestly positive.To summarize, we have found that — after fully accounting for the semi-classical regimeof validity — the Kerr area spectrum is asymptotically identical for the two distinctmethods of interest. (These being Hod’s [4] and Kunstatter’s [8], along with the essentialmodifications suggested by Maggiore [9] and Vagenas [13].) In this way, we have alsoconfirmed — now unambiguously — a universal form [13] for the Schwarzschild and Kerrspectra. The reader may be concerned about n sometimes measuring the area and other times, the arealseparation from extremality. However, in the true semi-classical regime — where M >> J and so A ≈ πM — this distinction is inconsequential. It should be noted that, because J and its corresponding quantum number are now being treated asparametrically small quantities, Eq.(25) in no way contradicts any of the spectra documented in [15, 22]. cknowledgments Research is financially supported by the University of Seoul. The author thanks EliasVagenas for useful discussions.
References [1] J.D. Bekenstein, Lett. Nuovo Cimento , 467 (1974);See, also, “Quantum Black Holes as Atoms”, arXiv:gr-qc/9710076 (1997).[2] J.D. Bekenstein, Phys. Rev. D , 2333 (1973).[3] See, for instance, M. Domagala and J. Lewandowski, Class. Quant. Grav. , 5233(2004) [arXiv:gr-qc/0407051].[4] S. Hod, Phys. Rev. Lett. , 4293 (1998) [arXiv:gr-qc/9812002].[5] H.-P. Nollert, Phys. Rev. D , 5253 (1993);N. Andersson, Class. Quant. Grav. , L61 (1993).[6] L. Motl, Adv. Theor. Math. Phys. , 1135 (2003) [arXiv:gr-qc/0212096];L. Motl and A. Neitzke, Adv. Theor. Math. Phys. , 307 (2003)[arXiv:hep-th/0301173].[7] S.W. Hawking, Nature , 30 (1974); Commun. Math. Phys. , 199 (1975).[8] G. Kunstatter, Phys. Rev. Lett. , 161301 (2003) [arXiv:gr-qc/0212014].[9] M. Maggiore, Phys. Rev. Lett. , 141301 (2008) [arXiv:0711.3145 (gr-qc)].[10] A.J.M. Medved, “A brief commentary on black hole entropy”, arXiv:gr-qc/0410086(2004).[11] J.D. Bekenstein and V. Mukhanov, Phys. Lett. B , 7 (1995) [gr-qc/9505012].[12] J.D. Bekenstein, Lett. Nuovo. Cim. , 737 (1972);Phys. Rev. D , 2333 (1973).[13] E.C. Vagenas, “Area spectrum of rotating black holes via the new interpretation ofquasinormal modes”, arXiv:0804.3264 [gr-qc] (2008).[14] J. Makela, P. Repo, M. Luomajoki and J. Piilonen, Phys. Rev. D , 024018 (2001)[arXiv:gr-qc/0012005].[15] G. Gour and A.J.M. Medved, Class. Quant. Grav. , 2261 (2003)[arXiv:gr-qc/0211089].[16] V.V. Kiselev, “Quantum spectrum of black holes”, arXiv:gr-qc/0509038 (2005).[17] U. Keshet and S. Hod, Phys. Rev. D , 061501 (2007) [arXiv:0705.1179 (gr-qc)];U. Keshet and A. Neitzke, “Asymptotic Spectroscopy of Rotating Black Holes”,arXiv:0709.1532 [hep-th] (2007). 718] E. Berti, V. Cardoso and S. Yoshida, Phys. Rev. D , 124018 (2004)[arXiv:gr-qc/0401052].[19] J.M. Bardeen, B. Carter and S.W. Hawking, Comm. Math. Phys. , 161 (1973).[20] See, for an overview and references, D.N. Page, New J. Phys. , 203 (2005)[arXiv:hep-th/0409024].[21] A. Barvinsky, S. Das and G. Kunstatter, Class. Quant. Grav. , 4845 (2001)[arXiv:gr-qc/0012066].[22] G. Gour and A.J.M. Medved, Class. Quant. Grav.20