Quantum Gravity

BLACK-HOLE SPECTROSCOPY AND ENTROPY 217violate the strong form of the Third Law; it just means that the system does notapproach a unique state for T → 0.The above discussion has been performed for a pure black hole without inclusionof matter degrees of freedom. In the presence of other variables, it isno longer possible to find simple solutions such as (7.49). One possible treatmentis to perform a semiclassical approximation as presented in Section 5.4.In this way, one can recover a functional Schrödinger equation for matter fieldson a black-hole background. For simple situations this equation can be solved(Demers and Kiefer 1996). Although the resulting solution is, of course, a purestate, the expectation value of the particle number operator exhibits a Planckiandistribution with respect to the Hawking temperature—this is how Hawkingradiation is being recovered in this approach. For this reason, one might evenspeculate that the information-loss problem for black holes is not a real problem,since only pure states appear for the full system. In fact, the mixed nature ofHawking radiation can be understood by the process of decoherence (Kiefer 2001,2004a). It is even possible that the Bekenstein–Hawking entropy (7.17) could becalculated from the decohering influence of additional degrees of freedom suchas the quasi-normal modes of the black hole; cf. the remarks in the next section.We finally mention that also loop quantum gravity was applied to sphericallysymmetric systems at the kinematical level (Bojowald 2004). One advantagecompared to the full theory is that the flux variables commute with each other;thus there exists a flux representation. It turns out that loop quantizing thereduced model and reducing the states of the full theory to spherical symmetrylead to the same result. It seems that singularities are avoided (Modesto 2004).This also follows if one uses geometrodynamical variables while using methodsfrom loop quantum gravity (Husain and Winkler 2005).7.3 Black-hole spectroscopy and entropyThe results of the last subsection indicate that black holes are truly quantumobjects. In fact, as especially Bekenstein (1999) has emphasized, they might playthe same role for the development for quantum gravity that atoms had playedfor quantum mechanics. In the light of this possible analogy, one may wonderwhether black holes possess a discrete spectrum of states similar to atoms. Argumentsin favour of this idea were given in Bekenstein (1974, 1999); cf. alsoMukhanov (1986). Bekenstein (1974) noticed that the horizon area of a (nonextremal)black hole can be treated in the classical theory as a mechanical adiabaticinvariant. This is borne out from gedanken experiments in which one shootsinto the hole charged particles (in the Reissner–Nordström case) or scalar waves(in the Kerr case) with appropriate energies. From experience with quantummechanics, one would expect that the corresponding quantum entity possesses adiscrete spectrum. The simplest possibility is certainly to have constant spacingbetween the eigenvalues, that is, A n ∝ n for n ∈ N. (In the Schwarzschild case,this would entail for the mass values M n ∝ √ n.) This can be tentatively con-