6.3 Why might logarithmic terms in the entropy be interesting?

By a logarithmic term we mean both the logarithmically UV-divergent term in the entropy and the UV-finite term, which scales logarithmically with respect to the characteristic size of the black hole. As we have seen, these terms are identical. However, after the UV divergences in the entropy have been renormalized, it is the UV finite term, which scales logarithmically, that will interest us here.

i) First of all, the logarithmic terms are universal and do not depend on the way the entropy was calculated and on the scheme in which the UV divergences have been regularized. This is in contrast to the power UV divergences in the entropy that depend both on the calculation procedure and on the regularization scheme.

ii) The logarithmic terms are related to the conformal anomalies. As the conformal anomalies play an important role in the modern theoretical models, any new manifestation of the anomalies merits our special attention. This may be even more important in view of ideas that the conformal symmetry may play a more fundamental role in nature than is usually thought. As is advocated by ’t Hooft in a number of recent papers [217, 218, 216] a crucial ingredient for understanding Hawking radiation and entropy is to realize that gravity itself is a spontaneously-broken conformal theory.

iii) For a large class of extremal black-hole solutions, which arise in supergravity theories considered as low energy approximations of string theory, there exists a microscopic calculation of the entropy. This calculation requires a certain amount of unbroken supersymmetry, so that the black holes in question are the Bogomol’nyi–Prasad–Sommerfeld (BPS) type solutions and use the conformal field theory tools, such as the Cardy formula. The Cardy formula predicts certain logarithmic corrections to the entropy (these corrections are discussed, in particular, in [35] and [201]). One may worry whether exactly the same corrections are reproduced in the macroscopic, field theoretical, computation of the entropy. This aspect was studied recently in [8] for black holes in 𝒩 = 4 supergravity and at least some partial (for the entropy due to matter multiplet of the supergravity) agreement with the microscopic calculation has indeed been observed.

iv) Speaking about the already renormalized entropy of black holes and taking into account the backreaction of the quantum matter on the geometry, the black-hole entropy can be represented as a series expansion in powers of Newton’s constant, 2 G ∕rg (the quantity 2 1∕rg, where rg = 2GM is the size of a black hole, is the scale of the curvature at the horizon; thus, the ratio G ∕r2g measures the strength of gravitational self-interaction at the horizon) or, equivalently, in powers of M 2PL∕M 2. In particular, for the Schwarzschild black hole of mass M in four spacetime dimensions, one finds

M 2 ( M 2 ) S = 4π --2--+ σ ln M + O --PL- . (300 ) M PL M 2
The logarithmic term is the only correction to the classical Bekenstein–Hawking entropy that is growing with mass M.

v) Although the logarithmic term is still negligibly small compared to the classical entropy for macroscopic black holes, it becomes important for small black holes especially at the latest stage of black hole evaporation. In particular, it manifests itself in a modification of the Hawking temperature as a function of mass M [103]. Indeed, neglecting the terms 2 O (M-P2L) in the entropy (300View Equation) one finds

∂S 2 −1 1∕TH = ---- = 8πM ∕M PL + σM (301 ) ∂M
so that the Hawking temperature TH ∼ M for small black holes. Depending on the sign of the coefficient σ in Eq. (300View Equation) there can be two different scenarios. If σ < 0, then the entropy S(M ) as function of mass develops a minimum at some value of Mmin ∼ MPL. For this value of the mass the temperature (301View Equation) becomes infinite. This is the final point (at least in this approximation) of the evaporation for black holes of mass M > Mmin. It is reached in finite time. Not worrying about exact numerical factors one has
dM ---- ∼ − TH4A+ ∼ − T4H M 2 (302 ) dt
for the evaporation rate. For large black holes (M0 ≫ Mmin) the evaporation time is tBH ∼ M 30∕M 4PL. This evaporation time can be obtained by solving Eq. (302View Equation) with the classical expression for the Hawking temperature TH ∼ M 2 ∕M PL, i.e., without the correction as in Eq. (301View Equation). Thus, if there is no logarithmic term in Eq. (300View Equation), any black hole evaporates in finite time. If the correction term is present, it becomes important for M0 ∼ Mmin. Then, assuming that Mmin ∼ MPL, one finds 5 4 tBH ∼ (M0 − MPL ) ∕M PL for the evaporation time. On the other hand, a black hole of mass M0 < Mmin, if it exists, evaporates down to zero mass in infinite time. Similar behavior is valid for black holes of arbitrary mass M0 if σ > 0 in Eqs. (300View Equation) and (301View Equation). The evaporation rate considerably slows down for small black holes since the Hawking temperature TH ∼ M for small M. The black hole then evaporates to zero mass in infinite time. Asymptotically, for large time t, the mass of the black hole decreases as −1∕5 M (t) ∼ t. Thus, the sub-Planckian black holes if (σ < 0) and any black holes if (σ > 0) live much longer than one would have expected if one used the classical expression for the entropy and for the Hawking temperature.

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