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Area spectrum and black hole thermodynamics

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Pith's one-line read Quantized horizon area with a step fixed by Landauer's principle yields logarithmic and inverse-area corrections to Hawking temperature and entropy for four black hole geometries.

arxiv 2504.12014 v2 pith:YBM3R27Z submitted 2025-04-16 gr-qc hep-th

Arpita Jana , Manjari Dutta , Sunandan Gangopadhyay This is my paper · ORCID
classification gr-qchep-th
keywords blackholeareacorrectedhawkingtemperaturethermodynamicschange
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The reading

Black holes are usually described by a smooth horizon and a temperature set by their mass. This paper explores what happens if the horizon area can only change in fixed packets, like the energy levels of an atom. For a Schwarzschild black hole, the area is tied to the mass, so discrete areas mean discrete masses. The size of the packet is a free number. The authors fix that number by borrowing Landauer's principle, which says erasing one bit of information costs at least a small amount of energy related to temperature. They assume that each quantum jump of the horizon erases one bit, which fixes the packet size.

With the packet size fixed, the change in mass between two adjacent levels, divided by the change in entropy, gives a temperature that differs slightly from the usual Hawking temperature. The paper repeats this for four geometries: Schwarzschild, a quantum-corrected version, a charged black hole, and a rotating black hole. In each case the new temperature, when put back into the first law of thermodynamics, gives an entropy with the usual area term plus logarithmic and inverse-area corrections.

The main caveat is internal consistency. The temperature is computed using a constant entropy gap between levels, but the corrected entropy derived from that temperature has a level-dependent gap at the same order of approximation. The paper does not check whether the two pictures agree. The relation used to fix the quantum-correction parameter in the quantum-corrected case is also imposed rather than derived.

Extended reading notes

Core claim

For a Schwarzschild black hole, area quantization plus the saturated Landauer principle gives the corrected temperature T = T_H (1 - alpha m_P^2/(64 pi M^2) + alpha^2 m_P^4/(2 (32)^2 pi^2 M^4)) and, through the first law, the corrected entropy S = A m_P^2/4 + (alpha/16) ln A + alpha^2/(64 A m_P^2) + constant. The analogous corrected temperature and entropy formulas for quantum-corrected Schwarzschild, Reissner-Nordstrom, and Kerr black holes are the central outputs.

Load-bearing premise

The corrected temperature is computed from the discrete first law T = Delta M / Delta S with Delta S taken as the constant alpha/4, from the bare area law. But the corrected entropy derived from that temperature, eq.(13), gives Delta S = alpha/4 + alpha/(16 n) at the same order in 1/M^2 as the claimed temperature correction. The paper never imposes Delta S consistency, so the temperature corrections in eqs.(11), (27), (39), (54) are not self-consistent with the entropy corrections in eqs.(13), (33), (43), (58).

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Assumptions & free parameters 2 free parameters · 7 assumptions · 0 invented entities

The central claim rests on the assumed area spectrum A(n) = alpha (n + 1/2) l_P^2, the constant entropy spacing derived from the bare area law, the applicability of the first law to discrete transitions, the saturated Landauer condition that fixes alpha, and, for the quantum-corrected case, the asymptotic-safety running Newton constant. The most fragile inputs are the constant Delta S = alpha/4 and the imposed relation omega = alpha/(16 pi).

free parameters (2)
  • alpha (area quantization spacing) = 4 ln 2, approx 2.7726, the saturation value
    Free coefficient in A(n) = alpha (n + 1/2) l_P^2. Fixed by imposing Delta E_BH = T_H Delta S = T_H alpha/4 equal to Landauer energy T_H ln 2 in eqs.(4)-(7). The logarithmic entropy coefficient alpha/16 depends directly on this choice.
  • omega (RG improvement parameter) = alpha/(16 pi) = ln 2/(4 pi), approx 0.0551; the plot in Fig. 1 uses 0.05
    In Sec. III, omega is fixed by comparing the quantization-corrected temperature eq.(27) with the RG-corrected Hawking temperature eq.(28). This is an imposed relation rather than an independent measurement.
assumptions (7)
  • domain assumption Horizon area is quantized as A(n) = alpha (n + 1/2) l_P^2 with positive integer n and large-n semi-classical limit.
    Assumed at the start of Sec. II, eq.(2), following Bekenstein and refs [13,17]. The entire mass spectrum, temperature and entropy corrections follow from this spectrum.
  • domain assumption Leading-order entropy of each area state is S = A/4, so adjacent states differ by Delta S = alpha/4.
    Used in eqs.(3)-(4) and as the denominator in all corrected-temperature derivations. The assumption that Delta S is constant for all n is load-bearing and is not checked against the corrected entropy.
  • domain assumption The first law of black hole thermodynamics applies to discrete transitions as T = Delta M / Delta S, and as T = dM/dS for the integral.
    Invoked in eqs.(4), (11), (12), (31). This treats a discrete quantum transition as an equilibrium thermodynamic process without further justification.
  • domain assumption Landauer's principle holds in its saturated form for erasing one bit of information by a black hole transition.
    Invoked in eqs.(1), (6)-(7). The equality Delta E_BH = Delta E_L is proposed, not derived, and it fixes alpha = 4 ln 2.
  • domain assumption The running Newton constant G(r) = G0/(1 + omega G0/r^2) and the associated lapse function describe the quantum-corrected Schwarzschild black hole.
    Borrowed from refs [24,30,31] in Sec. III, eqs.(18)-(20). The quantum-corrected section depends on this model.
  • domain assumption Standard Reissner-Nordstrom and Kerr horizon-area relations, with small-charge and small-J expansions, are valid.
    Used in Secs. IV and V, eqs.(34)-(51), to connect mass spectra to area quantization.
  • standard math Taylor expansions in 1/n and truncation at the stated orders are valid.
    Used pervasively, e.g., eqs.(9)-(10). Justified by n >> 1 in the semi-classical limit.

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Pith. "Pith review of Area spectrum and black hole thermodynamics." pith.science (2026). https://pith.science/paper/YBM3R27Z

@misc{pith2026250412014,
  author       = {Pith},
  title        = {Pith review of: Area spectrum and black hole thermodynamics},
  year         = {2026},
  howpublished = {\url{https://pith.science/paper/YBM3R27Z}},
  note         = {Machine review of arXiv:2504.12014}
}
read the original abstract

The role of horizon area quantization on black hole thermodynamics is investigated in this article. The coefficient appearing in the quantization of area is fixed by an appeal to the saturated form of the Landauer's principle. Then by considering transition between discrete states of the event horizon area which in turn is equivalent to transitions between discrete mass states of the black hole, the change in the mass can be obtained. The change in mass is then equated to the product of the Hawking temperature and change in entropy of the black hole between two consecutive discrete states applying the first law of black hole thermodynamics. This gives the corrected Hawking temperature. In particular, we apply this technique to the Schwarzschild black hole, the quantum corrected Schwarzschild black hole, the Reissner-Nordstr\"{o}m black hole which is a charged black hole, and the rotating Kerr black hole geometry, and obtain the corrected Hawking temperature in each of these cases. We then take a step forward by inserting this corrected Hawking temperature in the first law of black hole thermodynamics once again to calculate the entropy of the black hole in terms of the horizon area of the black hole. This leads to logarithmic and inverse corrections to the entropy of the black hole.

Figures

Figures reproduced from arXiv: 2504.12014 by the authors.

Figure 1
Figure 1. FIG. 1. Entropy versus area plot for the Schwarzschild and [PITH_FULL_IMAGE:figures/full_fig_p005_1.png] view at source ↗

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