REVIEW
General Logarithmic Corrections to Black Hole Entropy
High Energy Physics - Theory · Submitted 31 Oct 2001 · 31 references
Pith's verdict: Correct saddle-point derivation, useful new coefficients for AdS and RN in arbitrary dimensions, but the abstract's framing slightly oversells what the fluctuation argument delivers.
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Referee report
The paper
The paper claims
Logarithmic corrections to black hole entropy are argued to come from ordinary thermal fluctuations, so the coefficient in front of $k$ is fixed by specific heat and temperature rather than by any particular microscopic theory.
The reading
What carries the argument
The load-bearing device is the steepest-descent evaluation of the inverse Laplace transform of the partition function. The second derivative of the entropy with respect to inverse temperature, evaluated at the saddle, equals the mean-square energy fluctuation, which in turn equals $CT^2$. That identity converts a purely thermodynamic quantity into the prefactor of the density of states, and its logarithm is the correction.
Extended reading notes
Core claim
The central claim is a master formula for the leading correction to entropy from Gaussian fluctuations about equilibrium: $S = S_0 - 1/2$ times the logarithm of the product of specific heat and temperature squared, $CT^2$. The derivation runs from the canonical partition function through an inverse Laplace transform evaluated at the saddle point, where the second derivative of the entropy with respect to inverse temperature is identified with the energy variance and hence with $CT^2$. Applied to black holes, this yields corrections of the form $-k$ times the logarithm of the horizon area, with $k=3/2$ for BTZ, $k=d/(2(d-2))$ for $d$-dimensional anti-de Sitter Schwarzschild, and $k=(d-4)/(2(d-2))$ for $d$-dimensional Reissner–Nordström — the last vanishing in four dimensions. A second route, assuming an exact entropy function of the form $a$ times the inverse temperature to the power $m$ plus $b$ times the inverse temperature to the power $n$ (the conformal-field-theory form is the special case $m=n=1$), reproduces the same leading coefficient independently of $m$ and $n$, so the result does not rest on the CFT form being exactly right.
Load-bearing premise
The argument needs the black hole to be a canonical ensemble with positive specific heat and a partition function dominated by a stable Gaussian saddle point, which is exactly what fails for the ordinary four-dimensional asymptotically flat Schwarzschild black hole.
Editorial extensions
If this is right
- Any exact microstate counting for a BTZ black hole should reproduce the coefficient $-3/2$ in front of the logarithm of the entropy; the paper notes this has already been found by independent routes.
- For $d$-dimensional anti-de Sitter Schwarzschild black holes the coefficient is $d/(2(d-2))$, which is $1$ in four dimensions and approaches $1/2$ at large $d$.
- For four-dimensional Reissner–Nordström black holes the fluctuation correction vanishes, so a nonzero logarithmic term found by exact counting there would not be thermal in origin.
- Because the coefficient is negative, the corrected entropy is slightly super-additive: the entropy of two horizons exceeds the sum of their separate entropies.
- The whole treatment fails as the temperature goes to zero, so near-extremal black holes lie outside its reach, consistent with their being strongly quantum objects.
Reading between the lines
- If the logarithmic coefficient is fixed entirely by macroscopic data — specific heat and temperature — then it is a weak discriminator between candidate quantum gravity theories; only the sub-leading polynomial terms would carry microscopic information.
- The authors concede that four-dimensional asymptotically flat Schwarzschild has negative specific heat and so falls outside the canonical treatment, while spin-network counting does find a logarithmic term there; that suggests two genuinely different mechanisms, and a Schwarzschild black hole confined in a box, or one with high-temperature degrees of freedom included, is the natural place to see w
- Super-additivity of the corrected entropy is a concrete, if weak, statement about how horizons compose, and it could be tested against any framework that assigns entropy to a union of horizons.
- The prediction that the coefficient is independent of the exponents in the assumed entropy function suggests a broader check: pick any saddle-point entropy function and verify that only the second derivative at equilibrium matters.
Original abstract
We compute leading order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around equilibrium. When applied to black holes, these corrections are shown to be of the form $-k\ln(Area)$. For BTZ black holes, $k=3/2$, as found earlier. We extend the result to anti-de Sitter Schwarzschild and Reissner-Nordstrom black holes in arbitrary dimensions. Finally we examine the role of conformal field theory in black hole entropy and its corrections.
The compiled record
The paper's own sentences, arranged, kept beside the review and separate from it. Each label is drawn from a closed vocabulary; the number beside it is the probability the classifier gave that label. Numbers in the tables are the authors', checked by code. No sentence in this record was written by a model.
The abstract, sentence by sentence
We compute leading order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around equilibrium. When applied to black holes, these corrections are shown to be of the form $-k\ln(Area)$. For BTZ black holes, $k=3/2$, as found earlier. We extend the result to anti-de Sitter Schwarzschild and Reissner-Nordstrom black holes in arbitrary dimensions. Finally we examine the role of conformal field theory in black hole entropy and its corrections.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We compute leading order corrections to the entropy of any thermodynamic system due to small statistical fluctuations around equilibrium. | contribution 0.97 | 0.94 | 0.15 | asserted 1.00 |
| s1 | When applied to black holes, these corrections are shown to be of the form $-k\ln(Area)$. | result 1.00 | 0.86 | 0.69 | asserted 1.00 |
| s2 | For BTZ black holes, $k=3/2$, as found earlier. | result 0.51 | 0.07 | 0.09 | asserted 1.00 |
| s3 | We extend the result to anti-de Sitter Schwarzschild and Reissner-Nordstrom black holes in arbitrary dimensions. | contribution 0.95 | 0.90 | 0.13 | asserted 0.97 |
| s4 | Finally we examine the role of conformal field theory in black hole entropy and its corrections. | contribution 0.95 | 0.80 | 0.06 | unclear 0.87 |
What was examined, and what was not
- arXiv listing abstract v3
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- P abstract
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References
How later papers used it
63 later papers cite it
4 later papers by other authors are listed below: 4 compared their own results with it.
Compares with it 4 of 12 papers with this label
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Found that its corrected-entropy formula for r_+>r_e is the same formula as for the non-rotating BTZ black hole in this paper.
Logarithmic corrections to entropy for black holes with hyperbolic horizon (2008) · 0.89 · paraphrased: the licence does not allow quoting
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Said that its results in general differ from the earlier results reported in this paper.
Novel logarithmic corrections to black hole entropy (2021) · 0.90 · paraphrased: the licence does not allow quoting
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Found that in the grand canonical ensemble the entropy correction is proportional to lnS with a prefactor of 3, instead of 3/2 in the canonical ensemble calculated in this paper.
On Black Hole Entropy Corrections in the Grand Canonical Ensemble (2011) · 0.98 · paraphrased: the licence does not allow quoting
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Obtained a coefficient k=1 that matches exactly with the result found earlier for four-dimensional black holes in AdS at high temperatures in this paper.
Statistical ensembles and logarithmic corrections to black hole entropy (2023) · 1.00 · paraphrased: the licence does not allow quoting
How the literature uses it
60 arXiv papers cite this in 102 sentences
Papers whose main result relates to this one's
States the same result 1 paper
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We show that the Bekenstein-Hawking entropy associated with any black hole undergoes logarithmic corrections when small thermodynamic fluctuations around equilibrium are taken into account.
Leading Log Corrections to Bekenstein-Hawking Entropy · 0.96
Papers citing it · 50 sentences without a confident label
Uses its method or tool 2 papers
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Under the steepest decent method, the equation of corrected entropy takes the the following form [ref] [equation] By using the conditions $∂S̃/∂ξ=0$ and $∂^2 S̃/∂ξ^2>0$, the corrected entropy relation under the first-order corrections has been modified [ref] and by neglecting the higher order terms, the exact expression of entropy is expressed as [ref] [equation] where $ψ$ represents the logarithmic correction parameter.
Thermodynamics and logarithmic corrections of symmergent black holes · Thermal fluctuations · 0.80
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In what follows, we will describe a procedure to compute logarithmic corrections to the entropy of such systems from the knowledge of macroscopic response functions [ref] .
Statistical ensembles and logarithmic corrections to black hole entropy · Canonical ensemble · 0.93
Compares with it 13 papers
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This is the same form as for large 5DSAdS [ref] .
Logarithmic corrections to entropy for black holes with hyperbolic horizon · 5DTBH and Gauss-Bonnet black holes · 0.93
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Therefore, our results in general differ from the earlier results reported in [ref] .
Novel logarithmic corrections to black hole entropy · Microcanonical entropy and fluctuations · 0.90
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Hence we see that in the grand canonical ensemble, the entropy correction is proportional to $lnS$, but with a prefactor of $3$, instead of $3/2$ in the canonical ensemble, calculated in [ref] , [ref] .
On Black Hole Entropy Corrections in the Grand Canonical Ensemble · BTZ Black Hole in the Grand Canonical Ensemble · 0.98
Cites it as support 1 paper
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The latter is true if and only if, the coefficient $$ is positive, which is indeed required for the black hole to satisfy the generalised second law [ref] .
Thermal fluctuations of black holes with non-linear electrodynamics and charged Renyi entropy · Thermal fluctuations and stability · 0.98
Cites it as background 23 papers
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The Bekenstein–Hawking entropy $S_=A/4G$ [ref] requires a Hilbert space of dimension $e^A/4G$, and the microstates behind it have been identified only in special settings—string theory [ref] , loop quantum gravity [ref] —each of which also predicts a subleading logarithmic term $S=A/4G+αln(A/4G)+…$ [ref] .
From arithmetic spectra to a quantum-corrected black hole geometry · Introduction · 0.91
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For black holes in AdS, the effect of thermal fluctuations on entropy was computed in earlier works [ref] .
Novel logarithmic corrections to black hole entropy · Conclusions · 0.95
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The former was studied in [ref] and the latter has been extensively investigated by several authors (see, e.g [ref] , [ref] , [ref] ).
On Black Hole Entropy Corrections in the Grand Canonical Ensemble · Introduction · 0.82
Discussion
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