REVIEW
A Correspondence Principle for Black Holes and Strings
High Energy Physics - Theory · Submitted 13 Dec 1996 · 25 references
Pith's verdict: The durable part is the operational matching rule and the classification of which weak-coupling object dominates; the p ≤ 2 p-brane agreement leans on a continuation the authors themselves flag.
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Referee report
The paper
The paper claims
Black hole entropy and string entropy agree at one special coupling, the authors argue, up to a constant of order one.
The reading
What carries the argument
The machinery is a scale-matching computation. For a $d$-dimensional Schwarzschild hole, setting $r_0^2$ of order $r_0^{d-2}/G$ against the string mass relation $M_s^2 = N/alpha'$ with $G = g^2 alpha'$ gives $S_{bh} = r_0^{d-1}/G = O(1) sqrt{N}$, the same scaling as the free-string entropy. For charged holes the authors work in the string metric rather than the Einstein metric, since the dilaton is not constant; the horizon area in the Einstein metric is $A = 4 pi r_0^2 cosh gamma_1 cosh gamma_p$, and the left- and right-moving oscillator numbers $N_L, N_R$ built from $M^2 - p_{L,R}^2$ reproduce that same combination of boost parameters. For black p-branes the matching condition becomes $r_0 = alpha'^{1/2}(cosh alpha)^{1/2}$, and the paper compares the horizon entropy against two candidate weak-coupling states: a few long strings, with $S_1 = alpha'^{1/2} Delta E$, and a gas of $Q^2$ massless open-string species on the branes, with $S_2$ scaling as $(Delta E)^{p/(p+1)}$. The larger of the two tracks the black hole entropy in the near-extremal and far-from-extremal regimes respectively. A redshift analysis shows the large gravitational dressing near the horizon rescales both the local energy and the internal volume in compensating ways, leaving the entropy comparison intact.
Extended reading notes
Core claim
The claim is a two-part correspondence principle. First, when the curvature at the horizon in the string metric reaches the string scale, the typical black hole state becomes a typical state of strings and D-branes with the same charges and angular momentum. Second, the mass changes by at most a factor of order unity through the transition. The consequence the authors draw is that the black hole entropy must be comparable to the string entropy, not parametrically different from it. The reason this is not automatic is that a black hole's mass is fixed in Planck units while a string's mass is fixed in string units, so the two can agree at only one value of the coupling; the principle says that value is exactly where the horizon shrinks to string size. Carrying out the comparison case by case, the authors find $S_{bh} = A/4G$ and the string or D-brane entropy agree in their dependence on mass and charge, with only an order-one numerical mismatch that depends on precisely when the black hole forms. They also note that the local Hawking temperature at the matching point always comes out to be the string scale.
Load-bearing premise
The p-brane comparisons require the classical ten-dimensional black brane formulas to remain trustworthy all the way down to the coupling where the horizon curvature is of order the string scale, even though for $n>4$ the largest curvature sits away from the horizon and $alpha'$ corrections set in earlier; the authors say plainly that it is not clear why this is justified.
Editorial extensions
If this is right
- Black hole entropy acquires a statistical interpretation within string theory for neutral Schwarzschild holes in every dimension, not only for the supersymmetric charged cases where exact counting had been done.
- The identity of the typical microstate is fixed by the charge content: no large Ramond-Ramond charge gives a single long string, one large Ramond charge gives an interacting gas of open strings on D-branes, and two or more large Ramond charges give a free gas of moduli.
- The two ways of compactifying a black p-brane, an array of black holes versus a translationally invariant solution, are reproduced by two different D-brane configurations, coincident branes versus branes with uniformly spread Wilson lines, with the higher-entropy configuration matching in each regime.
- Following a state of fixed entropy across the transition, its mass changes only by an order-one factor, so black holes and strings can be viewed as two descriptions of the same object at different coupling.
- Because the correspondence connects non-BPS black holes to strings of different weakly coupled theories, it gives an approximate check of string duality away from the supersymmetric limit.
Reading between the lines
- If the correct matching surface is where curvature first reaches the string scale rather than the horizon itself, then for the $n>4$ p-branes the correspondence as stated is being applied at the wrong place, and a strong-curvature description of the intermediate region, not a classical continuation, is what the argument actually needs.
- The principle turns black hole entropy into a constraint on the density of states at a specific coupling rather than in an asymptotic limit, which is a sharper target than the degeneracy formulas usually compared against horizon areas.
- Since the local temperature at the matching point is always the string scale, the picture suggests that the information question should be posed as how a pure string state appears in the black hole phase, not as what happens at the endpoint of evaporation.
- The order-one ambiguity means the correspondence cannot by itself discriminate between weakly coupled string theories with different degeneracies; only a computation of the numerical coefficient, or of which subset of string states is realised, could do that.
Original abstract
For most black holes in string theory, the Schwarzschild radius in string units decreases as the string coupling is reduced. We formulate a correspondence principle, which states that (i) when the size of the horizon drops below the size of a string, the typical black hole state becomes a typical state of strings and D-branes with the same charges, and (ii) the mass does not change abruptly during the transition. This provides a statistical interpretation of black hole entropy. This approach does not yield the numerical coefficient, but gives the correct dependence on mass and charge in a wide range of cases, including neutral black holes.
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
For most black holes in string theory, the Schwarzschild radius in string units decreases as the string coupling is reduced. We formulate a correspondence principle, which states that (i) when the size of the horizon drops below the size of a string, the typical black hole state becomes a typical state of strings and D-branes with the same charges, and (ii) the mass does not change abruptly during the transition. This provides a statistical interpretation of black hole entropy. This approach does not yield the numerical coefficient, but gives the correct dependence on mass and charge in a wide range of cases, including neutral black holes.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | For most black holes in string theory, the Schwarzschild radius in string units decreases as the string coupling is reduced. | background 0.93 | 0.24 | 0.05 | asserted 1.00 |
| s1 | We formulate a correspondence principle, which states that (i) when the size of the horizon drops below the size of a string, the typical black hole state becomes a typical state of strings and D-branes with the same charges, and (ii) the mass does not change abruptly during the transition. | contribution 0.99 | 0.89 | 0.76 | asserted 0.98 |
| s2 | This provides a statistical interpretation of black hole entropy. | interpretation 0.99 | 0.76 | 0.29 | asserted 0.99 |
| s3 | This approach does not yield the numerical coefficient, but gives the correct dependence on mass and charge in a wide range of cases, including neutral black holes. | limitation 0.97 | 0.84 | 0.14 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
161 arXiv papers cite this in 243 sentences
Papers citing it · 173 sentences without a confident label
Contests it 1 paper
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Really, before one takes the logarithm, one would need to multiply $σ_ls(ϵ)$ by a constant with the dimensions of energy but, in [ref] no such constant is provided by the theory.
Entropy and quantum gravity · Introduction · contest 0.79
Builds on it 6 papers
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Nevertheless, it provides a convenient starting point to understand how the correspondence between black holes and fundamental strings, as proposed in [ref] , acquires new richness and intricacy when rotation is present.
The correspondence between rotating black holes and fundamental strings · Introduction · 0.84
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This correspondence has been generalized in the case of charged black holes [ref] .
Closed string transport coefficients and the membrane paradigm · Introduction · 0.87
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The following discussion can be understood as a generalization of the celebrated Black Hole - String correspondence[In this work we refer to the correspondence, as opposed to the transition, to emphasize the fact that we are not studying the details of the transition as a dynamical process, but focusing instead on the correspondence of the entropies of the black hole system and the tower system to obtain a microscopic interpretation of the entropy of the former in terms of the microstate counting of the latter.] [ref] (see also [ref] for recent discussions on the topic), which accounted for the entropy of general Schwarzschild black holes (up numerical prefactors, but crucially matching the area dependence) by the microscopic counting of the microstates associated to a free string in the limit of vanishing string coupling.
On the Origin of Species Thermodynamics and the Black Hole - Tower Correspondence · The Black Hole - Tower Correspondence: black hole entropy from non-gravitational thermodynamics · 0.85
Uses its method or tool 2 papers
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The main idea will be to use the correspondence principle formulated in [ref] , following earlier suggestions in [ref] .
How to Expose a Black Hole · Introduction · 0.90
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One can construct an electric NS-NS charged black hole from the Schwarzshild solution by using the solution generating method [ref] .
Closed string transport coefficients and the membrane paradigm · Black hole solutions · 0.87
Compares with it 4 papers
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Our first set of results concerns the impressive quantitative agreement between the results of Strominger and Vafa [ref] and subsequent authors for the entropy of extremal and near-extremal black holes and the original Hawking entropy formulae and also between the semi-qualitative results of Susskind [ref] and of Horowitz and Polchinski [ref] for the entropy of, say, Schwarzschild black holes and the original Hawking entropy formula for those.
Entropy and quantum gravity · Introduction · 0.85
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At this point it may be mentioned that appearance of string states in boundary theory also occurred in [ref] which corresponds to the Gross-Witten transition [ref] and which was identified with a crossover from supergravity black hole solution to string state [ref] .] Beyond the critical potential we encounter different possible situations depending on the position (expectation value of the Polyakov loop) of the extra saddle point.
Phase transitions in higher derivative gravity and gauge theory: R-charged black holes · Introduction and Summary · 0.80
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The chaotic nature of the decay process together with the non trivial dependence of final microstates gave rise to a greybody emission spectrum with an effective temperature $T_eff=T_H/√(N)$, which is different from the temperature of the string/BH transition [ref] .
The chaotic emergence of thermalization in highly excited string decays · Conclusion and future directions · 0.84
Cites it as support 2 papers
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This interpretation has been confirmed in many examples in string theory [ref] .
Symmetries in quantum field theory and quantum gravity · Introduction · 0.96
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Some evidence for this can be seen in the work of Horowitz and Polchinski [ref] , based on earlier work in [ref] , who found a string background, that involves a winding condensate near the Hagedorn inverse temperature $_Hag$ in the form of highly excited self-gravitating oscillating strings.
Off-Shell Strings II: Black Hole Entropy · After Tachyon Condensation · 1.00
Cites it as background 47 papers
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According to the correspondence principle [ref] , at some particular value of $g_s$, the description of the system changes from that of a black hole to that of a fundamental string.
How to Expose a Black Hole · Review of the correspondence principle · 0.86
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String theory represents probably the best candidate nowadays for such a theory and it has a general principle (known as the “String-Black holes correspondence principle” [ref] ) individuating those microstates.
Self-interacting fundamental strings and black holes · Introduction · 0.81
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Of particular interest for the discussion of [ref] is the nature of this CFT in the limit where $β$ approaches the inverse Hagedorn temperature of string theory in flat spacetime, $β_H$.
On Small Black Holes in String Theory · Introduction and summary · 0.81
Discussion
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