REVIEW
Quasilocal Thermodynamics of Dilaton Gravity coupled to Gauge Fields
General Relativity and Quantum Cosmology · Submitted 5 May 1995 · 30 references
Pith's verdict: A careful, self-contained extension of Brown–York quasilocal thermodynamics to dilaton gravity with Yang–Mills, axion, three- and four-form fields; the entropy and first-law results hold up, with one presentational gap at the horizon and no worked charged example.
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Referee report
The paper
The paper claims
For a broad class of dilaton-gravity theories coupled to gauge fields, the entropy of a stationary black hole is fixed by the horizon area weighted by the dilaton coupling, and the first law of thermodynamics holds as an integral over a fin
The reading
What carries the argument
The load-bearing device is the micro-canonical action: the ordinary action plus a boundary term $int_{partial Sigma} d^2x,(E N - J_a N^a + Q_A N V^A)$, chosen so that varying it fixes the extensive variables (energy, angular momentum, charges, boundary metric, dilaton, boundary gauge potentials) rather than their intensive conjugates. Because it differs from the total action only by boundary terms, it leaves the equations of motion alone; because its canonical form contains only constraints, it vanishes on a stationary classical solution except for whatever the excised horizon leaves behind. That leftover is the whole entropy calculation, and it is what makes the entropy formula a one-line consequence of the conical regularity conditions rather than a separate argument.
Extended reading notes
Core claim
The central claim is that the Brown-York quasilocal construction goes through, essentially unchanged in structure, for the most general four-dimensional action with at most two derivatives per term that couples a dilaton to gravity and to Yang-Mills, axion, three-form, and four-form gauge fields. Concretely: the quasilocal energy density $E$, the momentum density $J_a$, the stress $S_{ab}$, and the dilaton density $Y$ are read off from the boundary momentum $pi^{ij}$ and the dilaton momentum, with a reference-subtraction term $S_0$ that keeps the action linear in lapse and shift so that the extensive variables stay extensive. Conserved charges follow whenever a Killing vector on the boundary also leaves the dilaton constant along its orbits, giving a mass and an angular momentum from a flux integral over the two-surface $partial Sigma$; gauge charges follow from a covariantly constant Lie-algebra-valued scalar. The thermodynamic result is that, after excising a neighbourhood of the horizon and imposing the standard conical-regularity conditions, the micro-canonical action evaluated on the classical solution reduces to a single inner-boundary term, $-4pi int d^2x sqrt{sigma} f_{EH}(Psi)$, so that the entropy is that quantity with the sign flipped, and its variation gives the first law as a boundary integral $delta S = int_{B_{outer}} d^2x, beta,(delta E - omega^a delta J_a + V^A delta Q_A + tfrac{1}{2} S^{ab} delta sigma_{ab} + Y delta Psi + I^a_A delta W^a_A)$. With $f_{EH} = (16pi)^{-1}$ and no dilaton this collapses to the familiar quarter-area law and to the previously derived finite-system first law for vacuum and Einstein-Maxwell gravity.
Load-bearing premise
The entropy result rests entirely on excising a small neighbourhood of the event horizon and imposing the standard conical-singularity regularity conditions there, so that the entire micro-canonical action collapses to one inner-boundary term; if that excision bookkeeping misses a contribution in the presence of a dilaton-curvature coupling, the formula does not follow.
Editorial extensions
If this is right
- Entropy for any stationary black hole in this class of theories is $4pi int sqrt{sigma}, f_{EH}(Psi)$, so a conformal rescaling of the metric that removes the dilaton-curvature coupling changes the numerical value of the entropy — the frame matters.
- The first law holds for observers at a finite boundary, with $beta$ left inside the integral because temperature need not be constant over a finite enclosing surface.
- A four-form field strength acts as a cosmological constant whose value is a constant of integration set by the Gauss constraint, so the same formalism covers black holes with a dynamically determined cosmological term.
- Gauge-field contributions to the first law appear as $beta V^A delta Q_A$ only when $beta V^A$ is constant on the boundary and aligned with a gauge Killing scalar; otherwise the charge term does not factor out of the integral.
- For vanishing dilaton and axion with an Abelian gauge field, the results reduce to the known finite-system first law for vacuum and Einstein-Maxwell gravity.
Reading between the lines
- Because the entropy is weighted by $f_{EH}(Psi)$ evaluated on the horizon, two black holes with identical horizon area but different dilaton values at the horizon would carry different entropies — a discriminating statement that a concrete charged dilaton solution could test directly.
- The paper never checks the excision-plus-regularity step against an explicit dilaton black hole metric, so the natural next move is to evaluate $4pi int sqrt{sigma} f_{EH}$ on a known charged dilaton solution and compare with the horizon area in the appropriate frame.
- The formalism is restricted to actions with at most two derivatives, and the authors say so; higher-curvature corrections, which string theory certainly generates, are left as an open problem rather than absorbed.
- The treatment of conserved gauge charges requires a covariantly constant Lie-algebra-valued scalar, which most non-Abelian solutions do not possess — so the 'conserved colour charge' story is narrower than the general first law it feeds into.
Original abstract
We consider an Einstein-Hilbert-Dilaton action for gravity coupled to various types of Abelian and non-Abelian gauge fields in a spatially finite system. These include Yang-Mills fields and Abelian gauge fields with three and four-form field strengths. We obtain various quasilocal quantities associated with these fields, including their energy and angular momentum, and develop methods for calculating conserved charges when a solution possesses sufficient symmetry. For stationary black holes, we find an expression for the entropy from the micro-canonical form of the action. We also find a form of the first law of black hole thermodynamics for black holes with the gauge fields of the type considered here.
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 consider an Einstein-Hilbert-Dilaton action for gravity coupled to various types of Abelian and non-Abelian gauge fields in a spatially finite system. These include Yang-Mills fields and Abelian gauge fields with three and four-form field strengths. We obtain various quasilocal quantities associated with these fields, including their energy and angular momentum, and develop methods for calculating conserved charges when a solution possesses sufficient symmetry. For stationary black holes, we find an expression for the entropy from the micro-canonical form of the action. We also find a form of the first law of black hole thermodynamics for black holes with the gauge fields of the type considered here.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We consider an Einstein-Hilbert-Dilaton action for gravity coupled to various types of Abelian and non-Abelian gauge fields in a spatially finite system. | contribution 0.74 | 0.78 | 0.04 | asserted 0.91 |
| s1 | These include Yang-Mills fields and Abelian gauge fields with three and four-form field strengths. | contribution 0.60 | 0.15 | 0.03 | asserted 0.99 |
| s2 | We obtain various quasilocal quantities associated with these fields, including their energy and angular momentum, and develop methods for calculating conserved charges when a solution possesses sufficient symmetry. | contribution 0.90 | 0.96 | 0.18 | asserted 1.00 |
| s3 | For stationary black holes, we find an expression for the entropy from the micro-canonical form of the action. | result 0.99 | 0.96 | 0.21 | asserted 1.00 |
| s4 | We also find a form of the first law of black hole thermodynamics for black holes with the gauge fields of the type considered here. | result 0.98 | 0.95 | 0.24 | asserted 1.00 |
What was examined, and what was not
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References
How the literature uses it
42 arXiv papers cite this in 47 sentences
Papers citing it · 27 sentences without a confident label
Cites it as background 20 papers
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The idea stems from reconsidering the thermodynamics of asymptotically Anti-de Sitter (AdS) black holes in the context of a variable cosmological constant $Λ$ [ref] , which is treated as a thermodynamic pressure according to the following (perfect fluid) prescription:
Holographic dual of extended black hole thermodynamics · Introduction · 0.88
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Black Hole Chemistry [ref] – an extension of black hole thermodynamics in which the (negative) cosmological constant $Λ$ is identified with a (positive) thermodynamic pressure in $d$ spacetime dimensions via [ref] P = -Λ/8πG = (d-1)(d-2)/16πL^2 G – is 15 years old, though its roots extend further back [ref] .
Recent Developments in Holographic Black Hole Chemistry · Introduction · 0.99
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This has motivated further study into the thermodynamic phase structure of AdS black holes, where extensive effort has been devoted to understanding the extended phase space, in which the cosmological constant acts as a thermodynamic pressure [ref] via the identification [equation] which can be achieved, for example, with a gauge 3-form potential [ref] .
Critical Phenomena of Born-Infeld-de Sitter Black Holes in Cavities · The Action and Thermodynamic Quantities · 0.89
Discussion
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