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Gravitational entropy and thermodynamics away from the horizon
High Energy Physics - Theory · Submitted 27 Jan 2012
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim In this context, the Einstein equations are equivalent to the field theory thermodynamic relation TdS=dE+PdV supplemented by an equation of state.
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The abstract, sentence by sentence
We define, by an integral of geometric quantities over a spherical shell of arbitrary radius, an invariant gravitational entropy. This definition relies on defining a gravitational energy and pressure, and it reduces at the horizon of both black branes and black holes to Wald's Noether charge entropy. We support the thermodynamic interpretation of the proposed entropy by showing that, for some cases, the field theory duals of the entropy, energy and pressure are the same as the corresponding quantities in the field theory. In this context, the Einstein equations are equivalent to the field theory thermodynamic relation TdS=dE+PdV supplemented by an equation of state.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | We define, by an integral of geometric quantities over a spherical shell of arbitrary radius, an invariant gravitational entropy. | contribution 0.90 | 0.94 | 0.13 | asserted 1.00 |
| s1 | This definition relies on defining a gravitational energy and pressure, and it reduces at the horizon of both black branes and black holes to Wald's Noether charge entropy. | method 0.49 | 0.80 | 0.14 | asserted 1.00 |
| s2 | We support the thermodynamic interpretation of the proposed entropy by showing that, for some cases, the field theory duals of the entropy, energy and pressure are the same as the corresponding quantities in the field theory. | result 0.71 | 0.90 | 0.17 | asserted 0.96 |
| s3 | In this context, the Einstein equations are equivalent to the field theory thermodynamic relation TdS=dE+PdV supplemented by an equation of state. | result 0.59 | 0.76 | 0.39 | asserted 1.00 |
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The paper
Original abstract
We define, by an integral of geometric quantities over a spherical shell of arbitrary radius, an invariant gravitational entropy. This definition relies on defining a gravitational energy and pressure, and it reduces at the horizon of both black branes and black holes to Wald's Noether charge entropy. We support the thermodynamic interpretation of the proposed entropy by showing that, for some cases, the field theory duals of the entropy, energy and pressure are the same as the corresponding quantities in the field theory. In this context, the Einstein equations are equivalent to the field theory thermodynamic relation TdS=dE+PdV supplemented by an equation of state.
Human review
How the literature uses it
1 arXiv paper cite this in 2 sentences
Papers citing it
Uses its method or tool 1 paper
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Next we use Brown`s methods [ref] as applied in [ref] to show that the Lie derivative of the area of the $D-2$ hypersurface is canonically conjugate to the Wald entropy.
Wave function of the quantum black hole · Introduction · 0.95
What it cites · 11 sentences without a confident label
Uses the method or tool of 2 papers
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In these cases, we turn to the Brown–York tensor [ref] to define the energy density and pressure, $T^ab = lim_r→∞2/√(-h)(δI/δḣ_ab) ,$ where $I=∫d^Dx √(-g)L$ is the action of the gravitational theory with Lagrangian $L$ and a dot denotes a derivative with respect to $r$.
Quasilocal Energy and Conserved Charges Derived from the Gravitational Action · Gravitational entropy away from a horizon · 1.00
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In these cases, we turn to the Brown–York tensor [ref] to define the energy density and pressure, $T^ab = lim_r→∞2/√(-h)(δI/δḣ_ab) ,$ where $I=∫d^Dx √(-g)L$ is the action of the gravitational theory with Lagrangian $L$ and a dot denotes a derivative with respect to $r$.
A Stress Tensor for Anti-de Sitter Gravity · Gravitational entropy away from a horizon · 1.00
Cites as support 1 paper
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That this can be imposed at any radial position is guaranteed by the Einstein equations, which translates to [ref] $d/dr[√(-h)(K^t_t-K^x_x)]=0$ or $Θ̇_E=0$.
The Noether charge entropy in anti-deSitter space and its field theory dual · Thermodynamics in the bulk revisited · 0.96
Cites as background 14 papers
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Wald's Noether charge entropy (NCE) [ref] is a conserved geometric charge that is associated with black hole (BH) horizons.
Some Properties of Noether Charge and a Proposal for Dynamical Black Hole Entropy · Introduction · 0.98
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Wald's Noether charge entropy (NCE) [ref] is a conserved geometric charge that is associated with black hole (BH) horizons.
Black Hole Entropy is Noether Charge · Introduction · 0.98
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Wald's Noether charge entropy (NCE) [ref] is a conserved geometric charge that is associated with black hole (BH) horizons.
On Black Hole Entropy · Introduction · 0.98
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