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Thermodynamic Behavior of Field Equations for f(R) Gravity
General Relativity and Quantum Cosmology · Submitted 14 Dec 2006
abstract compiled · 2026-09-23 · no language-model review
The authors' main claim The appearance of the additional term is consistent with the argument recently given by Eling {\it et al.} (gr-qc/0602001) that the horizon thermodynamics is non-equilibrium one for the $f(R)$ gravity.
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The abstract, sentence by sentence
Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics both at event horizon of static spherically symmetric black holes and apparent horizon of Friedmann-Robertson-Walker (FRW) universe, which indicates intrinsic thermodynamic properties of spacetime horizon. In the present paper we deal with the so-called $f(R)$ gravity, whose action is a function of the curvature scalar $R$. In the setup of static spherically symmetric black hole spacetime, we find that at the event horizon, the field equations of $f(R)$ gravity can be written into a form $dE = TdS - PdV + Td\bar{S}$, where $T$ is the Hawking temperature and $S=Af'(R)/4G$ is the horizon entropy of the black hole, $E$ is the horizon energy of the black hole, $P$ is the radial pressure of matter, $V$ is the volume of black hole horizon, and $d\bar S$ can be interpreted as the entropy production term due to nonequilibrium thermodynamics of spacetime. In the setup of FRW universe, the field equations can also be cast to a similar form, $dE=TdS +WdV +Td\bar S$, at the apparent horizon, where $W=(\rho-P)/2$, $E$ is the energy of perfect fluid with energy density $\rho$ and pressure $P$ inside the apparent horizon. Compared to the case of Einstein's general relativity, an additional term $d\bar S$ also appears here. The appearance of the additional term is consistent with the argument recently given by Eling {\it et al.} (gr-qc/0602001) that the horizon thermodynamics is non-equilibrium one for the $f(R)$ gravity.
background contribution method result interpretation main claim
| # | sentence | role | the authors' own work | states the main conclusion | strength |
|---|---|---|---|---|---|
| s0 | Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics both at event horizon of static spherically symmetric black holes and apparent horizon of Friedmann-Robertson-Walker (FRW) universe, which indicates intrinsic thermodynamic properties of spacetime horizon. | background 1.00 | 0.05 | 0.06 | asserted 0.90 |
| s1 | In the present paper we deal with the so-called $f(R)$ gravity, whose action is a function of the curvature scalar $R$. | background 0.57 | 0.12 | 0.03 | asserted 0.79 |
| s2 | In the setup of static spherically symmetric black hole spacetime, we find that at the event horizon, the field equations of $f(R)$ gravity can be written into a form $dE = TdS - PdV + Td\bar{S}$, where $T$ is the Hawking temperature and $S=Af'(R)/4G$ is the horizon entropy of the black hole, $E$ is the horizon energy of the black hole, $P$ is the radial pressure of matter, $V$ is the volume of black hole horizon, and $d\bar S$ can be interpreted as the entropy production term due to nonequilibrium thermodynamics of spacetime. | result 0.99 | 0.95 | 0.51 | asserted 1.00 |
| s3 | In the setup of FRW universe, the field equations can also be cast to a similar form, $dE=TdS +WdV +Td\bar S$, at the apparent horizon, where $W=(\rho-P)/2$, $E$ is the energy of perfect fluid with energy density $\rho$ and pressure $P$ inside the apparent horizon. | result 0.99 | 0.91 | 0.20 | asserted 1.00 |
| s4 | Compared to the case of Einstein's general relativity, an additional term $d\bar S$ also appears here. | result 0.73 | 0.57 | 0.14 | asserted 1.00 |
| s5 | The appearance of the additional term is consistent with the argument recently given by Eling {\it et al.} (gr-qc/0602001) that the horizon thermodynamics is non-equilibrium one for the $f(R)$ gravity. | interpretation 0.99 | 0.10 | 0.18 | hedged possibility 0.96 |
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The paper
Original abstract
Recently it has shown that Einstein's field equations can be rewritten into a form of the first law of thermodynamics both at event horizon of static spherically symmetric black holes and apparent horizon of Friedmann-Robertson-Walker (FRW) universe, which indicates intrinsic thermodynamic properties of spacetime horizon. In the present paper we deal with the so-called $f(R)$ gravity, whose action is a function of the curvature scalar $R$. In the setup of static spherically symmetric black hole spacetime, we find that at the event horizon, the field equations of $f(R)$ gravity can be written into a form $dE = TdS - PdV + Td\bar{S}$, where $T$ is the Hawking temperature and $S=Af'(R)/4G$ is the horizon entropy of the black hole, $E$ is the horizon energy of the black hole, $P$ is the radial pressure of matter, $V$ is the volume of black hole horizon, and $d\bar S$ can be interpreted as the entropy production term due to nonequilibrium thermodynamics of spacetime. In the setup of FRW universe, the field equations can also be cast to a similar form, $dE=TdS +WdV +Td\bar S$, at the apparent horizon, where $W=(\rho-P)/2$, $E$ is the energy of perfect fluid with energy density $\rho$ and pressure $P$ inside the apparent horizon. Compared to the case of Einstein's general relativity, an additional term $d\bar S$ also appears here. The appearance of the additional term is consistent with the argument recently given by Eling {\it et al.} (gr-qc/0602001) that the horizon thermodynamics is non-equilibrium one for the $f(R)$ gravity.
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References
How the literature uses it
32 arXiv papers cite this in 50 sentences
Papers citing it · 33 sentences without a confident label
Uses its method or tool 1 paper
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In this section we follow Ref. [ref] to find a higher dimensional form of the Misner-Sharp energy [ref] inside the horizon of the static spherically symmetric black hole in $f(R)$ gravity.
Black hole solutions in f(R) gravity coupled with non-linear Yang-Mills field · First Law of Thermodynamics · 0.89
Compares with it 2 papers
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Reproducing this approach we aim to inspect the internal part of entropy functional and compare the result with the one obtained by other approaches [ref] .
Is The Internal Entropy of F(R)-Gravity Really An Entropy? · Internal Entropy · 1.00
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[ref] , the procedure of ref.
Thermodynamic approach to field equations in Lovelock gravity and f(R) gravity revisited · Introduction · 0.84
Cites it as background 11 papers
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Ever since the thermodynamics of $F(R)$-Gravity was studied [ref] , the additional effects, arising from the higher order members of the function $F(R)$, were identified as disipative nature of gravity caused by a new entropy member — internal entropy [ref] .
Is The Internal Entropy of F(R)-Gravity Really An Entropy? · Introduction · 0.80
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In general, the apparent horizon is not constant but changes with time [ref] .
Non-flat and non-Extensive Thermodynamic Effects of M{\o}ller tetradic theory of gravitation on cosmology · Thermodynamic behavior of the dust and radiation cases · 0.88
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The problem of entropy production for $f(R)$ gravity has been extensively discussed in [ref] .
Viscous Spacetime Fluid and Higher Curvature Gravity · Introduction · 1.00
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