Next Article in Journal
Complexity and the Emergence of Physical Properties
Next Article in Special Issue
Geometric Thermodynamics: Black Holes and the Meaning of the Scalar Curvature
Previous Article in Journal
Structure of a Global Network of Financial Companies Based on Transfer Entropy
 
 
Font Type:
Arial Georgia Verdana
Font Size:
Aa Aa Aa
Line Spacing:
Column Width:
Background:
Article

Is Gravity Entropic Force?

College of Physical Science and Technology, Hebei University, Baoding 071002, China
Entropy 2014, 16(8), 4483-4488; https://0-doi-org.brum.beds.ac.uk/10.3390/e16084483
Submission received: 5 June 2014 / Revised: 4 August 2014 / Accepted: 4 August 2014 / Published: 11 August 2014
(This article belongs to the Special Issue Entropy and Spacetime)

Abstract

:
If we assume that the source of thermodynamic system, ρ and p, are also the source of gravity, then either thermal quantities, such as entropy, temperature, and chemical potential, can induce gravitational effects, or gravity can induce thermal effects. We find that gravity can be seen as entropic force only for systems with constant temperature and zero chemical potential. The case for Newtonian approximation is discussed.

1. Introduction

It has been recognized that there are profound connections between gravity and thermodynamics [15]. Since then, these connections has been steadily becoming stronger. It has been shown that the entropy S can be taken as the Noether charge associated with the diffeomorphism invariance of the theory [6,7]. In [8], the Einstein equation has been derived from the first law of thermodynamics. This attempt also has been investigated in modified gravity theories [911] and has been revisited in [12]. In a wide class of spacetime, the field equations in both general relativity and Lovelock theories can be expressed as a thermodynamic identity near the horizon [1315] (see a review [16]). In [17], the generalized Tolman–Oppenheimer–Volkoff equation is derived by using the maximum entropy principle to a charged perfect fluid, impling that there are fundamental relationships between general relativity and ordinary thermodynamics. The equations of motion for modified gravity theories, such as F(R) gravity, the scalar-Gauss–Bonnet gravity, F( Entropy 16 04483f1) gravity, and the non-local gravity, are equivalent to the Clausius relation in thermodynamics [18]. In [19], the Einstein–Hilbert action can be constructed by minimizing the free energy. It was argued that the variation of the surface term evaluated on any null surface which acts a local Rindler horizon can be given a direct thermodynamic interpretation [20]. Gravity was explained as an entropic force caused by changes in the information associated with the positions of material bodies [21]. Possible modifications and extensions to this interesting idea were proposed [2225]. All these studies were based on some assumptions, such as Unruh temperature, horizon, null surfaces, and so on. In [26], The thermal entropy density has been obtained for any arbitrary spacetime without assuming a temperature or a horizon, implying that gravity possesses thermal effects, or, thermal entropy density possesses effects of gravity.
Here we generalize the results in [26] to the case of nonzero chemical potential. The results we obtained indicate that the changes of temperature, entropy, particles, and chemical potential will result in gravitational force, or gravitational force will induce changes of temperature, entropy, particles, and chemical potential.

2. Relations between Gravity and Thermodynamics

Both the energy density ρ and the pressure p play important roles in general relativity or thermodynamics. ρ and p are components of the stress-energy tensor in general relativity and are fundamental variables in thermodynamics. Let us begin with the first law of thermodynamics for fluids consisting of particles in curved spacetime
d E = T d S - p d V + μ d N .
where E, S, and N represent the total energy, entropy and particle number within the volume V, μ, T, and p are the chemical potential, the temperature, and the pressure of the perfect fluid, respectively. d V = h d 3 x with h the determinant of the spatial metric. We take c = G = 1 and use metric signature (−,+,+, +) throughout this paper. Rewriting (1) in terms of densities
d ( ρ V ) = T d ( s V ) - p d V + μ d ( n V ) ,
we can easily get
ρ d V + V d ρ = T V d s + T s d V - p d V + n μ d V + V μ d n ,
where s is the entropy density and n the particle number density. Applying Equation (1) to a unit volume, we have
d ρ = T d s + μ d n .
Combining Equations (3) and (4), we obtain the integrated form of Gibbs–Duhem relation [27]
T s = ρ + P - μ n .
For μ = 0, Equation (5) reduces to the thermal entropy density obtained in [26]. Now, considering the Einstein equation in [28]
R μ ν - 1 2 g μ ν R + Λ g μ ν = 8 π T μ ν ,
and the stress energy tensor of the perfect fluid
T μ ν = g μ ν p + ( ρ + p ) u μ u ν ,
we obtain
R - 4 Λ = - 8 π ( 3 p - ρ ) .
Using the 3 + 1 decomposition of Einstein equation, we derive [28,29]
n μ n ν R μ ν + 1 2 R - Λ = 8 π E ,
where nμ is the unit normal vector field to the 3 dimension hypersurfaces ∑ and = Γ2(ρ+p) − p with Γ the Lorentz factor. According to the scalar Gauss relation, one can get
R + K 2 - K i j K i j - 2 Λ = 16 π E ,
where is the Ricci scalar of the 3 dimension hypersurfaces ∑, Kij the extrinsic curvature tensor of ∑, and K the trace of the Kij . Combining Equations (8) and (10), we obtain the expression of ρ + p in general relativity [26]
ρ + p = 1 4 π ( 4 Γ 2 - 1 ) [ R + K 2 - K i j K i j - 1 2 R ] .
The four dimension Ricci scalar, R, can be decomposed as [29]
R = R + K 2 + K i j K i j - 2 N L m K - 2 N D i D i N ,
where m is the Lie derivative along m of any vector tangent to ∑ and Di is the Levi–Civita connection associated with the metric of the 3 dimension hypersurfaces ∑. Then Equation (11) can be expressed with three dimension spacial geometrical quantities as [26]
ρ + p = 1 8 π ( 4 Γ 2 - 1 ) [ R + K 2 - 3 K i j K i j + 2 N L m K + 2 N D i D i N ] .
In order to relate thermodynamics with gravity, we must suggest a hypothesis. We first review Newton’s equivalence principle so as to have a better understanding. When a particle falls freely in a gravitational field, the gravity is also the inertial force. This fact leads to Newton’s equivalence principle that the inertial and the gravitational mass of a particle are equal. Like the case of Newton’s equivalence principle, for any perfect fluid in spacetime the energy density ρ and the pressure p are both the sources of gravity and thermodynamic system. So we put forward a hypothesis that the source of thermodynamic system, ρ and p, are also the source of gravity, namely
( ρ + p ) gravitational source = ( ρ + p ) thermal source .
For radiation, this hypothesis holds [26]. In [8], the δQ of thermodynamic system was assumed as the energy flux and then the Einstein equation was obtained. Analogous assumptions have also been suggested in [17,26,30,31], though these assumptions have not been aware of by the authors. With this hypothesis (14) and combining Equations (5) and (13), we obtain
T s + μ n = 1 8 π ( 4 Γ 2 - 1 ) × [ R + K 2 - 3 K i j K i j + 2 N L m K + 2 N D i D i N ] .
The terms of the left-hand side of this equation are thermodynamical quantities, while the terms of the right-hand side of this equation are geometrical quantities of the spacetime. Equation (15) implies that gravity can induce thermal effects, or, thermal quantities, such as entropy, temperature, and chemical potential, can induce gravitational effects. To understand the physical significance of Equation (15) well, we consider the Newtonian approximation: gμν = ημν +hμν, with hμν ≪ 1. For matter with p ≃ 0 (the pressure of a body becomes important when its constituent particles are traveling at speeds close to that of light, which we can exclude from the Newtonian limit by hypothesis), such as dust or dark matter, we have the Poisson equation ∇2ϕ = 4πρ with ϕ = −h00/2. Taking into account Gibbs–Duhem relation (5) and the hypothesis (14), we obtain
2 ϕ = 4 π ( T s + μ n ) .
Equation (16) (or (15)) relates gravity with thermodynamics. Variations in temperature, entropy, chemical potential, and particle number will lead to a variation in the potential ϕ, and vice versa. Equation (16) (or (15)) also indicates that gravity is related to the entropy but is not entropic force. Only for systems with constant temperature and zero chemical potential, gravitational force is entropic force. These results confirm the arguments in [32] that experiments with ultra-cold neutrons in the gravitational field of Earth disprove the speculations on the entropic origin of gravitation. In [33], it also argued that the argument for the entropic origin of gravity is problematic.

3. Conclusions

We have shown that if we conjecture that the source of thermodynamic system, ρ and p, are also the source of gravity: (ρ + p)gravitational source = (ρ + p)thermal source, thermal quantities, such as entropy, temperature, and chemical potential, can induce gravitational effects, or gravity can induce thermal effects. For Newtonian approximation, the gravitational potential is related to the temperature, entropy, chemical potential, and particle number, which implies that gravity is entropic force only for systems with constant temperature and zero chemical potential. For general case, gravity is not an entropic force. Whether the results obtained here can be generalized to the case of modified gravity, such as F(R) gravity [10] and F( Entropy 16 04483f1) gravity [18], is worthy of investigation. All the analyses have been carried out without assuming a specific expression of temperature or horizon. For a static system at thermal equilibrium in general relativity, the temperature of the perfect fluid may take the form, T - g 00 = c o n s t ., which is called the Tolman temperature [3436]. Whether the temperature in Equation (1) can be taken as Tolman temperature is also worthy of further investigation. The results we obtained confirm that there is a profound connection between gravity and thermodynamics.

Acknowledgments

This study is supported in part by National Natural Science Foundation of China (Grant Nos. 11147028 and 11273010) and Hebei Provincial Natural Science Foundation of China (Grant No. A2014201068).

Conflicts of Interest

The author declares no conflict of interest.

References

  1. Cocke, W.J. A maximum entropy principle in general relativity and the stability of fluid spheres. Ann. Inst. Henri Poincaré 1965, 2, 283–306. [Google Scholar]
  2. Bekenstein, J.D. Black holes and entropy. Phys. Rev. D 1973, 7, 2333–2346. [Google Scholar]
  3. Hawking, S.W. Particle Creation by Black Holes. Commun. Math. Phys 1975, 43, 199–220. [Google Scholar]
  4. Davies, P.C.W. Scalar particle production in Schwarzschild and Rindler metrics. J. Phys. A 1975, 8, 609–616. [Google Scholar]
  5. Unruh, W.G. Notes on black hole evaporation. Phys. Rev. D 1976, 14. [Google Scholar] [CrossRef]
  6. Wald, R.M. Black hole entropy is the Noether charge. Phys. Rev. D 1993, 48, 3427–3431. [Google Scholar]
  7. Iyer, V.; Wald, R.M. Some properties of Noether charge and a proposal for dynamical black hole entropy. Phys. Rev. D 1994, 50, 846–864. [Google Scholar]
  8. Jacobson, T. Thermodynamics of Spacetime: The Einstein Equation of State. Phys. Rev. Lett 1995, 75, 1260–1263. [Google Scholar]
  9. Eling, C.; Guedens, R.; Jacobson, T. Non-equilibrium thermodynamics of spacetime. Phys. Rev. Lett 2006, 96, 121301. [Google Scholar]
  10. Elizalde, E.; Silva, P.J. F(R) gravity equation of state. Phys. Rev. D 2008, 78, 061501. [Google Scholar]
  11. Brustein, R.; Hadad, M. The Einstein equations for generalized theories of gravity and the thermodynamic relation delta Q = TδS are equivalent. Phys. Rev. Lett 2009, 103, 101301. [Google Scholar]
  12. Makela, J.; Peltola, A. Gravitation and thermodynamics: The Einstein equation of state revisited. Int. J. Mod. Phys. D 2009, 18, 669–689. [Google Scholar]
  13. Padmanabhan, T. Classical and quantum thermodynamics of horizons in spherically symmetric space-times. Class. Quan. Gravity 2002, 19, 5387–5408. [Google Scholar]
  14. Paranjape, A.; Sarkar, S.; Padmanabhan, T. Thermodynamic route to field equations in Lancos-Lovelock gravity. Phys. Rev. D 2006, 74, 104015. [Google Scholar]
  15. Kothawala, D.; Sarkar, S.; Padmanabhan, T. Einstein’s equations as a thermodynamic identity: The Cases of stationary axisymmetric horizons and evolving spherically symmetric horizons. Phys. Lett. B 2007, 652, 338–342. [Google Scholar]
  16. Padmanabhan, T. Thermodynamical Aspects of Gravity: New insights. Rep. Prog. Phys 2010, 73, 046901. [Google Scholar]
  17. Gao, S. A general maximum entropy principle for self-gravitating perfect fluid. Phys. Rev. D 2011, 84, 104023. [Google Scholar]
  18. Bamba, K.; Geng, C.Q.; Nojiri, S.; Odintsov, S.D. Equivalence of modified gravity equation to the Clausius relation. Europhys. Lett 2010, 89, 50003. [Google Scholar]
  19. Bracken, P. The Einstein–Hilbert Action Horizons and Connections with Thermodynamics. Adv. Stud. Theor. Phys 2012, 6, 83–93. [Google Scholar]
  20. Parattu, K.; Majhi, B.R.; Padmanabhan, T. Structure of the gravitational action and its relation with horizon thermodynamics and emergent gravity paradigm. Phys. Rev. D 2013, 87, 124011. [Google Scholar]
  21. Verlinde, E.P. On the Origin of Gravity and the Laws of Newton. J. High Energy Phys 4, 029. [CrossRef]
  22. Gao, C. Modified Entropic Force. Phys. Rev. D 2010, 81, 087306. [Google Scholar]
  23. Li, M.; Wang, Y. Quantum UV/IR Relations and Holographic Dark Energy from Entropic Force. Phys. Lett. B 2010, 687, 243–247. [Google Scholar]
  24. Cai, Y.-F.; Saridakis, E.N. Inflation in Entropic Cosmology: Primordial Perturbations and non-Gaussianities. Phys. Lett. B 2011, 697, 280–287. [Google Scholar]
  25. Hendi, S.H.; Sheykhi, A. Entropic Corrections to Einstein Equations. Phys. Rev. D 2011, 83, 084012. [Google Scholar]
  26. Yang, R. The thermal entropy density of spacetime. Entropy 2013, 15, 156–161. [Google Scholar]
  27. Gao, S.; Wald, R. Physical process version of the first law and the generalized second law for charged and rotating black holes. Phys. Rev. D 2001, 64, 084020. [Google Scholar]
  28. Wald, R.M. General Relativity; The University of Chicago Press: Chicago, IL, USA, 1984. [Google Scholar]
  29. Gourgoulhon, E. 3+1 Formalism and Bases of Numerical Relativity. 2007. arXiv:gr-qc/0703035. [Google Scholar]
  30. Fang, X.; Gao, S. General proof of the entropy principle for self-gravitating fluid in static spacetimes. 2013. arXiv:1311.6899. [Google Scholar]
  31. Green, S.R.; Schiffrin, J.S.; Wald, R.M. Dynamic and Thermodynamic Stability of Relativistic, Perfect Fluid Stars. Class. Quantum Gravity 2014, 31, 035023. [Google Scholar]
  32. Kobakhidze, A. Gravity is not an entropic force. Phys. Rev. D 2011, 83, 021502. [Google Scholar]
  33. Gao, S. Is gravity an entropic force. Entropy 2011, 13, 936–948. [Google Scholar]
  34. Tolman, R.C. On the Weight of Heat and Thermal Equilibrium in General Relativity. Phys. Rev 1930, 35. [Google Scholar] [CrossRef]
  35. Tolman, R.C.; Ehrenfest, P. Temperature Equilibrium in a Static Gravitational Field. Phys. Rev 1930, 36. [Google Scholar] [CrossRef]
  36. Rovelli, C.; Smerlak, M. Thermal time and the Tolman-Ehrenfest effect: Temperature as the “speed of time”. Class. Quantum Gravity 2011, 28, 075007. [Google Scholar]

Share and Cite

MDPI and ACS Style

Yang, R. Is Gravity Entropic Force? Entropy 2014, 16, 4483-4488. https://0-doi-org.brum.beds.ac.uk/10.3390/e16084483

AMA Style

Yang R. Is Gravity Entropic Force? Entropy. 2014; 16(8):4483-4488. https://0-doi-org.brum.beds.ac.uk/10.3390/e16084483

Chicago/Turabian Style

Yang, Rongjia. 2014. "Is Gravity Entropic Force?" Entropy 16, no. 8: 4483-4488. https://0-doi-org.brum.beds.ac.uk/10.3390/e16084483

Article Metrics

Back to TopTop