Now we derive the relation from (3.15) for the equilibrium process with . For clearness, we use the the static mass density to
replace V. Substituting (3.7), (3.8) and (3.9) into (3.15), we get dimensionless differential equation
The solution is given by
where parameters are defined by
(3.21) shows how the internal potentials influence the mass density. (3.21) reduces to (3.10) if .
4. Discussion and Conclusion
1) The above calculations show that functions of state consistent with relativity should include the influences of gravity. The energy-momentum tensor and geodesics connect the macro concepts with micro movements of particles.
2) In the case of ideal gas with at low temperature, (3.11)-(3.12) give the equation of state for the adiabatic monatomic gas
which is identical to the empirical law in thermodynamics. When , we have , so the adiabatic index is not a constant for large range of
temperature due to the relativistic effect. These results show the validity of (3.11)-(3.13) and the consistence with normal thermodynamics.
3) By (3.11), letting or , we get the Stefan-Boltzmann’s law . This means that the above results automatically include photons, and the Stefan-Boltzmann’s law is also valid for the ultra-relativistic particles.
4) In general relativity, all processes occur automatically, and is independent of any practical process. Of course, is related to the property of particles. Furthermore, equation of state (3.11)-(3.12) provides a singularity-free stellar structure in thermal equilibrium  .
5) In the case , the motion of the particles will slightly deviate from the geodesic. By (3.21) we find, leads to , which means the
zero temperature can not reach. The physical reason for such conclusion is unclear.
The author is grateful to Prof. Ta-Tsien, Prof. Tie-Hu Qin and Prof. Ji-Zong Li for encouragement.
Conflicts of Interest
The authors declare no conflicts of interest.
Stewart, G.J.M. (1977) On Transient Relativistic Thermodynamics and Kinetic Theory. Proceedings of the Royal Society London A, 357, 59-75.
Israel, W. and Stewart, J.M. (1979) Transient Relativistic Thermody-namics and Kinetic Theory. Annals of Physics, 118, 341-372.
Carter, B. (1991) Convective Variational Approach to Relativistic Thermodynamics of Dissipative Fluids. Proceedings of the Royal Society London A, 433, 45.
|||Lichnerowicz, A. (1967) Relativistic Hydrodynamics and Magnetohydrodynamics. Benjamin, New York.|
|||Andersson, N. and Comer, G.L. (2007) Relativistic Fluid Dynamics: Physics for Many Different Scales. Living Reviews in Relativity, 10, 1.|
Hiscock, W.A. and Lindblom, L. (1983) Stability and Causality in Dissipative Relativistic Fluids. Annals of Physics, 151, 466-496.
Gourgoulhon, E. (2006) An Introduction to Relativistic Hydrodynamics. EAS Publications Series, 21, 43.
|||Weinberg, S.L. (1972) Gravitation and Cosmology. Ch. 2.8, Ch. 5, Ch. 11, Wiley, New York.|
Burko, L.M. (1995) Answer to Question #10, Cooling and Expansion of the Universe. American Journal of Physics, 63, 1065-1066.
|||Keeports, D. (1995) Answer to Question #10, Cooling and Expansion of the Universe. American Journal of Physics, 63, 1067.|
Blau, S. (1995) What Happens to Energy in the Cosmic Expansion? American Journal of Physics, 63, 1066-1067.
Clifton, T. and Barrow, J.D. (2007) The Ups and Downs of Cyclic Universes. Physical Review D, 75, Article ID: 043515.
Gong, Y.G., Wang, B. and Wang, A.Z. (2007) Thermodynamical Properties of the Universe with Dark Energy. JCAP, 0701, 024.
Alcaniz, J.S.J. and Lima, A.S. (2005) Interpreting Cosmological Vacuum Decay. Physical Review D, 72, Article ID: 063516.
|||Izquierdo, G. and Pavón, D. (2006) Dark Energy and the Generalized Second Law. Physics Letters B, 633, 420-426.|
Clifton, T. and Barrow, J.D. (2006) Decaying Gravity. Physical Review D, 73, Article ID: 104022.
|||Rahvar, S. (2006) Cooling in the Universe.|
|||Gu, Y.Q. (2007) Structure of the Star with Ideal Gases.|
|||Gu, Y.Q. (2017) The Vierbein Formalism and Energy-Momentum Tensor of Spinors.|
Gu, Y.Q. (2007) A Cosmological Model with Dark Spinor Source. International Journal of Modern Physics A, 22, 4667-4678.
|||Gu, Y.Q. (2017) Natural Coordinate System in Curved Space-Time.|
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