A statistical mechanical problem in Schwarzschild spacetime

Abstract

We use Fermi coordinates to calculate the canonical partition function for an ideal gas in a circular geodesic orbit in Schwarzschild spacetime. To test the validity of the results we prove theorems for limiting cases. We recover the Newtonian gas law subject only to tidal forces in the Newtonian limit. Additionally we recover the special relativistic gas law as the radius of the orbit increases to infinity. We also discuss how the method can be extended to the non ideal gas case.

This is a preview of subscription content, log in to check access.

References

  1. 1.

    Hakim R. (1967). Remarks on relativistic statistical mechanics. I. J. Math. Phys. 8: 1315–1344

    Article  Google Scholar 

  2. 2.

    Horwitz L.P., Schieve W.C. and Piron C. (1981). Gibbs ensembles in relativistic classical and quantum mechanics. Ann. Phys. NY 137: 306–340

    Article  ADS  MathSciNet  Google Scholar 

  3. 3.

    Miller E.D. and Karsch F. (1981). Covariant structure of relativistic gases in equilibrium. Phys. Rev. D 24: 2564–2575

    Article  ADS  Google Scholar 

  4. 4.

    Droz-Vincent P. (1997). Direct interactions in relativistic statistical mechanics. Found. Phys. 27: 363–378

    Article  MathSciNet  Google Scholar 

  5. 5.

    Montesinos M. and Rovelli C. (2001). Statistical mechanics of generally covariant quantum theories: a Boltzmann-like approach. Class. Quantum Grav. 18: 555–569

    MATH  Article  ADS  MathSciNet  Google Scholar 

  6. 6.

    Manasse F.K. and Misner C.W. (1963). Fermi normal coordinates and some basic concepts in differential geometry. J. Math. Phys. 4: 735–745

    MATH  Article  MathSciNet  Google Scholar 

  7. 7.

    Parker L. and Pimentel L.O. (1982). Gravitational perturbation of the hydrogen spectrum. Phys. Rev. D 25: 3180–3190

    Article  ADS  Google Scholar 

  8. 8.

    Pauli W. (1981). Theory of relativity. Dover, New York, 139–141

    Google Scholar 

  9. 9.

    Chernikov N.A. (1964). Equilibrium distribution of the relativistic gas. Acta Phys. Pol. 26: 1069–1092

    MathSciNet  Google Scholar 

  10. 10.

    Misner C.W., Thorne J.A. and Wheeler K.S. (1973). Gravitation. Freeman, San Francisco

    Google Scholar 

  11. 11.

    Ehlers J. and Rudolph E. (1977). Dynamics of extended bodies in general relativity. Gen. Rel. Grav. 8: 197–217

    MATH  Article  MathSciNet  Google Scholar 

  12. 12.

    Parker L. (1980). One-electron atom as a probe of spacetime curvature. Phys. Rev. D 22: 1922–1934

    Article  ADS  Google Scholar 

  13. 13.

    Chernikov N.A. (1963). The relativistic gas in the gravitational field. Acta Phys. Pol. 23: 629–645

    MathSciNet  Google Scholar 

  14. 14.

    Sachs R.K. and Wu H. (1977). General Relativity for Mathematicians. Springer, Heidelberg

    Google Scholar 

  15. 15.

    Tolman R.C. (1934). Relativity, Thermodynamics and Cosmology. Clarendon, Oxford

    Google Scholar 

  16. 16.

    Klein O. (1949). On the thermodynamical equilibrium of fluids in gravitational fields. Rev. Mod. Phys. 21: 531–533

    MATH  Article  ADS  Google Scholar 

  17. 17.

    Martin-Löf A. (1979). Lecture Notes in Physics, 101. Springer, Heidelberg

    Google Scholar 

  18. 18.

    Bini D., Geralico A. and Jantzen R.T. (2005). Kerr metric, static observers and Fermi coordinates. Class. Quantum Grav. 22: 4279–4243

    Article  ADS  Google Scholar 

  19. 19.

    Horn R. and Johnson C. (1985). Matrix Analysis. Cambridge University, New York

    Google Scholar 

  20. 20.

    Li W.Q. and Ni W.T. (1979). Expansions of the affinity, metric and geodesic equations in Fermi normal coordinates about a geodesic. J. Math. Phys. 20: 1925–1929

    MATH  Article  ADS  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to David Klein.

Rights and permissions

Reprints and Permissions

About this article

Cite this article

Collas, P., Klein, D. A statistical mechanical problem in Schwarzschild spacetime. Gen Relativ Gravit 39, 737–755 (2007). https://doi.org/10.1007/s10714-007-0416-4

Download citation

Keywords

  • Ideal gas
  • Schwarzschild spacetime
  • Fermi coordinates
  • Statistical mechanics