Nonsaturation of Gravitational Forces
1.M. E. Fisher and D. Ruelle, J. Math. Phys. 7, 260 (1966).
2.F. J. Dyson and A. Lenard, J. Math. Phys. 8, 423 (1967).
3.A. M. Salzberg, J. Math. Phys. 6, 158 (1965).
4.It may be amusing to note that we do in fact obtain a proof of an inequality justifying this interpretation of the Pauli principle, following a rigorous treatment of the gravitational Hamiltonian (see Sec. II).
5.Note that in a d‐dimensional space, (7) would be replaced by the following estimate: so that gravitational forces just saturate for a one‐dimensional fermion system.
6.The same results remain true for particles with different masses if m is interpreted as the largest mass. The assumption of a common mass merely simplifies the writing of our equations.
7.Asymptotically, for N large enough, this can be improved into
8.A similar uncertainty relation dealing with the average squared two‐particle distance may be derived from a study of an N‐particle system with two‐body harmonic forces. See J.‐M. Lévy‐Leblond, Phys. Letters 26A, 540 (1968).
9.Corresponding to Footnote 7, for the coefficient in (36) may be asymptotically improved in
10.S. Chandrasekhar, Monthly Notices Roy. Astron. Soc. 91, 456 (1931).
10.See, for instance, S. Chandrasekhar, Introduction to the Study of Stellar Structure (The University of Chicago Press, Chicago, 1939),
10.or E. Schatzman, White Dwarfs (Interscience Publishers, Inc., New York, 1958).
11.L. Landau, Phys. Z. Sowjetunion 1, 285 (1932),
11.reprinted in Collected Papers of L. D. Landau, D. ter Haar, Ed. (Pergamon Press, Inc., New York, 1965), p. 60.
12.The nonlinear character of a completely relativistic theory of gravitation drastically modifies the saturation problem. For a very simple‐minded discussion of this point, see J.‐M. Lévy‐Leblond and P. Thurnauer, Am. J. Phys. 34, 1110 (1966).
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