Abstract
Using the Weyl-type canonical coordinates, an integration of Einstein's field equations in the cylindrosymmetric case considered by Kurşunoğlu is reexamined. It is made clear that the resulting metric is not describing the spacetime in a rotating frame, but in astatic cylindrical elastic medium. The conclusion of Kurşunoğlu that “for an observer on a rotating disk there is no way of escape from a curved spacetime” is therefore not valid. The metric in an empty rotating frame is found as a solution of Einstein's field equations, and is not orthogonal. It is shown that the corresponding orthogonal solution represents spacetime in an inertial frame expressed in cylindrical coordinates. Introducing a noncoordinate basis, the metric in a rotating frame is given the static form of Kurşunoğlu's solution. The essential role played by the nonvanishing structure coefficients in this case is made clear.
Similar content being viewed by others
References
B. Kurşunoğlu,Proc. Cambr. Phil. Soc. 47, 177 (1951).
H. Weyl,Ann. Phys. Lpz. 54, 117 (1917).
H. Levy,Nuovo Cimento 56B, 253 (1968).
F. J. Tipler,Phys. Rev. D 9, 2203 (1974).
M. M. Som, A. F. F. Teixeira, and I. Wolk,Gen. Rel. Grav. 7, 263 (1976).
A. Sloane,Aust. J. Phys. 31, 427 (1978).
W. B. Bonnor,J. Phys. A: Math. Gen. 13, 2121 (1980).
J. N. Islam,Proc. R. Soc. Lond. A 372, 111 (1980).
A. K. Raychauchury and S. N. Guha Thakurta,Phys. Rev. D 22, 802 (1980).
L. Marder,Proc. R. Soc. Lond. A 224, 524 (1958).
J. Horsky and J. Novotny,J. Phys. A: Math. Gen. 2, 251 (1969).
J. Horsky,Czech. J. Phys. B18, 569 (1968).
R. C. Tolman,Relativity, Thermodynamics and Cosmology (Clarendon Press, Oxford, 1934), §92.
W. Wilson,Phil. Mag. 40, 703 (1920).
R. J. Adler, M. J. Bazin, and M. Schiffer,General Relativity, 2nd ed. (McGraw-Hill, 1975), §4.2.
H. Arzeliès,Relativité généralisée, Fasc. 1 (Gauthier-Villars, 1961), §198.
M. P. Ryan, Jr. and L. C. Shepley,Homogeneous Relativistic Cosmologies (Princeton Univ. Press, 1975), Chapter 2.
C. Misner, K. Thorne, and J. Wheeler,Gravitation (Freeman, 1973), p. 204.
J. F. Corum,J. Math. Phys. 18, 770 (1977).
J. v. Weyssenhoff,Bull. Acad. Polon. Sc. Lett., Ser. A,1937, 252.
C. Møller,The Theory of Relativity, 2nd ed. (Clarendon Press, 1972), §8.13.
T. Levi-Civita,R. C. Accad. Lincei 28, 101 (1919).
Author information
Authors and Affiliations
Rights and permissions
About this article
Cite this article
Grøn, Ø. The metric in a static cylindrical elastic medium and in an empty rotating frame as solutions of Einstein's field equations. Found Phys 12, 509–520 (1982). https://doi.org/10.1007/BF00729998
Received:
Revised:
Issue Date:
DOI: https://doi.org/10.1007/BF00729998