Cylindrical gravitational waves with two degrees of freedom: An exact solution

T. Piran; P. N. Safier; J. Katz

doi:10.1103/PhysRevD.34.331

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Cylindrical gravitational waves with two degrees of freedom: An exact solution

T. Piran

P. N. Safier and J. Katz

Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel and The Institute for Advanced Study, Princeton, New Jersey 08540

Racah Institute of Physics, The Hebrew University, Jerusalem 91904, Israel

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Phys. Rev. D 34, 331 – Published 15 July, 1986

DOI: https://doi.org/10.1103/PhysRevD.34.331

Abstract

The exact two-parameter solution of Einstein’s equations described below represents ingoing and outgoing cylindrical gravitational waves with two degrees of polarization. It has been obtained from the Kerr metric by applying a well-known trick but, unlike the Kerr metric, it has no singularities.

References (10)

W. Kinnersley, in General Relativity and Gravitation, edited by G. Shaviv and J. Rosen (Wiley, New York, 1975), p. 109.
R. Kerr, Phys. Rev. Lett. 11, 237 (1963).
ã =0, which corresponds to the Schwarzschild solution, is uninterestingly pathological.
R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967).
P. Jordan, J. Ehlers, and W. Kundt, Akad. Wiss. Lit. Mainz, Abh. Math. Naturwiss. Kl. Jahrg. No. 2 (1960); A. S. Kompaneetz, Zh. Eksp. Theor. Fiz. 34, 953 (1958) [Sov. Phys. JETP 7, 659 (1958)].
A. Einstein and N. Rosen, J. Franklin Inst. 223, 43 (1937).
Y. Nutku and M. Halil, Phys. Rev. Lett. 39, 1379 (1977).
S. Chandrasekhar and V. Ferrari, Proc. Soc. R. London 396, 55 (1984). The Nutku-Halil solution is obtained by using what Chandrasekhar calls alpha, a wave function in which the role of t and r are interchanged. In our case alpha is simply $r^{2}$ , which is very different.
In spite of the lack of asymptotic flatness in the z direction, we can define null infinity for radially going rays.
See N. Rosen, Phys. Rev. 110, 291 (1958); see also J. Boardman and P. G. Bergmann, 115, 1318 (1959). The fastest way to obtain the result is to use the formulas of J. Katz, Class. Quantum Gravit. 2, 423 (1985). Notice that OMEGA, which diverges at ${scrI}^{+}$ and ${scrI}^{-}$ does not enter in the energy formula.

[c1] W. Kinnersley, in General Relativity and Gravitation, edited by G. Shaviv and J. Rosen (Wiley, New York, 1975), p. 109.

[c2] R. Kerr, Phys. Rev. Lett. 11, 237 (1963).

[c3] ã =0, which corresponds to the Schwarzschild solution, is uninterestingly pathological.

[c4] R. H. Boyer and R. W. Lindquist, J. Math. Phys. 8, 265 (1967).

[c5] P. Jordan, J. Ehlers, and W. Kundt, Akad. Wiss. Lit. Mainz, Abh. Math. Naturwiss. Kl. Jahrg. No. 2 (1960); A. S. Kompaneetz, Zh. Eksp. Theor. Fiz. 34, 953 (1958) [Sov. Phys. JETP 7, 659 (1958)].

[c6] A. Einstein and N. Rosen, J. Franklin Inst. 223, 43 (1937).

[c7] Y. Nutku and M. Halil, Phys. Rev. Lett. 39, 1379 (1977).

[c8] S. Chandrasekhar and V. Ferrari, Proc. Soc. R. London 396, 55 (1984). The Nutku-Halil solution is obtained by using what Chandrasekhar calls alpha, a wave function in which the role of t and r are interchanged. In our case alpha is simply $r^{2}$ , which is very different.

[c9] In spite of the lack of asymptotic flatness in the z direction, we can define null infinity for radially going rays.

[c10] See N. Rosen, Phys. Rev. 110, 291 (1958); see also J. Boardman and P. G. Bergmann, 115, 1318 (1959). The fastest way to obtain the result is to use the formulas of J. Katz, Class. Quantum Gravit. 2, 423 (1985). Notice that OMEGA, which diverges at ${scrI}^{+}$ and ${scrI}^{-}$ does not enter in the energy formula.

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Cylindrical gravitational waves with two degrees of freedom: An exact solution

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