Summary
Lichnerowicz (1960), by using a property of the Riemann tensor, has given the field equations, Rij=ϱwiwj, wiwi=0, ϱ being a scalar and has termed them as representing a state of « total radiation ». Recently Rao (1970) derived these field equations under a geometrical relation which as stated by him imposes on the Rieman tensor a severe restriction, and he obtained a class of exact solutions of the above field equations corresponding to the cylindrically symmetric space-time with two degrees of freedom. The present authors, in this paper, have obtained exact wave solutions of the field equations representing zero-rest-mass scalar field in the generalized Einstein-Rosen space-time, and it has been shown that by a suitable choice of the scalar field, the field equations considered by Lichnerowicz and Rao can be deduced without imposing any condition on the geometrical nature of the Riemann tensor. Finally a method is given by which most general solutions of these field equations, can be generated from those of the empty space field equations. It is further shown that the solutions of Rao (1960) are only a special case of those found in this paper.
Article PDF
Similar content being viewed by others
Avoid common mistakes on your manuscript.
References
A. Einstein -N. Rosen,On gravitational waves, J. Franklin Inst.,223 (1937), pp. 43–54.
N. Rosen,Some cylindrical gravitational waves, Bull. Res. Coun. Israel,3 (1954), p. 328.
W. B. Bonnor,Nonsingular fields in general relativity, J. Math. Mech.,6 (1957), p. 203.
J. Weber -J. A. Wheeler,Reality of cylindrical gravitational waves of Einstein and Rosen, Rev. mod. phys.,29 (1957), p. 509.
L. Marder,Gravitational waves in general relativity, I:Cylindrical waves, Proc. roy. soc., A344 (1958), p. 524.
A. S. Kompaneets,Strong gravitational waves in free space, Soviet physics JETP,34 (1958), pp. 659–660.
K. B. Lal -T. Singh,Cylindrical wave solutions of Einstein's field equations of general relativity containing zero-rest-mass scalar fields I, Tensor, N.S.,27 (1973), pp. 211–216.
A. I. Janis -E. T. Newman -J. Winicour,Reality of Schwarzschild singularity, Phys. Rev. Letters,20 (1968), pp. 878–880.
R. Courant - K. Friedrichs,Shock waves and supersonic flow, Interscience N. Y. (1948), p. 41.
F. A. E. Pirani,Invariant formulation of gravitational radiation theory, Phys. Review,105 (1957), p. 1089.
J. K. Rao,Cylindrically symmetric null fields in general relativity, Indian Jour. of Pure and Appl. Maths.,3 (1970), p. 365.
A. Lichnerowicz,Onde et radiations electromagnétiques et gravitationelles en rélativité générale, Ann. mat. pura appl., (IV),50 (1960), p. 450.
J. J. Stachel,Cylindrical gravitational news, Journal Maths. Phys., U.S.A.,7 (1966), pp. 1321–1331.
Author information
Authors and Affiliations
Additional information
Entrata in Redazione il 15 settembre 1976.
Rights and permissions
About this article
Cite this article
Lal, K.B., Khan, M.Q. Cylindrical wave solutions of the field equations representing zero-rest-mass scalar fields and those of lichnerowicz's « total radiation » in a generalized Einstein-Rosen space-time. Annali di Matematica 115, 181–192 (1977). https://doi.org/10.1007/BF02414716
Issue Date:
DOI: https://doi.org/10.1007/BF02414716