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The Noether conservation laws of some Vaidiya metrics. (English) Zbl 1186.83092
Summary: In this paper, we show that a large amount information can be extracted from a knowledge of the vector fields that leave the action integral invariant, viz., Noether symmetries. In addition to a larger class of conservation laws than those given by the isometries or Killing vectors, we may conclude what the isometries are and that these form a Lie subalgebra of the Noether symmetry algebra. We perform our analysis on versions of the Vaidiya metric yielding some previously unknown information regarding the corresponding manifold. Lastly, with particular reference to this metric, we show that the only variations on m(u) that occur are m=0, m=constant, m=u and m=m(u).
MSC:
83C57Black holes
94A17Measures of information, entropy
83C40Gravitational energy and conservation laws; groups of motions
References:
[1]Lindquist, R.W., Schwartz, R.A., Misner, C.W.: Vaidiya’s radiating Schwarzschild metric. Phys. Rev. 137(5B), B1364–B1368 (1965) · doi:10.1103/PhysRev.137.B1364
[2]Vaidiya, P.C.: Proc. Indian Acad. Sci. A33, 264 (1951)
[3]Finkelstein, D.: Phys. Rev. 110, 1965 (1958)
[4]Dwivedit, I.H., Joshi, P.S.: On the nature of naked singularities in Vaidiya spacetimes. Class. Quantum Gravity 6, 1599–1606 (1989) · Zbl 0695.53065 · doi:10.1088/0264-9381/6/11/013
[5]Ramachandra, B.S.: The Carter constant and Petrov classification of the Vaidiya-Einstein-Kerr spacetime. Gen. Relativ. Gravit. doi: 10.1007/s10714-009-0805-y
[6]Olver, P.: Application of Lie Groups to Differential Equations. Springer, New York (1986)
[7]Kobayashi, S.: Transformation Groups in Differential Geometry. Classics in Mathematics. Springer, Berlin (1972)
[8]Noether, E.: Invariante variationsprobleme. Nachr. Akad. Wiss. Gött. Math. Phys. Kl. 2, 235–257 (1918) (English translation in Transp. Theory Stat. Phys. 1(3), 186–207 (1971))
[9]Stephani, H.: Differential Equations: Their Solution Using Symmetries. Cambridge University Press, Cambridge (1989)