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Proper conformal symmetries of conformal symmetric space-times. (English) Zbl 0659.53029

This paper begins with a discussion of conformal Killing vector fields in a vacuum space-time. It can, however, be simplified by noting that it is a simple consequence of Brinkmann’s theorem (which says that if \(g_{ab}\) and \(\sigma g_{ab}\) are vacuum metrics then either \(\sigma =cons\tan t\) or \(g_{ab}\) and \(\sigma g_{ab}\) are pp waves) that the only vacuum space-times admitting proper (i.e. not homothetic or isometric) conformal Killing vectors are (certain of the) pp waves.
The main result of the paper concerns conformally symmetric space-times (where the Weyl tensor satisfies \(C_{abcd;e}=0)\) and it is proved that if such a space-time is not conformally flat and admits a proper conformal Killing vector \(\xi\) and if \(\xi\) also satisfies \({\mathcal L}_{\xi}G_{ab}=0,\) where \(G_{ab}\) is the Einstein tensor, then M is a “plane fronted gravitational wave with parallel rays”. The author begins the proof by showing that the Petrov type must be N (and his proof can be shortened by noting that this result follows immediately from his equation (13) and the well-known Bel criteria). From this point the remainder of the proof is somewhat vacuous since any conformally symmetric type N space-time necessarily admits a nonzero covariantly constant null bivector and thus is a “plane fronted gravitational wave with parallel rays” without any assumption regarding conformal vector fields or the Einstein tensor [the reviewer, J. Phys. A 10, 29-42 (1977; Zbl 0346.53011)]. One can say more; such space-times are (generalized) plane waves purely on account on the conformally symmetric (and type N) condition.
Reviewer: G.S.Hall

MSC:

53B30 Local differential geometry of Lorentz metrics, indefinite metrics
53B50 Applications of local differential geometry to the sciences
83C40 Gravitational energy and conservation laws; groups of motions

Citations:

Zbl 0346.53011
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References:

[1] DOI: 10.1063/1.1705267 · doi:10.1063/1.1705267
[2] DOI: 10.1007/BF01210929 · Zbl 0604.53038 · doi:10.1007/BF01210929
[3] DOI: 10.1063/1.527805 · Zbl 0614.53048 · doi:10.1063/1.527805
[4] DOI: 10.1088/0264-9381/4/1/016 · Zbl 0614.53053 · doi:10.1088/0264-9381/4/1/016
[5] DOI: 10.1098/rspa.1972.0042 · Zbl 0243.53030 · doi:10.1098/rspa.1972.0042
[6] Chaki M. C., Indian J. Math. 5 pp 113– (1963)
[7] DOI: 10.1063/1.1664886 · Zbl 0176.19402 · doi:10.1063/1.1664886
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