Abstract
Brinkmann [1] has shown that conformally related distinct Ricci flat solutions are pp-waves. Brinkmann’s result has been generalized to include the conformally invariant source terms. It has been shown that [4] if g ik and \(\overline g \) ik (=w −2 g ik ; w: a scalar function), are distinct metrics having the same Einstein tensor, G ik =\(\overline G \) ik , then both represent (generalized) pp-waves and w i is a null covariantly constant vector of g ik . Thus pp-waves are the only candidates which yield conformally related nontrivial solutions of G ik =T ik =\(\overline G \) ik , with T ik being conformally invariant source.
In this paper the functional form of the conformal factor for the conformally related pp-waves/generalized pp-waves has been obtained. It has been shown that the most general pp-wave, conformally related to ds 2=−2du[dv−mdy=Hdu]+P −2[dy 2+dz 2], turns out to be (au+b)−2 ds 2, where a,b are constants. Only in the special case when m=0, H=1, and P=P(y, z), the conformal factor is (au+b)−2 or (a(u+v)+b)−2.
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References
H W Brinkmann, Math. Ann. 94, 119 (1925)
J Ehlers and W Kundt, in Gravitation: An introduction to Current Research edited by L Witten (Wiley, New York, 1962)
V Daftardar and N Dadhich, General Relativ. Gravit. 26, 859 (1994) and the references therein
V Daftardar-Gejji, General Relativ. Gravit. 30, 695 (1998)
G S Hall and A D Rendall, J. Mall. Phys. 28, 1837 (1987)
H Stephani, General Relativity (Cambridge University Press, 1990)
R Maartens and S D Maharaj, Class. Quantum Gravit. 8, 503 (1991)
A A Coley and B O J Tupper, J. Math. Phys. 31, 649 (1990)
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Daftardar-Gejji, V. On conformally related pp-waves. Pramana - J Phys 56, 591–596 (2001). https://doi.org/10.1007/s12043-001-0083-z
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DOI: https://doi.org/10.1007/s12043-001-0083-z