Geodesic
The metric which allows the pure radiation field
Theoretical and applied mechanics, Volume 23 (1997) no. 1, p. 151 
Dragi Radojević. The metric which allows the pure  radiation field. Theoretical and applied mechanics, Volume 23 (1997) no. 1, p. 151 . http://geodesic.mathdoc.fr/item/TAM_1997_23_1_a8/
@article{TAM_1997_23_1_a8,
     author = {Dragi Radojevi\'c},
     title = {The metric which allows the pure  radiation field},
     journal = {Theoretical and applied mechanics},
     pages = {151 },
     year = {1997},
     volume = {23},
     number = {1},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/TAM_1997_23_1_a8/}
}
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See the original article notice from the eLibrary of Mathematical Institute of the Serbian Academy of Sciences and Arts source

We investigate a diagonal Riemannian metric. The metric represented by four unknown functions, two of them depend on $x^1$ only and two depend on $x^4$ only. We compose the Einstein tensor. The functions composing the metric tensor are determined from tho condition that the form of Einstein tensor correspond to the energy-momentum tensor of the pure radiation field. This is the special case of the electromagnetic field for which the electric and magnetic three-vectors are equal and perpendicular.