Static plane-symmetric scalar fields with a traceless energy-momentum tensor in general relativity

Kerr–Schild Symmetries

“…As is well known, the celebrated Kerr metric was in fact originally presented in its Kerr-Schild form [3], and the general Kerr-Schild vacuum solution was explicitly found [1,4]. The Ansatz was also succesfully applied to the Einstein-Maxwell equations [4,5] and to the case of null radiation [6]. The Kerr-Schild metrics were also analyzed on theoretical grounds, see for instance [7], and a review with the main results can be found in [8].…”

Exact solutions for the massless plane symmetric scalar field in general relativity, with cosmological constant

“…In certain cases, such as the plane-symmetric perfect fluid solution of Taub and Tabensky [8], a fluid equation of state can admit a formulation as a scalar potential. The solutions derived above can be converted to time-dependent metrics via the complex transformation u → iw and t → iv, as in Vaidya and Som [5]. Further, to avoid imaginary terms in the metric, one or more constants would also have to be chosen to be imaginary.…”

“…Einstein's equation with cosmological constant is G ab + g ab = κT ab (5) whereas the equation of the massless scalar field is…”

“…The method is exemplified in a paper by Vaidya and Som [5]. The solutions can be obtained via complex transformations, with the resulting spacetimes representing non-isotropic plane-symmetric cosmologies where the matter is given by the scalar field and cosmological constant term.…”

A Vaidya-type generalization of Kerr spacetime

“…When this tensor vanishes, the Kerr solution is recovered. The resulting current situation is neatly summarised in [2] as follows: "In the axisymmetric case, the complete solution was first found by Herlt [3], using a formalism developed by Vaidya [4,5]". The Herlt solution includes the so-called "radiating Kerr metric" constructed by Vaidya and Patel [6].…”