Symmetries, conservation laws, reductions, and exact solutions for the Klein–Gordon equation in de Sitter space–times

Abstract

In this paper, we complement the analysis involving the “fundamental” solutions of the Klein–Gordon equation in de Sitter space–times given by Yagdjian and A. Galstian (Comm. Math. Phys. 285, 293 (2009); Discrete and Continuous Dynamical Systems S, 2(3), 483 (2009)). Using the symmetry generators, we classify and reduce the underlying equations and show how this process may lead to exact solutions by quadratures.

Symmetry analysis of the Klein–Gordon equation in Bianchi I spacetimes

“…the symmetry classification of the geodesic equations of Riemannian spaces [3,4], the symmetry classification of the two and three dimensional Newtonian systems [5,6,7,8] and many others [9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25]. Furthermore, a symmetry analysis of wave equation in a power-law Bianchi III spacetime spacetime can be found in [26] and a symmetry analysis of the wave equation on static spherically symmetric spacetimes, with higher symmetries, was recently carried out in [27] In [28], it was proved that for a linear, in the derivatives, second order partial differential equation (PDE) the Lie point symmetries are related with the conformal algebra of the geometry defined by the PDE.…”

Classification of the Lie and Noether point symmetries for the Wave and the Klein–Gordon equations in pp-wave spacetimes

“…Furthermore, Lie point symmetries have also been used for the study of the geodesic equations and for the determination of exact solutions of the wave equation in various gravitation models [8,9,10,11,12]. As far as concerns the wave equation in Riemannian spacetimes, a symmetry analysis of wave equation in a power-law Bianchi III spacetime can be found in [13] and a symmetry analysis of the wave equation on static spherically symmetric spacetimes, with higher symmetries, was carried out in [14]. Recently, in [15], we started a research program where we performed the symmetry classification of the wave and the Klein-Gordon equation in Bianchi I spacetimes.…”

“…Point symmetries and potentials for the Klein-Gordon equation(4) in the isometry classes[11][12][13][14] …”

Noether symmetries of vacuum classes of pp-waves and the wave equation

“…The symmetries of the heat and Poisson equation in a general Riemannian space have been determined [10,11]. A symmetric analysis of the Shrödinger equation [12] and Klein Gordon equation [12,13] for various spacetimes has been carried out. Symmetry classifications for the wave equation on Bianchi III manifold, FRW universes and spherically symmetric static spacetimes can be found in [14,15], respectively.…”