Abstract
Algorithms to construct the optimal systems of dimension of at most three of Lie algebras are given. These algorithms are applied to determine the Lie algebra structure and optimal systems of the symmetries of the wave equation on static spherically symmetric spacetimes admitting G7 as an isometry algebra. Joint invariants and invariant solutions corresponding to three-dimensional optimal systems are also determined.
Highlights
It was shown in [1,2,3] that spherically symmetric spacetimes belong to one of the following four classes according to their isometries and metrics:
G6 corresponding to the static spacetimes Bertotti–Robinson and two other metrics of Petrov type
The reason is that a PDE with four independent variables can be reduced to an ordinary differential equation (ODE) using three-dimensional subalgebras satisfying the transversality condition with rank three [10]
Summary
It was shown in [1,2,3] that spherically symmetric spacetimes belong to one of the following four classes according to their isometries and metrics:. We continue the investigation started in [4] by finding the optimal system of subalgebras of dimension of at most three and the corresponding invariant solutions for spacetimes admitting G7 as isometry algebras. We can always construct a family of group invariant solutions obtained by using a subgroup of a symmetry group admitted by a given differential equation, as explained in [5]. The reason is that a PDE with four independent variables can be reduced to an ordinary differential equation (ODE) using three-dimensional subalgebras satisfying the transversality condition with rank three [10] This provides the non-trivial invariant solutions under a maximum number of symmetries. In order to find the general adjoint action of the semisimple part of L, we need to make a suitable change of basis depending on the root space decomposition or the Iwasawa decomposition of the semisimple part L based on the signature of the Killing form
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