P. P. Fiziev, D. V. Shirkov
The paper presents a generalization and further development of our recent
publications where solutions of the Klein-Gordon equation defined on a few
particular D = (2 + 1)-dim static space-time manifolds were considered. The
latter involve toy models of 2-dim spaces with axial symmetry, including
dimension reduction to the 1-dim space as a singular limiting case. Here the
non-static models of space geometry with axial symmetry are under
consideration. To make these models closer to physical reality, we define a set
of "admissible" shape functions \rho(t, z) as the (2 + 1)-dim Einstein
equations solutions in the vacuum space-time, in the presence of the
\Lambda-term, and for the space-time filled with the standard "dust". It is
curious that in the last case the Einstein equations reduce to the well-known
Monge-Amp`ere equation, thus enabling one to obtain the general solution of the
Cauchy problem, as well as a set of other specific solutions involving one
arbitrary function. A few explicit solutions of the Klein-Gordon equation in
this set are given. An interesting qualitative feature of these solutions
relates to the dimension reduction points, their classification, and time
behavior. In particular, these new entities could provide us with novel insight
into the nature of P- and T-violation, and of Big Bang. A short comparison with
other attempts to utilize dimensional reduction of the space-time is given.
View original:
http://arxiv.org/abs/1104.0903
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