A new perspective on the Ermakov-Pinney and scalar wave equations
The first part of the paper proves that a subset of the general set of Ermakov-Pinney equations
can be obtained by differentiation of a first-order non-linear differential equation. The second part
of the paper proves that, similarly, the equation for the amplitude function for the parametrix of
the scalar wave equation can be obtained by covariant differentiation of a first-order non-linear
equation. The construction of such a first-order non-linear equation relies upon a pair of auxiliary
1-forms (psi,rho). The 1-form psi satisfies the divergenceless condition div(psi) = 0, whereas the 1-form rho
fulfills the non-linear equation div(rho)+rho**2 = 0. The auxiliary 1-forms (psi,rho) are evaluated explicitly
in Kasner space-time, and hence also amplitude and phase function in the parametrix are obtained.
Thus, the novel method developed in this paper can be used with profit in physical applications.
This work is licensed under a Creative Commons Attribution 4.0 International License.
Letters in High Energy Physics (LHEP) is an open access journal published by Andromeda Publishing and Education Services. The articles in LHEP are distributed according to the terms of the creative commons license CC-BY 4.0. Under the terms of this license, copyright is retained by the author while use, distribution and reproduction in any medium are permitted provided proper credit is given to original authors and sources.
Terms of Submission
By submitting an article for publication in LHEP, the submitting author asserts that:
1. The article presents original contributions by the author(s) which have not been published previously in a peer-reviewed medium and are not subject to copyright protection.
2. The co-authors of the article, if any, as well as any institution whose approval is required, agree to the publication of the article in LHEP.