A Morse Theory for Massive Particles and Photons in General Relativity
Fabio Giannoni (Universita' di Camerino, Italy), Antonio Masiello, (Politecnico di Bari, Italy), Paolo Piccione (Universidade de Sao Paulo,, Brazil)
TL;DR
This paper develops a Morse Theory framework for analyzing timelike geodesics and light rays in General Relativity, utilizing an extension of the Fermat Principle and global analysis techniques.
Contribution
It introduces a novel Morse Theory approach for both massive particles and photons in General Relativity, extending existing methods to include light rays.
Findings
Morse inequalities for timelike geodesics derived
A limit process for light rays established
Framework applicable to global analysis in GR
Abstract
In this paper we develop a Morse Theory for timelike geodesics parameterized by a constant multiple of proper time. The results are obtained using an extension to the timelike case of the relativistic Fermat Principle, and techniques from Global Analysis on infinite dimensional manifolds. In the second part of the paper we discuss a limit process that allows to obtain also a Morse theory for light rays.
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