A conformally invariant variational problem for time-like curves

TL;DR

This paper investigates a conformally invariant variational problem for time-like curves in Einstein universes, characterizing stationary solutions and explicitly solving for certain cases using advanced mathematical functions.

Contribution

It introduces a conformally invariant variational problem for time-like curves and provides explicit solutions in four-dimensional Einstein universes using elliptic functions.

Findings

Stationary curves are confined to Einstein universes of dimensions 2, 3, or 4.

Linearly-full stationary curves in 4D Einstein universes can be explicitly integrated.

Solutions involve elliptic functions, integrals, and Jacobi's theta functions.

Abstract

We study the conformally invariant variational problem for time-like curves in the $n$-dimensional Einstein universe defined by the conformal strain functional. We prove that the stationary curves are trapped into an Einsetin universe of dimension $2$, $3$ or $4$. We study the linearly-full stationary curves in a four-dimensional Einstein universe and we show that they can be integrated by quadratures in terms of elliptic functions, elliptic integrals and Jacobi's theta functions.