Noether symmetries of the minimal surface Lagrangian for Gödel-type spacetimes

TL;DR

This paper addresses the Noether symmetries of the minimal surface Lagrangian in Gödel-type spacetimes by computing the symmetry generators for four classes I–IV. It applies the Noether framework to a three-variable minimal-surface problem with a constant-volume constraint, deriving a set of conserved currents tied to the spacetime's Killing structure. The main results show that the NS algebra matches the Killing algebra for most classes (five generators; special class I and Class IV yield seven and six generators, respectively), with explicit gauge vectors proportional to the corresponding Killing vectors. The conservation relations enable reduction of the minimal surface PDE to ODEs, illustrating the practical utility of Noether symmetries in analyzing minimal surfaces in rotating, causality-violating Gödel-type spacetimes.

Abstract

We investigate the Noether symmetries of the minimal surface Lagrangian for four classes of metrics in Gödel-type spacetimes. Then, calculating the Noether symmetries for all classes, namely, classes I, II, III and IV, we determine the conserved fields corresponding to each classes, allowing us to derive a comprehensive characterization of the minimal surface equations for Gödel-type spacetimes.