Explicit self-similar solutions of time-like extremal hypersurfaces in $\mathbb{R}^{1+3}$

TL;DR

This paper presents two explicit self-similar solutions for time-like extremal hypersurfaces in Minkowski spacetime and analyzes their stability properties through linearization.

Contribution

The work provides explicit self-similar solutions and investigates the eigenvalues of the linearized operator around these solutions, highlighting stability issues.

Findings

Two explicit self-similar solutions are constructed.

An unstable eigenvalue is identified in the linearized analysis.

Insights into the stability of these solutions are discussed.

Abstract

In this letter, two explicit self-similar solutions to a graph representation of time-like extremal hypersurfaces in Minkowski spacetime $\mathbb{R}^{1+3}$ are given. Meanwhile, there is an untable eigenvalue in the linearized time-like extremal hypersurfaces equation around two explicit self-similar solutions.