  In this article, we prove the global well-posedness in the critical Sobolev
space $H_{rad}^2\left(\mathbb{R}^2\right) \times H_{rad}^1
\left(\mathbb{R}^2\right)$ for the radial time-like extremal hypersurface
equation in $\left(1+3\right)$- dimensional Minkowski space-time. This is
achieved by deriving a new div-curl type lemma and combined it with energy and
``momentum" balance law to get some space-time estimates of the nonlinearity.
