  In \cite{ly, ly2}, Liu and the second author propose a definition of the
quasi-local mass and prove its positivity. This is demonstrated through an
inequality which in turn can be interpreted as a total mean curvature
comparison theorem for isometric embeddings of a surface of positive Gaussian
curvature. The Riemannian version corresponds to an earlier theorem of Shi and
Tam \cite{st}. In this article, we generalize such an inequality to the case
when the Gaussian curvature of the surface is allowed to be negative. This is
done by an isometric embedding into the hyperboloid in the Minkowski space and
a future-directed time-like quasi-local energy-momentum is obtained.
