  In this article, we study the small sphere limit of the Wang-Yau quasi-local
energy defined in [18,19]. Given a point $p$ in a spacetime $N$, we consider a
canonical family of surfaces approaching $p$ along its future null cone and
evaluate the limit of the Wang-Yau quasi-local energy. The evaluation relies on
solving an "optimal embedding equation" whose solutions represent critical
points of the quasi-local energy. For a spacetime with matter fields, the
scenario is similar to that of the large sphere limit found in [7]. Namely,
there is a natural solution which is a local minimum, and the limit of its
quasi-local energy recovers the stress-energy tensor at $p$. For a vacuum
spacetime, the quasi-local energy vanishes to higher order and the solution of
the optimal embedding equation is more complicated. Nevertheless, we are able
to show that there exists a solution which is a local minimum and that the
limit of its quasi-local energy is related to the Bel-Robinson tensor. Together
with earlier work [7], this completes the consistency verification of the
Wang-Yau quasi-local energy with all classical limits.
