  We present a brief but nearly self-contained proof of a formula for the
Weil-Petersson Hessian of the geodesic length of a closed curve (either simple
or not simple) on a hyperbolic surface. The formula is the sum of the integrals
of two naturally defined positive functions over the geodesic, proving
convexity of this functional over Teichmuller space (due to Wolpert (1987)). We
then estimate this Hessian from below in terms of local quantities and distance
along the geodesic. The formula extends to proper arcs on punctured hyperbolic
surfaces, and the estimate to laminations. Wolpert's result that the Thurston
metric is a multiple of the Weil-Petersson metric directly follows on taking a
limit of the formula over an appropriate sequence of curves. We give further
applications to upper bounds of the Hessian, especially near pinching loci, and
recover through a geometric argument Wolpert's recent result on the convexity
of length to the half-power.
