  We consider excited states in a CFT, obtained by applying a weak unitary
perturbation to the vacuum. The perturbation is generated by the integral of a
local operator $J^{(n)}$ of modular weight $n$ over a spacelike surface passing
through $x = 0$. For $\vert n \vert \geq 2$ the modular Hamiltonian associated
with a division of space at $x = 0$ picks up an endpoint contribution,
sensitive to the details of the perturbation (including the shape of the
spacelike surface) at $x = 0$. The endpoint contribution is a sum of light-ray
moments of the perturbing operator $J^{(n)}$ and its descendants. For
perturbations on null planes only moments of $J^{(n)}$ itself contribute.
