  We construct local, boost covariant boundary QFT nets of von Neumann algebras
on the interior of the Lorentz hyperboloid LH = {x^2 - t^2 > R^2, x>0}, in the
two-dimensional Minkowski spacetime. Our first construction is canonical,
starting with a local conformal net on the real line, and is analogous to our
previous construction of local boundary CFT nets on the Minkowski half-space.
This net is in a thermal state at Hawking temperature. Then, inspired by a
recent construction by E. Witten and one of us, we consider a unitary semigroup
that we use to build up infinitely many nets. Surprisingly, the one-particle
semigroup is again isomorphic to the semigroup of symmetric inner functions of
the disk. In particular, by considering the U(1)-current net, we can associate
with any given symmetric inner function a local, boundary QFT net on LH. By
considering different states, we shall also have nets in a ground state, rather
than in a KMS state.
