  In this paper we investigate quantum fields propagating on given, static,
spherically symmetric spacetimes, which are isometric to a part of the
Schwarzschild spacetime. Without specifying the internal geometry we show, that
there exist bounds on the energy densities of ground states of a quantum scalar
field on such spacetimes. The bounds (from above and below) come from the
so-called Quantum Energy Inequalities, and are centered around the energy
density of the Boulware state (the ground state for Schwarzschild spacetime).
The specific value of the bound from below depends critically on the distance
$\ell$ from the horizon, where the spacetimes of compact objects cease to be
isometric to the Schwarzschild spacetime. In the limit of small $\ell$ we
prove, that the energy densities of ground states cannot be below the Boulware
level.
