  Several approaches to Hawking radiation on Schwarzschild spacetime rely in
some way or another on the fact that the Kruskal manifold has two causally
disconnected exterior regions. We investigate the Hawking(-Unruh) effect for a
real scalar field on the $\RPthree$ geon: an inextendible, globally hyperbolic,
space and time orientable eternal black hole spacetime that is locally
isometric to Kruskal but contains only one exterior region. The
Hartle-Hawking-like vacuum~$\hhvacgeon$, which can be characterized
alternatively by the positive frequency properties along the horizons or by the
complex analytic properties of the Feynman propagator, turns out to contain
exterior region Boulware modes in correlated pairs, and any operator in the
exterior that only couples to one member of each correlated Boulware pair has
thermal expectation values in the usual Hawking temperature. Generic operators
in the exterior do not have this special form; however, we use a Bogoliubov
transformation, a particle detector analysis, and a particle
emission-absorption analysis that invokes the analytic properties of the
Feynman propagator, to argue that $\hhvacgeon$ appears as a thermal bath with
the standard Hawking temperature to any exterior observer at asymptotically
early and late Schwarzschild times. A~(naive) saddle-point estimate for the
path-integral-approach partition function yields for the geon only half of the
Bekenstein-Hawking entropy of a Schwarzschild black hole with the same ADM
mass: possible implications of this result for the validity of path-integral
methods or for the statistical interpretation of black-hole entropy are
discussed. Analogous results hold for a Rindler observer in a flat spacetime
whose global properties mimic those of the geon.
