We derive the gravitational field and the spacetime metric generated by
sources in quantum superposition of different locations. We start by working in
a Newtonian approximation, in which the effective gravitational potential is
computed as the expectation value of the gravitational potential operator in a
Gaussian distribution of width $R$ for the position of the source. The
effective gravitational potential is then covariantly uplifted to a fully
relativistic metric in general relativity, describing the spacetime generated
by averaging over the state of such sources. These results are then rederived
and extended by adopting an independent construction in terms of quantum
reference frames. We find three classes of quantum effective metrics which are
all asymptotically flat and reproduce the Schwarzschild metric at great
distances. The solutions differ, however, in the inner core. The quantum
uncertainty $\Delta r\sim R$ in the position of the source prevents the radius
of the transverse two-sphere to shrink to zero. Depending on the strength of
the quantum superposition effects, we have either a nonsingular black hole with
a “quantum hair” and an event horizon, a one-way wormhole with a critical
null throat or a traversable wormhole. We also provide a detailed study of the
geometric and thermodynamic properties of the spacetime structure for each of
these three families of models, as well as their phenomenology.
