  It is of great interest to explore matter in nontrivial quantum arrangements,
including Schrodinger cat-like states. Such states are sensitive to decoherence
from their environment. Recently, in Ref. [1] we computed the rate of
decoherence of a piece of superposed matter that primarily only interacts
gravitationally, a dark-matter-Schrodinger-cat-state (DMSCS), within the
nonrelativistic approximation. In this work we improve this to a general
relativistic analysis. We firstly derive a single particle relativistic
Schrodinger equation for a probe particle that passes through the DMSCS; the
interaction is provided by the weak field metric of general relativity from the
source. For a static DMSCS we find a neat generalization of our previous
results. We then turn to the interesting new case of a time dependent DMSCS,
which can be provided by a coherently oscillating axion field leading to
superposed time dependent oscillations in the metric; a truly quantum-general
relativistic phenomenon. We use scattering theory to derive the decoherence
rate in all these cases. When the DMSCS is in a superposition of distinct
density profiles, we find that the decoherence rate can be appreciable. We then
consider the novel special case in which the density is not in a superposition,
but the phase of its field oscillation is; this is a property that cannot be
decohered within the nonrelativistic framework. We find that if the probe
particle and/or the DMSCS's velocity dispersion is slow, then the rate of
decoherence of the phase is exponentially suppressed. However, if both the
probe and the DMSCS's velocity dispersion are relativistic, then the phase can
decohere more rapidly. As applications, we find that diffuse galactic axions
with superposed phases are robust against decoherence, while dense boson stars
and regions near black hole horizons are not, and we discuss implications for
experiment.
