  We find vacuum solutions such that massive gravitons are confined in a local
spacetime region by their gravitational energy in asymptotically flat
spacetimes in the context of the bigravity theory. We call such
self-gravitating objects massive graviton geons. The basic equations can be
reduced to the Schr\"odinger-Poisson equations with the tensor "wavefunction"
in the Newtonian limit. We obtain a non-spherically symmetric solution with
$j=2,\ell=0$ as well as a spherically symmetric solution with $j=0,\ell=2$ in
this system where $j$ is the total angular momentum quantum number and $\ell$
is the orbital angular momentum quantum number, respectively. The energy
eigenvalue of the Schr\"odinger equation in the non-spherical solution is
smaller than that in the spherical solution. We then study the perturbative
stability of the spherical solution and find that there is an unstable mode in
the quadrupole mode perturbations which may be interpreted as the transition
mode to the non-spherical solution. The results suggest that the
non-spherically symmetric solution is the ground state of the massive graviton
geon. The massive graviton geons may decay in time due to emissions of
gravitational waves but this timescale can be quite long when the massive
gravitons are non-relativistic and then the geons can be long-lived. We also
argue possible prospects of the massive graviton geons: applications to the
ultralight dark matter scenario, nonlinear (in)stability of the Minkowski
spacetime, and a quantum transition of the spacetime.
