Abstract
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.
Highlights
Recent observations found gravitational waves from black hole mergers in which a few percent of the energy of the system is radiated by the gravitational waves [1,2,3,4]
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 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 nonspherical solution
Summary
Recent observations found gravitational waves from black hole mergers in which a few percent of the energy of the system is radiated by the gravitational waves [1,2,3,4]. It seems possible that the gravitational waves are gravitationally bounded in a local region of the spacetime by their own gravitational energy This time-dependent self-gravitating object is called a gravitational geon, short for gravitational-electromagnetic entity, which was introduced by Wheeler [5]. We consider asymptotically flat spacetimes and construct a gravitational geon of massive gravitons. We can naturally expect that the massive graviton yields self-gravitating solutions which we call massive graviton geons. The analysis [27] is based on the linear perturbation theory around the homogeneous background It has not been cleared whether or not the massive gravitons yield a nonperturbative localized object. The monopole geon is the spherically symmetric configuration of the massive graviton which corresponds to the eigenstate of the zero total angular momentum. We find an octupole configuration of the massive graviton geon in Appendix B
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