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We study a new type of self-gravitating object, described by a soliton solution to the coupled system of the Einstein equation and a matter field equation. The solution describing the self-gravitating object is not static but, instead, is periodic, with both the spacetime geometry and the matter field oscillating in time. In particular, we show that a system described by a real scalar field g^;-m^2=0 admits such a solution, which is stable under perturbations. The existence of such objects could have important implications for astrophysics and cosmology.
Seidel et al. (1991) studied this question.