In what sense can it be said that spacetime possesses momentum? Can an experiment be envisaged to test this question?
In the case of an asymptotically flat spacetime (basically, one in which all of the matter is confined to a finite area of the spacetime), there is a quantity called the ADM Momentum that will tell you the total momentum contained in the spacetime, factoring both the matter and a contribution "from the geometry". There are other schemes that let you do stuff like this if you have a timelike or null killing vector defined on a timelike or null 3-d surface. Then, you can talk about the momentum (and energy) contained "within" this surface. There is a review article on the topic here