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Quantum gravity may be viewed as a bipartite matter-geometry system. Evolution of matter-geometry entanglement entropy is an interesting question for issues such as the emergence of QFT on curved spacetime from quantum gravity: emergence would require initial states to evolve to product states. We study this question in a cosmological model. We give numerical evidence that matter-geometry entanglement entropy increases indefinitely (with respect to a relational time) for apparently arbitrary initial states. This result raises the question of whether matter and geometry remain entangled in a large Universe.

By applying a parabolic-hyperbolic formulation of the constraints and superposing Kerr-Schild black holes, a simple method is introduced to initialize time evolution of binary systems. As the input parameters are essentially the same as those used in the post-Newtonian (PN) setup the proposed method interrelates various physical expressions applied in PN and in fully relativistic formulations. The global ADM charges are also determined by the input parameters, and no use of boundary conditions in the strong field regime is made.