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In this paper, we study a possibility where gravity and time emerge fromquantum matter. Within the Hilbert space of matter fields defined on a spatialmanifold, we consider a sub-Hilbert space spanned by states which areparameterized by spatial metric. In those states, metric is introduced as acollective variable that controls local structures of entanglement. Theunderlying matter fields endow the states labeled by metric with an unambiguousinner product. Then we construct a Hamiltonian for the matter fields that is anendomorphism of the sub-Hilbert space, thereby inducing a quantum Hamiltonianof the metric. It is shown that there exists a matter Hamiltonian that inducesthe general relativity in the semi-classical field theory limit. Although theHamiltonian is not local in the absolute sense, it has a weaker notion oflocality, called relative locality : the range of interactions is set by theentanglement present in target states on which the Hamiltonian acts. Ingeneral, normalizable states are not invariant under the transformationsgenerated by the Hamiltonian. As a result, a physical state spontaneouslybreaks the Hamiltonian constraint, and picks a moment of time. The subsequentflow of time can be understood as a Goldstone mode associated with the brokensymmetry. The construction allows one to study dynamics of gravity from theperspective of matter fields. The Hawking radiation corresponds to a unitaryevolution where entanglement across horizon is gradually transferred from colordegrees of freedom to singlet degrees of freedom. The underlying quantum statesremain pure as evaporating black holes keep entanglement with early Hawkingradiations in the singlet sector which is not captured by theBekenstein-Hawking entropy.

Emergent gravity from relatively local Hamiltonians and a possible resolution of the black hole information puzzle