Post-Newtonian Constraints on Semiclassical Gravity with Quantum Superpositions

TL;DR

The paper tests whether deterministic semiclassical gravity, sourcing the metric from quantum expectation values, yields a consistent post-Newtonian expansion for spatial quantum superpositions. It shows the Newtonian limit reproduces GR when states are normalized, but at first post-Newtonian order the theory generates state-dependent corrections that can vastly exceed GR predictions and lack Planck suppression. This reveals a fundamental inconsistency in the semiclassical approach and places strong constraints on such models, indicating that additional ingredients are needed to reconcile gravity with quantum matter. The results highlight post-Newtonian dynamics as a crucial probe of gravity-quantum interfaces and motivate exploring stochastic or genuinely quantum gravitational degrees of freedom.

Abstract

Semiclassical gravity, in which a classical spacetime is sourced by the quantum expectation value of the stress-energy tensor, is a standard framework for describing the gravitational interaction of quantum matter. In the nonrelativistic limit this approach leads to the Schrödinger-Newton equation, which is often assumed to be consistent at least in the weak-field regime. In this work, we reexamine this assumption for spatial quantum superpositions of massive particles. We show that, when the quantum state is properly normalized, no modification of the Newtonian gravitational potential arises at leading order. However, at first post-Newtonian order the semiclassical coupling generically produces state-dependent contributions involving the mass density and the mass current of the superposition. These terms have a parametric scaling which is different from that of the corresponding relativistic corrections and which does not have Planck mass suppression. Our results therefore impose a strong post-Newtonian consistency constraint on deterministic semiclassical gravity, indicating that sourcing the metric solely by expectation values is insufficient to recover a consistent relativistic weak-field expansion.