Post-Newtonian Constraints on Semiclassical Gravity with Quantum Superpositions

TL;DR

This paper investigates the validity of semiclassical gravity in quantum superpositions, revealing that at post-Newtonian order, the theory predicts state-dependent gravitational effects that challenge its consistency.

Contribution

It demonstrates that sourcing gravity solely from expectation values is insufficient at post-Newtonian order, highlighting the need for a more complete quantum treatment of gravity.

Findings

No leading-order modification of Newtonian potential for quantum superpositions.

State-dependent post-Newtonian contributions involve mass density and current.

Semiclassical coupling's parametric scaling differs from relativistic corrections.

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 Schr\"odinger-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.