When is the semiclassical Einstein equation a controlled approximation?
In plain words
A common shortcut treats spacetime classically and matter quantum mechanically, with spacetime curved by the average energy of the quantum fields. The shortcut fails when quantum fluctuations of that energy are large, and its precise range of validity is unknown.
Precise statement
Semiclassical gravity sets $G_{ab} = (8\pi G/c^4) \langle T_{ab}\rangle_{\omega}$ for a Hadamard state $\omega$. Determine conditions on $\omega$ and the spacetime (for example bounds on the fluctuation $\langle T_{ab} T_{cd}\rangle - \langle T_{ab}\rangle\langle T_{cd}\rangle$ relative to $\langle T_{ab}\rangle^2$, smeared over curvature-scale regions) under which solutions approximate the full quantum theory to leading order in $\hbar$, and give a well-posed initial value formulation that excludes higher-derivative runaway solutions. An answer is a theorem or a large-$N$ derivation with stated error bounds.
What would settle it
A proof, in a large-N or other controlled limit, that solutions of the semiclassical equations track expectation values of a quantum theory of metric and matter with explicit error bounds.
Status in the literature
Unverified note
Large-N arguments and order-reduction schemes exist; no general error bound is known as of 2026.