Self-consistent initial-data conjecture for semiclassical gravity

Let $\mathcal{C}$ be a Cauchy surface with classical initial data consisting of the induced metric $h_{ab}$, extrinsic curvature $K_{ab}$, and its first two normal-time derivatives $\,{\mathchar'26\mkern-12mud}\, K_{ab}$ and $\,{\mathchar'26\mkern-12mud}\,^2 K_{ab}$. Equivalently, prescribe the metric time derivatives $g_{ab}|_{\mathcal{C}}$, $\dot g_{ab}|_{\mathcal{C}}$, $g^{(2)}_{ab}|_{\mathcal{C}}$, and $g^{(3)}_{ab}|_{\mathcal{C}}$, together with the approximating quantities $\Sigma$, $\,{\mathchar'26\mkern-12mud}\,\Sigma$, $\,{\mathchar'26\mkern-12mud}\,^2\Sigma$, $U$, and $V$.

Self-consistent initial-data conjecture. Self-consistent initial data for the fourth-order time derivatives of the metric can be obtained from these classical initial data by solving the fourth-order terms, namely $\nabla_n^3K_{a'b'}\sim g^{(4)}_{ab}|_{\mathcal{C}}$, using the semiclassical Einstein equation. Furthermore, higher-order time derivatives of the induced metric can be obtained by formally differentiating the semiclassical Einstein equation and applying the same procedure.

This conjecture recasts the proposed zeroth-stage construction of the surface Hadamard condition for the initial-value problem of semiclassical gravity. It asserts that the missing fourth time derivative, and subsequently higher derivatives, can be determined recursively from classical initial data and the semiclassical Einstein equation; the supplied context does not establish the conjecture or give a resolution status.