Although this rewriting amounts to a saddle-level change of variables, it is operationally nontrivial: it recasts the intrinsically bilocal deformation into a covariant derivative expansion on the deformed geometry, yielding (i) a systematic perturbative control of subleading terms and (ii) a unified language for comparison with local stress-tensor flows of $T\overline{T}$-type. Once the Green’s-function prescription is fixed, scheme-dependent local counterterms are cleanly separated from the regulator-independent nonlocal kernel.

For dS and AdS gravities, the gauge-fixing can be found in the recent literature [36].

Solution of metric perturbation $h^{*}$ follows from a perturbative expansion around a vacuum Einstein background, which ensures the power counting $h_{\mu \nu }^{*}=O({\lambda }^0)$ so that higher-order metric terms do not affect the leading deformation. Total-derivative terms in $h$ are omitted, as they can be absorbed into boundary terms. Full derivations are provided in SM A [23] [see in particular Eq. (A5)]. $G_{\mu \nu \rho \sigma }$ is defined as the inverse of the gauge-fixed quadratic graviton operator obtained by expanding the gravitational action to second order around ${\widehat{\gamma }}_{\mu \nu }$. Ghosts contribute only to the one-loop determinant and do not affect the tree-level saddle equation; see SM A [23].

For generic curved background, one can define the reference background $\widehat{\gamma }$ satisfies (3) and the Green’s function can be defined in a classical way which highly depends on the boundary conditions. For simplicity, here we consider the Green’s function on a manifold without boundary, which exhibits exponential decay as the separation between the two points tends to infinity.

Since the deformation generates the intrinsically nonlocal kernel (10), the locality assumptions entering the Weinberg–Witten theorem do not apply, and the saddle-point correspondence to Einstein dynamics is therefore not in conflict with that theorem. Once the Green’s function is specified, the operator $1/\square$ is fixed up to local counterterms; in Minkowski signature, adopting the retarded prescription enforces causality.

Here we assume that the action does not include the Riemann curvature tensor ${\mathcal{R}}_{\nu \rho \sigma }^{\mu }$. In principle, ${\mathcal{A}}^{(\mu )}$ should be a local function of all independent invariants composed of $g^{\mu \nu }$, ${\mathcal{R}}_{\mu \nu }$, and $X_{\mu \nu }^{(i)}$.

The solution takes the form $${\mathcal{B}}_0=C(\varphi ){\prod }_{j=1}^d{\chi }_j^{\frac{1}{2}}=C(\varphi )\frac{\sqrt{\det (X_{\mu \nu })}}{\sqrt{\det (g_{\mu \nu })}},$$(19)which is analogous to the action of the Nambu-Goto string [43] up to a regular function $C(\varphi )$.

The coefficients of local terms such as $\int \sqrt{g}$ and $\int \sqrt{g}R$ are scheme-dependent and can be shifted by local counterterms; the nonlocal kernel is fixed once the Green’s function prescription is specified.