PropositionStatement: AI-adaptedProof: AI-generatedPipeline-generatedprecheck passverified 2026-10-22 (gpt-6.1-sol)

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Compact interior changes leave a fixed ADM integral unchanged

Physical scope

A fixed-chart equality of asymptotic limits under compactly supported metric changes; no field equation or positivity result is used.

Statement

Let $h$ and $h^{'}$ be two $C^{2}$ Riemannian metrics on the same end $E = {x \in R^{3} : r = ∣ x ∣ > R_{0}}$ , with the same fixed Cartesian chart and asymptotic time normalization stated below. Suppose $h = h^{'}$ outside a compact set $C \subset E$ and both fixed-chart ADM limits exist. Then $E_{A D M} (h; x) = E_{A D M} (h^{'}; x) .$ The symmetric tensors $K$ associated with the two data sets may differ; each need only satisfy the decay in the definition.

Statement references

A specified asymptotically Euclidean ADM end

Facts & Assumptions

Given: The same end chart and ADM formula for both metrics, a compact set $C \subset E$ , equality $h = h^{'}$ on $E ∖ C$ , and existence of both limits.

[F1]

The ADM integrand in the fixed chart is $(\partial_{j} h_{i j} - \partial_{i} h_{j j}) n^{i}$ on $S_{R}$ , with the same constant prefactor for both metrics; $K$ does not enter that formula (A specified asymptotically Euclidean ADM end).

[A1]

The two limits exist, and the metrics agree on the complement of the same compact set in the same chart, as stipulated.

Proof

Given: [F1], [A1], and the two metrics in the Statement.

1.1A1algebra

If $C$ is nonempty, continuity of $r$ on compact $C$ gives a finite $R_{C} = \max_{x \in C} r (x)$ . If $C$ is empty, set $R_{C} = R_{0}$ . For every $R > R_{C}$ , the sphere $S_{R}$ is disjoint from $C$ and lies in the open set $E ∖ C$ , on which $h = h^{'}$ . Since the metrics agree on this open set, their first Cartesian derivatives also agree at every point of $S_{R}$ .

2.1F1A1step 1.1algebra∎

By [F1] and step 1.1, the two integrands coincide pointwise on every $S_{R}$ with $R > R_{C}$ , so the two surface integrals are equal for all such $R$ . Taking the limits, which exist by [A1], gives equal limits. Multiplication by the common factor $c^{4} / (16 π G_{N})$ proves the equality of ADM energies.

Remarks

This is a fixed-chart statement. It makes no claim of diffeomorphism invariance, constraint preservation, completeness, or scalar-curvature equality.

Depends on

A specified asymptotically Euclidean ADM end