LemmaStatement: AI-adaptedProof: AI-generated

How statement and proof provenance work

The first chip identifies the source of the statement or construction; the second identifies the source of its local proof or verification.

Literature-sourced: the exact statement appears in a cited source; only wording and notation differ.
AI-adapted: a semantically identical restatement of literature-sourced material, modulo indexing, notation, and boundary cases adopted by the library.
AI-generated: a genuinely novel statement formulated by AI, with no source for the claim itself.

These labels describe origin, not correctness: citations and verification chips remain separate evidence.

The asymptotic metric flux of Kerr equals its mass parameter

Statement

The ADM energy of the specified constant $t$ Kerr end exists and equals $E_{A D M} = M$ .

Statement references

Asymptotic mass and axial angular-momentum conventions

The Boyer–Lindquist spatial end and differentiated expansion

Facts & Assumptions

[F1]

The following exact suppliers supply the geometric objects and earlier conclusions used here: Asymptotic mass and axial angular-momentum conventions The Boyer–Lindquist spatial end and differentiated expansion

Proof

**Given:**The ADM metric-flux convention and the differentiated expansion of [F1].

Goal: Evaluate the energy flux.

1.1givenF1

Put $k_{i j} = 2 M x_{i} x_{j} / r^{3}$ . Direct differentiation in dimension three gives $\partial_{j} k_{i j} = 2 M x_{i} / r^{3}$ and $\partial_{i} k_{j j} = - 2 M x_{i} / r^{3}$ . Thus their difference contracted with $n^{i}$ is $4 M / r^{2}$ . The $O_{2} (r^{- 2})$ remainder of the metric contributes $O (r^{- 3})$ to this contracted integrand.

2.1givenF1step 1.1∎

Integration over a Euclidean coordinate sphere gives $\int_{S_{r}} 4 M / r^{2} d S_{0} = 16 π M$ . The integrated remainder is $O (r^{- 1})$ and vanishes. Division by $16 π$ and the limit in the definition yield $E_{A D M} = M$ .

External input

The cited Kerr coordinate formulas provide background for the local calculation in this item.

Depends on

Used by · 2

Dependency tree · 2 levels

Each result is drawn at its shortest distance from this item. Arrows run from prerequisites to the items that use them; this item has a heavier outline. External source nodes link directly to the cited text. The view shows up to two dependency steps; open a prerequisite to continue.

External dependencies

The Kerr Metric · §2, equations (2.1), (2.5)–(2.8), PDF pp. 2–3

The review gives ingoing and Boyer–Lindquist Kerr forms, their coordinate transformation, and the asymptotic identification of mass M and angular momentum J=aM.

Hypotheses: The Kerr metric in the stated coordinate systems and their indicated domains.

Reading limits: Read the cited formulas and surrounding discussion. This is background for the Kerr construction, not a proof of the full indexed statement or its normalization-specific calculations.

Sources

The Kerr Metric · §2, equations (2.1), (2.5)–(2.8), PDF pp. 2–3