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The Kerr end is an ADM rest frame

Statement

The constant t Kerr end has vanishing ADM linear momentum and ADM rest mass MADM=M>0.

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 The asymptotic metric flux of Kerr equals its mass parameter Stationary Lorentzian data and tensor conventions

[F2]

Work with the Kerr exterior parameters M>0 and ∣a∣≤M of Kerr exterior tensor including the extremal parameter range. In particular, the mass parameter in the end expansion and energy calculation is positive.

Proof

Given: The asymptotic charge conventions, end expansions and energy calculation of [F1], with the positive Kerr mass parameter of [F2].

Goal: Distinguish rest mass from energy and show they agree here.

1.1givenF1

For the future unit normal n=N−1(∂t−βi∂i), choose K=12Lnh on tangent vectors. The stationary slicing identity is Kij=−(Diβj+Djβi)/(2N): it follows by writing ∂t=Nn+β in the Lie derivative of h, using L∂th=0 and Lβhij=Diβj+Djβi. The opposite convention has the opposite sign and the same estimates. By the end expansion ∂β=O(r−3) and Γ(h)β=O(r−4), so Kij=O(r−3) and tr⁡hK=O(r−3).

2.1givenF1F2step 1.1∎

The momentum surface integrand is O(r−3) while νh=O(1) and the area is O(r2). Each momentum limit is zero. Together with EADM=M from [F1] and M>0 from [F2], this gives MADM=M2=M. No positive-mass theorem, completeness of the interior slice, or horizon regularity of this slice is needed for this direct end computation.

External input

The cited Kerr review supplies the coordinate and parameter background stated in its reference record; the result above is derived under the declared hypotheses and dependencies.

Depends on

Used by · 1

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)–(2.9), PDF pp. 2–3

The ingoing and Boyer–Lindquist Kerr metric forms and their coordinate transformation are displayed; the review identifies M as the mass and the roots of the radial function as null horizon candidates.

Hypotheses: Kerr metric in the stated ingoing and Boyer–Lindquist coordinate domains.

Reading limits: Consulted the browser-extracted PDF text of §2, equations (2.1), (2.5)–(2.9) and adjacent discussion on PDF pages 2–3. This supplies coordinate and parameter background only; it is not used as a proof of vanishing ADM momentum or the normalization-specific charge calculation. Those conclusions follow from the local argument and declared mathematical suppliers. No full-review reading or reconstruction of its proofs is claimed.

Sources