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Positive stationary mass from the bifurcate vacuum boundary

Statement

Under exactly the article regular stationary-vacuum hypotheses, the original stationary end coefficient $M_{0}$ is strictly positive. More precisely the complete exterior section has ADM energy $E > 0$ , zero linear momentum, and $M_{0} = m_{K} = E$ . This conclusion assumes neither maximality nor axisymmetry.

Statement references

Regular non-degenerate stationary vacuum spacetime

Asymptotic mass and axial angular-momentum conventions

Facts & Assumptions

Given: The data and hypotheses of the Statement.

Goal: The asserted conclusion with its stated scope.

[F1]

The section satisfies complete AF initial-data assumptions, vacuum DEC, and trace decay of order three. Complete boundary initial-data and vanishing momentum interface Vacuum exterior-section constraints imply the dominant energy condition Extrinsic curvature decay on the original stationary end

[F2]

The into-exterior boundary expansions both vanish. Zero boundary expansions without maximality

[F3]

The exact boundary positive-energy input applies with $n = 3$ and $γ = 3$ . External boundary positive-energy theorem without maximality

[F4]

Ordinary stationary vacuum ADM and Komar masses agree and the latter is the original coefficient. External ADM–Komar equality on ordinary stationary vacuum ends Komar flux equals the original stationary mass coefficient

Proof

1.1F1F2F3algebra

Apply the boundary positive-energy input to $\overline{C}$ using completeness and smooth nonempty compact boundary, the decay with $q = 1$ , zero constraint densities, zero boundary expansions, and trace decay with $γ = 3 > 2$ . Every condition is supplied without a maximality assumption, hence $E > 0$ . The same initial-data interface gives $P = 0$ .

2.1F4algebra∎

The end is ordinary four-dimensional stationary vacuum, so ADM–Komar equality applies with the normalization fixed in the explicit end flux. Therefore $M_{0} = m_{K} = E > 0$ .

Depends on

Used by · 9

Dependency tree · 2 levels

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