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Maximality and smooth boundary in the ADM theorem

Definition

For the Radu ADM package let n be the future unit normal to Σ0, h=g∣TΣ0 and k(Y,V)=−g(∇Yn,V) for fields tangent to the slice. The additional original maximality and boundary condition is tr⁡hk=0on Σ1, and smooth extension of the induced geometry to Σ‾1=Σ1∪S0 within Σ0. The trace is hijkij. This concerns the original stationary slice; it does not assert existence of any transformed maximal slice or impose maximality as part of the nontrapping theorem.

Facts & Assumptions

[F1]

Radu small-ADM global and bifurcation package provides the source slice and its specified sphere. Spacelike sections, maximality and lapse–shift data provides its future unit normal, induced positive metric and the same negative-sign second fundamental form convention.

Justification

The normal and tangential derivatives along the supplied smooth hypersurface determine k and h. Positive definiteness gives the inverse used in the trace, and the zero trace is an unambiguous predicate; its zero value is unchanged if a source uses the opposite sign for the second fundamental form. Here the displayed sign is exactly the ADM source's convention. The specified smooth boundary extension is an additional regularity predicate, not a consequence of trace zero or an assertion that new hypersurfaces have been constructed.

Depends on

Used by · 3

Dependency tree · 2 levels

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External dependencies

Alexandru F. Radu, Rigidity of stationary black holes with small ADM angular momentum · §1.1 definitions of h and k, maximality equation and smooth-extension sentence immediately after SBS

For the future unit normal T_0 to Sigma^0, set h=g restricted to T Sigma^0 and k(Y,V)=-g(D_Y T_0,V) for tangent Y,V. The additional maximality hypothesis is tr_h k=0 on Sigma_1; its induced geometry extends smoothly to S_0 within Sigma^0. These are conditions on the given slice, not an existence assertion for another slice.

Hypotheses: The connected oriented time-oriented smooth stationary vacuum four-spacetime of signature (-,+,+,+) and the GR/SBS slice data of section 1.1: Sigma^0 is the supplied embedded spacelike partial Cauchy hypersurface, Sigma_1=Phi_0(E_1), and S_0=Phi_0({|x|=1}). Maximality and smooth extension are additional assumptions, separate from the later angular-momentum restriction and quantitative smallness conditions.

Reading limits: Freshly retrieved the indicated arXiv HTML version and read the exact formulation and adjacent hypothesis definitions. This citation attributes the locally transcribed definition; no theorem proof or supplier proof review was performed.

Sources