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A commuting axial action preserves the stationary field and its causal character

Statement

For the complete stationary flow $ϕ_{t}$ and commuting global axial isometries $ψ_{s}$ in the comparison data, $(ψ_{s})_{*} T = T$ for every $s$ and $g (T, T) \circ ψ_{s} = g (T, T)$ . Here vector-field pushforward is defined by $((ψ_{s})_{*} T)_{ψ_{s} (p)} = d ψ_{s} ∣_{p} (T_{p})$ for $p \in M$ ; the action map $ψ_{s}$ is a diffeomorphism with inverse $ψ_{- s}$ . In particular the timelike set, null set and ergoregion of the specified stationary field are preserved by this circle action. These identities use a supplied global action and assert no extension of a locally constructed axial field.

Facts & Assumptions

[F1]

The stationary and axial actions are globally defined commuting isometries. Smooth global data for the axisymmetric comparison

[F2]

The stationary generator is $T = {\partial_{t} ∣}_{t = 0} ϕ_{t}$ , and its metric norm determines its pointwise causal character. Stationary Lorentzian data and tensor conventions

[F3]

The ergoregion of the specified stationary field is ${p : g_{p} (T, T) > 0}$ ; points with $g (T, T) = 0$ are excluded. Staticity, axisymmetry and black-hole regularity hypotheses

[F4]

For smooth maps $F : M \to N$ and $G : N \to P$ , the chain rule gives $d (G \circ F)_{p} = d G_{F (p)} \circ d F_{p}$ for every $p \in M$ . The chain rule for differentials of smooth maps

Proof

Given: The data and assumptions in the Statement.

Goal: Establish the claimed conditional conclusion.

1.1F1F2F4given

By the chain rule [F4] applied to the smooth orbit curve $t \mapsto ϕ_{t} (p)$ and the smooth map $ψ_{s}$ , differentiate $ψ_{s} (ϕ_{t} (p)) = ϕ_{t} (ψ_{s} (p))$ at $t = 0$ . This yields $d ψ_{s} ∣_{p} (T_{p}) = T_{ψ_{s} (p)}$ at every point and every circle parameter. By the inline definition, this is $(ψ_{s})_{*} T = T$ . The differentiation is legitimate globally because both compositions are defined globally by hypothesis.

2.1F1step 1.1F2F3algebra∎

Since $ψ_{s}^{*} g = g$ , substitute the identity from step 1.1 into the metric pairing to obtain $g (T, T) (ψ_{s} (p)) = g (T, T) (p)$ . Each of the strict positive, strict negative and zero level sets is therefore invariant. Thus the axial action preserves the stationary field, the metric and the stated causal subsets, with the strict positive set being the ergoregion by [F3]. No extension of a local horizon symmetry is asserted.

Depends on

Used by · 1

Axisymmetric vacuum uniqueness with complete global hypotheses

Dependency tree · 2 levels

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