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Normal boost action of the Hawking flow

PDF p. 17: (3.9); PDF p. 18: (3.11)

Statement

Adopt the stationary vacuum model and regular stationary data, with $T ∣_{S} \equiv̸ 0$ , maximal original exterior section and mass normalized to one. Use the common reference collar and its smooth Lorentzian Harrison family $g_{q}$ , $q \in [0, 1]$ . Here $S$ is the spacelike bifurcation surface, $K$ is the fixed Hawking Killing field, $ϕ_{s}^{K}$ is its maximal flow, and $κ > 0$ is its fixed boost rate. On $S$ , let ${\overset{ˉ}{n}}_{q}$ be the future unit normal to the reference section and ${\overset{ˉ}{ν}}_{q}$ the unit normal to $S$ within that section pointing into the exterior. These form an orthonormal basis of the $g_{q}$ -normal plane of $S$ , with ${\overset{ˉ}{n}}_{q}$ timelike.

For every $p \in S$ , $q \in [0, 1]$ , and $s, α \in R$ , the flow fixes $p$ and

$d ϕ_{s}^{K} ∣_{p} (\cosh α {\overset{ˉ}{ν}}_{q} + \sinh α {\overset{ˉ}{n}}_{q}) = \cosh (α + κ s) {\overset{ˉ}{ν}}_{q} + \sinh (α + κ s) {\overset{ˉ}{n}}_{q} .$

In particular, $d ϕ_{s}^{K} ∣_{p} {\overset{ˉ}{ν}}_{q} = \cosh (κ s) {\overset{ˉ}{ν}}_{q} + \sinh (κ s) {\overset{ˉ}{n}}_{q}$ .

The original-section maximality premise is imposed on the actual physical exterior section $C = Σ_{0} \cap E$ for the original metric $g$ , so $H_{g} ∣_{C} = 0$ . Where this result uses the original reference section $Σ_{1}$ , it denotes the closure $B = C \cup S$ of this actual section. Maximality of a separately labelled chart-side region or another supplied section is not substituted for this premise. The spherical identification and the physical stationary end chart remain independent. Regular non-degenerate stationary vacuum spacetime Spacelike sections, maximality and lapse–shift data

Statement references

Stationary vacuum model and asymptotic normalization

Regular non-degenerate stationary vacuum spacetime

Spacelike sections, maximality and lapse–shift data

Mass-one convention and Harrison denominator

Lemma 3.3: common spacelike reference collar

Local Hawking and axial fields, boost and pole

Preservation of bifurcate horizons and surface gravity

Local and global flows generated by a vector field

The fundamental theorem on flows

Facts & Assumptions

[A1]

Adopt the stated stationary vacuum model, regular data, maximal exterior section and mass-one normalization. Stationary vacuum model and asymptotic normalization Regular non-degenerate stationary vacuum spacetime Spacelike sections, maximality and lapse–shift data Mass-one convention and Harrison denominator

[F1]

The common collar gives the stated normal fields, $K ∣_{S} = 0$ , and $\nabla_{{\overset{ˉ}{ν}}_{q}}^{(q)} K = κ {\overset{ˉ}{n}}_{q}$ on $S$ for the same $κ > 0$ . Lemma 3.3: common spacelike reference collar The local Hawking-field result identifies $K$ and its positive normalization. Local Hawking and axial fields, boost and pole The horizon-preservation result supplies $L_{K} g_{q} = 0$ on a neighborhood of $S$ for every $q \in [0, 1]$ . Preservation of bifurcate horizons and surface gravity

[F2]

For a Killing field, the metric-compatible connection satisfies $g_{q} (\nabla_{v}^{(q)} K, w) + g_{q} (v, \nabla_{w}^{(q)} K) = 0$ . Killing fields and conserved currents, and on-shell conservation of the Hilbert stress-energy tensor

[F3]

A local flow follows integral curves, and the maximal integral curve through a point is unique. Local and global flows generated by a vector field Through each point there is a unique maximal integral curve For a smooth vector field on a smooth manifold $M$ , the set of pairs $(t, x)$ for which the maximal integral curve from $x$ is defined at $t$ is open in $R \times M$ , and evaluation along those curves is the unique smooth maximal local flow. The fundamental theorem on flows

[F4]

If a Killing field vanishes on a spacelike surface and its normal derivative has boost matrix $κ (\begin{array}{cc} 0 & 1 \\ 1 & 0 \end{array})$ in a future-timelike and outward-spacelike orthonormal basis, its flow differential has the corresponding hyperbolic boost matrix wherever the flow is defined. Normal boost action and invariant two-forms at a fixed surface

Proof

Given: The model, collar, fields and parameter ranges in the Statement.

Goal: Prove the displayed normal action and its zero-rapidity specialization.

1.1A1F1F2

Fix $p \in S$ and $q \in [0, 1]$ , and put $B (v) = \nabla_{v}^{(q)} K ∣_{p}$ . Since $K$ vanishes along $S$ , $B (X) = 0$ for $X \in T_{p} S$ . By the Killing equation, $g_{q} (B v, X) = - g_{q} (v, B X) = 0$ , so $B$ preserves the normal plane. The collar gives $B {\overset{ˉ}{ν}}_{q} = κ {\overset{ˉ}{n}}_{q}$ . Write $B {\overset{ˉ}{n}}_{q} = a {\overset{ˉ}{n}}_{q} + b {\overset{ˉ}{ν}}_{q}$ ; skewness gives $a = 0$ and $b = κ$ . Thus $B$ has the matrix required in [F4] in the ordered basis $({\overset{ˉ}{n}}_{q}, {\overset{ˉ}{ν}}_{q})$ .

2.1F1F3F4step 1.1algebra∎

The constant curve at $p$ is an integral curve of $K$ on all of $R$ . By maximality and uniqueness it is the maximal integral curve through $p$ , with domain $I_{p} = R$ . Apply the maximal-flow theorem in [F3] to the smooth field $K$ on its open neighborhood of $S$ : its smooth maximal flow has open domain $D = {(t, x) : t \in I_{x}}$ and evaluates the maximal integral curves. Thus $(s, p) \in D$ and $ϕ_{s}^{K} (p) = p$ for every real $s$ . For each such $s$ , openness gives a neighborhood of $p$ on which the time- $s$ map is smooth, so $d ϕ_{s}^{K} ∣_{p}$ exists. Applying [F4] with the matrix from step 1.1 gives $d ϕ_{s}^{K} {\overset{ˉ}{ν}}_{q} = \cosh (κ s) {\overset{ˉ}{ν}}_{q} + \sinh (κ s) {\overset{ˉ}{n}}_{q}$ and $d ϕ_{s}^{K} {\overset{ˉ}{n}}_{q} = \sinh (κ s) {\overset{ˉ}{ν}}_{q} + \cosh (κ s) {\overset{ˉ}{n}}_{q}$ . Linearity and the hyperbolic addition formulas yield the displayed action for every real $α$ ; setting $α = 0$ gives the particular formula.

Source notes

These premises specify the generated conditional scope and are not attributed to the original article. The substantive local horizon conclusions are retained. The immutable original extraction is unchanged; in particular, original Theorem 1.1 retains its unresolved source-assessment status.

Depends on

Used by · 2

Dependency tree · 2 levels

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Sources

The uniqueness of smooth stationary black holes · PDF pp. 17–18, equations (3.9) and (3.11)