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The coefficient-one horizon generator inherits the positive lapse derivative

Statement

Work in the smooth time-oriented Lorentzian four-dimensional setting of signature $(-, +, +, +)$ with its Levi–Civita connection $\nabla$ . Let $S = \partial Σ$ be an embedded spacelike two-sphere bounding a smooth spacelike hypersurface $Σ$ , smooth up to $S$ , with future unit normal $n$ and unit normal $ν$ to $S$ within $Σ$ pointing into the exterior section. Let $T, K, Z$ be smooth Killing fields near $S$ , with $K = T + Ω Z$ , constant $Ω$ , and $K ∣_{S} = 0$ . Suppose $Z$ generates a smooth time-orientation-preserving circle action by isometries on an invariant neighborhood of $S$ and that this action preserves $S$ . Put $N = - g (T, n)$ . There is a circle fixed point $p \in S$ with $Z (p) = T (p) = 0$ and $\nabla_{v} Z = 0$ for every $v \in (T_{p} S)^{⊥}$ . At any such pole the normal boost coefficient of $K$ , specified by $\nabla_{ν} K = a_{p} n$ and $\nabla_{n} K = a_{p} ν$ at $p$ , is $a_{p} = ν (N) ∣_{p}$ . Thus it is positive if the inward lapse derivative is positive there.

Moreover, suppose two distinct smooth embedded null hypersurfaces $H_{+}, H_{-}$ meet along $S$ , with $T_{q} H_{+} \cap T_{q} H_{-} = T_{q} S$ for $q \in S$ . On these respective branches suppose there are smooth nowhere-zero null normal generator fields $L_{+}, L_{-}$ and smooth functions $s$ on $H_{+}$ and $t$ on $H_{-}$ satisfying $\nabla_{L_{+}} L_{+} = \nabla_{L_{-}} L_{-} = 0$ , $L_{+} (s) = L_{-} (t) = 1$ , and $s^{- 1} (0) = t^{- 1} (0) = S$ . Suppose $K = c K_{0}$ near $S$ for a real constant $c$ and a smooth field $K_{0}$ with $K_{0} ∣_{H_{+}} = s L_{+}$ , $K_{0} ∣_{H_{-}} = - t L_{-}$ , and $\nabla_{ν} K_{0} = n$ on $S$ in the specified future/exterior normal convention. Then $c = a_{p} = ν (N) ∣_{p}$ . If this derivative is positive, $c > 0$ , $K$ is nonzero on $H_{\pm} ∖ S$ , and the signed branch equations there are $\nabla_{K} K = c K$ , $d (g (K, K)) = - 2 c K^{♭}$ on $H_{+} ∖ S$ and $\nabla_{K} K = - c K$ , $d (g (K, K)) = 2 c K^{♭}$ on $H_{-} ∖ S$ , where $K^{♭} = g (K, \cdot)$ . These are the signed surface-gravity equations of the resulting bifurcate Killing horizon. They extend as zero equations at $S$ and do not determine a coefficient from the zero equation alone. No existence of the additional branch and profile data is asserted.

Statement references

Spacelike sections, maximality and lapse–shift data

Bifurcate Killing horizons, rotation and normalization

Periodic isometries and invariant normal averaging

Metric compatibility, torsion freedom and the metric-selected connection class

A circle action on a sphere has a pole with trivial normal action

A Killing field vanishing on a spacelike surface has a normal boost derivative

Affine generator profiles give constant signed acceleration

For a Killing field the acceleration and surface-gravity equations agree

Facts & Assumptions

[F1]

The lapse is $N = - g (T, n)$ , $g (n, n) = - 1$ , and the connection is metric compatible. Since $ν$ is tangent to the spacelike hypersurface and normal to its boundary, $(n, ν)$ is an orthonormal basis of the normal plane to $S$ . Spacelike sections, maximality and lapse–shift data Metric compatibility, torsion freedom and the metric-selected connection class

[F2]

The circle-action definition supplies the generator and invariant-submanifold conventions. Its choice of least period $2 π$ for a normalized action is not needed here: the circle-pole lemma requires a smooth time-oriented circle action by isometries on the embedded spacelike two-sphere. That lemma supplies a pole with $T = Z = 0$ and trivial normal derivative of $Z$ ; the vanishing Killing field lemma supplies the normal boost form for $K$ . Periodic isometries and invariant normal averaging A circle action on a sphere has a pole with trivial normal action A Killing field vanishing on a spacelike surface has a normal boost derivative

[F3]

The additional branch, nowhere-zero generator, affine geodesic, parameter, zero-set and $K_{0}$ -normalization conditions are hypotheses of the second conclusion, not consequences of the horizon definition. For nonzero constant $c$ , the affine-profile supplier gives opposite branch accelerations. For the ambient Killing field $K$ , the covector supplier converts those accelerations into ambient covector equations. The bifurcate-horizon definition supplies the signed surface-gravity terminology on the nonzero-generator parts. Bifurcate Killing horizons, rotation and normalization Affine generator profiles give constant signed acceleration For a Killing field the acceleration and surface-gravity equations agree

Proof

Given: The data and hypotheses in the Statement.

Goal: Establish the lapse/boost identity and its conditional signed horizon consequences.

1.1givenF1F2

The action is smooth, time-orientation preserving and isometric, and preserves the embedded spacelike two-sphere $S$ , so the circle-pole lemma applies. It gives a pole $p$ with $T (p) = Z (p) = 0$ and $\nabla_{ν} Z = 0$ . The vanishing Killing field lemma applies to $K ∣_{S} = 0$ and the orthonormal normal basis $(n, ν)$ from [F1], giving $\nabla_{ν} K = a_{p} n$ and $\nabla_{n} K = a_{p} ν$ at $p$ .

2.1F1step 1.1

Differentiate the lapse in the inward tangent direction at the boundary: $ν (N) = - g (\nabla_{ν} T, n) - g (T, \nabla_{ν} n) = - g (\nabla_{ν} T, n)$ at $p$ since $T (p) = 0$ . Also $\nabla_{ν} K = \nabla_{ν} T$ because $Ω$ is constant and $\nabla_{ν} Z = 0$ . Thus $ν (N) ∣_{p} = - g (a_{p} n, n) = a_{p}$ . This also proves the sign assertion.

3.1givenF1step 1.1step 2.1

Under the additional branch and profile hypotheses, the constant relation $K = c K_{0}$ gives $\nabla_{ν} K = c \nabla_{ν} K_{0} = c n$ at $p$ . Comparing with step 1.1 yields $c = a_{p} = ν (N) ∣_{p}$ for any real $c$ . In the positive-derivative case this proves $c > 0$ , so in particular $c \neq 0$ as required by the affine-profile supplier.

4.1givenF3step 3.1∎

In that positive case the profiles $K = c s L_{+}$ and $K = - c t L_{-}$ , the nowhere-zero generators and the stipulated zero sets make $K$ nonzero on $H_{\pm} ∖ S$ and tangent and normal to each null branch. These complements are dense because the transverse intersection $S$ is a two-surface in each three-dimensional branch. The branches are therefore Killing horizons for the same ambient Killing field $K$ , with bifurcation surface $S$ . All affine-profile hypotheses in [F3] now hold, so $\nabla_{K} K = c K$ on $H_{+}$ and $\nabla_{K} K = - c K$ on $H_{-}$ . The ambient covector identity in [F3] gives $d (g (K, K)) = - 2 c K^{♭}$ and $d (g (K, K)) = 2 c K^{♭}$ , respectively. On their nonzero parts these specify the stated signed coefficients; at $S$ , smoothness and $K = 0$ give only the zero equations, which alone select no coefficient.

Depends on

Used by · 1

Local Hawking and axial fields, boost and pole

Dependency tree · 2 levels

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