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Reflection parities of canonical section data and stationary lapse–shift

Statement

Assume the exact local boost and commuting-field hypotheses. In any adapted coordinate patch made $W$ -invariant as in the reflection-jet supplier, let $O_{0}$ be its domain, $S_{0} = S \cap O_{0}$ , $W (U, V, y) = (- U, - V, y)$ and $Q = {U = V = r} \cap O_{0}$ . Suppose this local canonical section is spacelike near $S_{0}$ . Choose a continuous future unit normal $n$ and put $h = g ∣_{Q}$ , $K (v, w) = g (\nabla_{v} n, w)$ , $N = - g (T, n)$ and $X = T - N n$ . Let $R = W ∣_{Q}$ , so $R (r, y) = (- r, y)$ .

On this patch the infinite-jet parities are $j_{S_{0}}^{\infty} R^{*} h = j_{S_{0}}^{\infty} h$ , $j_{S_{0}}^{\infty} R^{*} K = - j_{S_{0}}^{\infty} K$ , $j_{S_{0}}^{\infty} R^{*} N = - j_{S_{0}}^{\infty} N$ and $j_{S_{0}}^{\infty} R^{*} X = j_{S_{0}}^{\infty} X$ . Section-jet equality means equality of all component derivatives in $(r, y)$ along $r = 0$ , including all mixed normal and tangential orders. These conclusions hold locally near every point where the corresponding canonical section is supplied and spacelike. Here $K$ has the opposite sign from the library datum $k = - g (\nabla n, \cdot)$ .

If, in addition, a common smooth spacelike canonical section $Q_{g l}$ through all of $S$ and a smooth involution $R_{g l}$ on its neighborhood of $S$ are supplied, and near every point of $S$ they agree with the local sections and reflections just specified, then all four parity identities hold along the whole $S$ for the induced data on $Q_{g l}$ and pullback by $R_{g l}$ . This retains the whole-surface conclusion when those compatible data exist. Neither a common section nor compatibility of patch reflections is inferred from the boost formula alone, and no section beyond the supplied canonical neighborhoods is asserted.

Statement references

All-jet reflection parity from an exact local boost

Spacelike sections, maximality and lapse–shift data

Facts & Assumptions

Given: The data and hypotheses of the Statement.

Goal: The asserted conclusion with its stated scope.

[F1]

On each adapted $W$ -invariant patch the exact-boost supplier gives $j_{S_{0}}^{\infty} (W^{*} g) = j_{S_{0}}^{\infty} g$ and $j_{S_{0}}^{\infty} (W^{*} T) = j_{S_{0}}^{\infty} T$ , using its pointwise pullback conventions. Its common-involution conclusion requires a separately supplied compatible involution; no gluing of different patch reflections is part of the local conclusion. All-jet reflection parity from an exact local boost

[F2]

A spacelike section has induced metric, future unit normal and second fundamental form; these are obtained from the metric and its first derivatives. Spacelike sections, maximality and lapse–shift data

[F3]

The Levi–Civita connection of a smooth Lorentzian metric is torsion-free and metric compatible. Its coordinate coefficients satisfy $Γ^{a}_{b c} = \frac{1}{2} g^{a d} (\partial_{b} g_{d c} + \partial_{c} g_{b d} - \partial_{d} g_{b c})$ , with $g^{a d}$ the inverse-metric entries. This formula uses nondegeneracy, without a positive-definiteness hypothesis. Metric compatibility, torsion freedom and the metric-selected connection class

[F4]

The product, quotient and chain rules apply to smooth component functions; repeated application computes every finite-order derivative of a smooth composition. Sum, product, quotient, and chain rules from difference quotients The chain rule for total derivatives: D(g∘ f)(a)=Dg(f(a))∘ Df(a)

Proof

Given: The data in the Statement. Fix one adapted patch; in steps 1.1–4.1 write $S$ for its subset $S_{0}$ , and use $Q, R, W$ only on that patch.

Goal: Prove the four local section-jet parities and their conditional whole-surface version.

1.1F1F2F4algebra

Use the $W$ -invariant neighborhood and pullback conventions of F1. Write $A = d W = diag (- 1, - 1, 1, 1)$ , $\tilde{g} = W^{*} g$ and $\tilde{T} = W^{*} T$ . Since $A$ is constant and $W (Q) = Q$ , restricting the metric pullback to $T Q$ gives $\tilde{g} ∣_{T Q} = R^{*} h$ . Restriction to $Q = {U = V = r}$ differentiates component functions by $\partial_{r} = \partial_{U} + \partial_{V}$ and the tangential coordinate derivatives. Thus F1 gives equality of all section jets of $R^{*} h$ and $h$ . Inverse-matrix entries are cofactors divided by the nonzero determinant; consequently repeated product and quotient rules also give equality of all ambient jets of ${\tilde{g}}^{- 1}$ and $g^{- 1}$ .

2.1F1F2F4step 1.1algebra

Put $α = d (U - V)$ , which annihilates $T Q$ . For either metric $q = g$ or $q = \tilde{g}$ , the vector $q^{- 1} α$ is nonzero and orthogonal to the spacelike hyperplane $T Q$ , so $q^{- 1} (α, α) < 0$ . The two unit normals are therefore $\pm ν (q)$ , where $ν (q) = q^{- 1} α / \sqrt{- q^{- 1} (α, α)}$ . Locally near each point of $S$ , the given continuous future normal is $n = σ ν (g)$ with a constant sign $σ \in {1, - 1}$ ; hence it is smooth. Set $m_{x} = A n_{W (x)}$ along $Q$ . Because $W^{*} α = - α$ , the inverse-metric pullback formula gives $m = - σ ν (\tilde{g})$ . Step 1.1 and repeated differentiation of this smooth normalization imply $j_{S}^{\infty} m = - j_{S}^{\infty} n$ on $Q$ . This is a jet identity and does not assert an actual reflection isometry.

2.2F1F3F4step 1.1algebra

Let $ε_{U} = ε_{V} = - 1$ and $ε_{A} = 1$ , so $A^{i}_{j} = ε_{i} δ^{i}_{j}$ . The component pullback identities are ${\tilde{g}}_{i j} (x) = ε_{i} ε_{j} g_{i j} (W (x))$ and ${\tilde{g}}^{i j} (x) = ε_{i} ε_{j} g^{i j} (W (x))$ . Substitute these into the Lorentzian Christoffel formula in F3 and differentiate the composition with $W$ . Each derivative in coordinate $b$ contributes $ε_{b}$ , and the two factors of each contracted sign cancel, giving ${\tilde{Γ}}^{a}_{b c} (x) = ε_{a} ε_{b} ε_{c} Γ^{a}_{b c} (W (x))$ . Independently, that same Christoffel formula and step 1.1 show that every derivative of ${\tilde{Γ}}^{a}_{b c} - Γ^{a}_{b c}$ vanishes on $S$ : an order- $k$ derivative of its formula uses only inverse-metric derivatives through order $k$ and metric derivatives through order $k + 1$ . No Riemannian connection theorem is being extended to indefinite signature.

3.1F2F3F4step 1.1step 2.1step 2.2algebra

For smooth ambient fields $v, w$ , write $\tilde{v} = A v \circ W$ and $\tilde{w} = A w \circ W$ . Expanding the connection derivative in components gives $({\tilde{\nabla}}_{\tilde{v}} \tilde{w})^{a} = {\tilde{v}}^{b} \partial_{b} {\tilde{w}}^{a} + {\tilde{Γ}}^{a}_{b c} {\tilde{v}}^{b} {\tilde{w}}^{c} = ε_{a} (v^{b} \partial_{b} w^{a} + Γ^{a}_{b c} v^{b} w^{c}) \circ W$ by step 2.2 and $ε_{i}^{2} = 1$ . Thus the reflection covariance needed here has been derived directly. Extend $n$ smoothly off $Q$ as allowed by F2 and apply this identity in tangent directions to $Q$ . Together with the metric pullback it yields $(R^{*} K)_{x} (u, v) = {\tilde{g}}_{x} ({\tilde{\nabla}}_{u} m, v)$ for $u, v \in T_{x} Q$ . In section coordinates this expression uses only the metric, connection coefficients, the normal and its first tangential derivatives. Steps 1.1, 2.1 and 2.2 therefore imply $j_{S}^{\infty} R^{*} K = - j_{S}^{\infty} K$ . The sign comes from the normal reversal and requires no maximality condition.

4.1F1F2F4step 1.1step 2.1step 3.1algebra

The scalar lapse pullback is $R^{*} N = - \tilde{g} (\tilde{T}, m) ∣_{Q}$ . By F1 and step 2.1 its section jet is that of $- g (T, - n) = g (T, n) = - N$ . For the tangential shift, vector pullback along $R$ agrees with the restriction of the ambient pullback, since $d W$ maps $T Q$ to $T Q$ . Therefore $R^{*} X = \tilde{T} ∣_{Q} - (R^{*} N) m$ has the same section jet as $T - (- N) (- n) = T - N n = X$ . Together with steps 1.1 and 3.1 these are all four asserted parities, with precisely the stated local scope and the convention $K = - k$ .

5.1givenF1F2step 1.1step 2.1step 3.1step 4.1∎

Now let $S$ again denote the whole surface and suppose the compatible data $Q_{g l}, R_{g l}$ in the Statement are supplied. Their restrictions near each point agree with the local $Q, R$ . The induced metrics, second fundamental forms and lapse–shift fields therefore agree there as well: they are computed from the same ambient metric and field, and the future unit normal is unique. Pullback by $R_{g l}$ restricts to pullback by $R$ . Steps 1.1–4.1 make every derivative of each corresponding parity difference vanish along $S_{0}$ in that patch. Since these patches cover $S$ , all four global parity differences have zero infinite jet along $S$ . This proves the retained whole-surface conclusion under its explicit compatibility hypotheses, without constructing a common section or identifying distinct patch reflections.

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