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Stationary exact corrections preserve two prescribed gauges

Statement

Let $(M, g, T)$ be stationary Lorentzian data and let $E \subset M$ be open, equipped with the restricted smooth charts of $M$ . Only the smooth manifold and field $T$ enter the construction. A stationary region in $E$ means an open subset invariant under the local flow of $T ∣_{E}$ wherever that flow is defined. A smooth differential form $α$ is stationary when $L_{T} α = 0$ , and a smooth function $h$ is stationary when $T h = 0$ . Closed and exact mean respectively $d α = 0$ and $α = d β$ for a smooth form $β$ of one lower degree.

On $E$ let $Θ$ be a smooth closed two-form, $D$ a smooth function, $c \in R$ , and $V$ a smooth stationary one-form with $d V = Θ$ and $V (T) + D = c$ . For $j \in {H, \infty}$ let $U_{j} \subset E$ be disjoint stationary regions, and let $V_{j}$ be smooth stationary one-forms on $U_{j}$ with $d V_{j} = Θ ∣_{U_{j}}$ and $V_{j} (T) + D = c$ . Assume that every smooth closed one-form on each $U_{j}$ is exact. Supply smooth stationary functions $η_{j} : E \to [0, 1]$ with ${supp}_{E} η_{j} \subset U_{j}$ and $η_{j} = 1$ on prescribed open subsets $P_{j} \subset U_{j}$ , where ${supp}_{E} η_{j}$ is the closure in $E$ of ${η_{j} \neq 0}$ . Then there exist smooth stationary functions $h_{H}, h_{\infty}$ on $E$ such that $W = V + d h_{H} + d h_{\infty}$ is stationary, $d W = Θ$ , $W (T) + D = c$ , and $W ∣_{P_{j}} = V_{j} ∣_{P_{j}}$ . Thus the corrections are globally exact stationary one-forms and both prescribed gauges are retained.

For the boundary-extension conclusion, let $S \subset \partial_{M} E$ be a smooth embedded two-sphere in $M$ , with $\partial_{M} E$ the topological boundary. Suppose there is an open neighborhood $O \subset M$ of $S$ with $O \cap E \subset P_{H}$ and a smooth one-form ${\overline{V}}_{H}$ on $O$ agreeing with $V_{H}$ on $O \cap E$ . Then $W$ extends smoothly to $E \cup O$ by ${\overline{V}}_{H}$ on $O$ . Every specified local symmetry $L_{X} {\overline{V}}_{H} = 0$ for a smooth field $X$ on $O$ is inherited by the extension on $O$ . This includes stationarity on $O$ if $X = T ∣_{O}$ is among these fields; no symmetry in the cutoff transition regions is asserted.

Facts & Assumptions

[F1]

The stationary Lorentzian interface supplies the ambient smooth manifold $M$ and smooth field $T$ . Open subsets carry canonical restricted smooth structures, independently of the presenting atlas; these are used for $E$ , $U_{j}$ , $P_{j}$ , $O$ , $O \cap E$ and $E \cup O$ . Regions use the local-flow convention, without a completeness assumption. Stationary Lorentzian data and tensor conventions An open subset of a smooth manifold has a canonical restricted smooth structure Local and global flows generated by a vector field

[F2]

Smooth forms and their Lie derivatives have the usual alternating-tensor meanings; stationarity of forms is explicitly defined in the Statement by vanishing Lie derivative. A smooth differential k-form The Lie derivative of a differential form

[F3]

Closed means exterior derivative zero and exact means an exterior derivative of a smooth form. Exactness of every closed one-form on each $U_{j}$ is an explicit hypothesis, not inferred from stationarity. Closed and exact differential forms

[F4]

Support is the closure of the nonzero locus, as in the smooth partition-of-unity convention. The supplied cutoffs are required to be stationary; neither their existence nor this extra property is deduced from a partition-of-unity definition. Smooth partition of unity on a manifold with boundary Interior, closure, boundary, exterior, derived set and isolated point in a topological space

[F5]

For a smooth scalar $a$ and field $Y$ , $d a (Y) = Y a$ . The exterior derivative of a function is its differential

[F6]

The exterior derivative is local and satisfies $d^{2} = 0$ . The exterior derivative is local The exterior derivative squares to zero

[F7]

Cartan's formula is $L_{Y} α = d (ι_{Y} α) + ι_{Y} (d α)$ . Cartan's magic formula

[F8]

The boundary in the extension clause is the topological boundary in $M$ , and $S$ is an embedded smooth submanifold that is a two-sphere with its usual smooth structure. These hypotheses do not identify all of $\partial_{M} E$ with $S$ or assert that $E \cup S$ is a manifold with boundary. Interior, closure, boundary, exterior, derived set and isolated point in a topological space Embedded submanifolds and slice charts The n-sphere with its standard smooth atlas

[F9]

A smooth vector field $X$ acts on smooth functions by the product rule $X (a b) = a X b + b X a$ . A vector field acts as a derivation of smooth functions

Proof

Given: The smooth data, exactness assumptions, cutoffs and, for the last conclusion, the ambient neighborhood and extension in the Statement.

Goal: Construct the exact corrections, retain both gauges and normalization, and glue the asserted local extension.

1.1givenF1F3F5

For each $j \in {H, \infty}$ , the one-form $V_{j} - V ∣_{U_{j}}$ is closed because its exterior derivative is $Θ ∣_{U_{j}} - Θ ∣_{U_{j}} = 0$ . Exactness on $U_{j}$ supplies a smooth scalar $χ_{j}$ with $d χ_{j} = V_{j} - V ∣_{U_{j}}$ . The equal contraction constants give $T χ_{j} = d χ_{j} (T) = V_{j} (T) - V (T) = 0$ on $U_{j}$ . This requires no choice of a normalization for $χ_{j}$ and no completeness of $T$ .

2.1givenF1F2F4F5F6F7F9step 1.1

Define $h_{j} = η_{j} χ_{j}$ on $U_{j}$ and $h_{j} = 0$ on $E ∖ U_{j}$ . Every point of $E ∖ U_{j}$ lies outside the closed set ${supp}_{E} η_{j}$ , so it has an open neighborhood in $E$ on which $η_{j} = 0$ . On that neighborhood the two definitions of $h_{j}$ agree identically with zero, proving smoothness across the edge of $U_{j}$ . On $U_{j}$ , the directional product rule gives $T h_{j} = (T η_{j}) χ_{j} + η_{j} T χ_{j} = 0$ ; elsewhere it follows from the same local zero description. Thus $d h_{j} (T) = 0$ globally. Also $d (d h_{j}) = 0$ , and Cartan's formula yields $L_{T} (d h_{j}) = d (d h_{j} (T)) + ι_{T} d (d h_{j}) = 0$ . Hence each correction is exact and stationary.

3.1givenF1F2F4F6step 1.1step 2.1

Put $W = V + d h_{H} + d h_{\infty}$ . By step 2.1 it is smooth and stationary, $d W = d V = Θ$ , and $W (T) + D = V (T) + D = c$ . On $P_{j}$ the function $h_{j}$ equals $χ_{j}$ on an open set, so locality gives $d h_{j} = d χ_{j}$ there. The other correction vanishes on $U_{j}$ , since its support lies in the disjoint region. Consequently $W = V + d χ_{j} = V_{j}$ on $P_{j}$ . This proves both prescribed gauge identities without altering the normalization.

4.1givenF1F2F8step 3.1∎

In the boundary-extension setting, $O \cap E \subset P_{H}$ , so step 3.1 gives $W = V_{H} = {\overline{V}}_{H}$ on the entire open overlap $O \cap E$ . Define $\tilde{W} = W$ on $E$ and $\tilde{W} = {\overline{V}}_{H}$ on $O$ . The overlap equality makes this well-defined. Every point has an open neighborhood on which it is one of these smooth forms, so it is smooth on $E \cup O$ . On $O$ it is exactly ${\overline{V}}_{H}$ , and therefore has every stipulated symmetry $L_{X} {\overline{V}}_{H} = 0$ there. This proves the extension and symmetry conclusion on an actual neighborhood of the sphere.

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