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Normalized Ernst data and zero Mars–Simon survive stationary gauges

Statement

On a connected Lorentzian four-dimensional domain $U$ carrying the stationary Lorentzian data and conventions of the Ernst interface, including an orientation and a fixed time orientation, let two smooth Ernst potentials $σ, σ^{'}$ satisfy $d σ = d σ^{'} = 2 ι_{T} F$ for the same geometric Killing field $T$ , metric, orientation and curvature convention. Suppose both tend to one on the same supplied parametrized end, in the sense of the Ernst interface. Then $σ = σ^{'}$ . A passive stationary-coordinate change means a change of coordinates on this same domain representing the same geometric data, including $T$ , the time orientation and the end; it does not rescale $T$ or change the orientation.

Fix a real constant $M$ (the mass parameter in applications). On $N = {p \in U : 1 - σ (p) \neq 0}$ define $P = (1 - σ)^{- 1}$ , $y = Re P$ , $z = Im P$ , $∣ D z ∣_{g}^{2} : = g^{- 1} (d z, d z) = g^{a b} (\partial_{a} z) (\partial_{b} z)$ , and $B : = z^{2} + 4 M^{2} (y^{2} + z^{2}) ∣ D z ∣_{g}^{2}$ . The contraction uses the spacetime inverse Lorentzian metric, without conjugation; the notation $∣ D z ∣_{g}^{2}$ does not assert positivity. These are smooth scalar fields on $N$ . Passive stationary-coordinate changes leave $F^{2}$ on $U$ and $y, z, B$ on $N$ unchanged, at the same geometric points and with the same $M$ .

For precision, let $S = s (S^{2}) \subset U$ for any fixed smooth embedding $s$ , where $S^{2} = {x \in R^{3} : \sum_{j = 1}^{3} (x^{j})^{2} = 1}$ has its usual smooth structure. The sphere scalar conditions meant here are $- 4 M^{2} F^{2} = (1 - σ)^{4}$ at every point of $S$ , and $y (p) > 1 / 2$ for at least one $p \in S \cap N$ . Each condition, whenever satisfied, remains satisfied after a passive stationary-coordinate change. This assertion does not require that all of $S$ lie in $N$ .

If the Mars–Simon tensor $S$ in the same convention vanishes throughout $N$ , then $∣ (1 - σ) S (T, E_{α}, E_{β}, E_{γ}) ∣ = 0$ in every frame on $N$ ; in particular every positive perturbative-smallness threshold for this expression is satisfied there. This includes the case $N = U$ and $S = 0$ throughout $U$ . The lemma establishes neither tensor vanishing, constancy or a bound for $B$ , nor the other hypotheses of a global rigidity theorem.

Statement references

Stationary Lorentzian data and tensor conventions

Ernst sublevels and the Mars–Simon tensor interface

Killing fields of a Lorentzian metric

The inverse metric, musical isomorphisms and index raising

Smooth embeddings

Facts & Assumptions

[F1]

The normalized Ernst differential, the supplied end limit, the Killing form and the Mars–Simon convention are those of the Ernst interface. Ernst sublevels and the Mars–Simon tensor interface

[F2]

The geometric Killing field is fixed under the passive coordinate changes in the Statement. Killing fields of a Lorentzian metric

[F3]

The inverse metric is a bilinear form on covectors, with components $g^{a b}$ in the coordinate covector basis. This defines the contraction used in $B$ . The inverse metric, musical isomorphisms and index raising

[F4]

The sphere in the Statement is the image of a fixed smooth embedding of the usual unit two-sphere; changing coordinates does not change this map or its image as geometric data. Smooth embeddings

[F5]

The stationary Lorentzian data on $U$ include a fixed time orientation, so the Killing-field and inverse-metric interfaces in [F2] and [F3] apply. Stationary Lorentzian data and tensor conventions

Proof

Given: The hypotheses in the Statement.

Goal: Prove uniqueness, the specified scalar invariances and the conditional zero-component assertion.

1.1F1given

Put $h = σ - σ^{'}$ . The defining equations give $d h = 0$ . In a coordinate ball, integrate the derivative of $h$ along each straight coordinate segment from its center: that derivative is zero, so $h$ is constant on the ball. Thus $h$ is locally constant; on connected $U$ the nonempty level set through any point is both open and closed, and hence is all of $U$ . Both potentials tend to one on the same supplied end, so their constant difference tends to zero and is zero.

2.1F1F2F5step 1.1

In a passive coordinate change the metric, geometric orientation, time orientation and field $T$ remain fixed. The metric connection, its Killing form, the oriented Hodge dual and their contractions in the Ernst interface therefore represent the same geometric tensors. In particular $F^{2}$ is the same complex scalar. Together with step 1.1 this identifies the same $σ$ and the same locus $N$ in both descriptions. The reciprocal has a smooth nonzero denominator on $N$ , so $P$ , its real and imaginary parts $y, z$ , and their differentials are smooth and have the same geometric values there.

3.1F1F3F4step 2.1

Evaluate the inverse-metric bilinear form of [F3] on the same covector $d z$ twice. In coordinate components this is exactly $g^{a b} (\partial_{a} z) (\partial_{b} z)$ ; under a coordinate change the inverse Jacobians in the two covector components cancel the Jacobians in the inverse-metric components. Hence this contraction and the explicitly defined $B$ are smooth invariant scalars on $N$ , with no sign or constancy asserted. For the fixed sphere image of [F4], the scalars on both sides of $- 4 M^{2} F^{2} = (1 - σ)^{4}$ have identical values at every point of $S$ in either description. Likewise any witness $p \in S \cap N$ to $y (p) > 1 / 2$ remains the same witness. This proves precisely the two conditional sphere assertions.

4.1F1F2step 2.1∎

If $S = 0$ on $N$ , evaluation on $T$ and any three frame vectors gives zero by multilinearity at every point of $N$ . Multiplication by the smooth scalar $1 - σ$ and taking the complex absolute value still give zero. Consequently every positive smallness threshold is satisfied by this expression in every frame on $N$ , including on all of $U$ when $N = U$ . This conditional argument supplies no vanishing theorem or additional rigidity hypotheses.

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