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The mass factor in the recovered Ernst identity

Statement

Let $(M, g, N, o)$ be a smooth oriented, time-oriented Lorentzian four-manifold of signature $(-, +, +, +)$ , let $T$ be a smooth Killing field and let $m > 0$ be constant. Keep the manifold, orientation and time orientation fixed, and set $\hat{g} = m^{- 2} g$ and $\hat{T} = m T$ . Define $F_{a b} = \nabla_{a} T_{b}$ , $F = F + i ⋆_{g} F$ and the hatted fields by the same conventions. Squares denote complex bilinear contraction with both indices raised by the indicated metric, without conjugation or a factor of one half. Then $\hat{F} = m^{- 1} F$ and ${\hat{F}}_{\hat{g}}^{2} = m^{2} F_{g}^{2}$ on $M$ .

Let $U \subseteq M$ be a connected open set with a designated coordinate end $(t, x) : U_{e n d} \to I \times {∣ x ∣ > R_{0}}$ , where $I$ is a nonempty open interval, $R_{0} > 0$ , $T = \partial_{t}$ and $d t \land d x^{1} \land d x^{2} \land d x^{3}$ is positive. Assume a smooth normalized Ernst potential $σ$ is supplied on $U$ : $d σ = 2 ι_{T} F$ , $Re σ = - g (T, T)$ and $σ \to 1$ uniformly in $t \in I$ and the spatial directions as $∣ x ∣ \to \infty$ . For the hatted geometry use the same end subset with coordinates $(\hat{t}, x) = (t / m, x)$ . Then $\hat{σ} : = σ$ is its unique normalized Ernst potential on $U$ . Existence of the original global potential is a hypothesis, rather than a consequence of local closedness.

If a continuous extension of $σ$ to $U \cup B$ is supplied for a set $B \subseteq \overline{U} ∖ U$ , the same function extends $\hat{σ}$ there. Here $B$ may be empty. If $p \in U \cup B$ is a common zero of $T$ and a supplied smooth field $Z$ near $p$ , it remains the same common zero of $\hat{T}$ and $Z$ ; this includes a supplied axial field. Any supplied bifurcation sphere remains the same sphere. On $U \cup B$ , and hence on any such sphere contained there, the identity $- 4 {\hat{F}}_{\hat{g}}^{2} = (1 - \hat{σ})^{4}$ is equivalent to $- 4 m^{2} F_{g}^{2} = (1 - σ)^{4}$ . At such a common zero $p$ , the denominator $1 - σ (p)$ is nonzero, and the inequality $Re (1 / (1 - \hat{σ} (p))) > 1 / 2$ is exactly $Re (1 / (1 - σ (p))) > 1 / 2$ for the original geometry.

Statement references

Constant homothety preserves the original vacuum and maximal section

Vacuum Killing and Ernst differential identities

Killing two-form and normalized Ernst notation

Facts & Assumptions

[F1]

Constant homothety preserves the connection, Killing fields, supplied bifurcation sphere and original maximal section, and satisfies $⋆_{\hat{g}} = ⋆_{g}$ on two-forms. Constant homothety preserves the original vacuum and maximal section

[F2]

If $Ric (g) = 0$ , the vacuum Ernst one-form is closed and has local potentials; this supplies no global primitive or normalization by itself. Here the global normalized potential on the specified connected $U$ , and any boundary extension used for evaluation, are supplied hypotheses. Vacuum Killing and Ernst differential identities

[F3]

Use the Killing-form, complex bilinear contraction, coordinate-end and normalized-potential conventions; the normalization includes the real-part condition and the uniform end limit. Killing two-form and normalized Ernst notation

Proof

Given: The geometry, connected domain, designated end and supplied normalized potential in the Statement, with its optional extension and evaluation data when used.

Goal: Prove the scaling and potential transfer, and the equivalence of the displayed identities and inequalities on their stated domains.

1.1givenF1F3algebra

Lowering the hatted field gives ${\hat{T}}^{♭} = m^{- 1} T^{♭}$ . The common connection and Hodge map therefore give $\hat{F} = m^{- 1} F$ and $\hat{F} = m^{- 1} F$ . Since ${\hat{g}}^{- 1} = m^{2} g^{- 1}$ , raising the two indices in the full bilinear contraction multiplies the square by $m^{- 2} m^{4} = m^{2}$ . No complex conjugation is inserted.

2.1givenF2F3step 1.1algebra

The supplied global potential obeys $d σ = 2 ι_{T} F = 2 ι_{\hat{T}} \hat{F}$ , and $- \hat{g} (\hat{T}, \hat{T}) = - g (T, T) = Re σ$ . Thus $\hat{σ} : = σ$ meets both differential and real-part conditions on all of $U$ . The vacuum differential identity alone would only supply local potentials; the global one used here is the stated hypothesis.

3.1givenF3step 2.1

The new end chart has range $m^{- 1} I \times {∣ x ∣ > R_{0}}$ and $\hat{T} = \partial_{\hat{t}}$ . Its orientation is unchanged because $m > 0$ . For every $ε > 0$ , the original bound for all $t \in I$ and $∣ x ∣ > R$ is exactly the hatted bound for all $\hat{t} \in m^{- 1} I$ and $∣ x ∣ > R$ . Hence $\hat{σ}$ is normalized at this end. If $τ$ is any other hatted normalized potential on $U$ , then $d (τ - σ) = 0$ . On a coordinate ball, integration of this zero differential along straight coordinate segments makes the difference constant; it is therefore locally constant, and connectedness of $U$ makes it one constant on $U$ . Its limit zero on the designated end forces that constant to vanish.

4.1givenF1F3step 1.1step 3.1algebra∎

Equality on $U$ carries the supplied continuous extension of $σ$ to the hatted potential on $U \cup B$ ; it supplies their values at every stated evaluation point. The point $p$ is unchanged and ${\hat{T}}_{p} = m T_{p} = 0$ , while the unchanged $Z_{p}$ is zero. Any supplied bifurcation sphere is unchanged by [F1]. The tensor scaling from step 1.1 holds on all of $M$ , so substitution of $\hat{σ} = σ$ gives the equivalence of the two identities wherever the supplied potential or extension is defined. On $U$ the real-part condition is $Re σ = - g (T, T)$ , and continuity carries this equality to each point of $B$ because $B \subseteq \overline{U}$ . At the common zero $p$ , it gives $Re σ (p) = 0$ , hence $Re (1 - σ (p)) = 1$ and the denominator cannot vanish. The denominator is identical for both potentials, so their reciprocals, real parts and strict inequalities agree.

Depends on

Used by · 1

Conditional horizon recovery and Kerr implications

Dependency tree · 2 levels

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