Harmonic-map and Weyl reconstruction comparison data

Let $\mathrm{\Gamma }=\{\rho =0\}\subset {\mathbb{R}}^3$ be the rotation axis in cylindrical coordinates $(\rho ,z,\varphi )$, with $\varphi$ of period $2\pi$. A reconstructed vacuum-map datum consists of the ordered rods of the ordered finite vacuum rod datum, two smooth real axisymmetric functions $u,v$ on ${\mathbb{R}}^3\setminus \mathrm{\Gamma }$, and smooth functions $w,\alpha$ there, satisfying $\mathrm{\Delta }u=2e^{4u}\mathrm{\mid }\mathrm{\nabla }v{\mathrm{\mid }}^2$ and $\mathrm{\Delta }v+4\mathrm{\nabla }u\cdot \mathrm{\nabla }v=0$. Derivatives and norms here are Euclidean. Equivalently these are the harmonic-map equations for the explicitly specified target metric $d{u}^2+e^{4u}d{v}^2$; only the displayed equations are used in this contract.

Put $U=u+\log \rho$. Require the reconstruction equations $w_{\rho }=2\rho e^{4u}{v}_z$, $w_z=-2\rho e^{4u}{v}_{\rho }$, ${\alpha }_{\rho }=\rho [U_{\rho }^2-U_z^2+e^{4u}(v_{\rho }^2-v_z^2)]$, and ${\alpha }_z=2\rho [U_{\rho }{U}_z+e^{4u}{v}_{\rho }{v}_z]$, together with $w\to 0$ and $\alpha \to 0$ at infinity. Define the reconstructed metric by $\mathbf{g}=-e^{2U}d{\tau }^2+{\rho }^2{e}^{-2U}(d\varphi +wd\tau {)}^2+e^{-2U+2\alpha }(d{\rho }^2+d{z}^2)$ on ${\mathbb{R}}_{\tau }\times ({\mathbb{R}}^3\setminus \mathrm{\Gamma })$. The datum supplies $w,\alpha$; their global existence or a reduction of arbitrary stationary geometries is not part of the definition.

Ordered horizon rods and twist-jump labels

Stationary Lorentzian data and tensor conventions

Given: Functions satisfying the stated equations and normalization, with the inputs in [F1]–[F3].

Goal: Verify that their reconstructed tensor is a well-defined stationary Lorentzian metric off the axis, and that the normalized reconstruction is unique conditional on existence.

1.1 Let $f$ be either $w$ or $\alpha$, and fix any two angular values ${\varphi }_1,{\varphi }_2$. On $H$, put $q(\rho ,z)=f(\rho ,z,{\varphi }_1)-f(\rho ,z,{\varphi }_2)$. The right sides of the reconstruction equations are independent of $\varphi$, because $u,v,U$ are axisymmetric. Thus $q$ is smooth with $q_{\rho }=q_z=0$. By [F2], $q$ is constant on $H$. Along $(\rho ,z)=(R,0)$ with $R\to \mathrm{\infty }$, both terms tend to zero by the normalization, so this constant is zero. As the angles were arbitrary, $w$ and $\alpha$ are axisymmetric. [given, F1, F2, algebra]

2.1 On $\rho >0$, all exponential coefficients are smooth and positive. The angular one-form $d\varphi$ is globally defined off the axis: local angular coordinates differ by constant multiples of $2\pi$, so their differentials agree. The coframe $d\tau$, $d\varphi +wd\tau$, $d\rho$, $dz$ is linearly independent, and the displayed tensor is diagonal in it with exactly one negative coefficient. Thus it is a smooth nondegenerate Lorentzian metric globally off the axis. Its coefficients are independent of $\tau$ by construction and of $\varphi$ by step 1.1, so time translations and rotations preserve it; in particular ${\mathrm{\partial }}_{\tau }$ is Killing in the stationary-field convention. [given, F1, step 1.1, algebra]

3.1 For fixed $u,v$ and two supplied normalized reconstruction pairs $(w,\alpha )$ and $(\tilde{w},\tilde{\alpha })$, fix $\varphi$ and restrict each difference to $H$. The reconstruction equations give zero $\rho ,z$ derivatives, so [F2] makes each difference constant on $H$. Along $(R,0)$ the zero limits of both pairs force each constant to vanish. This holds for every $\varphi$, giving equality on the full off-axis domain. The reconstruction equalities are definite scalar predicates, and this argument establishes uniqueness conditional on existence without an integrability, global-existence or geometric-reduction assertion. [given, F2, step 2.1, algebra] ∎

HKWX §1 equation (1.2) and its following target-metric display identify the two stated PDEs as the axisymmetric harmonic-map equations from Euclidean ${\mathbb{R}}^3\setminus \mathrm{\Gamma }$ into the target with metric $d{u}^2+e^{4u}d{v}^2$. Equations (1.1), (1.3) and (1.4), and the paragraph after (1.4), supply the metric ansatz, reconstruction quadratures and zero normalization adopted here. This is an imported identification of the displayed equations, not a claimed local proof of a stationary-vacuum reduction or a global reconstruction theorem. The Lorentzian-signature, symmetry and conditional-uniqueness arguments above are local.