Stationary-axisymmetric geometry and circularity domain

Let $(M,g,N,o)$ be a smooth, Hausdorff, second-countable, boundaryless, oriented and time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$, with smooth future timelike time-orientation field $N$, and let $U\subseteq M$ be a nonempty connected open set. A stationary field on $U$ is a specified smooth Killing field $k$ that is timelike on a specified nonempty open subset of $U$; no normalization or global timelikeness is inferred.

Here $U(1)=\{z\in \mathbb{C}:\mathrm{\mid }z\mathrm{\mid }=1\}$ with complex multiplication and the smooth circle-group structure fixed in Local potentials and smooth circle-bundle connection data. An axial $U(1)$ action on $U$ is a specified smooth map $\rho :U(1)\times U\to U$ satisfying $\rho (1,p)=p$ and $\rho (z_1{z}_2,p)=\rho (z_1,\rho (z_2,p))$, with $\rho (z,\cdot {)}^{\ast }g=g$ for every $z$; effective means that only $z=1$ acts identically on all of $U$. Its generator is the smooth vector field $m$ defined by $m_p=\frac{d}{d\theta }{\mathrm{\mid }}_{\theta =0}\rho (e^{i\theta },p)$; the parameter has period $2\pi$. Its axis is the fixed-point set $\mathcal{A}:=\{p\in U:\rho (z,p)=p\text{ for every }z\in U(1)\}$. For this page, an axisymmetric region is a region supplied with such an effective action and satisfying $g(m,m)>0$ on $U\setminus \mathcal{A}$. A stationary field and axial generator commute when their Lie bracket vanishes; in coordinates this means ${\sum }_a(k^a{\mathrm{\partial }}_a{m}^b-m^a{\mathrm{\partial }}_a{k}^b)=0$ for every component $b$.

Define

$U_{\ast }:=\{p\in U\setminus \mathcal{A}:k_p,m_p\text{ are linearly independent and }{g}_p{\mathrm{\mid }}_{\mathrm{span}\{k_p,m_p\}}\text{ is nondegenerate}\}\mathrm{.}$

Because $M$ is Hausdorff, each fixed set $\{p:\rho (z,p)=p\}$ is closed in $U$: it is the preimage of the closed diagonal in $U\times U$ under $p\mapsto (\rho (z,p),p)$. Their intersection $\mathcal{A}$ is closed, so $U\setminus \mathcal{A}$ is open. In a chart, linear independence is witnessed by a nonzero $2\times 2$ minor of the $4\times 2$ coordinate matrix with columns $k$ and $m$, and nondegeneracy is $\det G\mathrm{\ne }0$ for the continuous Gram matrix $G_{ij}:=g(K_i,K_j)$, where $K_1:=k$ and $K_2:=m$. These are open conditions, so $U_{\ast }$ is open. On $U_{\ast }$ the matrix $G$ is invertible. Its inverse entries are smooth by the cofactor-over-determinant formula. The smooth map

${\mathrm{\Pi }}_p(v):={\sum }_{i,j=1}^2{K}_i(p)(G(p{)}^{-1}{)}_{ij}{g}_p(K_j(p),v)$

is a projection onto $\mathrm{span}\{k,m\}$, since it fixes each $K_i$ and $v-{\mathrm{\Pi }}_p(v)$ is orthogonal to both generators. Therefore

${\mathcal{D}}_p:=\ker {\mathrm{\Pi }}_p=\{v\in T_{p}M:g_p(v,k_p)=g_p(v,m_p)=0\}$

is a smooth rank-two distribution on $U_{\ast }$: the projection has rank two in each tangent space, and its kernel has dimension $4-2=2$. The kernel varies smoothly because, locally, a nonzero two-by-two minor lets one solve its two linear equations smoothly for two components in terms of the other two.

An immersed surface through $p$ here means the image of a smooth immersion $\sigma :S\to U_{\ast }$ from a two-dimensional smooth manifold $S$, with $p\in \sigma (S)$; immersion means that $d{\sigma }_q:T_{q}S\to T_{\sigma (q)}{U}_{\ast }$ is injective for every $q\in S$. Its tangent plane at $\sigma (q)$ is $d{\sigma }_q(T_{q}S)$. Circularity on $U_{\ast }$ means that through every point of $U_{\ast }$ there is such a local immersed surface with $d{\sigma }_q(T_{q}S)={\mathcal{D}}_{\sigma (q)}$ at every point of its domain.

These are supplied geometric data and definitions. The specified smooth isometric $U(1)$ action has smooth generator $m$ and flow $p\mapsto \rho (e^{i\theta },p)$. Differentiating $\rho (e^{i\theta },\cdot {)}^{\ast }g=g$ at $\theta =0$ gives ${\mathcal{L}}_{m}g=0$, so $m$ is an axial Killing field on $U$ under Killing fields of a Lorentzian metric. This axial field is additional data, not derived from $k$. The definition does not assume that $k$ and $m$ commute or that $\mathcal{D}$ is integrable. The set $U_{\ast }$ may be empty; no nonemptiness is assumed. No symmetry inheritance from matter equations, global smooth quotient or orbit-space boundary structure, or Einstein–Maxwell solution is asserted. Any use of the causal or horizon terms follows Causal solutions, Killing horizons, escape and metric extensions. A future spacetime theorem must state any extra asymptotic, horizon, axis-regularity, analyticity, connectedness and nondegeneracy hypotheses it uses.

Oriented Lorentzian fields, observers and coordinate conventions

Killing fields of a Lorentzian metric

Causal solutions, Killing horizons, escape and metric extensions

Local potentials and smooth circle-bundle connection data

$C^r$ and smooth maps between smooth manifolds

The Lie derivative of a tensor field

Directional derivatives and partial derivatives of a map $U\subseteq {\mathbb{R}}^m\to {\mathbb{R}}^n$