Smooth null screen and subextremal Kerr–Newman parameter family

Let $(\mathcal{M},g,N)$ be a time-oriented smooth Lorentzian four-manifold of signature $(-,+,+,+)$, with the metric-selected connection $\mathrm{\nabla }$ of Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields. Here $N$ is the supplied smooth future timelike vector field defining the time orientation; it is not a coordinate frame or a tetrad.

A smooth null hypersurface is a smooth embedded three-dimensional submanifold $H\subset \mathcal{M}$ such that, at every $p\in H$, the restriction of $g_p$ to $T_{p}H$ has a one-dimensional radical $L_p:=\{v\in T_{p}H:g_p(v,w)=0\text{ for every }w\in T_{p}H\}\mathrm{.}$ Write $i:H\hookrightarrow \mathcal{M}$ for the inclusion, and let $\pi :T\mathcal{M}\to \mathcal{M}$ be the tangent-bundle projection. A smooth vector field $V$ along $H$ means a smooth map $V:H\to T\mathcal{M}$ with $\pi \circ V=i$, equivalently a smooth section of $i^{\ast }T\mathcal{M}=\{(p,v):p\in H,v\in T_p\mathcal{M}\}\mathrm{.}$ For a smooth tangent field $X$ on $H$, the notation ${\mathrm{\nabla }}_{X}V$ means $(i^{\ast }\mathrm{\nabla }{)}_{X}V$, using the pullback connection of Pullback connection for the ambient metric-selected connection $\mathrm{\nabla }$; its prescription in a pulled-back frame and frame-independent well-definedness are given by Pullback connection is well defined and functorial. It agrees with restricting ${\mathrm{\nabla }}_{\tilde{X}}\tilde{V}$ for any local smooth ambient extensions when $V$ and $X$ are restricted to $H$. Such extensions exist in the slice coordinates of Embedded submanifolds and slice charts by extending coordinate coefficients smoothly off the slice. For tangent $X,Y$, extending their tangential coordinate coefficients independently of the transverse coordinate and setting their transverse components to zero makes the ambient bracket restrict to the intrinsic bracket on $H$. Restricting the ambient metric-compatibility and torsion-free identities therefore gives, for tangent $X,Y$ on $H$ and fields $V,W$ along $H$, $Xg(V,W)=g({\mathrm{\nabla }}_{X}V,W)+g(V,{\mathrm{\nabla }}_{X}W),{\mathrm{\nabla }}_{X}Y-{\mathrm{\nabla }}_{Y}X=[X,Y]\mathrm{.}$ Pullback functoriality along any integral curve $\gamma$ in $H$ identifies the curve derivative with the pullback of $i^{\ast }\mathrm{\nabla }$; accordingly ${\mathrm{\nabla }}_{k}k=0$ means the tangent field is parallel along its generator with this affine parameter.

A smooth affine null generator field on $H$ is a smooth vector field $k$ along $H$ such that $k_p$ is a nonzero future-directed section of $L_p$ and ${\mathrm{\nabla }}_{k}k=0$ along $H$. Thus $k$ is both tangent and normal to $H$, and its integral-curve parameter is affine.

For each $p\in H$, the screen space is the quotient vector space ${\mathcal{S}}_p:=k_p^{\perp }\mathrm{/}\mathbb{R}{k}_p,k_p^{\perp }:=\{v\in T_p\mathcal{M}:g_p(v,k_p)=0\}\mathrm{.}$ Its candidate screen metric is ${\bar{g}}_p([X],[Y]):=g_p(X,Y)$. The result The null screen quotient is a smooth positive two-plane ↗ establishes that these spaces form a smooth rank-two vector bundle over $H$ and that $\bar{g}$ is a well-defined positive-definite metric on it.

For smooth local screen sections represented by smooth fields $X,Y$ tangent to $H$, define the screen deformation form $B([X],[Y]):=g({\mathrm{\nabla }}_{X}k,Y)\mathrm{.}$ The same lemma establishes that this form is well defined on the quotient. Its expansion is $\theta :={\mathrm{tr}}_{\bar{g}}B$ and its shear is $\sigma :=B-\frac{1}{2}\theta \bar{g}$. In the area theorem below, $B$ is proved symmetric, so $\sigma$ is a symmetric trace-free form on each screen space.

Independently, let $\mathcal{P}:=\{(m,a,q)\in {\mathbb{R}}^3:m>0,m^2>a^2+q^2\}\mathrm{.}$ These are real mathematical parameters with no physical units assigned here. On this open parameter domain set $s:=\sqrt{m^2-a^2-q^2},r_{+}:=m+s,D:=r_{+}^2+a^2,$ $A:=4\pi D,j:=ma,{\kappa }_{\mathrm{K}\mathrm{N}}:=\frac{r_{+}-m}{D},{\mathrm{\Omega }}_{\mathrm{K}\mathrm{N}}:=\frac{a}{D},{\mathrm{\Phi }}_{\mathrm{K}\mathrm{N}}:=\frac{q{r}_{+}}{D}\mathrm{.}$ Because the radicand and $D$ are strictly positive on $\mathcal{P}$, these are smooth real-valued functions there: the repeated derivative formula for $x^{1\mathrm{/}2}$ on $(0,\mathrm{\infty })$ and the chain rule are given in Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model, Facts [F2]. This definition assigns no geometric or thermodynamic interpretation to these functions and asserts no event-horizon existence, formation, uniqueness or thermodynamic-identification theorem.

The screen construction is local and makes no claim that every null hypersurface is complete or has compact cross-sections. The parameter domain is the strictly subextremal open family only; its extremal boundary is handled separately by the exact supplier in the companion example.