Kerr-Newman metric and potential on the exact Boyer-Lindquist domain

Use the smooth covariant metric and signature conventions of Smooth time-oriented Lorentzian four-manifolds: the geometric objects of the model and the smooth real form spaces ${\mathrm{\Omega }}^k(U)$ of A smooth differential $k$-form. Geometric electric flux-length normalization means the convention in Geometric electrovac equations and flux normalization; its field equations are not hypotheses in this coordinate definition.

Let $M,a,q\in \mathbb{R}$ have units of metres. Write $\mathrm{\Sigma }(r,\theta ):=r^2+a^2{\cos }^2\theta ,\mathrm{\Delta }(r):=r^2-2Mr+a^2+q^2,R(r):=r^2+a^2,B(\theta ):={\sin }^2\theta \mathrm{.}$ Let $U$ be an open subset of the Boyer-Lindquist coordinate region $\{(x^0,r,\theta ,\varphi ):\mathrm{\Sigma }>0,\mathrm{\Delta }\mathrm{\ne }0,0<\theta <\pi \}$. Coordinates $x^0,r$ have units of metres and $\theta ,\varphi$ are dimensionless. On $U$ define $g=-\frac{\mathrm{\Delta }}{\mathrm{\Sigma }}(d{x}^0-aBd\varphi {)}^2+\frac{\mathrm{\Sigma }}{\mathrm{\Delta }}d{r}^2+\mathrm{\Sigma }d{\theta }^2+\frac{B}{\mathrm{\Sigma }}(Rd\varphi -ad{x}^0{)}^2,\mathcal{A}=-\frac{qr}{\mathrm{\Sigma }}(d{x}^0-aBd\varphi ),\mathcal{F}:=d\mathcal{A}\mathrm{.}$ The metric is the exact real symmetric tensor given by this display; no value on $\mathrm{\Delta }=0$, the axes $\theta =0,\pi$, or $\mathrm{\Sigma }=0$ is part of this definition. The scalar $q$ is a geometric electric flux-length parameter; no SI conversion is fixed in this Definition.

Well-definedness on the stated patch. In the coordinates $(x^0,r,y,\varphi )$ with $y=\cos \theta$ and $b=1-y^2$, the coordinate change is a smooth diffeomorphism because $dy\mathrm{/}d\theta =-\sin \theta \mathrm{\ne }0$. The metric matrix has $g_{00}=-\frac{\mathrm{\Delta }-a^{2}b}{\mathrm{\Sigma }},g_{03}=-\frac{ab(R-\mathrm{\Delta })}{\mathrm{\Sigma }},g_{33}=\frac{b(R^2-a^2\mathrm{\Delta }b)}{\mathrm{\Sigma }},g_{11}=\frac{\mathrm{\Sigma }}{\mathrm{\Delta }},g_{22}=\frac{\mathrm{\Sigma }}{b}\mathrm{.}$ The inverse has nonzero entries $g^{00}=-\frac{R^2-a^2\mathrm{\Delta }b}{\mathrm{\Sigma }\mathrm{\Delta }},g^{03}=-\frac{a(R-\mathrm{\Delta })}{\mathrm{\Sigma }\mathrm{\Delta }},g^{33}=\frac{\mathrm{\Delta }-a^{2}b}{\mathrm{\Sigma }\mathrm{\Delta }b},g^{11}=\frac{\mathrm{\Delta }}{\mathrm{\Sigma }},g^{22}=\frac{b}{\mathrm{\Sigma }}\mathrm{.}$ Multiplying the $0,3$ blocks and the reciprocal $1,1$ and $2,2$ entries gives $g{g}^{-1}=I_4$. The $t,\varphi$ block determinant is $-\mathrm{\Delta }b$, and the full determinant is $-{\mathrm{\Sigma }}^2$ in these coordinates. If $\mathrm{\Delta }>0$, the $t,\varphi$ block has one positive and one negative eigenvalue while the $r,\theta$ entries are positive. If $\mathrm{\Delta }<0$, $g_{33}>0$ and the $t,\varphi$ block has positive determinant, hence is positive definite, while $g_{11}<0$ and $g_{22}>0$. Thus $g$ is nondegenerate of signature $(-,+,+,+)$ on every component of $U$. Since this is a coordinate patch, a smooth time orientation may be supplied locally; the coordinate tensors and their closedness do not depend on that choice.

The potential is a smooth real one-form on $U$ because $\mathrm{\Sigma }>0$, so $\mathcal{F}$ is a smooth real two-form. The coordinate formula in The local coordinate formula for the exterior derivative defines $\mathcal{F}=d\mathcal{A}$ componentwise. Applying it again pairs each second partial of a potential coefficient with its reversed-order partial and opposite wedge sign. They cancel by Clairaut--Schwarz theorem for continuous second partial derivatives, since every coefficient is smooth; hence $d\mathcal{F}=d^2\mathcal{A}=0$. This definition is only on the pole-free Boyer-Lindquist patch and asserts no global potential, axis extension, horizon extension, or global topology.