Geometric electrovac equations and flux normalization

Inputs. Fix the integral domain $P=\mathbb{Q}[r,y,M,a,q]$ on the ordered variables $(r,y,M,a,q)$ and its field of fractions $K=\mathrm{Frac}(P)$. A run receives a symmetric $4\times 4$ matrix $g=(g_{ij})$ and a covector $A=(A_i)$ with entries in $K$, represented by finite sparse polynomial numerator-denominator pairs, together with a specified polynomial $w\in P$. It also receives the chart hypothesis that $\det g$ is nonzero in $K$ and that on the real open chart under consideration all input denominators are nonzero and $w=\sqrt{\mathrm{\mid }\det g\mathrm{\mid }}>0$. The coordinate derivations are ${\delta }_0={\delta }_3=0$, ${\delta }_1={\mathrm{\partial }}_r$, and ${\delta }_2={\mathrm{\partial }}_y$. The chart hypothesis is input; the procedure does not prove Lorentz signature or check the volume identity. For geometric interpretation use the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$, with $R^a{}_{bcd}$ the coefficient of $R({\mathrm{\partial }}_c,{\mathrm{\partial }}_d){\mathrm{\partial }}_b$ and ${\mathrm{Ric}}_{bd}:=R^a{}_{bad}$. Expanding this definition yields the coordinate Ricci schedule in Instruction 3.

Outputs. Return the finite tables of: the sixteen entries of $g{g}^{-1}-I$; the six independent components of $F_{ij}:={\delta }_i{A}_j-{\delta }_j{A}_i$; all sixteen raised components $F^{ij}$; the four Maxwell-density residuals ${\sum }_i{\delta }_i(w{F}^{ij})$; all sixty-four Christoffel entries; the sixteen Ricci entries ${\mathrm{Ric}}_{ij}={\sum }_k({\delta }_k{\mathrm{\Gamma }}^k{}_{ij}-{\delta }_j{\mathrm{\Gamma }}^k{}_{ik})+{\sum }_{k,l}({\mathrm{\Gamma }}^k{}_{kl}{\mathrm{\Gamma }}^l{}_{ij}-{\mathrm{\Gamma }}^k{}_{jl}{\mathrm{\Gamma }}^l{}_{ik});$ the scalar $F_{ij}{F}^{ij}$; the sixteen entries of ${\mathrm{Ric}}_{ij}-2(F_{ik}{F}_j{}^k-\frac{1}{4}{g}_{ij}{F}_{kl}{F}^{kl});$ the scalar curvature and the Ricci-square. For every output rational function, return its numerator-denominator pair, its exact numerator-zero flag, and the finite list of all divisor polynomials used.

Instructions.

1. Represent a polynomial by its finite set of exponent-vector/coefficient pairs. Addition merges equal exponent vectors, multiplication enumerates every pair of input monomials and collects equal exponent vectors, and ${\mathrm{\partial }}_r,{\mathrm{\partial }}_y$ differentiate each monomial by the integer-power rule of For a natural $n\ge 1$ the function $x\mapsto x^n$ is differentiable everywhere with derivative $\iota (n)x^{n-1}$; for $n=0$ it is the constant $1$, with derivative $0$; for a natural $n\ge 1$ the function $x\mapsto x^{-n}$ is differentiable at every $x\mathrm{\ne }0$ with derivative $-\iota (n)x^{-n-1}$; consequently every polynomial function is differentiable at every real, with the derivative computed term by term. Represent a rational function by $(p,d)$ with $p,d\in P$ and $d\mathrm{\ne }0$. Addition and multiplication use cross-products, and division replaces $(p,d)\mathrm{/}(u,v)$ by $(pv,du)$ only when $u\mathrm{\ne }0$ in $P$. Differentiation uses $(p^{\mathrm{\prime }}d-p{d}^{\mathrm{\prime }},d^2)$.

2. Compute $\det g$ by its twenty-four signed permutation products and form $g^{-1}$ from cofactors divided by $\det g$. Record and check all sixteen entries of $g{g}^{-1}-I$.

3. All indices in this instruction range over $\{0,1,2,3\}$, and $g^{ij}:=(g^{-1}{)}_{ij}$. Compute every $F_{ij}$, then raise indices by $F_j{}^k:={\sum }_l{g}^{kl}{F}_{jl}$ and $F^{ij}:={\sum }_{k,l}{g}^{ik}{g}^{jl}{F}_{kl}$. Compute all four ${\sum }_i{\delta }_i(w{F}^{ij})$. Compute every Christoffel entry by ${\mathrm{\Gamma }}^i{}_{jk}=\frac{1}{2}{\sum }_l{g}^{il}({\delta }_j{g}_{lk}+{\delta }_k{g}_{jl}-{\delta }_l{g}_{jk})$, then all sixteen Ricci entries by the displayed formula, with no symmetry-based component omitted. Compute $F_{ij}{F}^{ij}$, every Einstein residual, $g^{ij}{\mathrm{Ric}}_{ij}$ and $g^{ik}{g}^{jl}{\mathrm{Ric}}_{ij}{\mathrm{Ric}}_{kl}$ by their full finite contractions.

4. Record every polynomial divisor introduced by inversion, division, or a rational derivative in the admissibility list. Do not remove a divisor from that list when a later cancellation removes its factor from the displayed numerator or denominator. Return failure if any required divisor is the zero polynomial or an input denominator is zero.

A zero flag means exactly that the polynomial numerator is the zero element of $P$. No floating-point tolerance, parameter sample, convergence test, or choice of roots is part of the procedure. This is a finite mathematical procedure; its operational correctness, termination, and resource bounds are separate proved claims.

For every admissible input of Finite exact rational electrovac tensor verification, every returned rational numerator-denominator pair represents the corresponding displayed inverse, Maxwell, Christoffel, Ricci, scalar, or Einstein-residual expression in $K=\mathrm{Frac}(\mathbb{Q}[r,y,M,a,q])$. Its numerator-zero flag is true if and only if that rational function is identically zero in $K$. On any real open chart where the recorded divisors are nonzero and $w=\sqrt{\mathrm{\mid }\det g\mathrm{\mid }}$, a zero flag therefore certifies the corresponding exact coordinate identity. The signature and volume hypotheses are not certified by this lemma.

The lemma verifies the abstract procedure, not a particular run or a computer algebra library. A finite run may be recorded as additional exact-algebra evidence; it does not certify physical applicability or supply the signature and volume hypotheses.

For an input of Finite exact rational electrovac tensor verification with $N_{\mathrm{i}\mathrm{n}}$ input coefficient/exponent records, suppose the expanded fixed schedule has $K$ polynomial operations and every intermediate polynomial support has at most $R$ monomials. The algorithm terminates. Under elementary rational-coefficient arithmetic, sparse addition, multiplication, differentiation and sorted-list insertion perform at most $K(2R^2+2R)$ coefficient operations and at most that many ordered exponent-vector insertions; sorted-list insertion uses at most $R$ support comparisons per insertion. If every intermediate support is retained, memory is at most $(K+N_{\mathrm{i}\mathrm{n}})R$ coefficient/exponent records. These are algebraic-operation and record bounds in terms of actual intermediate support; the statement gives no polynomial input-size bound, bit-complexity bound, practical runtime estimate or optimality claim.

The script execution time stored with a finite check is an observation about that execution, not a consequence of this lemma.

Let $(M,g,N,o)$ be a smooth oriented, time-oriented Lorentzian four-manifold of signature $(-,+,+,+)$; $o$ is the manifold orientation, $N$ is a smooth future timelike field, and the metric volume form is chosen to agree with $o$. Let $\mathrm{\nabla }$ be the metric-selected connection, let $\mathcal{F}$ be a smooth real two-form on $M$, and use the curvature convention $R(X,Y)Z={\mathrm{\nabla }}_X{\mathrm{\nabla }}_{Y}Z-{\mathrm{\nabla }}_Y{\mathrm{\nabla }}_{X}Z-{\mathrm{\nabla }}_{[X,Y]}Z$. Define $Q_{ab}(\mathcal{F}):={\mathcal{F}}_{ac}{\mathcal{F}}_b{}^c-\frac{1}{4}{g}_{ab}{\mathcal{F}}_{cd}{\mathcal{F}}^{cd},$ where both raised indices use $g^{-1}$. A geometric electrovac datum with zero cosmological constant is the tuple $(M,g,N,o,\mathcal{F})$ satisfying $d\mathcal{F}=0,d({\star }_g\mathcal{F})=0,{\mathrm{Ric}}_{ab}=2Q_{ab}(\mathcal{F})$ at every point of $M$. Here ${\star }_g$ is the Lorentzian Hodge map determined by $g$ and $o$; on two-forms ${\star }_g^2=-1$. No material current, matter tensor, global potential, asymptotic flatness, positive mass, horizon, or extension condition is part of this definition.

For flux integration, an oriented embedded smooth sphere means a compact smooth two-dimensional manifold $S$ without boundary, homeomorphic to the unit sphere $\{x\in {\mathbb{R}}^3:x\cdot x=1\}$, with a specified orientation in the sense of Oriented smooth manifolds and oriented charts and a smooth inclusion $i:S\hookrightarrow M$. Here smooth embedding means an injective immersion that is a homeomorphism onto its image with the subspace topology (Smooth embeddings); identify $S$ with that image. The sphere orientation is specified independently of the spacetime orientation; for coordinate spheres used below it is the stated outward orientation.

For a smooth two-form $\eta$ on a neighborhood of $i(S)$, use the pullback convention of The pullback of a differential form, $(i^{\ast }\eta {)}_p(v,w)={\eta }_{i(p)}(d{i}_{p}v,d{i}_{p}w),p\in S,v,w\in T_{p}S\mathrm{.}$ By Pullback of forms is smooth functorial and preserves wedges, this is a smooth top form on $S$. Its support is the closure in $S$ of its nonzero locus (Compact support of a differential form), hence is closed in compact $S$ and compact by A closed subspace of a compact space is compact, and a finite union of compact subspaces is compact. Thus $i^{\ast }\eta \in {\mathrm{\Omega }}_c^2(S)$.

Write ${\mathcal{I}}_S$ for Integral of a compactly supported top form: for a subordinate smooth partition $({\rho }_j)$ and connected charts ${\varphi }_j$, it sets ${\mathcal{I}}_S(\omega )={\sum }_j{I}_{{\varphi }_j}({\rho }_j\omega )$, where $I_{{\varphi }_j}$ is the signed chart integral of the compactly supported coefficient. Only finitely many terms are nonzero. The value is independent of charts, partition and refinement by Independence of atlas, partition and refinement. Define the surface notation by ${\int }_S\eta :={\mathcal{I}}_S(i^{\ast }\eta )\mathrm{.}$ This agrees with Integration on an oriented embedded submanifold; a reparametrization here means an orientation-preserving diffeomorphism $F:T\to S$ of oriented smooth two-manifolds (a smooth bijection with smooth inverse and orientation-preserving differential). For such $F$, the same value is ${\mathcal{I}}_T((iF{)}^{\ast }\eta )$. The integration definition and its independence theorem assume ${\mathrm{A}\mathrm{C}}_{\omega }$, already retained in the geometric background below. For $\eta =\mathcal{F}$ and $\eta ={\star }_g\mathcal{F}$, smoothness of the given field and Hodge map makes both pulled-back top forms admissible; their integrals are finite real numbers. Define the geometric electric and magnetic flux parameters by $q_e(S):=\frac{1}{4\pi }{\int }_S{\star }_g\mathcal{F},q_m(S):=\frac{1}{4\pi }{\int }_S\mathcal{F}\mathrm{.}$ When $M$ and the coordinates have length units, these parameters have units of length. A magnetic flux need not have a global potential; no flux quantization is included. The background Lorentzian geometry and Hodge conventions carry the countable-choice assumption ${\mathrm{A}\mathrm{C}}_{\omega }$ of their named suppliers; the algebraic identities in this definition require no additional choice principle.

Let $M,q_e,q_m\in \mathbb{R}$ be length parameters and put $Q^2:=q_e^2+q_m^2$. Let $J\subset (0,\mathrm{\infty })$ be open and require $f(r):=1-\frac{2M}{r}+\frac{Q^2}{r^2}\mathrm{\ne }0(r\in J)\mathrm{.}$ On the smooth product ${\mathcal{M}}_J:={\mathbb{R}}_{x^0}\times J_r\times S^2$, use $S^2=\{(X,Y,Z)\in {\mathbb{R}}^3:X^2+Y^2+Z^2=1\}$ with its subspace topology (Euclidean spheres and closed balls as subspaces of ${\mathbb{R}}^n$). Specify the smooth structure by the sphere atlas of The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, Definition (b): for $\epsilon =\pm 1$, delete $p_{\epsilon }=(0,0,\epsilon )$ and use ${\sigma }_{\epsilon }(X,Y,Z)=(X,Y)\mathrm{/}(1-\epsilon Z)$. For $u=(u_1,u_2)$ and $h=u_1^2+u_2^2$, its inverse is ${\sigma }_{\epsilon }^{-1}(u)=(2u_1,2u_2,\epsilon (h-1))\mathrm{/}(1+h)$; the transition is $u\mapsto u\mathrm{/}h$ on $h>0$. These rational maps are smooth on their domains. The product atlas consists of $(x^0,r,p)\mapsto (x^0,r,{\sigma }_{\epsilon }(p))$ on $\mathbb{R}\times J\times (S^2\setminus \{p_{\epsilon }\})$; its transitions leave $x^0,r$ fixed and apply the displayed sphere transition. The product is Hausdorff and second countable by the subspace topology from ${\mathbb{R}}^2\times {\mathbb{R}}^3$, whose rational balls give a countable basis. These charts specify its maximal smooth atlas, including when $J$ is empty. The inclusion of $S^2$ into ${\mathbb{R}}^3$ is smooth with injective differential in these parametrizations; its induced unit round metric is ${\gamma }_{S^2}$ (The round metric on the sphere as an induced metric). Orient $S^2$ by its outward unit normal: for $p\in S^2$ and $u,v\in T_p{S}^2$, define $({\omega }_{S^2}{)}_p(u,v):=\det (p,u,v)\mathrm{.}$ Thus ${\omega }_{S^2}$ is the positive area form of ${\gamma }_{S^2}$, and in the polar chart $X=\sin \theta \cos \varphi$, $Y=\sin \theta \sin \varphi$, $Z=\cos \theta$ it is $\sin \theta d\theta \wedge d\varphi$. The dyonic RN datum is $g=-f(r)d{x}^0\otimes d{x}^0+f(r{)}^{-1}dr\otimes dr+r^2{\gamma }_{S^2},\mathcal{F}=-\frac{q_e}{r^2}d{x}^0\wedge dr+q_m{\omega }_{S^2}\mathrm{.}$ Here $x^0$ and $r$ have units of metres, angles on $S^2$ are dimensionless, and $M,q_e,q_m$ have units of metres. The metric components are dimensionless in length coordinates; the nonzero coordinate components of $\mathcal{F}$ have units inherited from their coordinate Jacobians. Use the orientation $d{x}^0\wedge dr\wedge {\omega }_{S^2}$.

The set $J$ may have several connected components. On $\{f>0\}$ choose the time orientation ${\mathrm{\partial }}_{x^0}$; on $\{f<0\}$ choose ${\mathrm{\partial }}_r$. These are smooth timelike fields on their respective open subsets because $g({\mathrm{\partial }}_{x^0},{\mathrm{\partial }}_{x^0})=-f<0$ when $f>0$ and $g({\mathrm{\partial }}_r,{\mathrm{\partial }}_r)=f^{-1}<0$ when $f<0$. Since $f$ has no zero on ${\mathcal{M}}_J$, this defines a smooth time orientation on the whole product. The sphere area form and the displayed two-form are global on $S^2$.

Let $N,S$ denote the north and south poles. On $U_N:=S^2\setminus \{S\}$ and $U_S:=S^2\setminus \{N\}$, respectively, local potential one-forms are ${\mathcal{A}}_N=-\frac{q_e}{r}d{x}^0+q_m(1-\cos \theta )d\varphi ,{\mathcal{A}}_S=-\frac{q_e}{r}d{x}^0-q_m(1+\cos \theta )d\varphi \mathrm{.}$ Each displayed potential extends smoothly over its named included pole: $(1-\cos \theta )d\varphi$ is smooth at $N$ and $(1+\cos \theta )d\varphi$ is smooth at $S$. Direct differentiation gives $d{\mathcal{A}}_N=d{\mathcal{A}}_S=\mathcal{F}$ on the respective patches. Their overlap difference is $2q_{m}d\varphi$; it is local potential data, not a global real gauge function on the entire sphere. The definition assumes no $U(1)$ bundle or charge quantization. To verify pole smoothness, use local coordinates $u=X,v=Y$ near each pole. There $d\varphi =(udv-vdu)\mathrm{/}(u^2+v^2)$. Near $N$, $1-\cos \theta =(u^2+v^2)\mathrm{/}(1+Z)$, so $(1-\cos \theta )d\varphi =(udv-vdu)\mathrm{/}(1+Z)$ is smooth. Near $S$, $1+\cos \theta =(u^2+v^2)\mathrm{/}(1-Z)$, so $(1+\cos \theta )d\varphi =(udv-vdu)\mathrm{/}(1-Z)$ is smooth. The denominators are nonzero on the respective named patches.

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.

Let $M,q_e,q_m\in \mathbb{R}$, $Q^2=q_e^2+q_m^2$, and let $J\subset (0,\mathrm{\infty })$ be open with $f(r)=1-2M\mathrm{/}r+Q^2\mathrm{/}{r}^2\mathrm{\ne }0$ on $J$. For the dyonic RN datum of Dyonic Reissner-Nordström metric and field domains, in a spherical-polar chart its Ricci components are ${\mathrm{Ric}}_{00}=f(\frac{f^{\mathrm{\prime }\mathrm{\prime }}}{2}+\frac{f^{\mathrm{\prime }}}{r})=\frac{f{Q}^2}{r^4},{\mathrm{Ric}}_{11}=-\frac{1}{f}(\frac{f^{\mathrm{\prime }\mathrm{\prime }}}{2}+\frac{f^{\mathrm{\prime }}}{r})=-\frac{Q^2}{f{r}^4},$ ${\mathrm{Ric}}_{22}=1-f-r{f}^{\mathrm{\prime }}=\frac{Q^2}{r^2},{\mathrm{Ric}}_{33}=(1-f-r{f}^{\mathrm{\prime }}){\sin }^2\theta =\frac{Q^2{\sin }^2\theta }{r^2},$ and all mixed components vanish. The scalar curvature is zero and ${\mathrm{Ric}}_{ab}{\mathrm{Ric}}^{ab}=4Q^4\mathrm{/}{r}^8$. The Kretschmann scalar is $R_{abcd}{R}^{abcd}=f^{\mathrm{\prime }\mathrm{\prime }2}+\frac{4(f^{\mathrm{\prime }}{)}^2}{r^2}+\frac{4(1-f{)}^2}{r^4}=\frac{48M^2}{r^6}-\frac{96M{Q}^2}{r^7}+\frac{56Q^4}{r^8}\mathrm{.}$ All formulas hold on every component of $J$, including $f<0$.

The calculation is a mathematical identity on $f\mathrm{\ne }0$. It makes no assertion at a coordinate root, at $r=0$, about a distributional source there, or about a maximal extension.

Let $M,a,q\in \mathbb{R}$ have units of metres, choose $R_0>4\mathrm{\mid }M\mathrm{\mid }$, and set $\mathcal{X}:={\mathbb{R}}_{x^0}\times (R_0,\mathrm{\infty }{)}_r\times S^2,R=r^2+a^2,b=1-Z^2,\mathrm{\Sigma }=r^2+a^2{Z}^2,\mathrm{\Delta }=r^2-2Mr+a^2+q^2,$ where $S^2=\{(X,Y,Z)\in {\mathbb{R}}^3:X^2+Y^2+Z^2=1\}$ has its induced unit round metric $\gamma$ and outward area form $\omega$. Specify the sphere smooth structure by The Schwarzschild exterior model datum: domain, metric, mass parameter and time orientation, Definition (b): on $S^2\setminus \{(0,0,\epsilon )\}$ use ${\sigma }_{\epsilon }(X,Y,Z)=(X,Y)\mathrm{/}(1-\epsilon Z)$, $\epsilon =\pm 1$. With $h=u_1^2+u_2^2$, the inverse is $u\mapsto (2u_1,2u_2,\epsilon (h-1))\mathrm{/}(1+h)$, and the transition on $h>0$ is $u\mapsto u\mathrm{/}h$. These maps are smooth on their domains. Give $\mathcal{X}$ the maximal atlas generated by the product charts $(x^0,r,p)\mapsto (x^0,r,{\sigma }_{\epsilon }(p))$; its transitions leave $x^0,r$ fixed and apply the displayed sphere transition. The product is Hausdorff and second countable as a subspace of ${\mathbb{R}}^5$, using rational balls for a countable basis. The inverse parametrizations make the sphere inclusion smooth with injective differential, so ${\gamma }_p(u,v)=u\cdot v$ and ${\omega }_p(u,v)=\det (p,u,v)$ are smooth tensors with the stated outward orientation. Write $\lambda :=XdY-YdX,\bar{g}:=-\frac{\mathrm{\Delta }-a^{2}b}{\mathrm{\Sigma }}(d{x}^0{)}^2+\frac{\mathrm{\Sigma }}{\mathrm{\Delta }}d{r}^2+\mathrm{\Sigma }\gamma +\frac{a^2(R+\mathrm{\Sigma }-\mathrm{\Delta })}{\mathrm{\Sigma }}{\lambda }^2-\frac{2a(R-\mathrm{\Delta })}{\mathrm{\Sigma }}d{x}^0\odot \lambda ,$ $\bar{\mathcal{A}}:=-\frac{qr}{\mathrm{\Sigma }}(d{x}^0-a\lambda ),\bar{\mathcal{F}}:=d\bar{\mathcal{A}}\mathrm{.}$ Here ${\lambda }^2=\lambda \otimes \lambda$ and $u\odot v=(u\otimes v+v\otimes u)\mathrm{/}2$. This is the full-sphere exterior extension of the pole-free datum in Kerr-Newman metric and potential on the exact Boyer-Lindquist domain. The coordinates $x^0,r$ and parameters $M,a,q$ have units of metres; sphere coordinates are dimensionless. The field $\bar{\mathcal{F}}$ is the geometric length-normalized two-form and no SI conversion is part of this definition.