Harrison electromagnetic form

PDF p. 6: §1.4; PDF p. 8: (2.6)

Use stationary Lorentzian data $(M,g,N,T)$ and a supplied normalized Ernst potential $\sigma =f+i\psi$ on an open set $U$, with $f=-g(T,T)$, Killing two-form $F_{ab}={\mathrm{\nabla }}_a{T}_b$, complex two-form $\mathcal{F}=F+i{\star }_{g}F$, and $d\sigma =2{\iota }_T\mathcal{F}$. Here ${\iota }_T\mathcal{F}$ is the one-form $v\mapsto \mathcal{F}(T,v)$, and the Hodge star uses the supplied orientation. Fix $q\in [0,1]$, put $B_q=q(1+\sigma )+(1-\sigma )$ and $D_q=B_q\overline{B_q}$, and use the Harrison one-form data on an open subset $V\subseteq U$ where $D_q>0$. In particular, $W_q$ is the supplied real one-form, $A_{q}v=v+D_q^{-1}{W}_q(v)T$, and $g_q=(D_q\mathrm{/}4)A_q^{\ast }g$ is the supplied Lorentzian metric. The pullback is fiberwise: $(A_q^{\ast }\mathcal{F})(u,v)=\mathcal{F}(A_{q}u,A_{q}v)$.

Define the complex Harrison electromagnetic two-form and its real part by

${\mathcal{H}}_q=-\sqrt{1-q^2}\frac{\overline{B_q}}{B_q}{A}_q^{\ast }\mathcal{F}=-\sqrt{1-q^2}\frac{\overline{B_q}}{B_q}(\mathcal{F}+\frac{W_q\wedge d\sigma }{2D_q}),F_q=\mathrm{Re}{\mathcal{H}}_q\mathrm{.}$

With the same orientation used for $g_q$ as for $g$, the associated identity is ${\star }_{g_q}{F}_q=\mathrm{Im}{\mathcal{H}}_q$. No Maxwell equation or existence of $W_q$ is asserted by this definition.

Stationary Harrison one-form and metric

Mass-one convention and Harrison denominator

Killing two-form and normalized Ernst notation

Complex self-dual forms and metric-dependent contraction

Geometric electrovacuum equations and charge convention

Given: The data, parameter and restrictions in the Definition.

Goal: Establish well-definedness, the wedge formula, and the Hodge identity.

1.1 Since $D_q=\mathrm{\mid }{B}_q{\mathrm{\mid }}^2>0$, the denominator $B_q$ is nonzero, and all coefficients are smooth; $\sqrt{1-q^2}$ is a real constant, including at the endpoints. Expanding the two arguments of the alternating form gives $A_q^{\ast }\mathcal{F}=\mathcal{F}+D_q^{-1}{W}_q\wedge {\iota }_T\mathcal{F}$: the term with two copies of $T$ vanishes. Substitution of $d\sigma =2{\iota }_T\mathcal{F}$ proves the second expression and shows that ${\mathcal{H}}_q$ and $F_q$ are smooth two-forms. [given, F1, F2, algebra]

2.1 At each point, the rank-one determinant identity gives $\det A_q=1+D_q^{-1}{W}_q(T)=d_q\mathrm{/}{D}_q>0$; thus $A_q$ preserves the specified orientation. Put $c=D_q\mathrm{/}4>0$. For real two-forms $\alpha ,\beta$, the defining Hodge pairing, $g_q=c{A}_q^{\ast }g$, and four-dimensional volume scaling yield $⟨A_q^{\ast }\alpha ,A_q^{\ast }\beta {⟩}_{g_q}=c^{-2}{A}_q^{\ast }⟨\alpha ,\beta {⟩}_g$ and ${\mathrm{v}\mathrm{o}\mathrm{l}}_{g_q}=c^2{A}_q^{\ast }{\mathrm{v}\mathrm{o}\mathrm{l}}_g$. Pulling back the defining wedge identity therefore gives ${\star }_{g_q}{A}_q^{\ast }\beta =A_q^{\ast }({\star }_g\beta )$; nondegeneracy of the wedge pairing gives uniqueness. Complex linearity and ${\star }_g\mathcal{F}=-i\mathcal{F}$ imply ${\star }_{g_q}{\mathcal{H}}_q=-i{\mathcal{H}}_q$. Writing ${\mathcal{H}}_q=F_q+i{G}_q$ with real $G_q$, comparison of real parts gives ${\star }_{g_q}{F}_q=G_q=\mathrm{Im}{\mathcal{H}}_q$. [given, F1, F2, F3, step 1.1, algebra] ∎

The cited article attributes the formulas. The justification uses the supplied geometric definitions and pointwise algebra, and does not use the article's assertion as a premise.