SI and geometrized Einstein–Maxwell parameters — Examples

Given: The conformal Maxwell covariance and Einstein-tensor transformation identities of Four-dimensional conformal Maxwell covariance and Conformal Einstein tensor transformation, on the smooth Lorentzian geometry of Oriented Lorentzian fields, observers and coordinate conventions. The inherited geometric setup carries ${\mathrm{A}\mathrm{C}}_{\omega }$ as in [A1].

A smooth positive conformal rescaling in four dimensions preserves the source-free Maxwell equations when $F=0$ and $j=0$. (Four-dimensional conformal Maxwell covariance)

For $\tilde{g}={\mathrm{\Omega }}^{2}g$ and $u={\mathrm{\Omega }}^{-1}d\mathrm{\Omega }$, the scalar curvature is $\tilde{S}={\mathrm{\Omega }}^{-2}(S-6\theta -6s)$, with $\theta =g^{ab}{\mathrm{\nabla }}_a{u}_b$ and $s=g^{ab}{u}_a{u}_b$. (Conformal Einstein tensor transformation)

The metric-selected connection has the Christoffel formula ${\mathrm{\Gamma }}^a{}_{bc}=\frac{1}{2}{g}^{ad}({\mathrm{\partial }}_b{g}_{dc}+{\mathrm{\partial }}_c{g}_{bd}-{\mathrm{\partial }}_d{g}_{bc})$. (Metric compatibility, torsion freedom and the metric-selected connection class)

The Einstein tensor is $G_{ab}={\mathrm{Ric}}_{ab}-\frac{1}{2}S{g}_{ab}$, so $g^{ab}{G}_{ab}=-S$ in four dimensions. (The Riemann curvature operator and the derived Ricci, scalar and Einstein tensors)

The Lorentzian spacetime is a smooth oriented and time-oriented four-manifold with arbitrary smooth real two-forms, so it admits the zero field $F=0$ on the stated domain. (Oriented Lorentzian fields, observers and coordinate conventions)

The coupled equation and quadratic tensor are those of Exact geometric electrovac normalization: $G_{ab}=(\kappa \mathrm{/}{\mu }_0){\mathcal{Q}}_{ab}(g,F)$ at $\mathrm{\Lambda }=0$; its quadratic definition gives $\mathcal{Q}(g,0)=0$ for either metric.

The inherited choice assumption is Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, not full Choice. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))

Given: The positive conversion parameters and $\mathcal{F}=\alpha F$ of SI and geometrized Einstein–Maxwell parameters, the Lorentzian coordinate convention of Oriented Lorentzian fields, observers and coordinate conventions, and the exact geometric electrovac normalization of Exact geometric electrovac normalization. The inherited geometric setup carries ${\mathrm{A}\mathrm{C}}_{\omega }$ as in [A1].

The definitions give $\alpha =\sqrt{\kappa \mathrm{/}(2{\mu }_0)}$, $\kappa =8\pi G\mathrm{/}{c}^4$, and ${\epsilon }_0{\mu }_0{c}^2=1$. (SI and geometrized Einstein–Maxwell parameters)

The same two-form is rescaled by $\mathcal{F}=\alpha F$ in the exact geometric Einstein normalization. (Exact geometric electrovac normalization)

The inherited choice assumption is Countable Choice ${\mathrm{A}\mathrm{C}}_{\omega }$, not full Choice. (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$))