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Full dyonic Reissner-Nordström electrovac verification

Statement

For arbitrary $M, q_{e}, q_{m} \in R$ , put $Q^{2} = q_{e}^{2} + q_{m}^{2}$ and $f (r) = 1 - 2 M / r + Q^{2} / r^{2}$ . On every open $J \subset (0, \infty)$ with $f \neq 0$ , the dyonic RN datum of Dyonic Reissner-Nordström metric and field domains is a geometric electrovac datum of Geometric electrovac equations and flux normalization on $R_{x^{0}} \times J \times S^{2}$ . In particular, its global two-form is smooth, $d F = d (⋆_{g} F) = 0$ , and ${Ric}_{a b} = 2 Q_{a b} (F)$ everywhere on that domain.

For arbitrary formal $c, μ_{0}, G_{N} > 0$ , set $α = \sqrt{4 π G_{N} / (μ_{0} c^{4})}$ , $F = F / α$ , and $T_{a b} : = μ_{0}^{- 1} Q_{g} (F)_{a b}$ . Then the same datum satisfies

$d F = d (⋆_{g} F) = 0, S = 0, G_{a b} = {Ric}_{a b} = \frac{8 π G_{N}}{c^{4}} T_{a b},$

where $S$ is scalar curvature and $G_{a b} = {Ric}_{a b} - \frac{1}{2} S g_{a b}$ . This is the formal SI-to-geometric tensor normalization of Maxwell stress tensor in an observer frame(d). It asserts neither measured parameter values nor a bundle realization for arbitrary flux. When the separate fixed-sector action hypotheses hold, $T$ is also the electromagnetic Hilbert tensor of that supplier.

Facts & Assumptions

Given: The dyonic RN datum, the Lorentzian Hodge convention and its curvature identities.

[F1]

Dyonic Reissner-Nordström metric and field domains: $F = - q_{e} r^{- 2} d x^{0} \land d r + q_{m} ω_{S^{2}}$ and the oriented sphere area form is global.

[F2]

Exact Reissner-Nordström Ricci and curvature identities: in a polar chart, ${Ric}_{00} = f Q^{2} / r^{4}$ , ${Ric}_{11} = - Q^{2} / (f r^{4})$ , ${Ric}_{22} = Q^{2} / r^{2}$ , ${Ric}_{33} = Q^{2} \sin^{2} θ / r^{2}$ , with scalar curvature zero.

[F3]

Lorentzian exterior calculus and Hodge star: on two-forms, $(⋆_{g} F)_{a b} = \frac{1}{2} ({v o l}_{g})_{a b c d} F^{c d}$ , $⋆_{g}^{2} = - 1$ , and $d^{2} = 0$ .

[F4]

Geometric electrovac equations and flux normalization: source-free geometric electrovac requires $d F = 0$ , $d (⋆_{g} F) = 0$ , and $Ric = 2 Q (F)$ , with $Λ = 0$ .

[F5]

The current draft Maxwell stress tensor in an observer frame(d) states $T_{E M} = μ_{0}^{- 1} Q_{g} (F)$ and, for its formal positive constants and $F = α F$ , proves $κ_{E} T_{E M} = 2 Q_{g} (F)$ . Its proof is AI-generated and not independently accepted; its Hilbert-tensor interpretation is conditional on the fixed-sector action-variation hypotheses of that item. The algebraic normalization is used only at this exact conditional interface.

[A1]

The coordinate domain has $r > 0$ and $f (r) \neq 0$ ; no positive-mass, horizon, asymptotic, global-potential or source assumption is added.

Proof

1.1F1A1algebra

The angular area form $ω_{S^{2}}$ is closed, and $d (r^{- 2}) = - 2 r^{- 3} d r$ . Therefore $d F = - q_{e} d (r^{- 2}) \land d x^{0} \land d r + q_{m} d ω_{S^{2}} = 0$ because the first term contains $d r \land d r$ and the second is a three-form on the two-dimensional sphere.

1.2F1F3A1algebra

With orientation $d x^{0} \land d r \land ω_{S^{2}}$ , the Hodge formula in [F3] gives $⋆_{g} F = q_{e} ω_{S^{2}} + q_{m} r^{- 2} d x^{0} \land d r$ . Its exterior derivative is zero by the same repeated- $d r$ computation, so the vacuum Maxwell equation holds on the whole $f \neq 0$ domain, including components with $f < 0$ .

1.3F1algebra

Direct coordinate contraction of [F1] gives $F_{a b} F^{a b} = 2 (q_{m}^{2} - q_{e}^{2}) / r^{4}$ and the only nonzero diagonal entries of $Q$ are $Q_{00} = f Q^{2} / (2 r^{4})$ , $Q_{11} = - Q^{2} / (2 f r^{4})$ , $Q_{22} = Q^{2} / (2 r^{2})$ , $Q_{33} = Q^{2} \sin^{2} θ / (2 r^{2})$ ; every mixed entry is zero.

2.1F2F4step 1.1step 1.2step 1.3algebra

Comparing [F2] and step 1.3 gives ${Ric}_{a b} = 2 Q_{a b} (F)$ in every polar chart. Both sides are smooth tensor fields on the product, so equality on the dense polar-coordinate locus extends to the two poles by continuity. Together with steps 1.1-1.2 and the definition [F4], this proves the claimed source-free geometric electrovac statement.

3.1F2F5step 1.1step 1.2step 2.1algebra∎

For the positive constant $α$ , put $F = F / α$ . Linearity of $d$ and of the Hodge star gives $d F = d (⋆_{g} F) = 0$ from steps 1.1–1.2. By [F2], $S = 0$ , so $G = Ric$ . By [F5] and step 2.1, $Ric = 2 Q_{g} (F) = κ_{E} μ_{0}^{- 1} Q_{g} (F) = κ_{E} T$ , exactly the normalized tensor equation. This uses the explicitly defined smooth tensor $T$ and does not require arbitrary dyonic flux to have a globally specified circle-bundle realization.

Remarks

This is a deduction about the mathematical datum. The SI interpretation of $M, q_{e}, q_{m}$ and any claim that nature realizes this electrovac are outside this mathematical theorem. The formal tensor normalization is proved above. Its Hilbert interpretation retains the separate fixed-sector action hypotheses; these mathematical deductions do not establish physical applicability.

Full dyonic Reissner-Nordström electrovac verification

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources