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Full harmonic Majumdar-Papapetrou Einstein-Maxwell verification

Statement

Every datum of Positive harmonic conformastatic electrovac data satisfies the source-free geometric Einstein-Maxwell equations of Geometric electrovac equations and flux normalization:

$d F_{σ} = 0, d (⋆_{g} F_{σ}) = 0, {Ric}_{a b} = 2 Q_{a b} (F_{σ}), Q_{a b} (F) : = F_{a c} F_{b}^{c} - \frac{1}{4} g_{a b} F_{c d} F^{c d} .$

More explicitly, with $q_{H} : = \sum_{i} H_{i}^{2}$ , the only nonzero independent raised field components are $F_{σ}^{i 0} = σ H_{i} / H^{2}$ and the quadratic-field components are

$F_{σ, a b} F_{σ}^{a b} = - \frac{2 q_{H}}{H^{4}}, Q_{00} = \frac{q_{H}}{2 H^{6}}, Q_{0 i} = 0, Q_{i j} = \frac{δ_{i j} q_{H} - 2 H_{i} H_{j}}{2 H^{2}} .$

Facts & Assumptions

Given: The data of Positive harmonic conformastatic electrovac data, its Lorentzian metric, and the geometric electrovac convention in Geometric electrovac equations and flux normalization.

[F1]

Positive harmonic conformastatic electrovac data supplies $H > 0$ , $Δ_{δ} H = 0$ , $g_{00} = - H^{- 2}$ , $g_{i j} = H^{2} δ_{i j}$ , $A_{σ} = σ (H^{- 1} - 1) d x^{0}$ , and $F_{i 0} = - σ H_{i} / H^{2}$ .

[F2]

The positive-scalar conformastatic chart tensor is Lorentzian supplies $g^{00} = - H^{2}$ , $g^{i j} = H^{- 2} δ_{i j}$ , $\sqrt{∣ \det g ∣} = H^{2}$ , and the future orthonormal frame $n = H \partial_{x^{0}}$ , $e_{i} = H^{- 1} \partial_{i}$ .

[F3]

Arbitrary-positive-H conformastatic Ricci and Einstein components states the curvature convention and gives $Γ^{0}_{0 i} = - H_{i} / H$ , $Γ^{i}_{00} = - H_{i} / H^{5}$ , and $Γ^{i}_{j k} = (δ_{i j} H_{k} + δ_{i k} H_{j} - δ_{j k} H_{i}) / H$ . Summing its coordinate Ricci formula gives $Γ^{α}_{α i} = 2 H_{i} / H$ , $Γ^{α}_{α 0} = 0$ , and its arbitrary-positive- $H$ Ricci table.

[F4]

Sourced Maxwell sign and current conservation proves $d (⋆_{g} F) = - ι_{v} {v o l}_{g}$ for $v^{b} = \nabla_{a} F^{a b}$ , so $v = 0$ implies the source-free second Maxwell equation.

[F5]

Lorentzian exterior calculus and Hodge star supplies $d^{2} = 0$ for smooth differential forms.

[A1]

The hypotheses of Positive harmonic conformastatic electrovac data hold on $D$ ; in particular $H$ is smooth, positive, time-independent, and Euclidean harmonic.

Proof

1.1givenF1F2F5algebra

Since $F_{σ} = d A_{σ}$ , the identity $d^{2} = 0$ in [F5] gives $d F_{σ} = 0$ . Raising the two indices of $F_{i 0} = - σ H_{i} / H^{2}$ with [F2] gives $F^{i 0} = σ H_{i} / H^{2}$ and $F^{0 i} = - σ H_{i} / H^{2}$ . Contracting the antisymmetric pair yields $F_{a b} F^{a b} = 2 \sum_{i} F_{i 0} F^{i 0} = - 2 q_{H} / H^{4}$ .

2.1givenF1F2F3A1step 1.1algebra

The last Christoffel term in $\nabla_{a} F^{a b} = \partial_{a} F^{a b} + Γ^{a}_{a c} F^{c b} + Γ^{b}_{a c} F^{a c}$ vanishes because $Γ^{b}_{a c}$ is symmetric in $a, c$ and $F^{a c}$ is antisymmetric. For $b = 0$ , the remaining terms are $\partial_{i} (σ H_{i} / H^{2}) + (2 H_{i} / H) (σ H_{i} / H^{2}) = σ Δ_{δ} H / H^{2} = 0$ , after the product rule cancels the two $q_{H} / H^{3}$ terms. For $b = i$ , $\partial_{0} F^{0 i} = 0$ , all $F^{j i} = 0$ , and the only possible trace term is $Γ^{α}_{α 0} F^{0 i} = 0$ . Thus $v^{b} = \nabla_{a} F^{a b} = 0$ for every $b$ , and [F4] gives $d (⋆_{g} F_{σ}) = 0$ .

2.2givenF1F2step 1.1algebra

The only nonzero contraction in $F_{0 c} F_{0}^{c}$ is $\sum_{i} F_{0 i} F_{0}^{i} = q_{H} / H^{6}$ , while $F_{i c} F_{j}^{c} = F_{i 0} g^{00} F_{j 0} = - H_{i} H_{j} / H^{2}$ . Combining these with $F_{a b} F^{a b} = - 2 q_{H} / H^{4}$ and $g_{00} = - H^{- 2}$ , $g_{i j} = H^{2} δ_{i j}$ gives $Q_{00} = q_{H} / (2 H^{6})$ , $Q_{0 i} = 0$ , and $Q_{i j} = (δ_{i j} q_{H} - 2 H_{i} H_{j}) / (2 H^{2})$ .

3.1givenF2F3step 1.1step 2.1step 2.2algebra∎

Substituting $Δ_{δ} H = 0$ into the arbitrary-positive- $H$ table [F3] gives ${Ric}_{00} = q_{H} / H^{6}$ , ${Ric}_{0 i} = 0$ , and ${Ric}_{i j} = (δ_{i j} q_{H} - 2 H_{i} H_{j}) / H^{2}$ . These are exactly twice the entries in step 2.2; together with the closedness in step 1.1 and source-free divergence in step 2.1 they are all three equations in Geometric electrovac equations and flux normalization.

Full harmonic Majumdar-Papapetrou Einstein-Maxwell verification

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources