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Exact smooth superharmonic counterpoised dust family

Statement

Assume ${A C}_{ω}$ as in The Axiom of Countable Choice ( ${A C}_{ω}$ ), as required by the currently inspected general geometric convention in Smooth formal matter, EM tensors and observer quantities. Let $D \subseteq R^{3}$ be open and let $H \in C^{\infty} (D, (0, \infty))$ satisfy $Δ H \leq 0$ . Use $M = R \times D$ , its coordinate orientation, and the time orientation for which $n : = H \partial_{0}$ is future, all as fixed in Positive-scalar conformastatic chart data and verified in The positive-scalar conformastatic chart tensor is Lorentzian. Use the metric $g_{H}$ and coordinates $x^{0} = c t, x^{i}$ of Positive-scalar conformastatic chart data. Fix formal constants $c, μ_{0}, G_{N} > 0$ , put $ϵ_{0} = (μ_{0} c^{2})^{- 1}$ , $κ = 8 π G_{N} / c^{4}$ , and define the positive charge-to-mass parameter $Q_{s} : = \sqrt{4 π ϵ_{0} G_{N}}$ using the positive root of Explicit rational and real power regimes. For a sign $s \in {- 1, + 1}$ set $A : = s \frac{c}{Q_{s}} H^{- 1} d x^{0}, F : = d A, U : = c H \partial_{0}, ρ : = - \frac{c^{2} Δ H}{4 π G_{N} H^{3}}, σ : = s Q_{s} ρ, j : = σ U, T_{m}^{a b} : = ρ U^{a} U^{b} .$ Let $a^{b} : = U^{a} \nabla_{a} U^{b}$ be the covariant four-acceleration of the timelike field $U$ . Then $U = c n$ is future timelike with $g_{H} (U, U) = - c^{2}$ , all displayed fields are smooth, and $ρ \geq 0$ . On ${ρ > 0}$ the charge-to-mass ratio is $σ / ρ = s Q_{s}$ ; no ratio is defined by division at $ρ = 0$ . The tuple satisfies the formal sourced Maxwell equations $d F = 0$ , $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , the dust matter exchange equation $\nabla_{a} T_{m}^{a b} = F^{b}_{c} j^{c}$ , and the formal Einstein equation with $Λ = 0$ , $G_{a b} = κ (T_{m, a b} + T_{a b}^{E M}),$ where $T^{E M}$ is exactly the tensor of Smooth formal matter, EM tensors and observer quantities. Its total stress satisfies WEC, NEC, SEC, and DEC. The coordinate four-acceleration of the static dust field is $a^{0} = 0$ , $a^{i} = - c^{2} H_{i} / H^{3}$ , and its Lorentz force per unit mass balances it wherever $ρ > 0$ ; the undivided exchange identity also holds where $ρ = 0$ .

This is a conditional mathematical solution family for the stated formal equations. It makes no claim of physical realization, equilibrium stability, or coupled-field existence for arbitrary initial data.

Facts & Assumptions

Given: The open set $D$ , smooth positive $H$ with $Δ H \leq 0$ , constants $c, μ_{0}, G_{N} > 0$ , sign $s \in {- 1, + 1}$ , and the coordinate data of the Statement.

[F1]

Smooth formal matter, EM tensors and observer quantities defines the pointwise formal sourced Maxwell equations $d F = 0$ , $\nabla_{a} F^{a b} = - μ_{0} j^{b}$ , the matter-exchange equation $\nabla_{a} T_{m}^{a b} = F^{b}_{c} j^{c}$ , and the formal Einstein equation $G_{a b} + Λ g_{a b} = κ (T_{m, a b} + T_{a b}^{E M})$ . It also defines a local potential by $d A = F$ . This current draft definition is used only at these exact equation interfaces.

[F2]

Explicit rational and real power regimes defines the unique positive square root and rational powers for positive real bases. Here $Q_{s}$ is the unique positive root of $4 π ϵ_{0} G_{N}$ , and negative integer powers of $H$ mean reciprocals.

[F3]

Charged dust balance equivalence and advected charge ratio defines $a^{b} : = U^{a} \nabla_{a} U^{b}$ ; on a region where $ρ > 0$ , the dust exchange equation is equivalent to continuity $\nabla_{a} (ρ U^{a}) = 0$ and force balance $ρ a^{b} = σ F^{b}_{c} U^{c}$ . Its statement makes no division-based conclusion at $ρ = 0$ . This is the current draft theorem and is used only on its stated positive-density domain.

[F4]

EM, dust and perfect-fluid energy inequalities: the Maxwell tensor and a charged-dust tensor with $ρ \geq 0$ satisfy WEC, NEC, SEC, and DEC, and their sum retains each condition. This is the current draft theorem; no general matter positivity is assumed.

[F5]

Positive-scalar conformastatic chart data defines $M$ , $g_{H}$ , $H_{i}$ , $q$ , and $Δ H$ , with coordinate orientation and time orientation selected by $n = H \partial_{0}$ .

[F6]

The positive-scalar conformastatic chart tensor is Lorentzian gives the inverse metric, nondegeneracy, the future unit vector $n = H \partial_{0}$ , the orthonormal spatial frame $e_{i} = H^{- 1} \partial_{i}$ , and their orientation.

[F7]

Arbitrary-positive-H conformastatic Ricci and Einstein components gives the coordinate connection and the arbitrary-positive- $H$ Ricci, scalar, and Einstein table in the curvature convention stated there.

[F8]

Existence and uniqueness of the metric-selected connection of a nondegenerate metric, and its extension to tensor fields: for a smooth nondegenerate metric, there is a unique torsion-free metric-compatible connection with the Christoffel formula in the statement; its induced tensor derivative has the displayed component rule, including $\nabla_{c} F^{a b} = \partial_{c} F^{a b} + Γ^{a}_{d c} F^{d b} + Γ^{b}_{d c} F^{a d}$ .

[F9]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : differentiable real functions obey the product rule, so differentiating $H \cdot H^{- 1} = 1$ gives the reciprocal derivative used below.

[F10]

The local coordinate formula for the exterior derivative: for a coordinate one-form $A = A_{a} d x^{a}$ , $(d A)_{a b} = \partial_{a} A_{b} - \partial_{b} A_{a}$ .

[F11]

The exterior derivative squares to zero: for every smooth differential form $A$ , $d (d A) = 0$ .

[F12]

Smooth formal matter, EM tensors and observer quantities defines $T_{E M}^{a b} = μ_{0}^{- 1} (F^{a c} F^{b}_{c} - \frac{1}{4} g^{a b} F_{c d} F^{c d})$ and in a positively oriented orthonormal tetrad uses $F_{0 i} = - E_{i} / c$ .

[F13]

Smooth formal matter, EM tensors and observer quantities defines charged dust by $T_{m}^{a b} = ρ U^{a} U^{b}$ and $j^{a} = σ U^{a}$ , with $ρ, σ \in C^{1}$ , $ρ \geq 0$ , and future-timelike $U$ satisfying $g (U, U) = - c^{2}$ .

[A1]

The adopted background includes only ${A C}_{ω}$ as stated in The Axiom of Countable Choice ( ${A C}_{ω}$ ). The explicit global coordinates on $R \times D$ and every finite component calculation below use no stronger choice principle.

Proof

1.1givenF2F5F6F13A1algebra

Since $H$ is smooth and positive, its reciprocal is smooth by Positive-scalar conformastatic chart data; $ϵ_{0} > 0$ and $G_{N} > 0$ make the positive root $Q_{s}$ well-defined by [F2]. The chart tensor is a smooth Lorentzian metric with the stated orientation by [F5] and [F6]. Thus $U = c n$ is future timelike, and the formulas define smooth $A, F, U, ρ, σ, j, T_{m}$ ; they have the charged-dust form [F13], and $Δ H \leq 0$ gives $ρ \geq 0$ .

1.2givenF1F6F9F10F11algebra

Write $f = H^{- 1}$ . Differentiating $H f = 1$ in each spatial coordinate and using [F9] gives $\partial_{i} f = - H_{i} / H^{2}$ . The coordinate formula [F10] therefore gives the only nonzero covariant components $F_{i 0} = - s (c / Q_{s}) H_{i} / H^{2}$ and $F_{0 i} = s (c / Q_{s}) H_{i} / H^{2}$ . Raising indices with [F6] gives $F^{i 0} = s (c / Q_{s}) H_{i} / H^{2}$ and $F^{0 i} = - F^{i 0}$ ; all $F^{i j}$ vanish. Finally [F11] gives $d F = d (d A) = 0$ .

1.3F7F8

The connection used in the formal equations is the unique torsion-free metric-compatible connection by [F8]. The coefficients and compatibility of the local connection in [F7] were established in that item, so the two connections coincide; hence its component table may be used in the sourced equations below.

2.1givenF7F8step 1.2algebra

By the tensor derivative formula [F8], $\nabla_{a} F^{a b} = \partial_{a} F^{a b} + Γ^{a}_{a c} F^{c b} + Γ^{b}_{a c} F^{a c}$ , and the final term is zero because $Γ^{b}_{a c}$ is symmetric in $a, c$ while $F^{a c}$ is antisymmetric. From [F7], $Γ^{a}_{a i} = 2 H_{i} / H$ and $Γ^{a}_{a 0} = 0$ . Hence $\nabla_{a} F^{a 0} = \partial_{i} (s \frac{c}{Q_{s}} \frac{H_{i}}{H^{2}}) + \frac{2 H_{i}}{H} s \frac{c}{Q_{s}} \frac{H_{i}}{H^{2}} = s \frac{c}{Q_{s}} \frac{Δ H}{H^{2}}$ , because the two terms proportional to $q / H^{3}$ cancel.

2.2givenF6F7F8F13step 1.2algebra

The field $U = c H \partial_{0}$ has $g (U, U) = - c^{2}$ by [F13]. Its divergence is $\nabla_{a} (ρ U^{a}) = \partial_{0} (ρ c H) + Γ^{a}_{a 0} ρ c H = 0$ , since all fields are static. Its acceleration is $a^{i} = (c H)^{2} Γ^{i}_{00} = - c^{2} H_{i} / H^{3}$ and $a^{0} = 0$ . Moreover $F^{i}_{0} = g^{i j} F_{j 0} = - s (c / Q_{s}) H_{i} / H^{4}$ , hence $σ F^{i}_{c} U^{c} = s Q_{s} ρ F^{i}_{0} c H = - ρ c^{2} H_{i} / H^{3} = ρ a^{i}$ ; the temporal components of both sides are zero. This undivided relation holds at every point, including $ρ = 0$ .

2.3givenF2F6F12step 1.2algebra

In the positively oriented orthonormal frame $n = H \partial_{0}$ , $e_{i} = H^{- 1} \partial_{i}$ from [F6], the electric components are $E_{i} = c F (e_{i}, n) = - s c^{2} H_{i} / (Q_{s} H^{2})$ and the magnetic components vanish because $F_{i j} = 0$ . In that frame $F_{\hat{0} \hat{i}} = - E_{i} / c$ , $F_{c d} F^{c d} = - 2 ∣ E ∣^{2} / c^{2}$ , $F^{\hat{0} c} F^{\hat{0}}_{c} = ∣ E ∣^{2} / c^{2}$ , and $F^{\hat{i} c} F^{\hat{j}}_{c} = - E_{i} E_{j} / c^{2}$ . Substitution into the stress-tensor formula [F12] yields $T_{E M}^{\hat{0} \hat{0}} = \frac{ϵ_{0} ∣ E ∣^{2}}{2} = \frac{ϵ_{0} c^{4} q}{2 Q_{s}^{2} H^{4}}$ , $T_{E M}^{\hat{0} \hat{i}} = 0$ , and $T_{E M}^{\hat{i} \hat{j}} = \frac{ϵ_{0} c^{4}}{2 Q_{s}^{2} H^{4}} (δ_{i j} q - 2 H_{i} H_{j})$ . Using $κ = 8 π G_{N} / c^{4}$ and $Q_{s}^{2} = 4 π ϵ_{0} G_{N}$ gives $κ T_{E M}^{\hat{0} \hat{0}} = q / H^{4}$ and $κ T_{E M}^{\hat{i} \hat{j}} = (δ_{i j} q - 2 H_{i} H_{j}) / H^{4}$ .

3.1givenF1F2F13step 2.1algebra

The right side of the time component of the sourced equation [F1] is $- μ_{0} j^{0} = - μ_{0} s Q_{s} ρ c H = s μ_{0} Q_{s} c^{3} Δ H / (4 π G_{N} H^{2})$ by the definitions in the Statement. The definitions give $Q_{s}^{2} = 4 π G_{N} / (μ_{0} c^{2})$ by [F2], so $μ_{0} Q_{s} c^{3} / (4 π G_{N}) = c / Q_{s}$ ; this agrees with step 2.1. For each spatial $i$ , $F^{0 i}$ is static, $F^{j i} = 0$ , $Γ^{a}_{a 0} = 0$ , and the symmetric-connection contraction is zero, so $\nabla_{a} F^{a i} = 0 = - μ_{0} j^{i}$ . Thus all sourced Maxwell components hold.

3.2givenF1F3F8F13step 2.2algebra

On the open set ${ρ > 0}$ , step 2.2 verifies the two equivalent conditions in [F3], so the matter exchange equation [F1] follows there. At a point with $ρ = 0$ , the already verified divergence in step 2.2 makes $U^{b} \nabla_{a} (ρ U^{a}) = 0$ , while $ρ a^{b} = 0$ and $σ F^{b}_{c} U^{c} = 0$ because $σ = s Q_{s} ρ = 0$ by [F13]. The product rule $\nabla_{a} (ρ U^{a} U^{b}) = U^{b} \nabla_{a} (ρ U^{a}) + ρ a^{b}$ therefore proves the exchange equation on the zero-density set too.

3.3givenF1F7F13step 2.3algebra

In the same frame, $T_{m}^{\hat{0} \hat{0}} = ρ c^{2}$ and $T_{m}^{\hat{0} \hat{i}} = T_{m}^{\hat{i} \hat{j}} = 0$ by [F13], so $κ T_{m}^{\hat{0} \hat{0}} = 8 π G_{N} ρ / c^{2} = - 2 Δ H / H^{3}$ by the density in the Statement. Consequently the total components are $κ (T_{m} + T_{E M})^{\hat{0} \hat{0}} = (q - 2 H Δ H) / H^{4}$ and $κ (T_{m} + T_{E M})^{\hat{i} \hat{j}} = (δ_{i j} q - 2 H_{i} H_{j}) / H^{4}$ , with zero mixed components. In an orthonormal frame these numerical values equal the corresponding covariant components (the two time-index signs cancel), and they are exactly $G (n, n)$ , $G (e_{i}, e_{j})$ and $G (n, e_{i})$ from [F7]. Thus the formal Einstein equation [F1] holds componentwise.

4.1givenF4F12F13step 1.1step 3.3∎

Since $ρ \geq 0$ , [F4] applies to the Maxwell and dust tensors [F12, F13] separately and states that their sum satisfies WEC, NEC, SEC, and DEC. This is a conditional mathematical consequence of the formal model equations and the pointwise tensor definitions; it is not an empirical claim.

Remarks

The calculation checks every component of the formal Maxwell, dust-exchange, and $Λ = 0$ Einstein equations. At $Δ H = 0$ , $ρ = σ = 0$ and the formulas reduce to the source-free Maxwell branch without division by the density. The result does not supply PDE existence, stability, or a physical postulate.

Exact smooth superharmonic counterpoised dust family

Statement

Facts & Assumptions

Proof

Remarks

Depends on

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