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Arbitrary-positive-H conformastatic Ricci and Einstein components — Examples

1 · Prerequisites

2 · Summary

Scope

These examples specialize the exact smooth family to a regular sphere and to a quantified dilute parameter range. The first checks the center values, proper mass and charge, and everywhere-positive density. The second records uniform metric and density comparison bounds and shows that, away from the origin, the static dust's nonzero electromagnetic force per unit mass equals its nonzero acceleration; at the origin both vanish and the central static worldline is geodesic. Both are exact mathematical calculations, not measurements or stability claims.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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Regular smooth counterpoised sphere

Example

Assume ${A C}_{ω}$ . Let $b > 0$ have length units, let $x = (x^{1}, x^{2}, x^{3}) \in R^{3}$ , and define its Euclidean radius by $r : = \sqrt{(x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2}}$ . Take $a = b$ and $s = + 1$ in the exact family of Exact smooth superharmonic counterpoised dust family, and define $H (x) = 1 + b / \sqrt{r^{2} + b^{2}}$ on all of $R^{3}$ . For the family field strength $F$ , set $n : = H \partial_{0}$ , $e_{i} : = H^{- 1} \partial_{i}$ , and define the static-frame electric components by $E_{i} : = c F (e_{i}, n)$ for $i = 1, 2, 3$ . This is an everywhere regular smooth solution with center values $H (0) = 2, \nabla H (0) = 0, Δ H (0) = - \frac{3}{b^{2}}, ρ (0) = \frac{3 c^{2}}{32 π G_{N} b^{2}}, S_{g} (0) = \frac{3}{4 b^{2}}, E_{i} (0) = 0.$ Its proper mass and signed charge are $c^{2} b / G_{N}$ and $Q_{s} c^{2} b / G_{N}$ , respectively. The model density is positive everywhere, so its topological support, defined as the closure in $R^{3}$ of ${x \in R^{3} : ρ (x) \neq 0}$ , is all of $R^{3}$ .

Facts & Assumptions

Given: ${A C}_{ω}$ , $a = b > 0$ , $s = + 1$ , the smooth family, the arbitrary-positive- $H$ curvature table, and the proper-integral result.

[F1]

Exact smooth superharmonic counterpoised dust family supplies the exact coupled metric and fields, including $A = s (c / Q_{s}) H^{- 1} d x^{0}$ , $U = c H \partial_{0}$ , and $ρ = - c^{2} Δ H / (4 π G_{N} H^{3})$ , with $Q_{s} > 0$ and the stated geometric conventions. The observer components used below are computed locally from this potential.

[F2]

Arbitrary-positive-H conformastatic Ricci and Einstein components gives $S_{g} = - 2 Δ H / H^{3}$ .

[F3]

Globally regular sphere, proper integrals and dilute bounds gives finite proper mass $c^{2} a / G_{N}$ and charge $s Q_{s} c^{2} a / G_{N}$ for the stated profile, under ${A C}_{ω}$ .

[F4]

The positive-root function is smooth with derivative $s' (q) = 1 / (2 s (q))$ for $q > 0$ (The positive square-root branch is smooth). The chain, product and quotient rules hold (The chain rule, in one line from Carathéodory: if $g$ is differentiable at $c$ and $f$ is differentiable at $g (c)$ , then $f \circ g$ is differentiable at $c$ with $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , 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 $g (c) \neq 0$ ).

[F5]

In a smooth coordinate chart, $d (A_{a} d x^{a}) = \sum_{a} d A_{a} \land d x^{a}$ (The local coordinate formula for the exterior derivative), so $(d A)_{i 0} = \partial_{i} A_{0} - \partial_{0} A_{i}$ .

[A1]

Assume the indexed Axiom ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ): every family $(X_{n})_{n \in N}$ of nonempty sets has a function $f$ on $N$ with $f (n) \in X_{n}$ for every $n$ . This is the assumption required by [F3]; no full Axiom of Choice is assumed.

Verification

1.1givenF3F4algebra

Put $Ψ = (r^{2} + b^{2})^{1 / 2}$ . Since $r^{2} + b^{2} \geq b^{2} > 0$ , [F4] and the chain rule give $\partial_{i} Ψ = x^{i} / Ψ$ . Differentiating $H = 1 + b / Ψ$ with the quotient rule in [F4] gives $H_{i} = - b x^{i} / Ψ^{3}$ , hence $H_{i} (0) = 0$ . At $r = 0$ , $Ψ = b$ and $H (0) = 2$ . The Laplacian in [F3] gives $Δ H (0) = - 3 b^{3} / b^{5} = - 3 / b^{2}$ .

1.2givenA1F1F3algebra

Applying [F3] with $a = b$ and $s = + 1$ yields $M_{p r o p} = c^{2} b / G_{N}$ and $Q_{p r o p} = Q_{s} c^{2} b / G_{N}$ . The exact-solution conclusion is [F1].

2.1step 1.1F1F2F4F5algebra

Substituting these values into [F1] gives $ρ (0) = 3 c^{2} / (4 π G_{N} b^{2} \cdot 2^{3}) = 3 c^{2} / (32 π G_{N} b^{2})$ , and [F2] gives $S_{g} (0) = - 2 (- 3 / b^{2}) / 2^{3} = 3 / (4 b^{2})$ . For the electric components, the metric in [F1] gives the future unit vector $n = H \partial_{0}$ and positively oriented orthonormal spatial vectors $e_{i} = H^{- 1} \partial_{i}$ by direct contraction. Since $A_{0} = s (c / Q_{s}) H^{- 1}$ and $A_{i} = 0$ , [F4]–[F5] give $F_{i 0} = - s (c / Q_{s}) H_{i} / H^{2}$ . Therefore $E_{i} : = c F (e_{i}, n) = c F_{i 0} = - s c^{2} H_{i} / (Q_{s} H^{2})$ , which vanishes at the center by step 1.1.

3.1givenF1step 1.1algebra∎

The formula in [F1] has $Δ H = - 3 b^{3} / (r^{2} + b^{2})^{5 / 2} < 0$ and $H > 0$ at every $x \in R^{3}$ , hence $ρ > 0$ throughout $R^{3}$ . The denominator never vanishes because $b > 0$ , so the metric, potential, and fields extend smoothly through $r = 0$ . The nonzero set of $ρ$ is therefore $R^{3}$ , whose closure is $R^{3}$ ; by the Statement's definition, this is its topological support.

Remarks

This is a regular ideal mathematical solution, not a reported object. The dust is held static by its electromagnetic force; no equilibrium stability, formation history, or real-particle composition is inferred.

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Quantified dilute exact coupled family

Example

Assume ${A C}_{ω}$ . Let $b > 0$ have length units and let $δ$ be dimensionless with $0 < δ \leq 1 / 10$ . Put $a = δ b$ and choose either charge sign $s \in {- 1, + 1}$ in the exact coupled family. For $x = (x^{1}, x^{2}, x^{3}) \in R^{3}$ , define the Euclidean radius $r : = \sqrt{(x^{1})^{2} + (x^{2})^{2} + (x^{3})^{2}}$ and $h_{0} (x) : = b / \sqrt{r^{2} + b^{2}}$ , so $H = 1 + δ h_{0}$ . Then $∣ g_{00} + (1 - 2 δ h_{0}) ∣ \leq 3 δ^{2} \leq \frac{3}{100}, ∣ g_{i j} - (1 + 2 δ h_{0}) δ_{i j} ∣ \leq δ^{2} \leq \frac{1}{100}, 0 \leq 1 - \frac{ρ}{ρ_{l i n}} \leq 3 δ \leq \frac{3}{10},$ where $ρ_{l i n} = 3 c^{2} δ b^{3} / [4 π G_{N} (r^{2} + b^{2})^{5 / 2}]$ . These are exact ideal-family bounds, not measured errors. On $M = R_{x^{0}} \times R^{3}$ , the static dust velocity field is $U = c H \partial_{0}$ . For each fixed $x \in R^{3}$ and $x_{*}^{0} \in R$ , define its static worldline by $γ_{x, x_{*}^{0}} : R \to M, γ_{x, x_{*}^{0}} (τ) : = (x_{*}^{0} + c H (x) τ, x) .$ This is a smooth future-directed timelike curve with $\dot{γ} = U$ and $g_{H} (\dot{γ}, \dot{γ}) = - c^{2}$ ; $τ \in R$ , measured in seconds, is its proper-time parameter in the adopted length-coordinate convention. Here a static worldline is freely falling exactly when it is a geodesic, $\nabla_{\dot{γ}} \dot{γ} = 0$ , with $\nabla$ the connection of the exact family. Define the model's electromagnetic force per unit mass by $f^{b} : = (σ / ρ) F^{b}_{c} U^{c}$ ; this is defined everywhere for this profile since $ρ > 0$ . For this profile, $H_{i} = - δ b x^{i} / (r^{2} + b^{2})^{3 / 2}$ , so when $r > 0$ both $a^{i}$ and $f^{i}$ are nonzero and $a^{i} = f^{i}$ . At $r = 0$ both vectors vanish, and the central static worldline is geodesic.

Facts & Assumptions

Given: ${A C}_{ω}$ , $b > 0$ , $0 < δ \leq 1 / 10$ , $a = δ b$ , $s \in {- 1, + 1}$ , and the exact family and dilute comparison of the corollary.

[F1]

Globally regular sphere, proper integrals and dilute bounds supplies $H = 1 + δ h_{0}$ , the exact metric and density profile, and the bounds $∣ H^{- 2} - (1 - 2 δ h_{0}) ∣ \leq 3 δ^{2}$ , $∣ H^{2} - (1 + 2 δ h_{0}) ∣ \leq δ^{2}$ , and $0 \leq 1 - ρ / ρ_{l i n} \leq 3 δ$ .

[F2]

Exact smooth superharmonic counterpoised dust family supplies $U = c H \partial_{0}$ , $a^{i} = - c^{2} H_{i} / H^{3}$ , and the identity $ρ a^{i} = σ F^{i}_{c} U^{c}$ .

[F3]

The product and quotient differentiation rules hold for differentiable real functions where the denominator is nonzero (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 $g (c) \neq 0$ ).

[F4]

Continuous real functions on a compact interval are Riemann integrable (A continuous function on $[a, b]$ is Riemann integrable, by Heine-Cantor and Riemann's criterion). If a function is continuous on $[u, v]$ , differentiable inside, and its derivative has an integrable extension, Newton--Leibniz gives its endpoint difference (Newton–Leibniz needs only continuity on $[a, b]$ , differentiability on $(a, b)$ , and a Riemann-integrable extension of the interior derivative). Pointwise order and nonnegativity are preserved by integrals on $u < v$ (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ ).

[A1]

$0 < δ \leq 1 / 10$ , so $3 δ^{2} \leq 3 / 100$ , $δ^{2} \leq 1 / 100$ , and $3 δ \leq 3 / 10$ .

Verification

1.1givenF1A1algebra

Put $z = δ h_{0}$ . Since $r^{2} + b^{2} \geq b^{2}$ and both square roots are positive, $0 < h_{0} \leq 1$ and $0 < z \leq δ$ . By the metric formula in [F1], $g_{00} = - (1 + z)^{- 2}$ , so the time-component discrepancy is $∣ (1 + z)^{- 2} - 1 + 2 z ∣$ .

2.1givenF3F4A1step 1.1algebra

Define $f (u) : = (1 + u)^{- 2} - 1 + 2 u$ for $u \geq 0$ . Twice applying the quotient/product rules [F3] gives $f (0) = f^{'} (0) = 0$ and $f^{''} (u) = 6 (1 + u)^{- 4}$ , between $0$ and $6$ . For fixed $z > 0$ , set $G_{z} (u) : = (z - u) f^{'} (u) + f (u)$ on $[0, z]$ . It is continuous with continuous derivative $G_{z}^{'} (u) = (z - u) f^{''} (u)$ , which is integrable by [F4]. Newton--Leibniz [F4] gives $f (z) = G_{z} (z) - G_{z} (0) = \int_{0}^{z} (z - u) f^{''} (u) d u .$ The integrand lies between $0$ and $6 (z - u)$ , both continuous comparison functions. Integral monotonicity [F4], and Newton--Leibniz for the polynomial primitive $6 z u - 3 u^{2}$ , give $0 \leq f (z) \leq 3 z^{2}$ . Thus the discrepancy in step 1.1 is at most $3 z^{2} \leq 3 δ^{2}$ ; [A1] further gives $3 δ^{2} \leq 3 / 100$ . At $z = 0$ the same sharper comparison is zero by direct evaluation, without applying the nondegenerate-interval theorem.

3.1step 1.1step 2.1F1A1algebra

The spatial bound in [F1] gives $∣ g_{i j} - (1 + 2 z) δ_{i j} ∣ \leq δ^{2}$ , since $δ_{i j}$ is $0$ or $1$ . The density identity in [F1] gives $0 \leq 1 - ρ / ρ_{l i n} \leq 3 δ$ . Applying [A1] and combining with step 2.1 yields the three numerical bounds in the Example.

4.1givenF1F2step 3.1algebra∎

For the worldline $γ_{x, x_{*}^{0}}$ defined in the Statement, $\dot{γ} = U$ , so its covariant acceleration is the supplier field $a^{b}$ . Here $ρ > 0$ by [F1] and $H_{i} = - δ b x^{i} / (r^{2} + b^{2})^{3 / 2}$ ; hence $a^{i} = - c^{2} H_{i} / H^{3}$ is nonzero when $r > 0$ , while $a^{i} = 0$ at $r = 0$ . The supplier also gives $a^{0} = 0$ . Dividing its all-component identity $ρ a^{b} = σ F^{b}_{c} U^{c}$ by positive $ρ$ gives $a^{b} = f^{b} = (σ / ρ) F^{b}_{c} U^{c}$ everywhere. Thus both vectors are nonzero for $r > 0$ and vanish at the center; only the noncentral static worldlines fail the geodesic definition of free fall.

Remarks

The numerical fractions are consequences of the ideal choice $δ \leq 1 / 10$ and are not experimental uncertainties. No measurement, statistical estimate, or general perturbation theorem is asserted.

Arbitrary-positive-H conformastatic Ricci and Einstein components — Examples

1 · Prerequisites

2 · Summary

Scope

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

Regular smooth counterpoised sphere

Example

Facts & Assumptions

Verification

Remarks

Quantified dilute exact coupled family

Example

Facts & Assumptions

Verification

Remarks

Sources