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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.

Quantified dilute exact coupled family

Example

Facts & Assumptions

Verification

Remarks

Depends on

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