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Bell congruence worked: internal material length gamma(t)L and constant chart separation

Example

Worked computation of the internal material length for the Bell congruence of Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (d) and the material metric of The material metric of a congruence: evolution of rest separations and preservation of the internal metric under Born rigidity. Fix $c > 0$ and $α > 0$ , let $J : = (- c / α, c / α)$ , put $q (t) : = \frac{1}{2} α t^{2}$ on $J$ and let $x_{ξ} (t) : = x^{- 1} (c t, q (t) + ξ, 0, 0)$ for $ξ$ in an open interval containing $[0, L]$ , $L > 0$ , in the inertial chart $x$ with $x^{0} = c t$ . Then $∣ q^{'} (t) ∣ = α ∣ t ∣ < c$ on the two-sided domain $J$ ; write $β (t) : = α t / c$ and $γ (t) : = (1 - β (t)^{2})^{- 1 / 2}$ .

(a) The translated congruence field is $n = γ (1, β, 0, 0)$ and the material field $e : = \partial_{ξ}$ is the constant $(0, 1, 0, 0)$ ; since $η (n, e) = γ β$ , the $h$ -length of $e$ is $∣ e ∣_{h} = \sqrt{η (e, e) + η (n, e)^{2}} = \sqrt{1 + γ^{2} β^{2}} = γ (t)$ (The material metric of a congruence: evolution of rest separations and preservation of the internal metric under Born rigidity (a), Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (d)). Hence the internal longitudinal material-metric length between labels $0$ and $L$ is $∣ e ∣_{h} L = γ (t) L > L$ for every $t \neq 0$ , with minimum $L$ at $t = 0$ , while the chart-frame separation is $q (t) + L - q (t) = L$ for every $t$ in $J$ .

(b) The longitudinal expansion of the translated congruence is $θ (s, s) = γ^{3} q^{'} q^{''} / c^{3} = γ^{3} α^{2} t / c^{3}$ , with unit rest vector $s = γ (β, 1, 0, 0)$ (Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (d)); it vanishes at $t = 0$ and is nonzero for every $t \neq 0$ , negative for $t < 0$ and positive for $t > 0$ . By The material metric of a congruence: evolution of rest separations and preservation of the internal metric under Born rigidity (b) the internal longitudinal material metric component is $G_{11} = h (e, e) = γ (t)^{2}$ and its derivatives in geometric proper length $σ = c τ$ are $d / d σ G_{11} = 2 θ (e, e)$ and $d / d τ G_{11} = 2 c θ (e, e)$ ; with horizontal unit direction $h (e) / γ = s$ , equivalently the fractional rate of change of the rest separation per unit proper time is $d / d τ \log γ = c θ (s, s) = γ^{3} α^{2} t / c^{2}$ , and $d (γ L) / d τ = γ^{4} L α^{2} t / c^{2}$ .

(c) Explicit check at $α t = 3 c / 5$ , i.e. $t = 3 c / (5 α)$ in $J$ : $β = 3 / 5$ , $γ = 5 / 4$ and $\sinh$ and $\cosh$ are not needed; the rest separation is $5 L / 4 > L$ , the internal metric component is $G_{11} = 25 / 16 > 1$ , the expansion is $θ (s, s) = (125 / 64) (3 α / (5 c^{2})) = 75 α / (64 c^{2})$ , and using [L7], the fractional rest-separation rate is $c θ (s, s) = 75 α / (64 c)$ . At $t = 0$ the rest separation takes its minimum $L$ and the expansion vanishes; the domain is the two-sided interval $(- c / α, c / α)$ , so both the decelerating branch $t < 0$ (negative expansion) and the accelerating branch $t > 0$ (positive expansion) are included.

(d) Scope. The length $γ (t) L$ is the integral of the local rest metric over the material-label segment at common flow parameter; no finite endpoint-hyperplane distance is inferred. The computation is exact kinematics of the specified family: it uses no elasticity, no material failure criterion, no force law, no measurement and no empirical claim, and it draws no conclusion about real threads; no choice principle is used.

Facts & Assumptions

Given: $c > 0$ , $α > 0$ , $J = (- c / α, c / α)$ , $q (t) = \frac{1}{2} α t^{2}$ on $J$ , $L > 0$ , an open label interval containing $[0, L]$ , an inertial chart $x$ with $x^{0} = c t$ , the worldlines $x_{ξ} (t) = x^{- 1} (c t, q (t) + ξ, 0, 0)$ , and $β = α t / c$ , $γ = (1 - β^{2})^{- 1 / 2}$ .

[L1]

Model and chart: $(A, V, η, C^{+})$ with coordinate identification $x (p)$ and $η = diag (- 1, 1, 1, 1)$ in the orthonormal frame (Oriented time-oriented affine Minkowski space (the spacetime primitives), Inertial frames and inertial charts of the affine Minkowski model).

[L2]

Derivatives: $q^{'} = α t$ , $q^{''} = α$ , and the chain rule plus the total chain rule differentiate composites of the data functions (The derivative $f^{'} (c) = \lim_{x \to c} \frac{f (x) - f (c)}{x - c}$ of $f : A \to R$ at a point $c \in A$ that is a limit point of $A$ , and differentiability on a set, 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 $(f \circ g)^{'} (c) = f^{'} (g (c)) g^{'} (c)$ , The chain rule for total derivatives: $D (g \circ f) (a) = D g (f (a)) \circ D f (a)$ ).

[L3]

Translated-congruence first-order data: the unit field is $n = γ (1, β, 0, 0)$ , the unit longitudinal rest vector is $s = γ (β, 1, 0, 0)$ , and $θ (s, s) = γ^{3} q^{'} q^{''} / c^{3}$ , nonzero exactly where $q^{'} q^{''} \neq 0$ (Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (d), Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity (c)).

[L4]

Material metric: for an injective $C^{2}$ immersion with $\partial_{s} Φ = n (Φ)$ and material directions $e_{I}$ , $G_{I J} = h (e_{I}, e_{J})$ and $\partial_{s} G_{I J} = 2 θ (e_{I}, e_{J})$ , i.e. $\partial_{τ} G_{I J} = 2 c θ (e_{I}, e_{J})$ with $τ = s / c$ ; $θ$ kills the $n$ -component, so $θ (e, e) = γ^{2} θ (s, s)$ when $h (e) = γ s$ (The material metric of a congruence: evolution of rest separations and preservation of the internal metric under Born rigidity (a),(b),(d)).

[L5]

Rest form: $h (e, e) = η (e, e) + η (n, e)^{2}$ for the first-order rest projection $h$ (Kinematics of an observer congruence: acceleration, expansion, shear, vorticity and Born rigidity (a), The material metric of a congruence: evolution of rest separations and preservation of the internal metric under Born rigidity).

[L6]

Proper time and four-velocity: the flow lines have four-velocity $c n$ with $η (c n, c n) = - c^{2}$ , and the flow parameter satisfies $σ = c τ$ for the proper time $τ$ , with $γ \geq 1$ and $γ = 1$ iff $β = 0$ (Four-velocity and four-acceleration of a future timelike worldline, Norms of four-velocity and four-acceleration, and the observer rest-space decomposition (a),(d), Proper time along a future timelike worldline, The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse).

[L7]

The natural logarithm is defined on positive real numbers and has derivative $1 / u$ (The natural logarithm as the inverse of the exponential function, The natural logarithm has derivative 1/x and equals the integral from 1 to x of 1/t); hence $d \log γ / d τ = (d γ / d τ) / γ$ by [L2]. This is the fractional rate of $γ L$ for constant $L > 0$ .

Verification

technique · direct

1.1L1L2givenalgebra

(Basic data.) On $J$ one has $q^{'} (t) = α t$ and $q^{''} (t) = α$ by [L2], so $∣ q^{'} (t) ∣ = α ∣ t ∣ < c$ exactly on the stated two-sided domain, and $β = α t / c \in (- 1, 1)$ ; hence $γ = (1 - β^{2})^{- 1 / 2}$ is well defined and positive, and $1 + γ^{2} β^{2} = 1 / (1 - β^{2}) = γ^{2}$ .

2.1L3L4L5L6step 1.1algebra

(The unit field and the material direction.) To realize the hypotheses of [L4], let $τ (t) = \int_{0}^{t} γ (v)^{- 1} d v$ and $σ = c τ$ . By [L6] its inverse $t (σ)$ is $C^{1}$ with $t^{'} = γ / c$ ; the smooth data bootstrap this inverse to smoothness. The map $Φ (σ, ξ, y, z) = x^{- 1} (c t (σ), q (t (σ)) + ξ, y, z)$ on the open product of the parameter image, the stated label interval and $R^{2}$ has the explicit inverse $(c τ (t), x^{1} - q (t), x^{2}, x^{3})$ , so is an injective smooth immersion onto an open set; its flow derivative is $\partial_{σ} Φ = γ (1, β, 0, 0) = n$ . By [L3] the translated congruence field is $n = γ (1, β, 0, 0)$ , and the material direction is $e = \partial_{ξ} Φ = (0, 1, 0, 0)$ ; then $η (n, e) = γ β$ , and by [L5] the rest form of $e$ is $h (e, e) = η (e, e) + η (n, e)^{2} = 1 + γ^{2} β^{2} = γ (t)^{2}$ , while $h (e) / γ = γ (β, 1, 0, 0) = s$ is the horizontal unit rest vector of [L3].

3.1L4L6step 1.1step 2.1algebra

(Clause (a): the material length.) By step 2.1 the $h$ -length of $e$ is $∣ e ∣_{h} = \sqrt{h (e, e)} = γ (t) > 0$ , so the internal longitudinal material-metric length between labels $0$ and $L$ at common flow parameter is $\int_{0}^{L} \sqrt{G_{11}} d ξ = γ (t) L$ by [L4]; since $γ^{2} = 1 / (1 - β^{2}) > 1$ for $t \neq 0$ by step 1.1 and [L6], this exceeds $L$ for every $t \neq 0$ and equals $L$ only at $t = 0$ (where $β = 0$ ). The chart separation is $(q (t) + L) - q (t) = L$ for every $t \in J$ .

3.2L3step 1.1step 2.1algebra

(Clause (b): the expansion and the metric component.) By [L3] the longitudinal expansion is $θ (s, s) = γ^{3} q^{'} q^{''} / c^{3} = γ^{3} α^{2} t / c^{3}$ , which vanishes at $t = 0$ and has the sign of $t$ elsewhere because $γ > 0$ ; and by step 2.1, $G_{11} = h (e, e) = γ (t)^{2}$ .

4.1L4L7step 2.1step 3.2algebra

(Clause (b): the rates.) By [L4], $d / d σ G_{11} = 2 θ (e, e)$ and $d / d τ G_{11} = 2 c θ (e, e)$ ; since $θ (e, e) = γ^{2} θ (s, s)$ by [L4] and step 2.1, using [L7], the fractional rest-separation rate is $d / d τ \log γ = \frac{1}{2} \partial_{τ} G_{11} / G_{11} = c θ (e, e) / γ^{2} = c θ (s, s) = γ^{3} α^{2} t / c^{2}$ , and multiplying by $γ L$ gives $d (γ L) / d τ = γ^{4} L α^{2} t / c^{2}$ .

5.1step 3.1step 3.2step 4.1algebra

(Clause (c): the numerical check.) At $t = 3 c / (5 α)$ : $β = 3 / 5$ and $γ = (1 - 9 / 25)^{- 1 / 2} = (16 / 25)^{- 1 / 2} = 5 / 4$ ; the rest separation is $γ L = 5 L / 4 > L$ and the metric component is $G_{11} = γ^{2} = 25 / 16 > 1$ by steps 2.1 and 3.2; the expansion is $θ (s, s) = γ^{3} α^{2} t / c^{3} = (125 / 64) \cdot α^{2} \cdot (3 c / (5 α)) / c^{3} = (125 \cdot 3 α) / (64 \cdot 5 c^{2}) = 75 α / (64 c^{2})$ , and the fractional rate is $c θ (s, s) = 75 α / (64 c)$ by step 4.1. At $t = 0$ the separation is $L$ and the expansion vanishes, and the two-sided domain contains both signs of $t$ , hence both signs of the expansion.

6.1L4step 3.1step 4.1step 5.1∎

(Clause (d): scope.) Everything computed is the material-label integral $γ (t) L$ of the rest metric at common flow parameter, with its exact local rates; no endpoint distance on a single simultaneity hyperplane, no elasticity, failure criterion, force law, measurement or empirical claim enters, and every step is finite-dimensional algebra or calculus from the cited suppliers, so no choice principle is used.

Bell congruence worked: internal material length gamma(t)L and constant chart separation

Example

Facts & Assumptions

Verification

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