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Accelerated Observers — Examples

1 · Prerequisites

2 · Summary

These leaf examples and counterexamples work the accelerated-observer page on exact rationals and logarithms. The Rindler radar example computes the complete round trips at $ρ_{m} = 2 ρ_{0}$ and $ρ_{m} = ρ_{0} / 2$ with reflection events $(3 ρ_{0} / 2, 5 ρ_{0} / 2)$ and $(3 ρ_{0} / 8, 5 ρ_{0} / 8)$ , both with radar distance $ρ_{0} \log 2$ but rest distances $ρ_{0}$ and $ρ_{0} / 2$ , and the reflector at $3 ρ_{0} / 2$ shows equal rest distance but strictly smaller radar distance, so radar distance is not a function of rest distance (Rindler radar distances worked: reflectors at twice and at half the observer's radius). The companion counterexample interchanges the two coaxial observers and computes the two radar distances $ρ_{0} \log 2$ and $2 ρ_{0} \log 2$ between the same pair of worldlines, refuting symmetry of the radar assignment (Rindler radar distance between two coaxial observers is not symmetric under interchange).

The horizon counterexample refutes three exact claims about the null plane $x^{1} = x^{0}$ : an explicit future timelike segment crosses it, causally related events lie on opposite sides, and the metric is everywhere the nondegenerate constant $diag (- 1, 1, 1, 1)$ . The full plane is the boundary of the causal past of each Rindler observer; only its half-plane $x^{1} = x^{0} \geq 0$ lies on the boundary of the Rindler chart image $W = {x^{1} > ∣ x^{0} ∣}$ . The surviving horizon statement restricts reachability of particular congruence worldlines (The Rindler causal horizon is not a boundary or singularity of spacetime). The Sagnac example specializes the guided-ring theorem to $Ω R = 3 c / 5$ : return times $5 π R / c$ and $5 π R / (4 c)$ , chart-time difference $15 π R / (4 c)$ , detector proper-time difference $3 π R / c$ , chart-time rest-line holonomy $15 π R / (8 c)$ , exactly half the chart-time difference, and its proper-time counterpart $3 π R / (2 c)$ , exactly half the detector proper-time difference, with the clock conversion factor $4 / 5$ between the two holonomies, and the inertial limit at $Ω = 0$ (Guided-ring Sagnac delay worked at angular velocity Omega R = 3c/5). The final example works the quadratic Bell congruence $q (t) = \frac{1}{2} α t^{2}$ on the two-sided subluminal domain $∣ t ∣ < c / α$ : $G_{11} = γ^{2}$ , rest separation $γ (t) L$ with minimum $L$ at $t = 0$ , expansion $γ^{3} α^{2} t / c^{3}$ of both signs, and the exact check $β = 3 / 5$ , $γ = 5 / 4$ , $G_{11} = 25 / 16$ , $θ (s, s) = 75 α / (64 c^{2})$ (Bell congruence worked: internal material length gamma(t)L and constant chart separation). Every item is a conditional model computation with ideal clocks, ideal light rays and ideal reflectors; none asserts a measurement or an empirical conclusion, and no item takes a supply from another item of this page.

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

None yet.

5 · Examples, experiments and counterexamples

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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.

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Rindler radar distances worked: reflectors at twice and at half the observer's radius

Physical scope

Worked numerical instance of the conditional Rindler radar protocol for two coaxial uniformly accelerated ideal observers with ideal clocks, ideal light rays and ideal reflectors in flat classical spacetime. No apparatus, measurement, statistical or empirical claim is made, and all idealizations of pthm-sr-rindler-horizons-and-radar are retained.

Example

Worked computation in the Rindler chart of The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance. Let $c > 0$ , fix $ρ_{0} > 0$ and take $y_{0} = z_{0} = 0$ ; let $O$ be the congruence worldline $ρ = ρ_{0}$ with Rindler time $θ = c τ / ρ_{0}$ , and take the emission event $e_{-}$ at $θ_{e} = 0$ , so that $e_{-} = x^{- 1} (0, ρ_{0}, 0, 0)$ in the inertial chart $x$ .

(a) Reflector at $ρ_{m} = 2 ρ_{0}$ . The future null line from $e_{-}$ of constant $V$ has $V = - ρ_{0}$ , and it meets $ρ = 2 ρ_{0}$ at the Rindler time $θ_{m} = \log 2$ ; using $\sinh (\log 2) = 3 / 4$ and $\cosh (\log 2) = 5 / 4$ (Addition formulas, identities, parity, and derivatives of the hyperbolic functions, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm) the reflection event is $q_{m} = x^{- 1} (3 ρ_{0} / 2, 5 ρ_{0} / 2, 0, 0)$ , and the return event is $e_{+} = x^{- 1} (ρ_{0} \sinh (2 \log 2), ρ_{0} \cosh (2 \log 2), 0, 0) = x^{- 1} (15 ρ_{0} / 8, 17 ρ_{0} / 8, 0, 0)$ at Rindler time $θ (e_{+}) = 2 \log 2$ . Thus $(θ_{e} + θ (e_{+})) / 2 = \log 2 = θ_{m}$ , the radar time assigned to $q_{m}$ is $(ρ_{0} / c) \log 2$ plus the base constant of $O$ 's proper time, and the radar distance is $ρ_{0} \log 2$ . The two worldlines $ρ = ρ_{0}$ and $ρ = 2 ρ_{0}$ are at rest distance $ρ_{0}$ in the slice $θ = \log 2$ ; since $e > 2$ gives $\log 2 < 1$ (The elementary numerical bound $2 < e < 3$ , Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm), the radar distance is strictly smaller than the rest distance.

(b) Reflector at $ρ_{m} = ρ_{0} / 2$ . The future null line from $e_{-}$ of constant $U$ has $U = ρ_{0}$ , and it meets $ρ = ρ_{0} / 2$ at the Rindler time $θ_{m} = \log 2$ ; using $\sinh (\log 2) = 3 / 4$ and $\cosh (\log 2) = 5 / 4$ the reflection event is $q_{m}^{'} = x^{- 1} (3 ρ_{0} / 8, 5 ρ_{0} / 8, 0, 0)$ and the return event is again $e_{+} = x^{- 1} (15 ρ_{0} / 8, 17 ρ_{0} / 8, 0, 0)$ at Rindler time $2 \log 2$ . The radar time is again $(ρ_{0} / c) \log 2$ plus the base constant and the radar distance is again $ρ_{0} \log 2$ , while the rest distance in the slice $θ = \log 2$ is now $ρ_{0} / 2$ ; since $e < 3 < 4$ gives $\log 2 > 1 / 2$ (The elementary numerical bound $2 < e < 3$ , Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm), the radar distance is now strictly larger than the rest distance.

(c) The two cases (a)-(b) give the same radar distance $ρ_{0} \log 2$ at distinct rest distances $ρ_{0}$ and $ρ_{0} / 2$ ; this demonstrates noninjectivity of the assignment as the reflector radius varies. To prove that radar distance is not determined by the rest distance alone, also take $ρ_{m} = 3 ρ_{0} / 2$ . With the same emission at $θ_{e} = 0$ , the reflection occurs at $θ_{m} = \log (3 / 2)$ , at $q_{m}^{''} = x^{- 1} (5 ρ_{0} / 8, 13 ρ_{0} / 8, 0, 0)$ , and the return occurs at $2 \log (3 / 2)$ , at $e_{+}^{''} = x^{- 1} (65 ρ_{0} / 72, 97 ρ_{0} / 72, 0, 0)$ . This reflector has rest distance $ρ_{0} / 2$ , equal to case (b), but radar distance $ρ_{0} \log (3 / 2) < ρ_{0} \log 2$ , because $3 / 2 < 2$ and $\log$ is strictly increasing. These equal inputs and unequal outputs prove the stated failure of dependence on rest distance alone. The original two cases both satisfy $θ_{m} = (θ_{e} + θ (e_{+})) / 2$ and have round-trip proper time $2 ρ_{0} \log 2 / c$ ; the third case satisfies the same midpoint identity and has round-trip proper time $2 ρ_{0} \log (3 / 2) / c$ .

(d) Scope. Conditional model computation with ideal clocks, ideal light rays and ideal reflectors in the flat classical model; no measurement, apparatus, source or statistical claim, and no empirical inference, is made, and the idealizations of The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (v) are retained.

Facts & Assumptions

Given: $c > 0$ , $ρ_{0} > 0$ , $y_{0} = z_{0} = 0$ , the inertial chart $x$ with $x^{0} = c t$ , the congruence worldlines $ρ = ρ_{0}$ (observer $O$ ) and $ρ = ρ_{m}$ (reflector) with Rindler time $θ = c τ / ρ_{0}$ on $O$ , and the emission event $e_{-} = x^{- 1} (0, ρ_{0}, 0, 0)$ at $θ_{e} = 0$ ; the radar protocol of The operational radar protocol of an ideal observer is applied with $O$ as observer.

[L1]

Rindler chart and congruence: $U = ρ e^{θ}$ and $V = - ρ e^{- θ}$ ; the future null lines in the plane of the two worldlines are the constant- $U$ lines and the constant- $V$ lines; the congruence worldlines have Rindler time $θ = c τ / ρ$ up to an additive constant (The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (i),(ii); Inertial frames and inertial charts of the affine Minkowski model, Oriented time-oriented affine Minkowski space (the spacetime primitives)).

[L2]

Coaxial radar geometry: for a reflector at $ρ_{m} \neq ρ_{0}$ , the reflection occurs at $θ_{m} = θ_{e} + ∣ \log (ρ_{m} / ρ_{0}) ∣$ , the return at $θ (e_{+}) = θ_{e} + 2 ∣ \log (ρ_{m} / ρ_{0}) ∣$ with midpoint $θ_{m}$ , the radar distance is $ρ_{0} ∣ \log (ρ_{m} / ρ_{0}) ∣$ , and the radar time is $(ρ_{0} / c) θ_{m}$ plus a constant (The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (iii), The operational radar protocol of an ideal observer).

[L3]

Rest distance in a slice: two congruence worldlines $ρ = ρ^{'}$ and $ρ = ρ^{''}$ at the same Rindler time $θ$ are at rest distance $∣ ρ^{'} - ρ^{''} ∣$ (The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (iii), Simultaneity slices and rest distance relative to a future unit timelike vector).

[L4]

Hyperbolic and logarithmic values: $\sinh u = (e^{u} - e^{- u}) / 2$ , $\cosh u = (e^{u} + e^{- u}) / 2$ , $\sinh (x + y) = \sinh x \cosh y + \cosh x \sinh y$ , $\cosh (x + y) = \cosh x \cosh y + \sinh x \sinh y$ and $\cosh u + \sinh u = e^{u}$ ; $2 < e < 3 < 4$ , so $\log 2 < \log e = 1 < \log 4 = 2 \log 2$ and therefore $1 / 2 < \log 2 < 1$ , and $\log$ is strictly increasing with $\log (3 / 2) < \log 2$ (The six hyperbolic functions and their natural domains, Addition formulas, identities, parity, and derivatives of the hyperbolic functions, The elementary numerical bound $2 < e < 3$ , Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm, The natural logarithm as the inverse of the exponential function).

Proof technique: direct.

Verification

1.1L1L2L3L4algebra

(Clause (a): the outer reflector.) With $ρ_{m} = 2 ρ_{0}$ and $θ_{e} = 0$ , [L2] gives $θ_{m} = \log 2$ , so the reflection event is $q_{m} = (2 ρ_{0} \sinh \log 2, 2 ρ_{0} \cosh \log 2, 0, 0) = (3 ρ_{0} / 2, 5 ρ_{0} / 2, 0, 0)$ by the values $\sinh \log 2 = 3 / 4$ , $\cosh \log 2 = 5 / 4$ of [L4], and the return occurs at $θ (e_{+}) = 2 \log 2$ at $e_{+} = (ρ_{0} \sinh (2 \log 2), ρ_{0} \cosh (2 \log 2), 0, 0) = (15 ρ_{0} / 8, 17 ρ_{0} / 8, 0, 0)$ using $\sinh (2 \log 2) = 2 \cdot (3 / 4) (5 / 4) = 15 / 8$ and $\cosh (2 \log 2) = (25 + 9) / 16 = 17 / 8$ of [L4]; the midpoint identity and the radar distance $ρ_{0} \log 2$ follow from [L2]. The rest distance in the slice $θ = \log 2$ is $∣ 2 ρ_{0} - ρ_{0} ∣ = ρ_{0}$ by [L3], and $ρ_{0} \log 2 < ρ_{0}$ because $\log 2 < 1$ by [L4].

2.1L2L3L4step 1.1algebra

(Clause (b): the inner reflector.) With $ρ_{m} = ρ_{0} / 2$ and $θ_{e} = 0$ , [L2] gives $θ_{m} = \log (ρ_{0} / ρ_{m}) = \log 2$ , so the reflection event is $q_{m}^{'} = (ρ_{0} / 2) (3 / 4, 5 / 4, 0, 0) = (3 ρ_{0} / 8, 5 ρ_{0} / 8, 0, 0)$ by [L4], and the return at $θ (e_{+}) = 2 \log 2$ is the same event $e_{+} = (15 ρ_{0} / 8, 17 ρ_{0} / 8, 0, 0)$ as in step 1.1; the radar distance is again $ρ_{0} \log 2$ by [L2], while the rest distance in the slice $θ = \log 2$ is $∣ ρ_{0} - ρ_{0} / 2 ∣ = ρ_{0} / 2$ by [L3]. Since $2 < e < 3 < 4$ implies $\log 2 > 1 / 2$ by [L4], the radar distance exceeds the rest distance in this case.

3.1L1L2L3L4step 1.1step 2.1algebra

(Clause (c): the third reflector and noninjectivity.) With $ρ_{m} = 3 ρ_{0} / 2$ and $θ_{e} = 0$ , [L2] gives $θ_{m} = \log (3 / 2)$ ; the hyperbolic values $\sinh \log (3 / 2) = 5 / 12$ and $\cosh \log (3 / 2) = 13 / 12$ give the reflection event $q_{m}^{''} = (3 ρ_{0} / 2) (5 / 12, 13 / 12, 0, 0) = (5 ρ_{0} / 8, 13 ρ_{0} / 8, 0, 0)$ , and the return at $2 \log (3 / 2)$ is $e_{+}^{''} = (ρ_{0} \sinh (2 \log (3 / 2)), ρ_{0} \cosh (2 \log (3 / 2)), 0, 0) = (65 ρ_{0} / 72, 97 ρ_{0} / 72, 0, 0)$ , using $\sinh (2 u) = 2 \sinh u \cosh u = 65 / 72$ and $\cosh (2 u) = \cosh^{2} u + \sinh^{2} u = 97 / 72$ at $u = \log (3 / 2)$ by [L4]; the radar distance is $ρ_{0} \log (3 / 2)$ , the rest distance in the slice is $ρ_{0} / 2$ by [L3] (the same as in step 2.1), and $ρ_{0} \log (3 / 2) < ρ_{0} \log 2$ because $3 / 2 < 2$ and $\log$ is strictly increasing by [L4]. Thus two distinct reflector radii ( $ρ_{0} / 2$ and $3 ρ_{0} / 2$ ) share the rest distance $ρ_{0} / 2$ but have different radar distances, and the two cases of steps 1.1 and 2.1 share the radar distance $ρ_{0} \log 2$ at rest distances $ρ_{0}$ and $ρ_{0} / 2$ .

4.1L1L2step 1.1step 2.1step 3.1∎

(Midpoint identities, round-trip times and scope.) In each of the three cases the reported $θ_{m}$ is the midpoint $(θ_{e} + θ (e_{+})) / 2$ by [L2], and the round-trip proper time of $O$ is $ρ_{0} (θ (e_{+}) - θ_{e}) / c$ , equal to $2 ρ_{0} \log 2 / c$ in cases (a) and (b) and to $2 ρ_{0} \log (3 / 2) / c$ in case (c). All quantities are conditional model computations with ideal clocks, light rays and reflectors; no apparatus, measurement or empirical claim is made, and no choice principle was used.

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Rindler radar distance between two coaxial observers is not symmetric under interchange

Physical scope

Conditional model computation of the radar protocol for two coaxial uniformly accelerated ideal observers in flat classical spacetime with ideal clocks, ideal light rays and ideal reflectors. The asymmetry exhibited is a property of the two observers' radar assignments; no physical anisotropy, apparatus, measurement, statistical or empirical claim is made.

Statement refuted

For every pair of future timelike worldlines $y$ and $y^{'}$ of the model for which the radar protocol of The operational radar protocol of an ideal observer applies with $y$ as observer and $y^{'}$ as reflector, the radar distance assigned by $y$ to the reflection event on $y^{'}$ equals the radar distance assigned by $y^{'}$ to the reflection event on $y$ ; that is, the radar distance between two observers is symmetric under interchange of observer and reflector.

Facts & Assumptions

Given: The inertial chart $x$ with $x^{0} = c t$ of the flat model; the Rindler congruence data of The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance; the coaxial worldlines $O$ ( $ρ = ρ_{0}$ ) and $R$ ( $ρ = 2 ρ_{0}$ ) with the same $y = z = 0$ coordinates, $ρ_{0} > 0$ , $c > 0$ ; and the radar protocol of The operational radar protocol of an ideal observer applied with each worldline in turn as the observer.

[L1]

Rindler chart and congruence: $U = ρ e^{θ}$ , $V = - ρ e^{- θ}$ ; the future null lines in the plane of the two worldlines are the constant- $U$ and constant- $V$ lines; the congruence worldlines have proper acceleration $c^{2} / ρ$ and Rindler time $θ = c τ / ρ$ up to an additive constant (The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (i),(ii); Proper time along a future timelike worldline).

[L2]

Radar protocol: the radar distance assigned by the observer is $c (τ (s_{+}) - τ (s_{-})) / 2$ , half the round-trip proper time of the observer multiplied by $c$ , given emission event $e_{-}$ , reflection event $q$ , and reception event $e_{+}$ with $q - e_{-}$ and $e_{+} - q$ future null (The operational radar protocol of an ideal observer).

[L3]

The radar round trip of the coaxial Rindler family: from $O$ at Rindler time $θ_{e}$ to the reflector at $ρ_{m}$ , the reflection occurs at $θ_{m} = θ_{e} + ∣ \log (ρ_{m} / ρ_{0}) ∣$ and the return at $θ_{e} + 2 ∣ \log (ρ_{m} / ρ_{0}) ∣$ , giving radar distance $ρ_{0} ∣ \log (ρ_{m} / ρ_{0}) ∣$ for the observer at $ρ_{0}$ (The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (iii)).

[L4]

Hyperbolic values: $\sinh u = (e^{u} - e^{- u}) / 2$ , $\cosh u = (e^{u} + e^{- u}) / 2$ , $\cosh^{2} u - \sinh^{2} u = 1$ and $\cosh u + \sinh u = e^{u}$ , so at $u = \log 2$ one has $\sinh \log 2 = 3 / 4$ , $\cosh \log 2 = 5 / 4$ , $\sinh (2 \log 2) = 15 / 8$ and $\cosh (2 \log 2) = 17 / 8$ (The six hyperbolic functions and their natural domains, Addition formulas, identities, parity, and derivatives of the hyperbolic functions, The natural logarithm as the inverse of the exponential function, Order, continuity, range, and the product, quotient, and reciprocal laws for the natural logarithm).

[L5]

Coordinates and chart: points are determined by their inertial coordinates and $x^{0} = c t$ (Inertial frames and inertial charts of the affine Minkowski model, Oriented time-oriented affine Minkowski space (the spacetime primitives)).

Counterexample

1.1L1L3L4algebra

(Radar distance from $O$ to $R$ .) By [L3] applied with observer radius $ρ_{0}$ and reflector radius $ρ_{m} = 2 ρ_{0}$ , emitted from $O$ at $θ_{e} = 0$ , the reflection event on $R$ is at Rindler time $\log 2$ and the return event on $O$ at Rindler time $2 \log 2$ , so the round trip takes $2 ρ_{0} \log 2 / c$ of $O$ 's proper time and the radar distance assigned by $O$ is $D_{1} = ρ_{0} \log 2$ ; by [L1] and [L4] the reflection event is $x^{- 1} (ρ_{0} \cdot 2 \cdot (3 / 4), ρ_{0} \cdot 2 \cdot (5 / 4), 0, 0) = x^{- 1} (3 ρ_{0} / 2, 5 ρ_{0} / 2, 0, 0)$ and the reception event is $x^{- 1} (ρ_{0} \sinh (2 \log 2), ρ_{0} \cosh (2 \log 2), 0, 0) = x^{- 1} (15 ρ_{0} / 8, 17 ρ_{0} / 8, 0, 0)$ .

1.2L1L4L5algebra

(Radar distance from $R$ to $O$ .) Now take $R$ as observer, emitting at its Rindler time zero from $r_{-} = x^{- 1} (0, 2 ρ_{0}, 0, 0)$ . The future null line from $r_{-}$ toward $O$ is the constant- $U$ line with $U = 2 ρ_{0}$ (the constant- $V$ line leaves the radius increasing), and it meets $ρ = ρ_{0}$ at the unique Rindler time $θ$ with $ρ_{0} e^{θ} = 2 ρ_{0}$ , i.e. $θ = \log 2$ , at the reflection event $q_{m}^{'} = x^{- 1} (ρ_{0} \sinh \log 2, ρ_{0} \cosh \log 2, 0, 0) = x^{- 1} (3 ρ_{0} / 4, 5 ρ_{0} / 4, 0, 0)$ by [L4]. The return leg is the constant- $V$ line through $q_{m}^{'}$ , with $V = - ρ_{0} e^{- \log 2} = - ρ_{0} / 2$ , and it reaches $R$ at the unique Rindler time $θ$ with $- (2 ρ_{0}) e^{- θ} = - ρ_{0} / 2$ , i.e. $e^{- θ} = 1 / 4$ and $θ = 2 \log 2$ , at the event $x^{- 1} (2 ρ_{0} \sinh (2 \log 2), 2 ρ_{0} \cosh (2 \log 2), 0, 0) = x^{- 1} (15 ρ_{0} / 4, 17 ρ_{0} / 4, 0, 0)$ by [L4].

2.1L1L2step 1.2algebra

(The two assignments.) By [L2] and [L1], the round trip of step 1.2 lasts $(2 \log 2)$ in Rindler time of $R$ , whose proper time satisfies $d τ = ρ_{m} d θ / c = (2 ρ_{0} / c) d θ$ , so the elapsed proper time of $R$ is $4 ρ_{0} \log 2 / c$ and the radar distance assigned by $R$ is $D_{2} = c \cdot (1 / 2) \cdot 4 ρ_{0} \log 2 / c = 2 ρ_{0} \log 2$ .

3.1L1L2L4step 1.1step 2.1∎

(Asymmetry and scope.) Steps 1.1 and 2.1 give $D_{1} = ρ_{0} \log 2$ and $D_{2} = 2 ρ_{0} \log 2$ , and $\log 2 > \log 1 = 0$ with $ρ_{0} > 0$ makes $D_{1} > 0$ , so $D_{2} = 2 D_{1} \neq D_{1}$ and the radar distance between the two worldlines depends on which worldline is the observer; both reflection events lie in the causal future of their emission events by step 1.2 and [L1], and the protocol's symmetric half-split convention of [L2] was retained. This is a property of the two radar assignments, not a physical anisotropy; no measurement, statistical or empirical claim is made, and no choice principle was used.

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The Rindler causal horizon is not a boundary or singularity of spacetime

Physical scope

Counterexample within the flat classical model: the full null causal-horizon plane is not a spacetime boundary or singularity; only its specified half-plane belongs to the Rindler wedge boundary, and the only horizon statement concerning it is the causal restriction of reachability of particular congruence worldlines. No physical interpretation of the witnesses beyond the model's causal reachability is asserted, and no measurement, statistical or empirical claim is made.

Statement refuted

In the oriented time-oriented affine Minkowski space with the inertial chart $x = (x^{0}, x^{1}, x^{2}, x^{3})$ , the null plane $H : = {x^{1} = x^{0}}$ is a boundary of the model in the following three senses: (i) no future causal curve crosses from ${x^{1} > x^{0}}$ to ${x^{1} < x^{0}}$ ; (ii) $H$ separates all causal relations, so no event of ${x^{1} > x^{0}}$ is causally related to any event of ${x^{1} < x^{0}}$ ; and (iii) the metric degenerates on $H$ , so that $H$ is a singular boundary of the spacetime on which the model fails.

Here $H$ is the full causal-past boundary of the accelerated congruence; only its half-plane ${x^{1} = x^{0} \geq 0}$ belongs to the boundary of the right wedge $W = {x^{1} > ∣ x^{0} ∣}$ .

Facts & Assumptions

Given: A fixed real $c > 0$ ; the oriented time-oriented affine Minkowski space $(A, V, η, C^{+})$ with inertial chart $x$ , $H = {x^{1} = x^{0}}$ , and the right wedge $W = {x^{1} > ∣ x^{0} ∣}$ with the Rindler map $Φ$ defined in [L3].

[L1]

Causal classification and orientation: a displacement $w \in V$ is timelike iff $η (w, w) < 0$ , future timelike iff $w \in C^{+}$ , and future causal iff $w \in {\overset{ˉ}{C}}^{+} ∖ {0}$ , where ${\overset{ˉ}{C}}^{+} : = cl (C^{+})$ ; in every inertial chart $C^{+} = {w : w^{0} > ∣ w_{s} ∣}$ and ${\overset{ˉ}{C}}^{+} = {w : w^{0} \geq ∣ w_{s} ∣}$ , with $∣ w_{s} ∣ : = \sqrt{(w^{1})^{2} + (w^{2})^{2} + (w^{3})^{2}}$ . For events $p, q \in A$ , $p \leq q$ iff $q - p \in {\overset{ˉ}{C}}^{+}$ and $p < q$ iff $q - p \in C^{+}$ (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones, The closed future cone in inertial coordinates: description, finite sums, and the coordinate criterion for causal reachability (a),(c), Causal reachability, causal futures and pasts, causal diamonds, and reachability by future causal paths in affine Minkowski space (a)).

[L2]

The affine segment $s \mapsto p + s (q - p)$ , $s \in [0, 1]$ , is a $C^{1}$ curve of the model whose tangent, including its one-sided endpoint tangents, is the constant displacement $q - p$ (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, Definition, “Derivative and regularity of the parametrization”). A $C^{1}$ curve $z : I \to A$ on an interval $I \subseteq R$ is a future timelike worldline when every tangent is future timelike (the same definition, “Worldlines”).

[L3]

Metric and chart: the matrix of $η$ in every inertial chart is the constant $diag (- 1, 1, 1, 1)$ , and $Φ (θ, ρ, y, z) = (ρ \sinh θ, ρ \cosh θ, y, z)$ with $θ \in R$ , $ρ > 0$ is a smooth bijection onto $W$ with smooth inverse (Inertial frames and inertial charts of the affine Minkowski model, Oriented time-oriented affine Minkowski space (the spacetime primitives), Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (b)).

[L4]

Rindler causal criteria: define $U (q) : = x^{0} (q) + x^{1} (q)$ and $V (q) : = x^{0} (q) - x^{1} (q)$ for $q \in A$ . For fixed $ρ_{1} > 0$ and $y_{1}, z_{1} \in R$ , the congruence worldline is $O_{1} (θ) : = x^{- 1} (Φ (θ, ρ_{1}, y_{1}, z_{1}))$ , $θ \in R$ . An event $q$ reaches this worldline, meaning $q \leq O_{1} (θ)$ for some $θ$ , iff $V (q) < 0$ ; it is reached by this worldline, meaning $O_{1} (θ) \leq q$ for some $θ$ , iff $U (q) > 0$ (The Rindler chart: causal horizons of a uniformly accelerated observer and the restrictions of radar time and distance (ii)).

Counterexample

Steps 1.1–1.3 use events with exact rational inertial coordinates; step 1.4 supplies an inertial worldline. The failed conclusions (i)-(iii) are refuted in the order stated.

1.1L1L2L3algebra

(The segment and the crossing.) With $p = x^{- 1} (- 1, 0, 0, 0)$ , $q = x^{- 1} (3, 1, 0, 0)$ and $y (s) = p + s (q - p)$ , the tangent is the constant displacement $(4, 1, 0, 0)$ by [L2] and [L3]; its $η$ -square is $- 16 + 1 = - 15 < 0$ with time component $4 > 0$ , so every tangent is future timelike by [L1] and $y$ is a future timelike curve; the coordinate $x^{0} - x^{1}$ equals $- 1 + 3 s$ along $y$ , which is $- 1$ at $s = 0$ , $2$ at $s = 1$ , and vanishes exactly at $s = 1 / 3$ , giving the crossing event $x^{- 1} (1 / 3, 1 / 3, 0, 0) \in H$ .

1.2L1L3algebra

(Causal relations across and on $H$ .) By [L1], $p \leq q$ because $(4, 1, 0, 0) \in C^{+}$ ; $r = x^{- 1} (0, 1, 0, 0) < x^{- 1} (1, 1, 0, 0)$ because the displacement $(1, 0, 0, 0)$ is future timelike; and $x^{- 1} (1, 1, 0, 0) < x^{- 1} (2, 3 / 2, 0, 0)$ because $(1, 1 / 2, 0, 0)$ has $η$ -square $- 1 + 1 / 4 < 0$ and positive time component; the middle event lies on $H$ , the first and third do not.

1.3L3

(Nondegeneracy and the map domain.) By [L3] the metric is the constant nondegenerate $diag (- 1, 1, 1, 1)$ everywhere, in particular on $H$ and at the crossing event $x^{- 1} (1 / 3, 1 / 3, 0, 0)$ ; the Rindler map is a bijection onto $W$ with domain $ρ > 0$ , so its coordinate domain omits $ρ = 0$ and its image omits the full plane $H$ , whose half-plane $x^{1} = x^{0} \geq 0$ is one part of the wedge boundary $x^{1} = ∣ x^{0} ∣ \geq 0$ .

1.4L1L2L3algebra

(A second crossing by an inertial worldline.) The worldline $z : R \to A$ , $z (t) = x^{- 1} (c t, 0, 0, 0)$ , is $C^{\infty}$ and future timelike: its tangent is $c e_{0}$ with $η = - c^{2} < 0$ and time component $c > 0$ in the chosen chart, so by [L1] and [L2] it is regular and future timelike; $z (0)$ lies on $H$ , $x^{1} > x^{0}$ for $t < 0$ and $x^{1} < x^{0}$ for $t > 0$ , so $z$ crosses $H$ while remaining defined for every $t$ , refuting (i) again.

2.1L1step 1.1step 1.2

(Refutation of (i) and (ii).) Step 1.1 exhibits a future timelike curve crossing $H$ and a pair of causally related events on opposite sides of $H$ , refuting both the no-crossing claim (i) and the separation claim (ii); step 1.2 exhibits causal relations between events of $H$ and events of each side, refuting the reading of $H$ as causally isolating.

2.2L3L4step 1.3

(Refutation of (iii) and what remains true.) Step 1.3 shows the metric is everywhere nondegenerate on the model, so the degeneracy claim (iii) is false and the failure of the Rindler map at $ρ = 0$ is a property of the coordinate map alone; by [L4] the surviving correct statement is the conditional causal restriction of reachability of the congruence worldlines (events with $V \geq 0$ cannot reach them), which is not a statement about a boundary or singularity of the spacetime.

3.1L1L2step 2.1step 2.2∎

(Scope.) The witnesses are explicit affine-model events, segments and worldlines with computed $η$ -signs; no physical interpretation, measurement or empirical claim is made, and no choice principle was used.

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Guided-ring Sagnac delay worked at angular velocity Omega R = 3c/5

Physical scope

Worked numerical instance of the conditional guided-ring Sagnac model for an ideal rigidly rotating ring, an adopted ideal circular guide and an ideal clock detector in flat classical spacetime. No optical fiber, medium, apparatus, interference phase, measurement, statistical or empirical claim is made, and all idealizations of pthm-sr-guided-ring-sagnac-delay are retained.

Example

Worked instance of The guided-ring Sagnac delay: return times and receiver proper-time difference for counter-propagating guided signals on a rotating ring. Let $R > 0$ and $Ω = 3 c / (5 R)$ , so that $Ω R = 3 c / 5$ , $∣ Ω ∣ R < c$ and $γ = (1 - 9 / 25)^{- 1 / 2} = 5 / 4$ . Work in the inertial chart with $x^{0} = c t$ and let the ring have its plane $x^{3} = 0$ and let the emission event be $E = x^{- 1} (0, R, 0, 0)$ , where the material point labelled $φ = 0$ has $ϕ_{0} = 0$ .

(a) Return times. The co-rotating guided signal has $ϕ_{+} (t) = c t / R$ and the emitting material point has $ϕ (t) = Ω t = 3 c t / (5 R)$ ; the signal meets that point when $c t / R = 3 c t / (5 R) + 2 π$ , i.e. $t_{1} = 5 π R / c$ (the co-rotating relative angular rate is $2 c / (5 R)$ ); the counter-rotating signal $C_{-}$ , $ϕ_{-} (t) = - c t / R$ , meets it when $- c t / R = 3 c t / (5 R) - 2 π$ , i.e. $t_{2} = 5 π R / (4 c)$ , with signed relative angular rate $- 8 c / (5 R)$ and magnitude $8 c / (5 R)$ . The reception events are $e_{1} = x^{- 1} (5 π R, R \cos (3 π), R \sin (3 π), 0) = x^{- 1} (5 π R, - R, 0, 0)$ and $e_{2} = x^{- 1} (5 π R / 4, R \cos (3 π / 4), R \sin (3 π / 4), 0) = x^{- 1} (5 π R / 4, - R / \sqrt{2}, R / \sqrt{2}, 0)$ , where the angle values use the parity, addition and Pythagorean identities for sine and cosine (Pi as twice the smallest positive zero of cosine, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine).

(b) Differences and the factor-2 holonomy check. The chart-time difference of the two reunions is $t_{1} - t_{2} = 5 π R / c - 5 π R / (4 c) = 15 π R / (4 c) = 4 π Ω R^{2} / (c^{2} - Ω^{2} R^{2})$ , and the detector proper-time difference is $(t_{1} - t_{2}) \cdot (4 / 5) = 3 π R / c = 4 π Ω R^{2} / (c^{2} \sqrt{1 - Ω^{2} R^{2} / c^{2}})$ . The ring rest-line chart-time holonomy of The guided-ring Sagnac delay: return times and receiver proper-time difference for counter-propagating guided signals on a rotating ring (iv) is $Δ t_{h} = 15 π R / (8 c)$ , exactly half of $t_{1} - t_{2}$ , and the corresponding receiver proper-time holonomy is $(4 / 5) \cdot 15 π R / (8 c) = 3 π R / (2 c)$ , exactly half of $3 π R / c$ ; this verifies the factor-2 relation of that clause. The ambient rest distribution of the rotating congruence is nowhere integrable while the fixed-height-cylinder-restricted one-dimensional rest distribution is locally integrable with the nonzero global holonomy $15 π R / (8 c)$ (Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (c)).

(c) Inertial limit. For $Ω = 0$ with the same $R$ the two return times are both $2 π R / c$ and both differences vanish.

(d) Scope. Conditional model computation with the adopted ideal circular guide, ideal signals and the ideal clock detector of The guided-ring Sagnac delay: return times and receiver proper-time difference for counter-propagating guided signals on a rotating ring (ii); no interference phase, apparatus, medium, measurement, statistical or empirical claim is made, and the idealizations of that theorem are retained.

Facts & Assumptions

Given: $c > 0$ , $R > 0$ , $Ω = 3 c / (5 R)$ , so $Ω R = 3 c / 5$ and $γ = (1 - Ω^{2} R^{2} / c^{2})^{- 1 / 2} = 5 / 4$ ; the inertial chart $x$ with $x^{0} = c t$ ; the ring data, guided curves $C_{\pm}$ with $ϕ_{\pm} (t) = \pm c t / R$ plus the material label, and the ideal clock detector of The guided-ring Sagnac delay: return times and receiver proper-time difference for counter-propagating guided signals on a rotating ring clauses (ii)-(iv).

[L1]

Guided-ring theorem: on the ring with $∣ Ω ∣ R < c$ , $C_{+}$ meets the worldline $W_{ϕ_{0}}$ exactly at $t = 2 π k R / (c - Ω R)$ and $C_{-}$ exactly at $t = 2 π k R / (c + Ω R)$ , the smallest positive times being $t_{1}, t_{2}$ ; the chart-time difference is $4 π Ω R^{2} / (c^{2} - Ω^{2} R^{2})$ and the detector proper-time difference is that divided by $γ$ ; the ring rest-line chart-time holonomy over one circuit is $2 π Ω R^{2} / (c^{2} - Ω^{2} R^{2})$ and the receiver proper-time holonomy is that divided by $γ$ (The guided-ring Sagnac delay: return times and receiver proper-time difference for counter-propagating guided signals on a rotating ring (iii),(iv)).

[L2]

Ring cylinder data: the material point at chart angle $ϕ (t) = ϕ_{0} + Ω t$ and radius $R$ has coordinates $(c t, R \cos ϕ, R \sin ϕ, x_{0}^{3})$ , and the induced metric on the cylinder is $- (c^{2} - Ω^{2} R^{2}) d t^{2} + 2 Ω R^{2} d t d φ + R^{2} d φ^{2}$ with $φ = ϕ - Ω t$ (The guided-ring Sagnac delay: return times and receiver proper-time difference for counter-propagating guided signals on a rotating ring (i),(iv), Fermi-Walker transport preserves the metric, and explicit rigid and non-rigid observer congruences (c), Inertial frames and inertial charts of the affine Minkowski model, Oriented time-oriented affine Minkowski space (the spacetime primitives)).

[L3]

Proper time of the ring clock satisfies $d τ = d t / γ$ , with the base constant free (The guided-ring Sagnac delay: return times and receiver proper-time difference for counter-propagating guided signals on a rotating ring (i), Proper time along a future timelike worldline, The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse).

[L4]

Trigonometric values: $π$ is twice the first positive zero of $\cos$ and the first positive zero of $\sin$ is $π$ ; addition, parity and Pythagorean identities; quarter-turn shifts, so that $\sin (3 π) = 0$ , $\cos (3 π) = - 1$ and $\cos (3 π / 4) = - \sin (π / 4) = - \sqrt{2} / 2$ , $\sin (3 π / 4) = \cos (π / 4) = \sqrt{2} / 2$ with $\sin^{2} (π / 4) + \cos^{2} (π / 4) = 1$ and $\sin (π / 4) = \cos (π / 4) > 0$ (Pi as twice the smallest positive zero of cosine, Pi is the first positive zero of sine, The addition formulas for sine and cosine, Parity and the Pythagorean identity for sine and cosine, Quarter-turn values and shifts by pi/2 and pi).

Proof technique: direct.

Verification

1.1L1L2givenalgebra

(The numerical data and return times.) With $Ω R = 3 c / 5$ one has $∣ Ω ∣ R < c$ and $γ = (1 - 9 / 25)^{- 1 / 2} = (16 / 25)^{- 1 / 2} = 5 / 4$ ; the theorem [L1] gives $t_{1} = 2 π R / (c - Ω R) = 2 π R / (c (1 - 3 / 5)) = 5 π R / c$ and $t_{2} = 2 π R / (c + Ω R) = 2 π R / (c (1 + 3 / 5)) = 5 π R / (4 c)$ , both positive; these are the reunion times of the two signals with the emitting material point, after one circuit relative to it.

1.2L1L2

(Clause (b): the integrability statement.) By [L1] and [L2] the ambient rest distribution of the rotating congruence is nowhere integrable for $Ω \neq 0$ , while the fixed-height-cylinder rest distribution is locally integrable with nonvanishing global holonomy $15 π R / (8 c)$ ; the return-time computation above used only the guided curves and the material worldline.

2.1L2L4step 1.1algebra

(The reception events.) At $t_{1}$ the material point's angular label is $ϕ (t_{1}) = Ω t_{1} = (3 c / (5 R)) (5 π R / c) = 3 π$ , so $e_{1} = x^{- 1} (5 π R, R \cos 3 π, R \sin 3 π, 0) = x^{- 1} (5 π R, - R, 0, 0)$ by [L4]; at $t_{2}$ , $ϕ (t_{2}) = Ω t_{2} = (3 c / (5 R)) (5 π R / (4 c)) = 3 π / 4$ , so $e_{2} = x^{- 1} (5 π R / 4, - R / \sqrt{2}, R / \sqrt{2}, 0)$ by [L4]. The guided angles $ϕ_{+} (t_{1}) = c t_{1} / R = 5 π$ and $ϕ_{-} (t_{2}) = - c t_{2} / R = - 5 π / 4$ agree with the material angles modulo $2 π$ : $5 π = 3 π + 2 π$ and $- 5 π / 4 = 3 π / 4 - 2 π$ .

2.2L1L3step 1.1algebra

(Clause (b): differences and the factor-two relation.) By step 1.1, $t_{1} - t_{2} = 5 π R / c - 5 π R / (4 c) = 15 π R / (4 c)$ , and the theorem's formula $4 π Ω R^{2} / (c^{2} - Ω^{2} R^{2})$ gives $4 π (3 c / (5 R)) R^{2} / (c^{2} (16 / 25)) = 12 π c R \cdot 25 / (5 \cdot 16 c^{2}) = 15 π R / (4 c)$ , in agreement; the detector proper-time difference is $(15 π R / (4 c)) \cdot (4 / 5) = 3 π R / c$ by [L3], matching $4 π Ω R^{2} / (c^{2} \sqrt{1 - Ω^{2} R^{2} / c^{2}}) = 15 π R / (4 c) \cdot (4 / 5) = 3 π R / c$ . The rest-line chart-time holonomy is $Δ t_{h} = 2 π Ω R^{2} / (c^{2} - Ω^{2} R^{2}) = \frac{1}{2} \cdot 15 π R / (4 c) = 15 π R / (8 c)$ , exactly half of $t_{1} - t_{2}$ , and the receiver proper-time holonomy is $Δ t_{h} / γ = (4 / 5) (15 π R / (8 c)) = 3 π R / (2 c)$ , exactly half of the detector proper-time difference.

2.3L1step 1.1

(Clause (c): the inertial limit.) At $Ω = 0$ the theorem's formulas give $t_{1} = t_{2} = 2 π R / (c)$ and vanishing chart-time difference, proper-time difference and holonomies; the rest lines are the $t = c o n s t$ circles.

3.1L1step 2.2step 2.3∎

(Clause (d): scope.) Every quantity is a conditional model computation for the adopted guide, signals and detector; no medium, apparatus, interference phase, measurement or empirical claim enters, and no choice principle was used.

Accelerated Observers — Examples

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

5 · Examples, experiments and counterexamples

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

Example

Facts & Assumptions

Verification

Rindler radar distances worked: reflectors at twice and at half the observer's radius

Example

Facts & Assumptions

Verification

Rindler radar distance between two coaxial observers is not symmetric under interchange

Statement refuted

Facts & Assumptions

Counterexample

The Rindler causal horizon is not a boundary or singularity of spacetime

Statement refuted

Facts & Assumptions

Counterexample

Guided-ring Sagnac delay worked at angular velocity Omega R = 3c/5

Example

Facts & Assumptions

Verification

Sources