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Same-endpoint clock comparison: the inertial clock ages most

Physical scope

Ideal clocks and prescribed histories in the flat classical special-relativity model: the reading of an ideal clock between two fixed endpoint events along a piecewise C^1 future timelike worldline, compared with the reading along the straight inertial worldline between the same events. Conditional on the named postulates (ideal clocks with no acceleration-dependent term, prescribed histories, flat spacetime, fixed c). Excludes real clocks and apparatus, calibration and resolution effects, gravity, quantum phenomena, and any empirical or statistical claim.

Statement

Adopt as physical assumptions the postulates Adopted inertial covariance of the model equations, Adopted ideal-clock and ideal-vacuum-light interpretation and Adopted free trajectories of massive particles and the clock hypothesis, and let $c$ and the SI conventions be those of SI units and coordinate conventions of the special-relativity framework. Let $p$ and $q$ be events with $q - p$ future timelike, fix reals $a < b$ , and put $T : = \sqrt{- η (q - p, q - p)} > 0$ , and let $x_{s} : [a, b] \to A$ be the straight (inertial) future timelike worldline from $p$ to $q$ , $x_{s} (s) : = p + \frac{s - a}{b - a} (q - p)$ . Let $x : [a, b] \to A$ be a piecewise $C^{1}$ future timelike worldline from $p$ to $q$ in the weak sense of Curves, regular causal curves and future timelike worldlines in affine Minkowski space, with total proper time $Δ τ (x)$ defined by Total proper time of a piecewise C^1 future timelike worldline. Then, as conditional statements about ideal clocks of the adopted model and not about any actual apparatus:

(i) An ideal clock carried along $x_{s}$ from $p$ to $q$ reads between the parameter values $a$ and $b$ the time $c^{- 1} T$ .

(ii) An ideal clock carried along $x$ from $p$ to $q$ reads $Δ τ (x)$ , and $Δ τ (x) \leq c^{- 1} T$ , with equality if and only if $x$ is a strictly increasing reparametrization of the straight segment from $p$ to $q$ in the sense of The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space (ii).

(iii) If $x$ is piecewise inertial (Inertial and piecewise inertial worldlines of the affine Minkowski model) and its image is not contained in the straight segment from $p$ to $q$ , then $Δ τ (x) < c^{- 1} T$ strictly; so among ideal clocks carried between the same two events along future timelike piecewise $C^{1}$ histories of the weak class of Curves, regular causal curves and future timelike worldlines in affine Minkowski space (including its ordinary Riemann special case (H)) the straight inertial clock records the largest reading, and every history whose image is not the straight segment records a strictly smaller one.

(iv) The two numbers $Δ τ (x)$ and $c^{- 1} T$ are determined by the parametrized histories and the model's conversion constant alone: the integrand $\sqrt{- η (x^{'}, x^{'})}$ of Total proper time of a piecewise C^1 future timelike worldline involves no chart, so the comparison is a definite pair of clock readings attached to the same pair of endpoint events and involves no simultaneity convention.

Scope and qualification. The statement inherits the idealizations of its postulates (ideal pointlike clocks with no acceleration-dependent reading term, prescribed histories, flat classical spacetime, the fixed conversion constant $c$ ); it asserts no measurement outcome, contains no statistical inference, and carries no claim that real clocks or the physical world are exactly described by the model. The conclusion is conditional on the named postulates and does not follow from the geometry alone.

Facts & Assumptions

Given: The adopted postulates Adopted inertial covariance of the model equations (A1), Adopted ideal-clock and ideal-vacuum-light interpretation (A2) and Adopted free trajectories of massive particles and the clock hypothesis (A3); the SI conventions of SI units and coordinate conventions of the special-relativity framework with the fixed conversion constant $c > 0$ ; events $p, q$ with $q - p$ future timelike, $T = \sqrt{- η (q - p, q - p)} > 0$ ; the straight worldline $x_{s} (s) = p + \frac{s - a}{b - a} (q - p)$ on $[a, b]$ , $a < b$ ; and a weak piecewise $C^{1}$ future timelike worldline $x : [a, b] \to A$ from $p$ to $q$ with total proper time $Δ τ (x)$ .

[A1]

Inertial covariance (Adopted inertial covariance of the model equations): the oriented time-oriented affine Minkowski space is the spacetime of the framework, and the model's equations are covariant under the affine maps whose linear part preserves $η$ , has determinant $+ 1$ and maps $C^{+}$ onto $C^{+}$ (Oriented time-oriented affine Minkowski space (the spacetime primitives)).

[A2]

Ideal clock reading rule (Adopted ideal-clock and ideal-vacuum-light interpretation (i)): along a curve whose coordinate representation in an inertial chart is $C^{1}$ with everywhere future timelike derivative, an ideal clock reads between parameters $a < b$ the value $c^{- 1} \int_{a}^{b} \sqrt{- η (x^{'} (s), x^{'} (s))} d s$ , with no additional acceleration-dependent term.

[A3]

Clock hypothesis for the general worldline class (Adopted free trajectories of massive particles and the clock hypothesis (ii)): an ideal clock carried along a future timelike worldline of the model, including an accelerated one and a weak piecewise $C^{1}$ one with finitely many kinks, reads between two parameter values the proper-time difference $Δ τ$ of Total proper time of a piecewise C^1 future timelike worldline (for weak piecewise $C^{1}$ histories), with no term depending on acceleration, jerk or any higher derivative; the reading does not depend on an increasing regular change of parameter.

[L4]

Endpoint bound and flat maximum (The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space): for the weak piecewise future timelike $x$ from $p$ to $q$ , $c Δ τ (x) \leq T$ ; equality holds exactly for the strictly increasing reparametrizations of the straight segment in the sense of that theorem's clause (ii); and if the image of $x$ is not contained in the straight segment, in particular if $x$ is piecewise inertial with piece tangents not all positive multiples of one another, then $c Δ τ (x) < T$ , while a piecewise inertial itinerary whose tangents are all positive multiples of one another attains the bound.

[L5]

Proper-time definition and chart independence (Total proper time of a piecewise C^1 future timelike worldline, Proper time along a future timelike worldline, Curves, regular causal curves and future timelike worldlines in affine Minkowski space): $Δ τ (x) = c^{- 1} \sum_{i} \int_{(s_{i - 1}, s_{i})} \sqrt{- η (x^{'}, x^{'})}$ is defined through the intrinsic integrand $\sqrt{- η (x^{'}, x^{'})}$ , is independent of the inertial chart and of the admissible partition, and for the straight worldline with constant tangent $d / (b - a)$ equals $c^{- 1} \sqrt{- η (d, d)}$ , the straight-segment value of Proper time along a future timelike worldline.

[L6]

Bilinearity and roots (Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space, Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ): $η$ is bilinear, so $- η (λ d, λ d) = λ^{2} (- η (d, d))$ for real $λ$ , and $\sqrt{λ^{2} T^{2}} = λ T$ for every real $λ > 0$ ; every nonnegative real has a unique nonnegative square root.

[L7]

Piecewise inertial histories (Inertial and piecewise inertial worldlines of the affine Minkowski model): a piecewise inertial worldline is a weak piecewise $C^{1}$ future timelike worldline with constant future timelike tangents on the open pieces.

Proof

technique · direct

1.1givenA1A3L5L7

The straight worldline $x_{s}$ has constant tangent $w : = \frac{1}{b - a} (q - p)$ , which is future timelike because $q - p$ is future timelike and $\frac{1}{b - a} > 0$ ; hence $x_{s}$ is a $C^{1}$ future timelike worldline from $p$ to $q$ , and $x$ is a weak piecewise $C^{1}$ future timelike worldline from $p$ to $q$ by hypothesis, so both are histories of the class covered by the clock postulates [A2] and [A3].

2.1step 1.1A2L6algebra

By the reading rule [A2] applied to the $C^{1}$ history $x_{s}$ , the ideal clock reads $c^{- 1} \int_{a}^{b} \sqrt{- η (x_{s}^{'}, x_{s}^{'})} d s = c^{- 1} (b - a) \sqrt{- η (w, w)}$ ; by bilinearity [L6], $- η (w, w) = \frac{1}{(b - a)^{2}} (- η (q - p, q - p)) = \frac{T^{2}}{(b - a)^{2}}$ , so $\sqrt{- η (w, w)} = \frac{T}{b - a}$ and the reading is $c^{- 1} T$ ; this is (i).

2.2step 1.1A3L5

By the clock hypothesis [A3] for the weak piecewise class, the ideal clock carried along $x$ from $p$ to $q$ reads $Δ τ (x)$ , which by [L5] equals the intrinsic proper-time value $c^{- 1} \sum_{i} \int_{(s_{i - 1}, s_{i})} \sqrt{- η (x^{'}, x^{'})}$ .

3.1step 2.1step 2.2L4

Endpoint comparison: by [L4] the endpoint bound gives $c Δ τ (x) \leq T$ , equivalently $Δ τ (x) \leq c^{- 1} T$ because $c > 0$ ; and equality holds if and only if $x$ is a strictly increasing reparametrization of the straight segment from $p$ to $q$ in the sense of The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space (ii); this proves (ii).

4.1step 3.1L4L7

Strict case: if $x$ is piecewise inertial with piece tangents not all positive multiples of one another, then its image is not contained in the straight segment and [L4] gives $c Δ τ (x) < T$ , that is $Δ τ (x) < c^{- 1} T$ . Since the straight clock reads $c^{- 1} T$ by (i) and every weak piecewise history reads at most $c^{- 1} T$ by (ii) with equality only for the straight-segment reparametrizations, the straight inertial clock records the largest reading among the histories of the class, and every history whose image is not the straight segment records strictly less — for a continuous history from $p$ to $q$ contained in the straight segment has the whole segment as image, so 'image not contained' and 'image not the straight segment' agree; this proves (iii).

5.1step 3.1step 4.1L5L6

Chart independence: the proper-time integrand $\sqrt{- η (x^{'}, x^{'})}$ is formed from $η$ and the tangent alone, and [L5] records that $Δ τ (x)$ is independent of the inertial chart and of the partition; likewise $T = \sqrt{- η (q - p, q - p)}$ is the interval of the intrinsic displacement, and the rest-frame computation of [L4] realizes it as the coordinate value in any inertial frame with first vector $(q - p) / T$ , so both numbers are determined by the parametrized histories and the fixed constant $c$ ; since the two readings are attached to the same events $p$ and $q$ , no simultaneity convention enters. This is (iv).

6.1step 2.1step 2.2step 3.1step 5.1A1A2A3L4L5∎

Scope: the deduction used the adopted postulates [A1]-[A3] and the mathematical endpoint theorem [L4]; the conclusion is conditional on those idealizations (ideal clocks without an acceleration-dependent term, prescribed histories, flat spacetime, fixed $c$ ), asserts no measurement outcome and no statistical inference, and claims nothing about real clocks or about nature; no choice principle is used beyond the published suppliers.

Same-endpoint clock comparison: the inertial clock ages most

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources