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The piecewise inertial twin reunion: conditional differential aging

Physical scope

The idealized twin-reunion scenario of the flat classical special-relativity model: two ideal clocks set to zero at their common departure event, one inertial and one on a two-leg piecewise inertial history with an instantaneous ideal turn, compared at their common reunion event. Conditional on the named postulates (ideal clocks with no acceleration-dependent term, prescribed histories, flat spacetime, fixed c); the turn idealizes a finite-duration turnaround. Excludes real clocks and apparatus, gravity, quantum effects, and every measurement or statistical claim.

Statement

Thought experiment: the piecewise inertial twin reunion of the adopted model, with its conditional conclusion. Adopt 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. Fix an inertial frame with chart $x$ , a duration $T > 0$ in seconds and a speed $v$ with $0 < ∣ v ∣ < c$ ; put $γ : = (1 - v^{2} / c^{2})^{- 1 / 2}$ . Define two histories (parameter $t$ in seconds, with $x^{0} = c t$ ):

stay-at-home: $x_{s t a y} (t) : = x^{- 1} (c t, 0, 0, 0)$ for $t \in [0, 2 T]$ ;
traveler: $x_{t r a v e l} (t) : = x^{- 1} (c t, v t, 0, 0)$ for $t \in [0, T]$ and $x_{t r a v e l} (t) : = x^{- 1} (c t, v (2 T - t), 0, 0)$ for $t \in [T, 2 T]$ .

Carry ideal clocks along these histories, with reading maps $C_{s t a y}, C_{t r a v e l} : [0, 2 T] \to R$ in seconds, whose increments obey the clock postulates. Set both to zero at the common departure event $p = x^{- 1} (0, 0, 0, 0)$ : $C_{s t a y} (0) = C_{t r a v e l} (0) = 0$ . This initial setting is part of the preparation; the postulates fix increments and allow additive offsets.

Then:

(i) $x_{s t a y}$ is the inertial future timelike worldline from $p : = x^{- 1} (0, 0, 0, 0)$ to $q : = x^{- 1} (2 c T, 0, 0, 0)$ , and $x_{t r a v e l}$ is a piecewise inertial future timelike worldline from $p$ to $q$ whose two pieces have the constant tangents $(c, v, 0, 0)$ and $(c, - v, 0, 0)$ , with the corner $r : = x^{- 1} (c T, v T, 0, 0)$ ; the image of $x_{t r a v e l}$ contains the three events $p, r, q$ , which are not collinear in the affine model, so the image of $x_{t r a v e l}$ is not contained in the straight segment from $p$ to $q$ .

(ii) Conditional on the postulates, an ideal clock carried from $p$ to $q$ along $x_{s t a y}$ reads $2 T$ , while an ideal clock carried from $p$ to $q$ along $x_{t r a v e l}$ reads $2 T / γ = 2 T \sqrt{1 - v^{2} / c^{2}}$ ; hence at the reunion event $q$ the traveling clock is behind the stay-at-home clock by $2 T (1 - 1 / γ) = 2 T (1 - \sqrt{1 - v^{2} / c^{2}}) > 0$ .

(iii) The conclusion is a comparison of two clock readings at the common events $p$ and $q$ ; it does not depend on a simultaneity convention or on the inertial frame used to describe the histories, because both readings are proper-time integrals of the two histories.

(iv) Scope and idealization: the scenario is a conditional model statement; the instantaneous turn at $r$ idealizes a turnaround of finite duration, the clocks are ideal with no acceleration-dependent reading term, spacetime is flat, and no gravitational, quantum, apparatus or statistical effect is modeled; no measurement is reported. The conclusion is conditional on the adopted postulates and does not assert that nature is exactly described by the model.

Facts & Assumptions

Given: The adopted 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; the SI conventions of SI units and coordinate conventions of the special-relativity framework with fixed $c > 0$ ; an inertial frame with chart $x$ ; a duration $T > 0$ and a speed $v$ with $0 < ∣ v ∣ < c$ ; $γ = (1 - v^{2} / c^{2})^{- 1 / 2}$ ; the two histories $x_{s t a y}$ and $x_{t r a v e l}$ of the statement, with $p = x^{- 1} (0, 0, 0, 0)$ , $r = x^{- 1} (c T, v T, 0, 0)$ and $q = x^{- 1} (2 c T, 0, 0, 0)$ ; their ideal-clock reading maps $C_{s t a y}, C_{t r a v e l} : [0, 2 T] \to R$ in seconds, set to $C_{s t a y} (0) = C_{t r a v e l} (0) = 0$ at $p$ .

[A1]

Inertial covariance (Adopted inertial covariance of the model equations, Oriented time-oriented affine Minkowski space (the spacetime primitives)): the oriented time-oriented affine Minkowski space is the spacetime of the framework, and the model's equations are covariant under the affine maps preserving $η$ , orientation and the future cone.

[A2]

Ideal-clock reading rule (Adopted ideal-clock and ideal-vacuum-light interpretation (i)): along a $C^{1}$ history with everywhere future timelike derivative the ideal clock's reading increment between parameters $a < b$ is $C (b) - C (a) = c^{- 1} \int_{a}^{b} \sqrt{- η (x^{'}, x^{'})} d s$ , with no acceleration-dependent term; the reading map $C$ is specified up to an additive constant.

[A3]

Clock hypothesis for the piecewise 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 a weak piecewise $C^{1}$ one with finitely many kinks, has reading increment $C (b) - C (a) = Δ τ_{[a, b]}$ of Total proper time of a piecewise C^1 future timelike worldline, with no term depending on acceleration, jerk or higher derivatives.

[L4]

Inertial and piecewise inertial worldlines (Inertial and piecewise inertial worldlines of the affine Minkowski model, Curves, regular causal curves and future timelike worldlines in affine Minkowski space, 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 timelike set has exactly two convex components, described in orthonormal coordinates, Inertial frames and inertial charts of the affine Minkowski model): a curve with constant tangent $d$ along an interval is inertial when $d$ is future timelike; a two-piece curve with constant future timelike tangents $d_{1}, d_{2}$ joining at a corner is a piecewise inertial weak piecewise $C^{1}$ future timelike worldline; and in the chart a coordinate vector $(w^{0}, w_{s})$ is future timelike exactly when $w^{0} > ∣ w_{s} ∣$ .

[L5]

Proper times of straight and piecewise straight histories (Proper time along a future timelike worldline, Total proper time of a piecewise C^1 future timelike worldline, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space): a straight history with constant tangent $w$ over a parameter interval of length $L$ has proper time $c^{- 1} L \sqrt{- η (w, w)}$ ; a two-leg history has total proper time the sum of its leg values; and the values depend only on the intrinsic integrand $\sqrt{- η (x^{'}, x^{'})}$ and the parameter interval, not on the inertial chart.

[L6]

Same-endpoint clock comparison (Same-endpoint clock comparison: the inertial clock ages most): for a weak piecewise future timelike history from $p$ to $q$ with $q - p$ future timelike, the clock reading along the straight history is $c^{- 1} \sqrt{- η (q - p, q - p)}$ and is the largest among the readings of the class, with strict inequality for every history whose image is not the straight segment; the conclusion is conditional on the same postulates.

[L7]

Endpoint theorem (The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space): for the weak piecewise future timelike class, $c Δ τ \leq \sqrt{- η (q - p, q - p)}$ with the stated equality and strictness clauses.

[L8]

Roots, monotonicity and order (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ , Squaring is monotone on the nonnegatives, Ordered field): $a \geq 0$ has a unique nonnegative square root; for $x, y \geq 0$ , $x < y ⟺ x^{2} < y^{2}$ ; and $0 < ∣ v ∣ < c$ gives $0 < v^{2} / c^{2} < 1$ .

Proof

technique · direct

1.1givenL4algebra

The stay-at-home history has parameter path $(c t, 0, 0, 0)$ , hence constant tangent $(c, 0, 0, 0)$ with $c > 0 = ∣ 0 ∣$ , so it is future timelike and inertial, running from $p$ at $t = 0$ to $q$ at $t = 2 T$ . The traveler's first piece has parameter path $(c t, v t, 0, 0)$ on $[0, T]$ , hence constant tangent $(c, v, 0, 0)$ with $c > ∣ v ∣$ , and its second piece has parameter path $(c t, v (2 T - t), 0, 0)$ on $[T, 2 T]$ , hence constant tangent $(c, - v, 0, 0)$ with $c > ∣ - v ∣$ ; at $t = T$ both formulas give the point $(c T, v T, 0, 0)$ , so the two pieces join at the corner $r$ and define a two-leg piecewise inertial future timelike worldline from $p = x^{- 1} (0, 0, 0, 0)$ at $t = 0$ to $q = x^{- 1} (2 c T, 0, 0, 0)$ at $t = 2 T$ .

2.1step 1.1L4algebra

The three events $p, r, q$ have coordinate columns $(0, 0, 0, 0)$ , $(c T, v T, 0, 0)$ , $(2 c T, 0, 0, 0)$ ; the affine line through $p$ and $q$ consists of the columns $λ (2 c T, 0, 0, 0)$ with second coordinate $0$ , and $r$ has second coordinate $v T \neq 0$ because $v \neq 0$ and $T > 0$ , so $r$ is not on that line; as affine charts preserve collinearity, $p, r, q$ are not collinear, and since the image of $x_{t r a v e l}$ contains them it is not contained in the straight segment from $p$ to $q$ .

2.2givenstep 1.1A2L5algebra

By the reading rule [A2] applied to the $C^{1}$ history $x_{s t a y}$ , $C_{s t a y} (2 T) - C_{s t a y} (0) = c^{- 1} \int_{0}^{2 T} \sqrt{- η ((c, 0, 0, 0), (c, 0, 0, 0))} d t = c^{- 1} \cdot 2 T \cdot c = 2 T$ , in agreement with the straight-segment value of [L5]. Since its given initial reading is zero, $C_{s t a y} (2 T) = 2 T$ .

2.3givenstep 1.1A3L5L8algebra

By the clock hypothesis [A3], $C_{t r a v e l} (2 T) - C_{t r a v e l} (0) = Δ τ (x_{t r a v e l})$ of Total proper time of a piecewise C^1 future timelike worldline; the two legs have constant tangents $(c, v, 0, 0)$ and $(c, - v, 0, 0)$ with the same interval value $- c^{2} + v^{2} = - c^{2} (1 - v^{2} / c^{2})$ , so by [L5] each leg contributes $c^{- 1} T \sqrt{c^{2} - v^{2}} = T \sqrt{1 - v^{2} / c^{2}} = T / γ$ . Since its given initial reading is zero, the reunion reading is $C_{t r a v e l} (2 T) = 2 T / γ = 2 T \sqrt{1 - v^{2} / c^{2}}$ .

3.1step 2.1step 2.2step 2.3L6L7L8algebra

Comparison: by [L8] $0 < v^{2} / c^{2} < 1$ , so $0 < 1 - v^{2} / c^{2} < 1$ and $\sqrt{1 - v^{2} / c^{2}} < 1$ by strict monotonicity of squaring on the nonnegatives, whence $2 T / γ < 2 T$ and the traveler is behind by $2 T (1 - 1 / γ) = 2 T (1 - \sqrt{1 - v^{2} / c^{2}}) > 0$ ; this is the strict case of the clock-comparison theorem [L6], consistent with step 2.1, since the image of $x_{t r a v e l}$ is not the straight segment.

4.1givenstep 2.2step 2.3step 3.1L5

With the given zero initial settings, both reunion readings equal proper-time integrals of the respective histories; the integrand $\sqrt{- η (x^{'}, x^{'})}$ is built from $η$ and the tangent alone, and [L5] records that the values do not depend on the inertial chart. The comparison uses the common events $p$ and $q$ and their specified clock settings, so no simultaneity convention or preferred inertial frame enters, which is (iii).

5.1step 3.1step 4.1A1A2A3L4L5L6∎

Scope: the instantaneous turn at $r$ idealizes a turnaround of finite duration, the clocks are ideal with no acceleration-dependent reading term, spacetime is flat and $c$ is fixed; the conclusion is a conditional statement about ideal clocks of the adopted model, asserts no measurement outcome and no statistical inference, and claims nothing about real clocks, gravity or quantum effects; no choice principle is used beyond the published suppliers. This is (iv).

The piecewise inertial twin reunion: conditional differential aging

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources