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Proper Time Comparisons

1 · Prerequisites

2 · Summary

This page compares elapsed proper times of worldlines with the same endpoints in the flat classical model. The mathematical layer starts from the reversed Cauchy-Schwarz inequality for same-orientation timelike vectors, with its exact equality case, and its consequence the reversed triangle inequality and the convexity of the future timelike cone, including finite sums (The reversed Cauchy-Schwarz inequality for future timelike vectors, The reversed triangle inequality and convexity of the future timelike cone); it fixes the vocabulary of inertial and piecewise inertial worldlines of the affine model (Inertial and piecewise inertial worldlines of the affine Minkowski model), constructs an inertial frame whose time axis is a prescribed future timelike direction (Existence of an inertial frame whose time axis is a prescribed future timelike direction), and proves that the displacement of any weak piecewise $C^{1}$ future timelike worldline is future timelike (The displacement of a future timelike worldline is future timelike).

The central mathematical theorem is the endpoint bound and the flat maximum: for a weak piecewise $C^{1}$ future timelike worldline between events $p$ and $q$ , $c Δ τ \leq \sqrt{- η (q - p, q - p)}$ , with equality exactly for the strictly increasing reparametrizations of the straight segment from $p$ to $q$ , strict inequality whenever the image leaves that segment, and the straight value attained by every future-directed traversal of the segment (The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space). The regularity class is the weak piecewise one, whose improper piecewise integrals are supplied by the worldlines-and-clocks page; the ordinary Riemann case (H) is included as a special case, and no curvature statement is made here.

The physical layer is conditional on the adopted postulates: an ideal clock carried between the same two events along any future timelike history of the class reads no more than the straight inertial clock, with equality only for the straight segment and strict inequality whenever the image is not the segment (Same-endpoint clock comparison: the inertial clock ages most). The piecewise inertial twin reunion is then computed as a conditional thought experiment: two ideal clocks prepared at one event, compared at their common reunion event, read $2 T$ and $2 T / γ$ (The piecewise inertial twin reunion: conditional differential aging). The final remark delimits the interpretation: acceleration is not an aging term, the two histories are not related by an inertial change of description, and the traveler's momentary-frame bookkeeping is reconciled by the simultaneity gap $2 T v^{2} / c^{2}$ , with no empirical content asserted (Acceleration is not an aging term, and the twin comparison is not symmetric: scope caveats).

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

DefinitionDefinition: AI-adaptedProof: Not applicableprecheck passOpen item page →

Inertial and piecewise inertial worldlines of the affine Minkowski model

Definition

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski space (Oriented time-oriented affine Minkowski space (the spacetime primitives)), let $I$ be an order-convex subset of $R$ with at least two elements (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length), and let curves $x : I \to A$ and their derivatives be as in Curves, regular causal curves and future timelike worldlines in affine Minkowski space: the displacement $x (s + h) - x (s)$ is taken in the affine structure of Affine spaces over a real vector space and displacement vectors, and $x^{'} (s) \in V$ , when it exists, is the limit of the difference quotients, read in the coordinate calculus of The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral in any inertial chart (Inertial frames and inertial charts of the affine Minkowski model, The affine atlas makes the Minkowski model a smooth 4-manifold diffeomorphic to R^4).

(a) Inertial worldlines

A curve $x : I \to A$ is inertial (also called uniform or straight) when there are a parameter value $s_{0} \in I$ , an event $p \in A$ and a vector $d \in V$ , with $d$ future timelike (Time orientation, future and past timelike/null/causal vectors, and the closed cones), such that

$x (s) = p + (s - s_{0}) d for every s \in I .$

The affine action is defined for every real parameter and every vector, so the display defines a map $I \to A$ (Affine spaces over a real vector space and displacement vectors); its image is $x [I] = {p + λ d : λ = s - s_{0}, s \in I}$ , and a restriction of an inertial worldline to a subinterval $J \subseteq I$ with at least two elements is inertial with the same $d$ : for any $t_{0} \in J$ its formula is $x (s) = x (t_{0}) + (s - t_{0}) d$ on $J$ . A singleton restriction is a map but is not a curve or worldline under the cited definition. Its tangent on $I$ is the constant vector $d$ : in the chart $x_{A}$ of an inertial frame the coordinate path is the affine function $s \mapsto x_{A} (p) + (s - s_{0}) E^{- 1} (d)$ , whose difference quotients equal the constant $E^{- 1} (d)$ (A finite list $v : n \to V$ is an ordered basis if and only if every $x \in V$ equals $\sum_{i < n} λ_{i} v_{i}$ for exactly one $λ : n \to F$ ; those scalars are the coordinates of $x$ in that ordered basis), so $x^{'} (s) = d$ at every $s \in I$ by the difference-quotient definition in Curves, regular causal curves and future timelike worldlines in affine Minkowski space; the same computation gives $C^{k}$ regularity for every $k \geq 1$ . Since the constant tangent $d$ is future timelike by hypothesis, an inertial curve is a future timelike worldline of the model (Timelike, null, spacelike and causal vectors of a Lorentz vector space, Curves, regular causal curves and future timelike worldlines in affine Minkowski space (Worldlines)). If two inertial worldlines $x (s) = p + (s - s_{0}) d$ and $y (t) = p^{'} + (t - t_{0}) d^{'}$ have the same image and the same direction of traversal, that is $d^{'} = α d$ with $α > 0$ , then $p^{'}$ lies in the image of $x$ , so $p^{'} = p + μ d$ for some real $μ$ , and

$y (t) = x (s_{0} + μ + α (t - t_{0})) for every t in the domain of y,$

an increasing affine reparametrization. Here an increasing affine reparametrization means composition with a bijection of parameter domains of the form $t \mapsto α t + β$ with $α > 0$ . Conversely, composing an inertial worldline with such a bijection gives an inertial worldline with the same image. This computation uses only the affine axioms and $d \neq 0$ .

(b) Piecewise inertial worldlines

Let $a < b$ be real numbers with $[a, b] \subseteq I$ . A curve $x : [a, b] \to A$ is piecewise inertial when there are a finite partition $a = s_{0} < s_{1} < \dots < s_{m} = b$ , events $p_{1}, \dots, p_{m} \in A$ and future timelike vectors $d_{1}, \dots, d_{m} \in V$ such that

$x (s) = p_{i} + (s - s_{i - 1}) d_{i} for every s \in [s_{i - 1}, s_{i}] and every i = 1, \dots, m,$

and consecutive pieces agree at the partition points, that is $p_{i + 1} = p_{i} + (s_{i} - s_{i - 1}) d_{i} = x (s_{i})$ for $i = 1, \dots, m - 1$ . The joining condition makes the piecewise formulas define a single function on $[a, b]$ , whose tangent on each open piece $(s_{i - 1}, s_{i})$ is the constant $d_{i}$ and whose finite one-sided difference-quotient tangents at an interior partition point $s_{i}$ are $d_{i}$ from the left and $d_{i + 1}$ from the right; at the endpoints $a$ and $b$ the one-sided quotients are $d_{1}$ and $d_{m}$ . Hence a piecewise inertial curve is a piecewise $C^{1}$ future timelike worldline on $[a, b]$ in the weak sense of Curves, regular causal curves and future timelike worldlines in affine Minkowski space — no continuous extension of the derivative to the closed pieces is required, and the tangents are future timelike at every smooth point and on both sides of every corner — and it is $C^{1}$ on $[a, b]$ if and only if $d_{1} = d_{2} = \dots = d_{m}$ . The events $x (s_{i})$ with $0 < i < m$ are the corners of the itinerary, and the itinerary is a two-leg itinerary when $m = 2$ .

(c) Scope

The definition asserts the existence of no worldlines beyond the explicitly parametrized straight and piecewise straight ones, and it attaches no clock reading to any curve: readings of ideal clocks are the subject of the model's postulates, not of this definition. No choice principle is used, since every object is finitely parametrized and every operation is finite-dimensional.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The reversed Cauchy-Schwarz inequality for future timelike vectors

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski space and let $v, w \in V$ be future timelike vectors. Put $a : = \sqrt{- η (v, v)} > 0$ and $b : = \sqrt{- η (w, w)} > 0$ , the timelike lengths of $v$ and $w$ . Then:

(i) $- η (v, w) \geq a b > 0$ .

(ii) Equality $- η (v, w) = a b$ holds if and only if $v$ and $w$ are positive proportional, that is $w = t v$ for some real $t > 0$ , or $v = t w$ for some real $t > 0$ .

(iii) Consequently $- η (v, w) > a b$ whenever $v$ and $w$ are not positive proportional, and $η (v, w) < 0$ in every case.

(iv) For two past timelike vectors $v, w$ the same inequality $- η (v, w) \geq \sqrt{- η (v, v)} \sqrt{- η (w, w)}$ holds, with equality if and only if they are positive proportional, because $- v$ and $- w$ are future timelike and $η (- v, - w) = η (v, w)$ .

The statement uses only $η$ and the future cone $C^{+}$ and involves no chart, frame or choice principle.

Facts & Assumptions

Given: The oriented time-oriented affine Minkowski space $(A, V, η, C^{+})$ and future timelike vectors $v, w \in V$ , with $a : = \sqrt{- η (v, v)} > 0$ and $b : = \sqrt{- η (w, w)} > 0$ .

[F1]

Inertial frames and charts: at every event $o \in A$ there is an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ ; its ordered basis is $η$ -orthonormal with $η (e_{a}, e_{b}) = diag (- 1, 1, 1, 1)_{a b}$ , its first vector $e_{0}$ lies in $C^{+}$ , and $E (u) : = \sum_{a} u^{a} e_{a}$ is a linear isomorphism $E : R^{4} \to V$ , so every vector $u \in V$ has unique coordinates $u = u^{0} e_{0} + u_{s}$ with $u_{s} = u^{1} e_{1} + u^{2} e_{2} + u^{3} e_{3}$ (Existence of future-oriented inertial frames at every event, Inertial frames and inertial charts of the affine Minkowski model).

[F2]

Cone coordinates: in an $η$ -orthonormal basis whose first vector is timelike the timelike set is the disjoint union $T = T_{+} \cup T_{-}$ with $T_{+} = {u : u^{0} > ∣ u_{s} ∣}$ and $T_{-} = {u : u^{0} < - ∣ u_{s} ∣}$ , and $T_{+}$ , $T_{-}$ are exactly the two connected components of $T$ ; the time orientation selects one component $C^{+}$ and $C^{-} : = T ∖ C^{+}$ is the other, so with the basis of [F1], where $e_{0} \in C^{+}$ , one has $C^{+} = T_{+}$ and $C^{-} = T_{-} = - C^{+}$ ; hence $u$ is future timelike if and only if $u^{0} > ∣ u_{s} ∣$ , and $u$ is past timelike if and only if $- u$ is future timelike (The timelike set has exactly two convex components, described in orthonormal coordinates, Time orientation, future and past timelike/null/causal vectors, and the closed cones).

[F3]

Spatial and bilinear structure: for a frame of [F1], the spatial three-space $S : = span (e_{1}, e_{2}, e_{3}) = {z \in V : η (e_{0}, z) = 0}$ carries the restriction $δ : = η ∣_{S}$ , which is positive definite, and $∣ z ∣ : = \sqrt{δ (z, z)}$ is a norm on $S$ with $(e_{1}, e_{2}, e_{3})$ a $δ$ -orthonormal basis; since $η$ is bilinear, for all $u = u^{0} e_{0} + u_{s}$ and $u^{'} = u^{' 0} e_{0} + u_{s}^{'}$ one has $η (u, u^{'}) = - u^{0} u^{' 0} + δ (u_{s}, u_{s}^{'})$ , and in particular $η (u, u) = - (u^{0})^{2} + ∣ u_{s} ∣^{2}$ (The Euclidean spatial structure and cross product of an inertial frame, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space).

[L4]

Finite Cauchy-Schwarz inequality: for reals $a_{0}, \dots, a_{n - 1}$ and $b_{0}, \dots, b_{n - 1}$ one has $(\sum_{k < n} a_{k} b_{k})^{2} \leq (\sum_{k < n} a_{k}^{2}) (\sum_{k < n} b_{k}^{2})$ and $∣ \sum_{k < n} a_{k} b_{k} ∣ \leq \sqrt{\sum_{k < n} a_{k}^{2}} \sqrt{\sum_{k < n} b_{k}^{2}}$ ; equality holds in the first display if and only if the two lists are proportional in the sense that there is a pair $(λ, μ) \neq (0, 0)$ of reals with $λ a_{k} = μ b_{k}$ for every $k < n$ (The Cauchy-Schwarz inequality for finite sums).

[L5]

Square roots and monotonicity of squaring: every real $c \geq 0$ has a unique $\sqrt{c} \geq 0$ with $(\sqrt{c})^{2} = c$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ); for $x, y \geq 0$ one has $x \leq y ⟺ x^{2} \leq y^{2}$ and $x < y ⟺ x^{2} < y^{2}$ (Squaring is monotone on the nonnegatives); for $c > 0$ the root notation is the rational power $c^{1 / 2}$ (Rational powers $a^{r}$ of a positive base).

[L6]

Order arithmetic of $R$ : a product of two positive reals is positive and a product of a positive and a negative real is negative (Sign rules for products and monotonicity of multiplication); inequalities may be added, and a constant may be added to both sides of an inequality (Order is preserved by adding a constant and by adding inequalities); and for reals $x, y$ exactly one of $x < y$ , $x = y$ , $x > y$ holds, a product of reals vanishes only if a factor vanishes, and $x \geq 0$ means $x > 0$ or $x = 0$ (Ordered field).

Proof

technique · direct

1.1givenF1F3choose

Fix an event $o \in A$ and an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with its basis and chart, and write $v = v^{0} e_{0} + v_{s}$ , $w = w^{0} e_{0} + w_{s}$ with spatial parts $v_{s}, w_{s} \in S$ ; the coordinates are unique by [F1] and the spatial parts lie in $S$ by the definition of $S$ in [F3].

2.1step 1.1F2F3L5L6

Since $v$ and $w$ are future timelike, [F2] gives $v^{0} > ∣ v_{s} ∣$ and $w^{0} > ∣ w_{s} ∣$ ; as $∣ v_{s} ∣, ∣ w_{s} ∣ \geq 0$ by [F3], this yields $v^{0} > 0$ , $w^{0} > 0$ , and by strict monotonicity of squaring on the nonnegatives $(v^{0})^{2} > ∣ v_{s} ∣^{2}$ and $(w^{0})^{2} > ∣ w_{s} ∣^{2}$ , so $a^{2} = (v^{0})^{2} - ∣ v_{s} ∣^{2} > 0$ and $b^{2} = (w^{0})^{2} - ∣ w_{s} ∣^{2} > 0$ , whence $a = \sqrt{(v^{0})^{2} - ∣ v_{s} ∣^{2}} > 0$ and $b = \sqrt{(w^{0})^{2} - ∣ w_{s} ∣^{2}} > 0$ by uniqueness of the nonnegative root.

2.2step 1.1F3L4algebra

By bilinearity [F3], $- η (v, w) = v^{0} w^{0} - δ (v_{s}, w_{s})$ ; applying the finite Cauchy-Schwarz inequality [L4] to the coordinates of $v_{s}$ and $w_{s}$ in the $δ$ -orthonormal basis $(e_{1}, e_{2}, e_{3})$ gives $∣ δ (v_{s}, w_{s}) ∣ \leq ∣ v_{s} ∣ ∣ w_{s} ∣$ , hence $δ (v_{s}, w_{s}) \leq ∣ v_{s} ∣ ∣ w_{s} ∣$ and $- η (v, w) \geq x$ for $x : = v^{0} w^{0} - ∣ v_{s} ∣ ∣ w_{s} ∣$ .

3.1step 2.1L6algebra

The algebraic identity $x^{2} - ((v^{0})^{2} - ∣ v_{s} ∣^{2}) ((w^{0})^{2} - ∣ w_{s} ∣^{2}) = (v^{0} ∣ w_{s} ∣ - w^{0} ∣ v_{s} ∣)^{2} \geq 0$ holds by expansion, so $x^{2} \geq a^{2} b^{2}$ ; moreover $x = w^{0} (v^{0} - ∣ v_{s} ∣) + ∣ v_{s} ∣ (w^{0} - ∣ w_{s} ∣)$ is a sum of a positive real and a nonnegative real, so $x > 0$ and in particular $x \geq 0$ .

4.1step 2.1step 2.2step 3.1L5L6

Since $x \geq 0$ and $a b > 0$ with $x^{2} \geq (a b)^{2}$ , monotonicity of squaring on the nonnegatives [L5] gives $x \geq a b$ , and with step 2.2 this yields $- η (v, w) \geq a b > 0$ , which is (i); equivalently $η (v, w) \leq - a b < 0$ .

5.1step 2.2step 3.1step 4.1algebra

Assume now $- η (v, w) = a b$ . Then the chain $a b = - η (v, w) \geq x \geq a b$ of steps 2.2 and 4.1 is equality at both places, so $δ (v_{s}, w_{s}) = ∣ v_{s} ∣ ∣ w_{s} ∣$ , and from $x = a b$ and the identity of step 3.1 we get $(v^{0} ∣ w_{s} ∣ - w^{0} ∣ v_{s} ∣)^{2} = x^{2} - (a b)^{2} = 0$ , that is $v^{0} ∣ w_{s} ∣ = w^{0} ∣ v_{s} ∣$ .

6.1step 2.1step 5.1L6

Case $v_{s} = 0$ : then $v^{0} ∣ w_{s} ∣ = w^{0} ∣ v_{s} ∣ = 0$ by step 5.1, and $v^{0} > 0$ , so $∣ w_{s} ∣ = 0$ and hence $w_{s} = 0$ ; therefore $v = v^{0} e_{0}$ and $w = w^{0} e_{0}$ with $v^{0}, w^{0} > 0$ , and $v = (v^{0} / w^{0}) w$ exhibits $v$ as a positive multiple of $w$ .

6.2step 2.1step 5.1L6

Case $w_{s} = 0$ : symmetrically $w^{0} ∣ v_{s} ∣ = v^{0} ∣ w_{s} ∣ = 0$ with $w^{0} > 0$ gives $v_{s} = 0$ , and $v^{0} > 0$ gives $w = (w^{0} / v^{0}) v$ , so $w$ is a positive multiple of $v$ .

6.3step 5.1L4L5L6algebra

Case $v_{s} \neq 0$ and $w_{s} \neq 0$ : the equality $δ (v_{s}, w_{s}) = ∣ v_{s} ∣ ∣ w_{s} ∣ > 0$ of step 5.1 is the equality case of the finite Cauchy-Schwarz inequality [L4] for two nonzero lists, so there are reals $(λ, μ) \neq (0, 0)$ with $λ v_{s} = μ w_{s}$ ; here $λ \neq 0$ (otherwise $w_{s} = 0$ ) and $μ \neq 0$ (otherwise $v_{s} = 0$ ), so $v_{s} = c w_{s}$ with $c : = μ / λ$ , and then $δ (v_{s}, w_{s}) = c ∣ w_{s} ∣^{2}$ while $∣ v_{s} ∣ = ∣ c ∣ ∣ w_{s} ∣$ , so $c ∣ w_{s} ∣^{2} = ∣ c ∣ ∣ w_{s} ∣^{2}$ with $∣ w_{s} ∣^{2} = δ (w_{s}, w_{s}) > 0$ ; hence $c = ∣ c ∣ \geq 0$ , and $c > 0$ because $c ∣ w_{s} ∣^{2} = ∣ v_{s} ∣ ∣ w_{s} ∣ > 0$ ; finally $v^{0} ∣ w_{s} ∣ = w^{0} ∣ v_{s} ∣ = w^{0} c ∣ w_{s} ∣$ with $∣ w_{s} ∣ > 0$ gives $v^{0} = c w^{0}$ , so $v = v^{0} e_{0} + c w_{s} = c (w^{0} e_{0} + w_{s}) = c w$ with $c > 0$ .

7.1step 6.1step 6.2step 6.3

The three cases of steps 6.1-6.3 exhaust the possibilities, so equality $- η (v, w) = a b$ forces $v$ and $w$ to be positive proportional; this is the forward implication of (ii).

8.1step 2.1F3L5algebra

Conversely, if $v = t w$ with $t > 0$ , then bilinearity gives $- η (v, w) = - t η (w, w) = t b^{2}$ and $- η (v, v) = - t^{2} η (w, w) = t^{2} b^{2}$ , so $a = t b$ by uniqueness of the nonnegative square root, whence $- η (v, w) = t b \cdot b = a b$ ; the case $w = t v$ is identical, and together with step 7.1 this proves (ii).

9.1step 4.1step 7.1step 8.1

For (iii): if $v$ and $w$ are not positive proportional then equality fails by (ii), so $- η (v, w) > a b$ by step 4.1, while $η (v, w) < 0$ holds in every case again by step 4.1.

10.1step 4.1step 7.1step 8.1F2F3∎

For (iv): if $v, w$ are past timelike then $- v, - w$ are future timelike by [F2], with $- η (- v, - v) = - η (v, v)$ , $- η (- w, - w) = - η (w, w)$ and $η (- v, - w) = η (v, w)$ by bilinearity [F3]; steps 4.1, 7.1 and 8.1 applied to $- v, - w$ therefore give $- η (v, w) = - η (- v, - w) \geq a b$ , with equality exactly when $- v, - w$ are positive proportional, that is exactly when $v, w$ are positive proportional.

CorollaryStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The reversed triangle inequality and convexity of the future timelike cone

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski space.

(a) Sum of two future timelike vectors. If $v, w \in V$ are future timelike, then $v + w$ is future timelike and $\sqrt{- η (v + w, v + w)} \geq \sqrt{- η (v, v)} + \sqrt{- η (w, w)},$ with equality if and only if $v$ and $w$ are positive proportional.

(b) Convexity and concavity along segments. If $v, w$ are future timelike, then $t v + (1 - t) w$ is future timelike for every real $t \in [0, 1]$ , so the set of future timelike vectors is convex; moreover $\sqrt{- η (t v + (1 - t) w, t v + (1 - t) w)} \geq t \sqrt{- η (v, v)} + (1 - t) \sqrt{- η (w, w)}$ for every $t \in [0, 1]$ , with equality for $0 < t < 1$ if and only if $v$ and $w$ are positive proportional.

(c) Finite sums. If $k \geq 1$ and $v_{1}, \dots, v_{k} \in V$ are future timelike, then $v_{1} + \dots + v_{k}$ is future timelike and $\sqrt{- η (\sum_{i = 1}^{k} v_{i}, \sum_{i = 1}^{k} v_{i})} \geq \sum_{i = 1}^{k} \sqrt{- η (v_{i}, v_{i})},$ with equality if and only if all $v_{i}$ are positive multiples of one single future timelike vector.

The inequalities are those of the model's $η$ and are independent of any chart; they are the reversed (Lorentzian) forms of the triangle inequality of the spatial Euclidean structure.

Facts & Assumptions

Given: The oriented time-oriented affine Minkowski space $(A, V, η, C^{+})$ .

[F1]

Inertial frames and cone coordinates: at every event there is an inertial frame, and in its coordinates $u = u^{0} e_{0} + u_{s}$ with spatial part $u_{s} \in S = span (e_{1}, e_{2}, e_{3})$ one has $η (u, u^{'}) = - u^{0} u^{' 0} + δ (u_{s}, u_{s}^{'})$ for all $u, u^{'} \in V$ and $u \in C^{+} ⟺ u^{0} > ∣ u_{s} ∣$ (Inertial frames and inertial charts of the affine Minkowski model, The timelike set has exactly two convex components, described in orthonormal coordinates, Time orientation, future and past timelike/null/causal vectors, and the closed cones, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space).

[F2]

The spatial form $δ$ is positive definite on $S$ , $(e_{1}, e_{2}, e_{3})$ is $δ$ -orthonormal, $∣ z ∣ : = \sqrt{δ (z, z)}$ is the induced norm, and for $z, z^{'} \in S$ one has $δ (z, z^{'}) = \sum_{i = 1}^{3} z^{i} z^{' i}$ and $∣ z ∣ = \sqrt{(z^{1})^{2} + (z^{2})^{2} + (z^{3})^{2}}$ ; the norm satisfies $∣ z ∣ \geq 0$ , $∣ z ∣ = 0 ⟺ z = 0$ , $∣ λ z ∣ = ∣ λ ∣ ∣ z ∣$ and the triangle inequality $∣ z + z^{'} ∣ \leq ∣ z ∣ + ∣ z^{'} ∣$ (The Euclidean spatial structure and cross product of an inertial frame, The inner-product norm is definite, homogeneous, and satisfies the triangle inequality).

[F3]

Reversed Cauchy-Schwarz inequality (The reversed Cauchy-Schwarz inequality for future timelike vectors): if $v, w$ are future timelike and $a : = \sqrt{- η (v, v)}$ , $b : = \sqrt{- η (w, w)}$ , then $- η (v, w) \geq a b > 0$ , and equality $- η (v, w) = a b$ holds if and only if $v$ and $w$ are positive proportional, that is $w = t v$ or $v = t w$ for some real $t > 0$ .

[L4]

[L5]

Order arithmetic of $R$ : inequalities may be added and a constant may be added to both sides of an inequality (Order is preserved by adding a constant and by adding inequalities); for $c > 0$ one has $a < b ⟺ a c < b c$ , and a product of two positive reals is positive (Sign rules for products and monotonicity of multiplication); and for reals $x, y$ exactly one of $x < y$ , $x = y$ , $x > y$ holds, while a product vanishes only if a factor vanishes (Ordered field).

[L6]

Induction principle: if a property $P$ satisfies $P (1)$ and $P (k) \Rightarrow P (k + 1)$ for every integer $k \geq 1$ , then $P (k)$ holds for every integer $k \geq 1$ (The principle of mathematical induction, applied to the shift $k \mapsto k - 1$ of the natural-number form).

Proof

technique · direct

1.1givenF1choose

Fix an event $o \in A$ and an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ with its coordinates; by [F1] every vector $u$ is written uniquely as $u = u^{0} e_{0} + u_{s}$ with $u_{s} \in S$ , the pairing is $η (u, u^{'}) = - u^{0} u^{' 0} + δ (u_{s}, u_{s}^{'})$ , and $u \in C^{+} ⟺ u^{0} > ∣ u_{s} ∣$ .

2.1step 1.1F1F2L5

Let $v, w$ be future timelike. Then $(v + w)^{0} = v^{0} + w^{0} > ∣ v_{s} ∣ + ∣ w_{s} ∣ \geq ∣ v_{s} + w_{s} ∣ = ∣ (v + w)_{s} ∣$ : the first inequality adds the two strict inequalities $v^{0} > ∣ v_{s} ∣$ and $w^{0} > ∣ w_{s} ∣$ that [F1] gives from future timelikeness, the second is the triangle inequality of the spatial norm, and the displayed equalities are additivity of coordinates; hence $v + w$ is future timelike.

3.1step 2.1F1F3L5algebra

Let $a : = \sqrt{- η (v, v)}$ , $b : = \sqrt{- η (w, w)}$ and $C : = \sqrt{- η (v + w, v + w)}$ , which is a positive real by step 2.1; bilinearity and symmetry of $η$ give $C^{2} = - η (v + w, v + w) = - η (v, v) - 2 η (v, w) - η (w, w) = a^{2} + b^{2} + 2 (- η (v, w))$ , and $- η (v, w) \geq a b$ by the reversed Cauchy-Schwarz inequality, so multiplying by the positive number $2$ and adding the constant $a^{2} + b^{2}$ yields $C^{2} \geq a^{2} + b^{2} + 2 a b = (a + b)^{2}$ .

3.2step 2.1F2L4L5

For $t \in (0, 1)$ the vectors $t v$ and $(1 - t) w$ are future timelike: $(t v)^{0} = t v^{0} > t ∣ v_{s} ∣ = ∣ t v_{s} ∣$ because $t > 0$ preserves the strict inequality and $∣ t v_{s} ∣ = t ∣ v_{s} ∣$ by the coordinate formula $∣ z ∣ = \sqrt{(z^{1})^{2} + (z^{2})^{2} + (z^{3})^{2}}$ of [F2] together with uniqueness of the nonnegative root, and symmetrically $(1 - t) w^{0} > (1 - t) ∣ w_{s} ∣ = ∣ (1 - t) w_{s} ∣$ ; for $t = 0$ and $t = 1$ the vector $t v + (1 - t) w$ equals $w$ respectively $v$ , which are future timelike.

3.3step 2.1L5algebra

For the base case $k = 1$ of (c): a future timelike vector $v_{1}$ is its own sum, $\sqrt{- η (v_{1}, v_{1})} \geq \sqrt{- η (v_{1}, v_{1})}$ holds with equality, and $v_{1} = 1 \cdot v_{1}$ exhibits $v_{1}$ as a positive multiple of the single future timelike vector $u : = v_{1}$ , so the equality characterization of (c) is satisfied for $k = 1$ .

4.1step 3.1L4

Since $C \geq 0$ and $a + b \geq 0$ with $C^{2} \geq (a + b)^{2}$ , monotonicity of squaring on the nonnegatives [L4] gives $C \geq a + b$ : this is the inequality of (a).

4.2step 3.2F1F3L4algebra

For $t \in (0, 1)$ , (a) applied to the future timelike vectors $t v$ and $(1 - t) w$ of step 3.2 gives $t v + (1 - t) w$ future timelike and $\sqrt{- η (t v + (1 - t) w, t v + (1 - t) w)} \geq \sqrt{- η (t v, t v)} + \sqrt{- η ((1 - t) w, (1 - t) w)} = t \sqrt{- η (v, v)} + (1 - t) \sqrt{- η (w, w)}$ , the two displayed equalities using bilinearity and uniqueness of the nonnegative root; this is the inequality of (b) on the open interval.

5.1step 4.1F3L4L5algebra

Equality holds in (a) if and only if $C = a + b$ , which by monotonicity of squaring on the nonnegatives [L4] is equivalent to $C^{2} = (a + b)^{2}$ , hence, subtracting the common term $a^{2} + b^{2}$ from the two displays of step 3.1 and multiplying by $1 / 2$ , to $- η (v, w) = a b$ , and by the equality case of the reversed Cauchy-Schwarz inequality [F3] to $v$ and $w$ being positive proportional; this proves (a).

5.2step 4.1L5algebra

Induction step, membership and inequality: assume $k \geq 1$ and that the claim of (c) holds for the integer $k$ ; let $v_{1}, \dots, v_{k + 1}$ be future timelike and put $s : = v_{1} + \dots + v_{k}$ , which is future timelike by the induction hypothesis; applying (a) of steps 2.1 and 4.1 to the future timelike vectors $s$ and $v_{k + 1}$ gives $s + v_{k + 1} = v_{1} + \dots + v_{k + 1}$ future timelike and $\sqrt{- η (s + v_{k + 1}, s + v_{k + 1})} \geq \sqrt{- η (s, s)} + \sqrt{- η (v_{k + 1}, v_{k + 1})} \geq \sum_{i = 1}^{k} \sqrt{- η (v_{i}, v_{i})} + \sqrt{- η (v_{k + 1}, v_{k + 1})}$ , the second inequality being the induction hypothesis.

6.1step 4.2step 5.1L5

Equality in the interior inequality of (b) for $0 < t < 1$ holds if and only if $t v$ and $(1 - t) w$ are positive proportional, by the equality case of (a) established in step 5.1 applied to the interior inequality of (b) of step 4.2; since $t > 0$ and $1 - t > 0$ , this is equivalent to $v$ and $w$ being positive proportional; at $t = 0$ and $t = 1$ the inequality reduces to an identity $b \geq b$ or $a \geq a$ , and the statement makes no interior equality claim there, so (b) is proved.

6.2step 5.2step 5.1F1L4L5algebra

Induction step, equality characterization: with the notation of step 5.2, if equality holds for the $k + 1$ vectors, then both inequalities in the chain of step 5.2 are equalities; the second equality means that the equality case of the induction hypothesis applies to $v_{1}, \dots, v_{k}$ , so there are a future timelike $u$ and reals $c_{1}, \dots, c_{k} > 0$ with $v_{i} = c_{i} u$ and $s = (\sum_{i = 1}^{k} c_{i}) u$ ; the first equality is the equality case of (a) for $s$ and $v_{k + 1}$ , so $s$ and $v_{k + 1}$ are positive proportional, and since $s \neq 0$ this gives $v_{k + 1} = c_{k + 1} u$ for some real $c_{k + 1} > 0$ , so all $k + 1$ vectors are positive multiples of the single future timelike vector $u$ ; conversely, if $v_{i} = c_{i} u$ with $c_{i} > 0$ and $u$ future timelike for $i = 1, \dots, k + 1$ , then $v_{1} + \dots + v_{k + 1} = (\sum_{i} c_{i}) u$ and bilinearity with uniqueness of the nonnegative root give $\sqrt{- η (\sum_{i} v_{i}, \sum_{i} v_{i})} = (\sum_{i} c_{i}) \sqrt{- η (u, u)} = \sum_{i} \sqrt{- η (v_{i}, v_{i})}$ , so equality holds.

7.1step 3.3step 5.2step 6.2L6∎

By [L6], applied to the property " $v_{1}, \dots, v_{k}$ future timelike implies the conclusions of (c) for these vectors", the base case of step 3.3 and the induction steps 5.2 and 6.2 prove (c) for every integer $k \geq 1$ : membership and inequality from step 5.2, and the equality characterization from step 6.2.

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

Existence of an inertial frame whose time axis is a prescribed future timelike direction

Statement

Let $(A, V, η, C^{+})$ be the oriented time-oriented affine Minkowski space, let $d \in V$ be a future timelike displacement and put $T : = \sqrt{- η (d, d)} > 0$ and $n : = d / T$ . Then:

(i) $η (n, n) = - 1$ and $n$ is future timelike, so $n$ is a future unit timelike vector.

(ii) For every event $o$ there is an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ of the model with $e_{0} = n$ .

(iii) For the inertial chart $x$ of any such frame one has $x (o + λ d) = (λ T, 0, 0, 0)$ for every real $λ$ ; in particular $x (o) = 0$ and $x (o + d) = (T, 0, 0, 0)$ . Consequently, for any two events $p, q$ with $q - p = d$ , the chart of such a frame represents the displacement $q - p$ by the coordinate vector $(T, 0, 0, 0)$ : the spatial coordinates of $p$ and $q$ agree and their time coordinates differ by $T$ .

Facts & Assumptions

Given: The oriented time-oriented affine Minkowski space $(A, V, η, C^{+})$ and a future timelike displacement $d \in V$ , with $T : = \sqrt{- η (d, d)} > 0$ and $n : = d / T$ .

[F1]

Inertial frames and charts: for every event $o$ and every $η$ -orthonormal ordered basis $(f_{0}, f_{1}, f_{2}, f_{3})$ of $V$ with future first vector and positive orientation, $(o; f_{0}, \dots, f_{3})$ is an inertial frame; its chart is $x (r) = E^{- 1} (r - o)$ with $E (u) = \sum_{a} u^{a} e_{a}$ the frame's linear isomorphism, and every vector $u$ has unique coordinates $u = u^{0} e_{0} + u_{s}$ , $u_{s} \in S = span (e_{1}, e_{2}, e_{3})$ (Inertial frames and inertial charts of the affine Minkowski model, Existence of future-oriented inertial frames at every event).

[F2]

Cone coordinates: in an inertial frame with first vector in $C^{+}$ one has $C^{+} = {u : u^{0} > ∣ u_{s} ∣}$ and $C^{-} = {u : u^{0} < - ∣ u_{s} ∣}$ , and $η (u, u) = - (u^{0})^{2} + ∣ u_{s} ∣^{2}$ ; the restriction of $η$ to $S$ is positive definite, $(e_{1}, e_{2}, e_{3})$ is orthonormal for it, and $∣ z ∣^{2} = δ (z, z)$ for $z \in S$ (The timelike set has exactly two convex components, described in orthonormal coordinates, Time orientation, future and past timelike/null/causal vectors, and the closed cones, The Euclidean spatial structure and cross product of an inertial frame).

[F3]

The passive boost $B (β)$ is defined for every parameter $β \in R^{3}$ with $∣ β ∣ < 1$ (The passive Lorentz boost with parameter beta); it is a proper orthochronous Lorentz map with inverse $B (- β)$ (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta), Linear and affine Lorentz maps, and the Lorentz group); and if $u$ is future timelike with $η (u, u) = - 1$ and coordinates $(u^{0}, u_{s})$ , then the parameter $β : = u_{s} / u^{0}$ satisfies $∣ β ∣ < 1$ and $B (β) u = f_{0}$ , where $f_{0}$ is the first vector of the frame used for the coordinates (The passive boost is a proper orthochronous Lorentz map with inverse B(-beta) (iv)).

[F4]

Lorentz maps preserve the metric and the orientation class: for a linear Lorentz map $L$ one has $η (L v, L w) = η (v, w)$ for all $v, w$ ; the composition and the inverse of Lorentz maps are Lorentz maps, and the proper orthochronous ones form a group; if $L$ is proper, then the oriented volume form $ϵ$ of The oriented volume form epsilon and the permutation symbols of an inertial frame is preserved, $ϵ (L v_{0}, \dots, L v_{3}) = ϵ (v_{0}, \dots, v_{3})$ ; and $ϵ$ takes the value $1$ on every positively oriented $η$ -orthonormal basis, so a tuple of four vectors with $ϵ$ -value $1$ is positively oriented (Lorentz maps are isomorphisms of determinant ±1 that preserve causal character and permute the cones; the Lorentz sets are groups, The oriented volume form is independent of the positively oriented inertial frame).

[L5]

Square roots and monotonicity: every real $c \geq 0$ has a unique $\sqrt{c} \geq 0$ with $(\sqrt{c})^{2} = c$ (Square roots exist: a unique $\sqrt{a} \geq 0$ with $(\sqrt{a})^{2} = a$ ; the positives are ${x^{2} : x \neq 0}$ ); for $x, y \geq 0$ one has $x < y ⟺ x^{2} < y^{2}$ (Squaring is monotone on the nonnegatives).

[L6]

Order arithmetic of $R$ : multiplication of an inequality by a positive real preserves it, and a product of positives is positive (Sign rules for products and monotonicity of multiplication); a constant may be added to both sides and inequalities may be added (Order is preserved by adding a constant and by adding inequalities); and the order is trichotomous and transitive, with $1 > 0$ (Ordered field).

Proof

technique · direct

1.1givenF1L5L6algebra

By bilinearity of $η$ and $T^{2} = - η (d, d)$ one has $η (n, n) = η (d, d) / T^{2} = (- T^{2}) / T^{2} = - 1$ , so $n$ is timelike and $η (n, n) = - 1$ ; moreover $d$ is nonzero because it is timelike, so $n = d / T$ is nonzero.

2.1step 1.1givenF1F2L5L6

Fix an event $o$ and an inertial frame $(o; f_{0}, f_{1}, f_{2}, f_{3})$ with chart $y$ , and write $n = n^{0} f_{0} + n_{s}$ in its coordinates; since $n$ is timelike, [F2] puts it in $C^{+}$ or $C^{-}$ , and since $n = d / T$ with $d^{0} > ∣ d_{s} ∣ \geq 0$ and $T > 0$ its first coordinate satisfies $n^{0} = d^{0} / T > 0$ , so $n$ lies in $C^{+} = {u : u^{0} > ∣ u_{s} ∣}$ and is future timelike; in particular $n^{0} > ∣ n_{s} ∣ \geq 0$ .

3.1step 2.1F2F3L5L6algebra

From $1 = - η (n, n) = (n^{0})^{2} - ∣ n_{s} ∣^{2}$ we get $(n^{0})^{2} = 1 + ∣ n_{s} ∣^{2} > ∣ n_{s} ∣^{2}$ ; setting $β : = n_{s} / n^{0}$ (the coordinate column, identified with the vector $β^{1} f_{1} + β^{2} f_{2} + β^{3} f_{3}$ of $S$ ) one has $∣ β ∣^{2} = δ (β, β) = ∣ n_{s} ∣^{2} / (n^{0})^{2} < 1$ , hence $∣ β ∣ < 1$ by strict monotonicity of squaring on the nonnegatives and $∣ β ∣ \geq 0$ ; thus the passive boost $B (β)$ is defined.

4.1step 3.1F3

By [F3] (iv), applied to the future timelike unit vector $n$ with coordinates $(n^{0}, n_{s})$ , the boost satisfies $B (β) n = f_{0}$ ; and $B (β)$ is proper orthochronous with inverse $B (β)^{- 1} = B (- β)$ .

5.1step 4.1F1F4

Define $e_{0} : = n = B (β)^{- 1} f_{0}$ and $e_{i} : = B (β)^{- 1} f_{i}$ for $i = 1, 2, 3$ ; then $e_{a} = B (β)^{- 1} f_{a}$ for all $a$ , so $η (e_{a}, e_{b}) = η (f_{a}, f_{b}) = diag (- 1, 1, 1, 1)_{a b}$ because a Lorentz map preserves $η$ , and the ordered basis $(e_{0}, e_{1}, e_{2}, e_{3})$ is $η$ -orthonormal with future first vector $e_{0} = n$ ; it is positively oriented because $B (β)^{- 1}$ is proper and preserves the oriented volume form, so $ϵ (e_{0}, \dots, e_{3}) = ϵ (f_{0}, \dots, f_{3}) = 1$ for the positively oriented basis $(f_{a})$ , whence the basis $(e_{a})$ is positively oriented. Therefore $(o; e_{0}, e_{1}, e_{2}, e_{3})$ is an inertial frame with $e_{0} = n$ , which is (ii).

6.1step 5.1F1algebra

For the chart $x = E^{- 1} (\cdot - o)$ of any inertial frame $(o; e_{0}, \dots, e_{3})$ with $e_{0} = n$ one has $E (1, 0, 0, 0) = e_{0} = n$ and $d = T n$ , so $x (o + λ d) = E^{- 1} (λ d) = λ T E^{- 1} (n) = λ T (1, 0, 0, 0) = (λ T, 0, 0, 0)$ for every real $λ$ , since $E^{- 1}$ is linear; in particular $x (o) = 0$ and $x (o + d) = (T, 0, 0, 0)$ , which is (iii).

7.1step 6.1F1algebra∎

Consequently, for events $p, q$ with $q - p = d$ , linearity of $E^{- 1}$ gives $x (q) - x (p) = E^{- 1} (q - p) = (T, 0, 0, 0)$ , so the spatial coordinates of $p$ and $q$ agree and their time coordinates differ by $T$ ; together with steps 1.1, 2.1, 5.1 and 6.1 this proves (i), (ii) and (iii).

LemmaStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The displacement of a future timelike worldline is future timelike

Statement

Let $c > 0$ and let $x : [a, b] \to A$ be a future timelike worldline with $a < b$ (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length) that is piecewise $C^{1}$ with partition $a = s_{0} < s_{1} < \dots < s_{m} = b$ in the weak sense of Curves, regular causal curves and future timelike worldlines in affine Minkowski space, with future timelike interior tangents and finite one-sided difference-quotient tangents. Put $p : = x (a)$ , $q : = x (b)$ and $d : = q - p$ . Then $d$ is future timelike. More precisely, writing $ξ^{a} : = x_{A}^{a} \circ x$ for the coordinate path of the chart $x_{A}$ of any inertial frame, one has $ξ^{0} (b) - ξ^{0} (a) > 0 and ∣ ξ_{v e c} (b) - ξ_{v e c} (a) ∣ < ξ^{0} (b) - ξ^{0} (a),$ where $∣ \cdot ∣$ is the Euclidean spatial norm of The Euclidean spatial structure and cross product of an inertial frame. Consequently $p \neq q$ , the number $\sqrt{- η (d, d)}$ is a positive real number, and in every inertial chart the time component of $d$ is positive and dominates the Euclidean norm of its spatial part, which is exactly the coordinate description of future timelikeness in that chart. Completeness of the real numbers is inherited from the improper-integrability supplier; no compactness assumption beyond the given compact interval or choice principle is used.

Facts & Assumptions

Given: A real $c > 0$ ; an inertial frame with chart $x_{A}$ and coordinate path $y : = ξ = (y^{0}, z)$ of a piecewise $C^{1}$ future timelike worldline $x : [a, b] \to A$ , $a < b$ , with partition $a = s_{0} < \dots < s_{m} = b$ , future timelike interior tangents and finite one-sided difference-quotient tangents (Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length, Oriented time-oriented affine Minkowski space (the spacetime primitives)).

[F1]

The weak piecewise class (Curves, regular causal curves and future timelike worldlines in affine Minkowski space): $x$ is $C^{1}$ on every open piece $(s_{i - 1}, s_{i})$ with future timelike tangent $x^{'}$ there, and the finite one-sided difference-quotient derivatives at both ends of every piece exist; no continuous extension of the open-piece derivative to the closed piece is assumed.

[F2]

Improper integrability and endpoint increments (Improper integrability and endpoint increments of weak piecewise future causal curves): for the piece $(α, β) = (s_{i - 1}, s_{i})$ with the notation $\int_{(α, β)} g = \lim_{u ↓ α} \int_{u}^{r} g + \lim_{v ↑ β} \int_{r}^{v} g$ of that lemma, one has $y^{0}^{'} \geq ∣ z^{'} ∣ \geq 0$ and $y^{0}^{'} > 0$ on $(α, β)$ ; every coordinate derivative is absolutely improperly Riemann integrable with signed improper integral its endpoint increment, so $\int_{(α, β)} y^{j}^{'} = y^{j} (β) - y^{j} (α)$ in the oriented convention of The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ; the improper values are additive under subdivision; and $D = x (b) - x (a)$ is nonzero future causal, with $D^{0} > 0$ and $∣ D_{s p a t i a l} ∣ \leq D^{0}$ in every inertial chart.

[F3]

Chart metric and cone coordinates (Inertial frames and inertial charts of the affine Minkowski model, The timelike set has exactly two convex components, described in orthonormal coordinates, The Euclidean spatial structure and cross product of an inertial frame, Lorentz forms of inertia (3,1,0) on a 4-dimensional real vector space): in the chart of an inertial frame (first basis vector future) a vector $v$ with coordinates $(v^{0}, v_{s})$ is future timelike exactly when $v^{0} > ∣ v_{s} ∣$ , and $- η (v, v) = (v^{0})^{2} - ∣ v_{s} ∣^{2}$ ; $η$ is bilinear.

[L4]

Norm inequalities (The inner-product norm is definite, homogeneous, and satisfies the triangle inequality, For $a \leq b$ and $f : [a, b] \to R^{m}$ integrable when $a < b$ , $∥ \int_{a}^{b} f ∥_{2} \leq \int_{a}^{b} ∥ f ∥_{2}$ ; for $a < b$ , $∥ f ∥_{2}$ is integrable, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral): the spatial Euclidean norm satisfies the triangle inequality, finite sums included; and for vector-valued functions on an interval, $∥ \int_{c}^{d} f ∥ \leq \int_{c}^{d} ∥ f ∥$ whenever $f$ is Riemann integrable.

[L5]

Monotonicity and positivity of the Riemann integral (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , A continuous $f \geq 0$ on $[a, b]$ with $\int_{a}^{b} f = 0$ is identically $0$ , The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ): if $g_{1} \leq g_{2}$ pointwise on $[c, d]$ then $\int_{c}^{d} g_{1} \leq \int_{c}^{d} g_{2}$ ; and if a nonnegative continuous real function on $[c, d]$ has positive value at some point, then its integral over $[c, d]$ is positive.

[L6]

Order arithmetic of $R$ (Ordered field): the order is transitive, and a finite sum of positive reals is positive.

[L7]

Integral linearity (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ): for integrable real functions on a compact truncation, $\int (f - g) = \int f - \int g$ . If the two improper integrals are finite, passing to their one-sided limits proves the same identity for their difference.

Proof

technique · direct

1.1givenF1F3

Fix the inertial frame and write $y = ξ = x_{A} \circ x = (y^{0}, z)$ with $z = (y^{1}, y^{2}, y^{3})$ ; the partition has $m \geq 1$ pieces, and on each open piece $(s_{i - 1}, s_{i})$ the tangent $x^{'}$ is future timelike, so by [F3] $y^{0}^{'} (u) > ∣ z^{'} (u) ∣ \geq 0$ at every $u \in (s_{i - 1}, s_{i})$ .

2.1step 1.1F1F2

The hypotheses of [F2] are satisfied: $x$ is $C^{1}$ on the open pieces, the one-sided difference-quotient derivatives are finite, and the interior tangents are future timelike, hence in particular nonzero future causal; therefore clauses (a)-(d) of [F2] apply to each piece $(α, β) = (s_{i - 1}, s_{i})$ .

3.1step 2.1F2L5L7

Per piece: the difference $g_{i} : = y^{0}^{'} - ∣ z^{'} ∣$ is continuous and positive on $(α, β)$ by step 1.1, so for any $α < u < v < β$ its integral over $[u, v]$ is positive by [L5]; the improper integral $\int_{(α, β)} g_{i}$ dominates that truncated integral, and the additive decomposition of [F2] gives $\int_{(α, β)} g_{i} = \int_{(α, β)} y^{0}^{'} - \int_{(α, β)} ∣ z^{'} ∣$ , so $\int_{(α, β)} ∣ z^{'} ∣ < \int_{(α, β)} y^{0}^{'} = y^{0} (β) - y^{0} (α)$ with the last equality from [F2]; in particular $y^{0} (β) - y^{0} (α) > 0$ .

3.2step 2.1F2L4

Per piece the spatial increment satisfies $∣ z (β) - z (α) ∣ = ∥ \int_{(α, β)} z^{'} ∥ \leq \int_{(α, β)} ∣ z^{'} ∣$ : for $α < u < v < β$ the vector norm inequality of [L4] gives $∥ \int_{u}^{v} z^{'} ∥ \leq \int_{u}^{v} ∣ z^{'} ∣$ , and letting $u ↓ α$ , $v ↑ β$ with the endpoint-increment limits of [F2] and continuity of the norm yields the claim.

4.1step 3.2F2L4L6

Summing the finitely many pieces, with $Δ_{i} z : = z (s_{i}) - z (s_{i - 1})$ : $∣ z (b) - z (a) ∣ \leq \sum_{i = 1}^{m} ∣ Δ_{i} z ∣ \leq \sum_{i = 1}^{m} \int_{(s_{i - 1}, s_{i})} ∣ z^{'} ∣ < \sum_{i = 1}^{m} \int_{(s_{i - 1}, s_{i})} y^{0}^{'} = y^{0} (b) - y^{0} (a)$ , where the first inequality is the finite triangle inequality [L4], the second is step 3.2, the strict inequality is step 3.1 together with positivity of the finite sum of the positive differences $\int_{(s_{i - 1}, s_{i})} (y^{0}^{'} - ∣ z^{'} ∣)$ [L6], and the final equality is additivity over the partition from [F2]; in particular $y^{0} (b) - y^{0} (a) > 0$ and $∣ z (b) - z (a) ∣ < y^{0} (b) - y^{0} (a)$ .

5.1step 4.1F3L4

Consequently, in this chart the displacement $d = q - p$ has coordinates $(y^{0} (b) - y^{0} (a), z (b) - z (a))$ with positive time component strictly dominating the spatial norm, so $d$ is future timelike by the coordinate criterion [F3]; since the inertial chart was arbitrary, the same conclusion holds in every inertial chart. Hence $p \neq q$ ( $d \neq 0$ because it is timelike), and by bilinearity [F3] one has $- η (d, d) = (y^{0} (b) - y^{0} (a))^{2} - ∣ z (b) - z (a) ∣^{2} > 0$ , so $\sqrt{- η (d, d)}$ is a positive real number, and the time component of $d$ dominates in every inertial chart — the coordinate description of future timelikeness.

6.1step 5.1∎

The argument uses only the weak piecewise class on the compact interval $[a, b]$ , truncated Riemann integrals and their one-sided limits, and the finite sum over the $m$ pieces: completeness of $R$ is used by [F2] to obtain finite improper limits from monotone bounded truncation integrals, no compactness assumption beyond the interval $[a, b]$ is made, and no choice principle is used.

TheoremStatement: AI-adaptedProof: AI-adaptedprecheck passOpen item page →

The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space

Statement

(i) Endpoint bound. $c Δ τ (x) \leq T$ .

(ii) Equality case. $c Δ τ (x) = T$ if and only if $x$ is a strictly increasing reparametrization of the straight segment from $p$ to $q$ : there is a continuous strictly increasing surjection $σ : [a, b] \to [0, 1]$ with $σ (a) = 0$ , $σ (b) = 1$ , $C^{1}$ on every open piece of a finite weak piecewise partition of $[a, b]$ , with $σ^{'} (s) > 0$ at every point of each open piece and finite positive one-sided difference-quotient derivatives at both endpoints of every piece, without any requirement that $σ^{'}$ extend continuously there, such that $x (s) = p + σ (s) (q - p)$ for every $s \in [a, b]$ ; equivalently, every open-piece tangent $x^{'} (s)$ is a positive multiple of $q - p$ .

(iii) Strictness. If the image of $x$ is not contained in the straight segment ${p + λ (q - p) : 0 \leq λ \leq 1}$ , then $c Δ τ (x) < T$ . In particular, if $x$ is piecewise inertial and its piece tangents $d_{1}, \dots, d_{m}$ are not all positive multiples of one another (equivalently, its image is not contained in the straight segment), then the inequality is strict, while a piecewise inertial itinerary whose piece tangents are all positive multiples of one another has the straight segment as image and attains the bound.

(iv) Straight worldlines attain the bound. For the worldline $x_{s} (s) : = p + \frac{s - a}{b - a} (q - p)$ one has $c Δ τ (x_{s}) = T$ ; more generally every worldline whose image is the straight segment and which is traversed in the future direction attains the bound.

(v) Scope. The theorem concerns the affine Minkowski model with a fixed positive constant $c$ ; it is a statement about the weak piecewise $C^{1}$ future timelike class, including the ordinary Riemann special case (H); it does not identify proper time with any clock reading, and it asserts nothing about maximality of geodesics in curved spacetimes or in Lorentzian manifolds with different global structure.

Facts & Assumptions

Given: A real $c > 0$ ; a weak piecewise $C^{1}$ future timelike worldline $x : [a, b] \to A$ , $a < b$ , with partition $a = s_{0} < \dots < s_{m} = b$ ; $p = x (a)$ , $q = x (b)$ , $d = q - p$ , $T = \sqrt{- η (d, d)}$ .

[F1]

Endpoint displacement (The displacement of a future timelike worldline is future timelike): $d = q - p$ is future timelike, so $T > 0$ , and in every inertial chart the time component of $d$ is positive and dominates the norm of its spatial part.

[F2]

Rest frame of the displacement (Existence of an inertial frame whose time axis is a prescribed future timelike direction, Inertial frames and inertial charts of the affine Minkowski model): $n : = d / T$ is future unit timelike, for every event $o$ there is an inertial frame with first vector $n$ , and for the chart $ξ$ of any such frame $ξ (o + λ d) = (λ T, 0, 0, 0)$ for every real $λ$ ; in particular with $o : = p$ one has $ξ (p) = (0, 0, 0, 0)$ and $ξ (q) = (T, 0, 0, 0)$ .

[F3]

Weak piecewise structure and proper time (Curves, regular causal curves and future timelike worldlines in affine Minkowski space, Improper integrability and endpoint increments of weak piecewise future causal curves, Total proper time of a piecewise C^1 future timelike worldline, The derivative and the Riemann integral of a vector-valued function: an intrinsic derivative and a componentwise integral, The integral with oriented limits: $\int_{a}^{a} f : = 0$ and $\int_{b}^{a} f : = - \int_{a}^{b} f$ ): $x$ is $C^{1}$ on each open piece with future timelike interior tangent and has finite one-sided difference-quotient tangents; on a piece $(α, β)$ with coordinate path $y = (y^{0}, z)$ one has $y^{0}^{'} > ∣ z^{'} ∣ \geq 0$ , the density $g = \sqrt{(y^{0}^{'})^{2} - ∣ z^{'} ∣^{2}} = \sqrt{- η (x^{'}, x^{'})}$ has a finite positive improper integral $\int_{(α, β)} g$ in the two-sided convention, the signed improper integral of $y^{0}^{'}$ over the piece equals $y^{0} (β) - y^{0} (α)$ , improper values are additive under subdivision, and the finite piece sums are independent of the admissible partition; $Δ τ (x) = c^{- 1} \sum_{i} \int_{(s_{i - 1}, s_{i})} g$ , the based proper-time function is strictly increasing, and under the corner hypothesis (H) the improper values coincide with the ordinary Riemann integrals, agreeing with Proper time along a future timelike worldline in the globally $C^{1}$ case.

[F4]

Cone coordinates and metric (Inertial frames and inertial charts of the affine Minkowski model, The timelike set has exactly two convex components, described in orthonormal coordinates, The Euclidean spatial structure and cross product of an inertial frame, Timelike, null, spacelike and causal vectors of a Lorentz vector space, Time orientation, future and past timelike/null/causal vectors, and the closed cones): in an inertial chart a vector with coordinates $(v^{0}, v_{s})$ is future timelike exactly when $v^{0} > ∣ v_{s} ∣$ , and $- η (v, v) = (v^{0})^{2} - ∣ v_{s} ∣^{2}$ ; a constant-tangent worldline on $[a, b]$ with tangent the future timelike $w$ has total proper time $c^{- 1} (b - a) \sqrt{- η (w, w)}$ .

[F5]

Piecewise inertial histories (Inertial and piecewise inertial worldlines of the affine Minkowski model): on each closed piece of a finite partition the curve is affine with a constant future timelike tangent, and consecutive pieces share their corner events.

[L5]

Square roots and monotonicity (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): every real $c \geq 0$ has a unique $\sqrt{c} \geq 0$ with $(\sqrt{c})^{2} = c$ ; for $x, y \geq 0$ one has $x \leq y ⟺ x^{2} \leq y^{2}$ .

[L6]

Monotonicity and positivity of the Riemann integral (If $f \leq g$ on $[a, b]$ and both are integrable then $\int_{a}^{b} f \leq \int_{a}^{b} g$ ; and $m (b - a) \leq \int_{a}^{b} f \leq M (b - a)$ , A continuous $f \geq 0$ on $[a, b]$ with $\int_{a}^{b} f = 0$ is identically $0$ ): $g_{1} \leq g_{2}$ pointwise on $[u, v]$ implies $\int_{u}^{v} g_{1} \leq \int_{u}^{v} g_{2}$ , and a nonnegative continuous function on $[u, v]$ that is positive somewhere has positive integral.

[L7]

Zero derivative gives a constant (A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant): a continuous real function on an order-convex interval whose derivative vanishes at every interior point is constant.

[L8]

Order arithmetic (Ordered field, Intervals of $R$ : the nine order-convex forms, nondegeneracy, and length): the order is transitive, a finite sum of nonnegative reals is nonnegative and vanishes only if every term vanishes, and a finite sum of positive reals is positive.

[L9]

Intermediate value theorem (Intermediate value theorem, by bisection with a canonical left-half rule: a continuous function on $[a, b]$ takes every value between $f (a)$ and $f (b)$ ): a continuous real function on $[a, b]$ assumes every value between its endpoint values.

[L10]

Integral linearity (Integrable functions on $[a, b]$ form a set closed under sums and scalar multiples, and $\int_{a}^{b} (λ f + μ g) = λ \int_{a}^{b} f + μ \int_{a}^{b} g$ ): on every compact truncation $\int (f - g) = \int f - \int g$ . When the two improper values are finite, their one-sided limits give the same identity for the improper integral of their difference.

Proof

technique · direct

1.1givenF1F2choose

By [F1] $d$ is future timelike and $T > 0$ ; put $n : = d / T$ and fix, by [F2], the inertial frame with origin $p$ and first vector $n$ , with chart $ξ$ and coordinate path $y = ξ \circ x = (y^{0}, z)$ ; then $ξ (p) = (0, 0, 0, 0)$ and $ξ (q) = (T, 0, 0, 0)$ , so $y (a) = (0, 0, 0, 0)$ and $y (b) = (T, 0, 0, 0)$ .

2.1step 1.1F3F4L5

On every open piece $(s_{i - 1}, s_{i})$ the tangent is future timelike, so [F3] and [F4] give $y^{0}^{'} > ∣ z^{'} ∣ \geq 0$ ; the density is $g_{i} = \sqrt{(y^{0}^{'})^{2} - ∣ z^{'} ∣^{2}}$ with finite positive improper integral, and setting $f_{i} : = g_{i}$ and $h_{i} : = y^{0}^{'} - f_{i}$ one has $h_{i} \geq 0$ and $f_{i} \leq y^{0}^{'}$ pointwise on the piece, because $0 \leq (y^{0}^{'})^{2} - ∣ z^{'} ∣^{2} \leq (y^{0}^{'})^{2}$ and $f_{i}, y^{0}^{'} \geq 0$ with monotonicity of the square root [L5].

3.1step 2.1F3L6

Endpoint bound: $c Δ τ (x) = \sum_{i = 1}^{m} \int_{(s_{i - 1}, s_{i})} f_{i} \leq \sum_{i = 1}^{m} \int_{(s_{i - 1}, s_{i})} y^{0}^{'} = y^{0} (b) - y^{0} (a) = T$ , where the first equality is the definition of $Δ τ$ in [F3], the inequality is termwise monotonicity of the improper integrals from step 2.1 together with positivity of each term, the middle equality combines the endpoint increments of [F3] for $y^{0}^{'}$ with additivity under the partition, and the last equality is $y^{0} (b) - y^{0} (a) = ξ^{0} (q) - ξ^{0} (p) = T$ from step 1.1; this is (i).

4.1step 2.1step 3.1L6L8L10

Assume now $c Δ τ (x) = T$ . Then by step 3.1 and additivity, $\sum_{i = 1}^{m} \int_{(s_{i - 1}, s_{i})} h_{i} = 0$ with every term $\geq 0$ by step 2.1, so by [L8] every term vanishes; since each $h_{i}$ is continuous and nonnegative on the open piece, $h_{i} \equiv 0$ there: if $h_{i} (u_{0}) > 0$ at some point, continuity gives $h_{i} \geq h_{i} (u_{0}) / 2 > 0$ on a compact interval around $u_{0}$ inside the piece, whose ordinary integral is then positive by [L6], and the improper integral dominates it, a contradiction.

5.1step 4.1F3L7algebra

Consequently $f_{i}^{2} = (y^{0}^{'})^{2}$ on each piece, so $∣ z^{'} ∣^{2} = (y^{0}^{'})^{2} - f_{i}^{2} = 0$ and hence $z^{'} = 0$ there; each coordinate of $z$ is continuous on the closed piece with vanishing derivative on its interior, so by [L7] each coordinate is constant on the closed piece, and since consecutive pieces share their corner values, $z \equiv z_{0}$ for a single $z_{0} \in R^{3}$ on all of $[a, b]$ .

6.1step 5.1F2algebra

Hence for every $s \in [a, b]$ the coordinates of $x (s)$ are $(y^{0} (s), z_{0})$ ; by the inverse chart formula $ξ^{- 1} (u) = p + E (u)$ with $E (u) = \sum_{a} u^{a} e_{a}$ from [F2] and step 1.1, $x (s) = p + \frac{y^{0} (s)}{T} d + w_{0}$ with $w_{0}$ the vector whose coordinates are $(0, z_{0})$ , and evaluating at $s = a$ , where $x (a) = p$ and $y^{0} (a) = 0$ , gives $w_{0} = 0$ ; thus $x (s) = p + σ (s) d$ for $σ (s) : = \frac{y^{0} (s)}{T}$ , and $σ$ is continuous with $σ (a) = 0$ and $σ (b) = 1$ .

7.1step 6.1F3F4algebra

Properties of $σ$ : it is $C^{1}$ on every open piece with $σ^{'} = y^{0}^{'} / T > 0$ by step 2.1; its one-sided difference-quotient derivatives at the piece endpoints are the one-sided quotients of $y^{0}$ divided by $T$ , hence finite and positive because the corresponding one-sided tangent of $x$ is future timelike, so its time component strictly dominates the spatial norm [F3, F4]; and it is strictly increasing: for $s < t$ in $[a, b]$ the restricted worldline $x ∣_{[s, t]}$ is again weak piecewise future timelike, and its proper time satisfies $Δ τ_{[s, t]} (x) = c^{- 1} \sum \int_{(induced pieces)} f$ with $f = y^{0}^{'}$ on all open pieces by step 5.1, so by the endpoint increments and additivity of [F3] this equals $c^{- 1} (y^{0} (t) - y^{0} (s))$ , while the based proper-time function is strictly increasing by [F3]; hence $y^{0} (t) - y^{0} (s) > 0$ and $σ (t) > σ (s)$ .

8.1step 6.1step 7.1L9algebra

By [L9], continuity and the endpoint values $σ (a) = 0$ , $σ (b) = 1$ imply that $σ$ assumes every value of $[0, 1]$ ; strict monotonicity keeps its values in $[0, 1]$ . This proves the forward implication of (ii): the $σ$ of step 6.1 and step 7.1 is a continuous strictly increasing surjection $[a, b] \to [0, 1]$ with $σ (a) = 0$ , $σ (b) = 1$ , $C^{1}$ on the open pieces of the partition of $x$ , with $σ^{'} > 0$ there and finite positive one-sided quotient derivatives, and $x (s) = p + σ (s) d$ . The tangent form of the statement is immediate in this direction, since $x^{'} (s) = σ^{'} (s) d$ with $σ^{'} (s) > 0$ at every differentiable point of every open piece.

9.1step 7.1step 8.1F3F4L7algebra

Conversely, suppose $x (s) = p + σ (s) (q - p) = p + σ (s) d$ for a $σ$ as in (ii) with respect to a finite weak piecewise partition of $[a, b]$ ; then on each open piece the tangent is $σ^{'} (s) d$ with $σ^{'} (s) > 0$ , so $x$ is again a weak piecewise future timelike worldline with that partition, its coordinate path in the chart of step 1.1 is $y^{0} = T σ$ , $z = 0$ , and its density is $f = \sqrt{(y^{0}^{'})^{2}} = y^{0}^{'} = T σ^{'}$ ; therefore $c Δ τ (x) = \sum \int f = \sum \int y^{0}^{'} = y^{0} (b) - y^{0} (a) = T (σ (b) - σ (a)) = T$ , using for the middle equality the endpoint increments and additivity of [F3] applied to $x$ with that partition; this is the reverse implication of (ii), and the stated equivalence with 'every open-piece tangent is a positive multiple of $q - p$ ' holds because the representation gives such tangents, and conversely such tangents have vanishing spatial coordinates in the rest frame of step 1.1, so [L7] makes the continuous spatial coordinates constant on each closed piece; the shared corner values and $z (a) = 0$ make $z = 0$ throughout. Then $σ = y^{0} / T$ has precisely the positivity, endpoint and surjectivity properties established in steps 7.1 and 8.1.

10.1step 9.1F4F5algebra

For (iii): by (ii), $c Δ τ (x) = T$ forces $x$ to be a reparametrization with $σ$ valued in $[0, 1]$ , so the image of $x$ is contained in ${p + λ d : 0 \leq λ \leq 1}$ ; hence if the image is not contained in the straight segment the inequality is strict. If $x$ is piecewise inertial with constant piece tangents $d_{1}, \dots, d_{m}$ , then: if all $d_{i}$ are positive multiples of one another, say $d_{i} = α_{i} u$ with $u$ future timelike and $α_{i} > 0$ , then all legs lie on the line through $p$ in direction $u$ and the parameter $σ$ defined by $x = p + σ d$ is continuous, piecewise affine and strictly increasing with image $[0, 1]$ , so the image is exactly the straight segment and the bound is attained by step 9.1; conversely, if the image is contained in the straight segment, each leg's tangent is parallel to $d$ and future timelike, hence a positive multiple of $d$ by [F4], so all $d_{i}$ are positive multiples of one another; this proves (iii).

10.2step 9.1F4F5algebra

For (iv): $x_{s} (s) = p + \frac{s - a}{b - a} d$ has constant tangent $\frac{1}{b - a} d$ , a future timelike vector, so it is a weak piecewise (indeed $C^{1}$ ) future timelike worldline, and $σ (s) = \frac{s - a}{b - a}$ satisfies the conditions of (ii): it is continuous, strictly increasing, maps $[a, b]$ onto $[0, 1]$ , is $C^{1}$ with positive constant derivative, and $x_{s} = p + σ d$ ; by step 9.1, $c Δ τ (x_{s}) = T$ . More generally, any weak piecewise future timelike worldline whose image is the straight segment from $p$ to $q$ and which is traversed in the future direction satisfies the representation of (ii) by the argument in step 9.1 and hence attains the bound.

11.1step 10.2F3∎

Clause (v) is a scope statement: every step above is an argument in the oriented time-oriented affine Minkowski model with the fixed positive constant $c$ , within the weak piecewise $C^{1}$ future timelike class; the corner hypothesis (H) is the special case in which the improper values equal the ordinary Riemann integrals and the total proper time agrees with Proper time along a future timelike worldline for globally $C^{1}$ histories by [F3], so that case is included; no clock reading, curved-spacetime maximality or choice principle is introduced, the only analytic inputs being the compact-interval improper-integral and one-dimensional differentiability suppliers listed above.

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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.

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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).

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Acceleration is not an aging term, and the twin comparison is not symmetric: scope caveats

Physical scope

Interpretation and delimitation of the proper-time comparison in the flat classical special-relativity model: the absence of an acceleration term in the ideal clock hypothesis, the geometric asymmetry of the two histories, and the simultaneity bookkeeping of the traveler's two momentary inertial rest frames. Conditional on the named postulates; excludes real-clock error models, apparatus, gravity and quantum effects, and asserts no measured outcome.

Scope caveats for the twin reunion

This remark delimits the interpretation of the proper-time comparison of the twin reunion (The piecewise inertial twin reunion: conditional differential aging) in the adopted model, with the ideal clock assumptions of Adopted ideal-clock and ideal-vacuum-light interpretation and Adopted free trajectories of massive particles and the clock hypothesis and the SI conventions of SI units and coordinate conventions of the special-relativity framework. It proves nothing new; it records what the earlier items do and do not assert.

Acceleration is not an additional aging term

In the model the reading of an ideal clock between two events of its history is the proper-time integral of its worldline, by clause (i) of Adopted ideal-clock and ideal-vacuum-light interpretation for $C^{1}$ histories and by clause (ii) of Adopted free trajectories of massive particles and the clock hypothesis for the weak piecewise $C^{1}$ class (Total proper time of a piecewise C^1 future timelike worldline); no term depending on the four-acceleration (Four-velocity and four-acceleration of a future timelike worldline) or on any higher derivative of the history appears, and the clock hypothesis is precisely the exclusion of such terms. The corner of the piecewise inertial history is an idealization: a physical turnaround occupies a finite parameter interval, the worldline is then $C^{1}$ or smooth there, and its proper time is still the integral. The endpoint bound and strictness clause of The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space apply to every weak piecewise future timelike history with the same endpoints $p, q$ . Thus any smoothing of the corner that remains in that class, preserves $p, q$ , and retains at least one event outside the straight segment still gives a strictly smaller reading than the stay-at-home clock. No existence or convergence claim for a smoothing family, or equality with the sharp-turn value $2 T / γ$ , is asserted.

The comparison is not symmetric in the model

One history of the reunion is the straight segment from $p$ to $q$ ; the other contains the corner event $r$ of The piecewise inertial twin reunion: conditional differential aging, and the three events $p, r, q$ are not collinear: in the chart of that item their coordinate columns are $(0, 0, 0, 0)$ , $(c T, v T, 0, 0)$ and $(2 c T, 0, 0, 0)$ with $v T \neq 0$ . Every change of inertial chart is an affine bijection (Inertial charts are related by proper orthochronous affine Lorentz maps, and every such map is a chart transition) and maps collinear triples to collinear triples, so no change of inertial frame maps the image of the stay-at-home history onto the image of the traveler's history, nor fixes both endpoints and exchanges the two histories. The two twins are therefore not related by an inertial change of description, and the difference of readings is a model consequence of the different histories, not a consequence of who is 'really moving': the symmetric time-dilation relation of Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation (i) concerns the two frames, while the reunion comparison concerns the two clock readings at the single common event $q$ .

Synchronization bookkeeping at the turn

In each momentary inertial rest frame of the traveler the stay-at-home clock runs slow by the factor $1 / γ$ , and the traveler may legitimately apply the time-dilation relation using successive pairs of simultaneous events, one on each clock history in that momentary frame (Coordinate formulas of a boost: clock rates, rod lengths, simultaneity and velocity transformation (i), Simultaneity slices and rest distance relative to a future unit timelike vector). But the two legs use different inertial frames. With $γ = (1 - v^{2} / c^{2})^{- 1 / 2}$ and the turn event $r$ of The piecewise inertial twin reunion: conditional differential aging, choose both rest-frame charts with origin $p$ as in The piecewise inertial twin reunion: conditional differential aging; the outbound chart $x^{'}$ then satisfies $x^{' 0} = γ (x^{0} - \frac{v}{c} x^{1})$ , so its slice through $r$ has $x^{' 0} = γ c T (1 - v^{2} / c^{2}) = c T / γ$ and meets the stay-at-home worldline $x^{1} = 0$ at $x^{0} = \frac{1}{γ} \cdot \frac{c T}{γ} = c T (1 - v^{2} / c^{2})$ , that is at coordinate time $t = T (1 - v^{2} / c^{2}) = T / γ^{2}$ ; the inbound rest frame's chart $x^{''}$ satisfies $x^{'' 0} = γ (x^{0} + \frac{v}{c} x^{1})$ , and its slice through $r$ meets the same worldline at $x^{0} = c T (1 + v^{2} / c^{2})$ , that is at coordinate time $t = T (1 + v^{2} / c^{2})$ . The two slices differ by $2 T v^{2} / c^{2}$ in coordinate time, and the identity $2 T = \frac{T}{γ^{2}} + \frac{2 T v^{2}}{c^{2}} + \frac{T}{γ^{2}},$ which only uses $1 / γ^{2} = 1 - v^{2} / c^{2}$ , inserts this gap into the traveler's two leg-by-leg accounts and reproduces the stay-at-home reading $2 T$ at the reunion, in agreement with the conditional comparison result Same-endpoint clock comparison: the inertial clock ages most. The naive reciprocity statement 'each twin judges the other's clock to run slow' therefore compares different pairs of events for the two travelers — events simultaneous in different momentary frames — and does not apply to the reunion comparison, which concerns the two clock readings at the single common event $q$ .

Empirical qualification

Everything above is a conditional statement about ideal clocks in the adopted model: the clocks are ideal with no acceleration-dependent reading term, the histories are prescribed, spacetime is flat and $c$ is fixed (Adopted free trajectories of massive particles and the clock hypothesis, Adopted ideal-clock and ideal-vacuum-light interpretation). No measurement, apparatus, calibration, sampling or uncertainty model is involved, and no claim is made about real clocks or about nature; real clock comparisons would additionally require apparatus and uncertainty analysis and are outside this remark.

5 · Examples, experiments and counterexamples

None yet.

Proper Time Comparisons

1 · Prerequisites

2 · Summary

3 · Logical flowchart

4 · Definitions, postulates, theorems and proofs

Inertial and piecewise inertial worldlines of the affine Minkowski model

Definition

(a) Inertial worldlines

(b) Piecewise inertial worldlines

(c) Scope

The reversed Cauchy-Schwarz inequality for future timelike vectors

Statement

Facts & Assumptions

Proof

The reversed triangle inequality and convexity of the future timelike cone

Statement

Facts & Assumptions

Proof

Existence of an inertial frame whose time axis is a prescribed future timelike direction

Statement

Facts & Assumptions

Proof

The displacement of a future timelike worldline is future timelike

Statement

Facts & Assumptions

Proof

The endpoint bound for proper time and the maximality of the straight worldline in affine Minkowski space

Statement

Facts & Assumptions

Proof

Same-endpoint clock comparison: the inertial clock ages most

Statement

Facts & Assumptions

Proof

The piecewise inertial twin reunion: conditional differential aging

Statement

Facts & Assumptions

Proof

Acceleration is not an aging term, and the twin comparison is not symmetric: scope caveats

Scope caveats for the twin reunion

Acceleration is not an additional aging term

The comparison is not symmetric in the model

Synchronization bookkeeping at the turn

Empirical qualification

5 · Examples, experiments and counterexamples

Sources