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Four-velocity norm and four-acceleration orthogonality for the model's clocks

Physical scope

Ideal clocks and massive test particles in the flat classical special-relativity model: four-velocity of squared norm -c^2 (dimensionless U/c unit timelike), four-acceleration orthogonal to four-velocity, vanishing four-acceleration for free particles, and subluminal speed relative to every ideal observer. Conditional on the named postulates; excludes interactions and force laws, gravity, quantum effects, real apparatus corrections and all empirical claims.

Statement

Adopt the postulates of the model: Adopted inertial covariance of the model equations and Adopted ideal-clock and ideal-vacuum-light interpretation of the spacetime-primitives page, together with Adopted free trajectories of massive particles and the clock hypothesis of this page, and let $c$ and the SI coordinates be those of SI units and coordinate conventions of the special-relativity framework. Let $x : I \to A$ be the history of an ideal clock of the model, a $C^{1}$ future timelike worldline ( $C^{2}$ where acceleration is used), with proper time $τ$ , four-velocity $U = d x / d τ$ and, when $x$ is $C^{2}$ , four-acceleration $A = d U / d τ$ (Four-velocity and four-acceleration of a future timelike worldline). Then:

(i) the reading of the ideal clock between parameter values $s_{1} < s_{2}$ equals $τ (s_{2}) - τ (s_{1})$ ;

(ii) $η (U, U) = - c^{2}$ everywhere, so $U$ is future timelike with velocity units, and the dimensionless vector $U / c$ is future unit timelike;

(iii) if $x$ is $C^{2}$ at $s$ , then $η (U (s), A (s)) = 0$ and $A (s)$ is zero or spacelike, with the unit of acceleration;

(iv) if the ideal clock is carried by a massive test particle subject to no interaction, and $x$ is also that particle's worldline (a free clock in this theorem), then its four-velocity is constant along the worldline and $A = 0$ identically;

(v) for an ideal observer at the same event, described by a future unit timelike vector $n = U_{o} / c$ with $η (n, n) = - 1$ , the clock's speed relative to the observer is $∣ w ∣ = c \sqrt{1 - γ^{- 2}}$ with $γ = - η (n, U / c) \geq 1$ , so it is strictly less than $c$ , and $γ = 1$ exactly when the clock's four-velocity equals the observer's.

Scope and qualification. This is a conditional statement about the adopted model; it asserts no measurement outcome, contains no statistical inference, and inherits the idealizations of the postulates (ideal point particles, ideal clocks, flat spacetime, no gravitational or quantum effects).

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; and a $C^{1}$ future timelike worldline $x : I \to A$ that is the history of an ideal clock, $C^{2}$ where acceleration is used.

[A1]

Free trajectories: a massive test particle subject to no interaction has as its worldline a straight future timelike line $x (s) = p + (s - s_{0}) v$ with $v$ future timelike; interactions and force laws are not part of the assumption (Adopted free trajectories of massive particles and the clock hypothesis, clause (i)).

[A2]

Clock hypothesis: an ideal clock carried along a future timelike worldline reads, between parameter values, the proper-time difference of Proper time along a future timelike worldline (Adopted free trajectories of massive particles and the clock hypothesis, clause (ii); Adopted ideal-clock and ideal-vacuum-light interpretation, clause (i)).

[A3]

The model's postulates hold: the affine Minkowski space is the spacetime, and the model equations are covariant under inertial changes of description (Adopted inertial covariance of the model equations); ideal clocks and ideal light are as stated in Adopted ideal-clock and ideal-vacuum-light interpretation.

[L1]

Units (SI units and coordinate conventions of the special-relativity framework): the four-velocity $U = d x / d τ$ has unit m/s and the four-acceleration m/s $^{2}$ ; the interval carries m $^{2}$ .

[L2]

Four-velocity and four-acceleration (Four-velocity and four-acceleration of a future timelike worldline): $U = x^{'} / τ^{'}$ and $A = U^{'} / τ^{'}$ ; the four-velocity of the observer's own straight worldline with future unit timelike $n$ is $c n$ .

[L3]

Mathematical content (Norms of four-velocity and four-acceleration, and the observer rest-space decomposition): $η (U, U) = - c^{2}$ ; $η (U, A) = 0$ and $A$ zero or spacelike when $x$ is $C^{2}$ ; for future unit timelike $n$ and $u = U / c$ , with $γ = - η (n, u)$ , $γ \geq 1$ , $∣ w ∣ = c \sqrt{1 - γ^{- 2}} < c$ for $w = c h (u) / γ$ , and $γ = 1$ if and only if $u = n$ ; $w$ is the velocity of a worldline with four-velocity $c u$ relative to $n$ .

[L4]

Proper-time derivative and straight-line computation (The proper-time function is a strictly increasing C^1 parametrization with C^1 inverse, Proper time along a future timelike worldline, Curves, regular causal curves and future timelike worldlines in affine Minkowski space): $τ^{'} = c^{- 1} \sqrt{- η (x^{'}, x^{'})}$ , and for the straight worldline $x (s) = p + (s - s_{0}) v$ the difference quotients are identically $v$ , so $x^{'} = v$ and $τ^{'} = c^{- 1} \sqrt{- η (v, v)}$ is constant.

Proof

technique · direct

1.1givenA2L4

Clause (i): by the clock hypothesis [A2] the reading of the ideal clock between $s_{1}$ and $s_{2}$ is the proper-time difference $τ (s_{2}) - τ (s_{1})$ of [L2]'s and [L4]'s definitions for a $C^{1}$ future timelike history; this is an adopted identification of clock readings with the mathematical proper time, not a geometric consequence.

1.2givenL1L3

Clause (ii): $η (U, U) = - c^{2}$ is clause (a) of the mathematical decomposition theorem [L3], and the unit of $U$ is m/s by [L1]; $U$ is future timelike by that theorem, and $U / c$ has squared norm $- 1$ , so it is the future unit timelike vector of the model.

1.3givenL1L3

Clause (iii): if $x$ is $C^{2}$ at $s$ , then $η (U (s), A (s)) = 0$ and $A (s)$ is zero or spacelike, both by clause (b) of [L3]; the unit of $A$ is m/s $^{2}$ by [L1].

1.4givenA1L2L4

Clause (iv): the clock's carrier is a massive test particle subject to no interaction, and its worldline is the same $x$ by the hypothesis of this clause. Apply [A1] to that particle: there are $p \in A$ and a future timelike $v \in V$ with $x (s) = p + (s - s_{0}) v$ . Thus $x$ is $C^{2}$ , including the one-sided derivatives at included endpoints, so four-acceleration is defined everywhere; by [L4] $x^{'} = v$ and $τ^{'} = c^{- 1} \sqrt{- η (v, v)}$ is a positive constant. Hence $U = x^{'} / τ^{'} = v / τ^{'}$ is constant, $U^{'} = 0$ , and $A = U^{'} / τ^{'} = 0$ identically.

1.5givenL2L3

Clause (v): let $n : = U_{o} / c$ be the observer's future unit timelike vector, so that the observer's four-velocity is $c n$ by [L2]; the dimensionless four-velocity of the clock is $u = U / c$ , and clause (d) of [L3] gives $γ = - η (n, u) \geq 1$ , speed $∣ w ∣ = c \sqrt{1 - γ^{- 2}} < c$ relative to the observer, and $γ = 1$ if and only if $u = n$ ; multiplying by $c$ this last equivalence reads $U = U_{o}$ , the clock's four-velocity equal to the observer's.

2.1step 1.5A3L3

The interpretation of $w$ as the clock's speed relative to the observer uses the adopted idealization of observers as future unit timelike vectors and the identification in clause (d) of [L3]; the covariance postulate [A3] licenses treating observers described in different inertial charts as the same model objects, but no statement about actual apparatus or measured speeds is made.

3.1step 1.1step 1.2step 1.3step 1.4step 1.5A1A2A3∎

The conclusion is conditional on [A1]-[A3]: each clause is either the adopted clock/free-trajectory assumption or a clause of the mathematical theorem [L3], and no empirical premise, statistical inference or measurement claim is used; the idealizations (point particles, ideal clocks, flat classical spacetime) are inherited by every clause.

Remarks

What is assumed and what is derived. Clauses (i) and (iv) rest on the adopted free-trajectory and clock assumptions; clauses (ii), (iii) and (v) are the mathematical content of Norms of four-velocity and four-acceleration, and the observer rest-space decomposition read through the model's interpretation. The theorem asserts what follows from the adopted model, not that ideal clocks or free particles exist as described in nature.
The carrier hypothesis in clause (iv). The ideal-clock reading rule alone supplies no trajectory law. Clause (iv) applies the free-particle postulate to a massive test particle carrying the clock on the same worldline; it does not infer straight motion from the reading rule or extend the particle postulate to arbitrary clock histories.
The speed bound is relative to an ideal observer. The bound $∣ w ∣ < c$ is a statement about the model's ideal observers $n$ ; it is not a measurement result, and no statement is made about the accuracy of real clocks or the outcome of any experiment.

Four-velocity norm and four-acceleration orthogonality for the model's clocks

Statement

Facts & Assumptions

Proof

Remarks

Depends on

Dependency tree · 2 levels

Sources