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Ideal-clock readings for a flat reunion path

Physical scope

Conditional ideal-clock calculation for two specified piecewise-inertial flat worldlines that reunite; it is a model prediction, not an experimental report.

Example

Choose an inertial frame $(o; e_{0}, e_{1}, e_{2}, e_{3})$ and its inertial chart $χ$ from Inertial frames and inertial charts of the affine Minkowski model, with coordinates $(x^{0}, x^{1}, x^{2}, x^{3}) = (c t, x, y, z)$ . Use the flat specialization of the prescribed background: $M = A$ with its affine smooth structure, $g$ the metric induced by $η$ on tangent vectors, and $N = e_{0}$ the constant future time-axis field. Thus $g_{a b} = diag (- 1, 1, 1, 1)$ in $χ$ . Let $T > 0$ seconds and $0 < v < c$ . A lab clock has chart path $χ \circ x_{L} (t) = (c t, 0, 0, 0)$ for $0 \leq t \leq 2 T$ . A traveler has chart path $χ \circ x_{R} (t) = (c t, v t, 0, 0)$ on $[0, T]$ and $χ \circ x_{R} (t) = (c t, v (2 T - t), 0, 0)$ on $[T, 2 T]$ . The traveler reverses direction at $t = T$ and reunites with the lab at $t = 2 T$ .

Facts & Assumptions

Given: The paths above, $T > 0$ , $0 < v < c$ , and the ideal-clock adoption of Ideal clocks on a prescribed Lorentzian background.

[F1]

On a future timelike path, the adopted ideal-clock reading is $d τ = c^{- 1} \sqrt{- g (T, T)} d s$ (Ideal clocks on a prescribed Lorentzian background).

[F2]

For a piecewise- $C^{1}$ future timelike worldline, proper-time increments are defined as the sum of the integrals over its pieces (Timelike worldlines, proper time, length and four-velocity on a Lorentzian four-manifold).

Verification

1.1givenF1algebra

For the lab, $d x^{0} / d t = c$ and $d x / d t = 0$ , so $- g (d x / d t, d x / d t) = c^{2}$ and the clock density is $1$ . Its elapsed reading is therefore $Δ τ_{L} = 2 T$ .

2.1givenF1F2algebra∎

On either traveler leg, $d x^{0} / d t = c$ and $∣ d x / d t ∣ = v$ , so $- g (d x / d t, d x / d t) = c^{2} - v^{2} > 0$ . The density is the constant $\sqrt{1 - v^{2} / c^{2}}$ on both legs. Additivity across the single corner follows from the piecewise definition [F2], giving $Δ τ_{R} = 2 T \sqrt{1 - v^{2} / c^{2}} < 2 T = Δ τ_{L}$ . This is a deduction from the ideal-clock model and specified paths, not a measured result.

Ideal-clock readings for a flat reunion path

Example

Facts & Assumptions

Verification

Depends on

Dependency tree · 2 levels

Sources