SCIENCE · INVESTIGATE AND EXPLAIN

Integrate elapsed clock time over a complete worldline

Proper time is the time accumulated by a clock along its actual trajectory. In flat spacetime, dτ=dt sqrt(1−v(t)²) with c=1. A constant-speed segment gives Δτ=Δt/γ, but a changing velocity requires integration or a sum over segments; one maximum speed does not specify the whole clock reading.

Learning outcome: Reconstruct both positions and endpoint agreement, integrate each proper-time density independently and compare an average-velocity estimate with the actual integral.

Starting recap

Name event pairs, frame signs and c=1 units; retain simultaneity, worldline integration, four-momentum and unused evidence boundaries.

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Try

At a=0.6, the piecewise case accumulates 8, while the varying-velocity integral gives about 9.362509. Both laboratory durations are 10 and final displacements are zero. Their maximum displacements are 3 and 1.5 respectively.

Open the experiment

Predict

Can zero average velocity be substituted for the instantaneous velocity inside the proper-time integral?

Your prediction and reason

Experiment

Reconstruct both positions and endpoint agreement, integrate each proper-time density independently and compare an average-velocity estimate with the actual integral.

Integrate the clock along its actual path

Worked teaching model · synthetic values · Original flat-spacetime round trip lasts laboratory T=10. Case0 moves+v for first half and−v for second, ideal instantaneous turn. Case1 v(t)=a(1−2t/T). Both return to the starting position. Proper time is ∫sqrt(1−v²)dt; no thrust, finite-acceleration mechanism or gravity supplied.

Maximum speed magnitude: 0.6 c Worldline: piecewise0 / linear velocity1: 1 case

Reset keeps your prediction, captures and practice history.

Proper time is a worldline integral; the two equal-duration round trips have different speed histories and clock readings.

Original flat-spacetime round trip lasts laboratory T=10. Case0 moves+v for first half and−v for second, ideal instantaneous turn. Case1 v(t)=a(1−2t/T). Both return to the starting position. Proper time is ∫sqrt(1−v²)dt; no thrust, finite-acceleration mechanism or gravity supplied.

Original conditional ledger; reference units retain their distinct meanings
Quantity	Value	Unit
Laboratory elapsed time	10	reference time
Traveling clock proper time	9.362509	reference time
Stationary minus traveling elapsed time	0.6374908	reference time
Final displacement	0	reference length
Maximum displacement	1.500000	reference length
Maximum speed magnitude	0.6000000	c

These original virtual events and collision records are not an actual clock, accelerator, photon-source or practical experiment. No physical apparatus operation is assigned.

Proper time is a worldline integral; the two equal-duration round trips have different speed histories and clock readings.

Observation to attach to a capture

Captures preserve recipes and replay their calculated results. They are synthetic model evidence, not physical measurements.

Understand

1. Define events, frames and conserved quantities

Two original round trips begin and end at the same laboratory position and last T=10. The first moves at +a for half the time and −a for half, with an ideal instantaneous reversal. Its proper time is T sqrt(1−a²). The reversal is a mathematical limiting construction, not a claim of a physically instantaneous acceleration.

The second has v(t)=a(1−2t/T). Integrating position gives x(t)=a(t−t²/T), which also returns to zero. Substituting u=1−2t/T in the proper-time integral gives τ=(T/2)[sqrt(1−a²)+asin(a)/a]. The arcsine is in radians. This formula is derived from the full speed history, not an average-velocity shortcut.

Both have zero net displacement, yet neither moving clock accumulates the stationary laboratory time T. The gradual-velocity case spends more of the duration at lower speed and accumulates more proper time than the fixed-magnitude case. A single global inertial rest frame does not follow the entire turning trajectory; this distinction resolves inappropriate symmetric-clock comparisons.

2. Reconstruct the original case

3. Predict and investigate

Can zero average velocity be substituted for the instantaneous velocity inside the proper-time integral?

Reconstruct both positions and endpoint agreement, integrate each proper-time density independently and compare an average-velocity estimate with the actual integral.

Transfer the event construction

Use a=0.8 in the piecewise case. What proper time is accumulated, and why is the net displacement insufficient to infer it?

Retain the chosen events, frame convention and intermediate reasoning before opening public feedback.

Compare a worked response

Proper time is 10sqrt(1−0.64)=6. The same zero displacement can be produced by many speed histories with different integrals; endpoint displacement alone does not determine accumulated clock time.

Primary mechanism context. Checked 31 October 2026. Original derivations, records and diagrams; no physical apparatus operation or actual independent clock/particle measurement is claimed. Retain first reasoning and public-feedback exposure in the course evidence workspace when registered.

Apply

Public practice checks a specific claim. It does not establish independent mastery.

Use a=0.8 in the piecewise case. What proper time is accumulated, and why is the net displacement insufficient to infer it?

Your explanation and limitations

Continue

Completion requires a saved prediction, at least one capture and a reflection. Presence checks do not grade your reasoning.

Skipping retains experiments and awards no mastery.

Previous: Construct the endpoint events that actually measure length Next: Derive velocity addition from transformed differentials

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