Simultanéité, mesure de longueur et transformation de vitesse
Introduit| English | Sens |
|---|---|
| proper length/ˈprɒpə leŋθ/ | longueur propre |
| spacetime interval/ˈspeɪstaɪm ˈɪntəvl/ | intervalle spatio-temporel |
A decision before an answer
- Two observers can disagree about which of two events happened first, and both can be right.
- Your goal: Transform event separations and identify proper length versus simultaneous endpoint readings.
Transform separations
- Use the Lorentz transformation with declared frame motion: if frame S′ moves at +v along x relative to S, then x′=γ(x−vt) and t′=γ(t−vx/c²) with γ=1/sqrt(1−v²/c²). Apply it to event separations Δx and Δt, not only to single coordinates.
- Two events simultaneous in S (Δt=0) at different places have Δt′=−γvΔx/c²: the event further in the +x direction happens earlier in S′. Simultaneity is frame-dependent, not a bookkeeping error.
To obtain the contracted length of a moving rod, a lab observer must record the two endpoint positions:
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Length in a frame is the endpoint separation at equal time in that frame; Δt=0 in the lab gives L=L0/γ.
Measure length in one frame
- A moving rod length must be read from endpoint positions at the same lab time (Δt=0). Then Δx′=γΔx gives L=L0/γ with L0 the rest-frame (proper) length.
- Reading the endpoints simultaneously in the rod frame instead does not produce the lab length; the two reading conditions answer different questions. State the frame and the reading condition before quoting any length.
A ship moves at 0.5c and launches a probe forward at 0.5c relative to the ship. The probe has lab speed:
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(0.5+0.5)/(1+0.25)=0.8; the denominator keeps the result below c.
Classify the interval
- The interval s²=c²Δt²−Δx² (mostly-minus convention, declared once) is invariant. s²>0 is timelike: a frame exists with the events co-located, and Δτ=s/c is the proper time between them.
- s²<0 is spacelike: a frame exists with the events simultaneous, and sqrt(−s²) is the proper distance; s²=0 is lightlike. Timelike order is the same in every frame; spacelike order can swap, and no signal can join the events because |Δx/Δt|>c.
Events A and B are 900 m apart in the lab and 2 μs apart in time (cΔt=600 m using c=3×10⁸ m/s). Since 600<900 the interval is spacelike; the simultaneity frame moves at v=c²Δt/Δx=(2/3)c and the proper distance is sqrt(900²−600²)=670.82 m. A 0.5c probe from a 0.5c ship has lab speed (0.5+0.5)c/(1+0.25)=0.8c, where γ=5/3. E=5 GeV with p=3 GeV/c gives rest energy sqrt(25−9)=4 GeV.
Two lab events have Δx=900 m and Δt=2 μs. Using c=3×10⁸ m/s, the frame in which they are simultaneous moves at ____ c (give the exact fraction).
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Set Δt′=0 in t′=γ(t−vx/c²): v=c²Δt/Δx=(600/900)c=2c/3.
Add velocities and check invariants
- Collinear velocities add as u=(u′+v)/(1+u′v/c²); the denominator keeps u<c for any u′,v<c, and u′=c gives u=c exactly.
- Energy and momentum form a four-vector with invariant E²−p²c²=m²c⁴: compute it in any convenient frame. Check the nonrelativistic limit, where the addition rule reduces to u=u′+v and the invariant reduces to the rest energy.
Quoting L=L0/γ without the simultaneous-endpoint condition, or adding velocities as u=u′+v at relativistic speeds. A simultaneity difference is physical, not a mistake.
Quelle réponse correspond à ce cas ?
Transformez les séparations d'événements et identifiez la longueur propre versus les lectures simultanées des extrémités
If two events are spacelike separated, every inertial frame agrees on which event happened first.
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Spacelike order can swap between frames; only timelike or lightlike order is invariant.
Keep the distinctions
- spacetime interval 时空间隔 — The invariant combination c²Δt²−Δx² under the declared sign convention, whose sign classifies an event separation.
- proper length 固有长度 — The endpoint separation measured in the object's rest frame; any other frame must read both endpoints simultaneously in its own time.
- Transform event separations and identify proper length versus simultaneous endpoint readings.
- Apply relativistic velocity addition and check a four-vector invariant.
- Classify the interval sign to order events and rule out causal contact.
Match each term with its precise meaning in this lesson.
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Keep the distinctions stated in the teaching example.
Put this lesson’s reasoning or event sequence in order.
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The order follows the stated process; check each stage before the next.