RA.2 · Simultaneity, length measurement and velocity transformation

GRE · GRE Subject Test · GRE 物理 · 知识点 42

42.1

同时性、长度测量与速度变换

Two observers can disagree about which of two events happened first, and both can be right.

Prerequisites: 22.

Transform event separations and identify proper length 固有长度 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

English	中文	拼音
proper length/ˈprɒpə leŋθ/	固有长度	gù yǒu cháng dù

42.2

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.

42.3

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.

42.4

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.

42.5

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 each result in the nonrelativistic limit, where the addition rule reduces to u=u′+v and the invariant reduces to the rest energy.

42.6

Worked method

Two lab events have Delta x = 900 m and Delta t = 2 microseconds. Their spacetime interval 时空间隔 is spacelike because c Delta t = 600 m is less than Delta x. A frame making them simultaneous satisfies

$$\Delta t'=\gamma(\Delta t-v\Delta x/c^2)=0,\quad v=c^2\Delta t/\Delta x=2c/3.$$

Their separation in that frame is $\sqrt{\Delta x^2-c^2\Delta t^2}=670.8$ m. No frame can put these spacelike events at the same position.

Simultaneity, length measurement and velocity transformation: GRE original diagram — Simultaneity, length measurement and velocity transformation: original GRE teaching diagram.

词汇训练

English	中文	拼音
spacetime interval/ˈspeɪstaɪm ˈɪntəvl/	时空间隔	shí kōng jiàn gé

42.7

Check conditions and vocabulary

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.

spacetime interval: The invariant combination c²Δt²−Δx² under the declared sign convention, whose sign classifies an event separation.

proper length: The endpoint separation measured in the object's rest frame; any other frame must read both endpoints simultaneously in its own time.