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Time dilation · Δt = γ·Δt0
Time dilation · Δt = γ·Δt0
动钟变慢 Δt=γΔt0 · γ=1/√(1−β²) · γ-β曲线 · 光钟
Moving clocks run slow — a light clock and the γ-vs-β curve, exactly
Length contraction · L = L0 / γ
Length contraction · L = L0 / γ
沿运动 L=L0/γ · 横向不变 · 缩短的尺/飞船
Moving objects contract along the motion — transverse unchanged, exactly
Relativistic velocity addition · u′ = (u+v)/(1+uv/c²)
Relativistic velocity addition · u′ = (u+v)/(1+uv/c²)
u'=(u+v)/(1+uv/c²) · 永不超c · 0.9c⊕0.9c=0.994c
Two speeds combine below c — never the naive sum, exactly
Lorentz transformation · x' = γ(x − vt)
Lorentz transformation · x' = γ(x − vt)
x'=γ(x−vt) · t'=γ(t−vx/c²) · 事件变换 · vs伽利略
How an event's coordinates change between inertial frames — exactly
Relativity of simultaneity · Δt' = −γvΔx/c²
Relativity of simultaneity · Δt' = −γvΔx/c²
Δt'=−γvΔx/c² · 一系同时另一系不同时 · 前钟滞后
Two events at the same time in one frame need not be simultaneous in another
Spacetime interval · s² = −(cΔt)² + Δx²
Spacetime interval · s² = −(cΔt)² + Δx²
s²=−(cΔt)²+Δx² · 类时/类空/类光 · boost下不变
The one number every inertial frame agrees on — and what it means causally
Relativistic momentum · p = γmv
Relativistic momentum · p = γmv
p=γmv≠mv · v→c发散 · vs经典
Why momentum diverges near the speed of light
Relativistic energy · E = γmc²
Relativistic energy · E = γmc²
E=γmc² · 静能mc² · E²=(pc)²+(mc²)²
Rest energy, total energy, and E² = (pc)² + (mc²)²
Relativistic kinetic energy · KE = (γ−1)mc²
Relativistic kinetic energy · KE = (γ−1)mc²
KE=(γ−1)mc² · 低速→½mv² · 近c发散
Where the classical ½mv² breaks down near light speed
Minkowski spacetime diagram
Minkowski spacetime diagram
ct-x · 世界线 · 45°光锥 · boost倾斜轴 arctan(β)
Light cones and the tilted axes of a boosted frame
Twin paradox · T_trav = T_home / γ
Twin paradox · T_trav = T_home / γ
旅行者少老 1/γ · 掉头换系不对称 · 时空图
The traveling twin returns younger — the turnaround breaks the symmetry, exactly
Ladder paradox · length contraction vs. simultaneity
Ladder paradox · length contraction vs. simultaneity
谷仓系装得下/杆系装不下 · 同时性相对性化解
The ladder fits in the barn frame but not its own — simultaneity resolves it, exactly