Galilean and Special Relativity伽利略与狭义相对论

A railway carriage moves at high constant speed past a platform. A lamp is fixed at the exact centre of the carriage. At one instant, in the carriage frame, the lamp emits a single flash that travels to detectors at the front and rear walls.一节车厢以高速匀速掠过站台。车厢正中央固定一盏灯。在车厢系中的某一时刻，灯发出一次闪光，传向前后壁上的探测器。

Spaceship $A$ moves at $0.60c$ due east relative to a space station. Take east as positive and use the relativistic velocity-addition formula throughout.飞船 $A$ 相对空间站以 $0.60c$ 向正东运动。取东为正，全题使用相对论速度叠加公式。

Cosmic-ray muons are created high in the atmosphere and travel downward at $v = 0.98c$. In their own rest frame the muon population decays with a mean lifetime $\tau = 2.2\ \mathrm{\mu s}$, following $N = N_0\,e^{-t/\tau}$, where $t$ is the time elapsed in the muon frame. A detector at the top of a mountain and a detector $2000\ \mathrm{m}$ lower at its base both count the muon flux.宇宙射线 μ 子在高空大气中产生，以 $v = 0.98c$ 向下运动。在其自身静止系中，μ 子总数按平均寿命 $\tau = 2.2\ \mathrm{\mu s}$ 衰变，遵循 $N = N_0\,e^{-t/\tau}$，其中 $t$ 为 μ 子系中经历的时间。一台探测器位于山顶，另一台位于其下方 $2000\ \mathrm{m}$ 的山脚，二者都测量 μ 子通量。

Two events $P$ and $Q$ are recorded in an inertial frame $S$. To keep the numbers simple the separations are quoted as $c\,\Delta t$ and $\Delta x$ in the same unit (light-microseconds, where $1\ \mathrm{l.\mu s} = c\times 1\ \mathrm{\mu s}$). A second frame $S'$ moves at $0.60c$ along the $x$-axis ($\gamma = 1.25$).两个事件 $P$ 与 $Q$ 记录于惯性系 $S$ 中。为使数字简单，间隔以同一单位（光微秒，$1\ \mathrm{l.\mu s} = c\times 1\ \mathrm{\mu s}$）给出 $c\,\Delta t$ 与 $\Delta x$。第二个参考系 $S'$ 沿 $x$ 轴以 $0.60c$ 运动（$\gamma = 1.25$）。

Muons are created $4.5\ \mathrm{km}$ above the ground and travel straight down at $v = 0.995c$, for which $\gamma = 10.0$. Their proper (rest-frame) lifetime is $\Delta t_0 = 2.2\ \mathrm{\mu s}$.μ 子在地面上方 $4.5\ \mathrm{km}$ 处产生，以 $v = 0.995c$ 竖直向下运动，对应 $\gamma = 10.0$。其固有（静止系）寿命为 $\Delta t_0 = 2.2\ \mathrm{\mu s}$。

In an inertial frame $S$, event $E$ occurs at position $x = 900\ \mathrm{m}$ at time $t = 2.0\ \mathrm{\mu s}$ (with the origin event at $x = 0$, $t = 0$). A frame $S'$ moves at $v = 0.60c$ along the $+x$ direction, with $\gamma = 1.25$ and origins coinciding at $t = t' = 0$. Take $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$.在惯性系 $S$ 中，事件 $E$ 发生于位置 $x = 900\ \mathrm{m}$、时刻 $t = 2.0\ \mathrm{\mu s}$（原点事件在 $x = 0$、$t = 0$）。参考系 $S'$ 沿 $+x$ 方向以 $v = 0.60c$ 运动，$\gamma = 1.25$，且 $t = t' = 0$ 时原点重合。取 $c = 3.00\times 10^{8}\ \mathrm{m\,s^{-1}}$。

A spacecraft travels from Earth to a star $6.0$ light-years away and back, moving at a constant $0.60c$ on each leg ($\gamma = 1.25$). Ignore the short turn-around. Distances are quoted in the Earth frame.一艘飞船从地球飞往 $6.0$ 光年外的一颗恒星再返回，每段行程都以恒定的 $0.60c$ 运动（$\gamma = 1.25$）。忽略短暂的折返过程。距离以地球系给出。