📚 A.5 Relativity (HL only) | A.5 相对论（仅限HL）
Special relativity, proposed by Einstein in 1905, revolutionized our understanding of space and time. It addresses inertial frames moving at constant velocity and resolves inconsistencies between mechanics and electromagnetism. This HL topic covers Galilean relativity, the Michelson-Morley experiment, the postulates of special relativity, consequences such as time dilation and length contraction, Lorentz transformations, and relativistic dynamics including mass-energy equivalence.
1905年爱因斯坦提出的狭义相对论彻底改变了我们对时空的认知。它研究匀速运动的惯性系，解决了力学与电磁学之间的矛盾。本HL课题涵盖伽利略相对性、迈克尔逊-莫雷实验、狭义相对论基本假设，以及时间膨胀、长度收缩等推论，还包括洛伦兹变换和相对论动力学（如质能等价）。
1. Galilean Relativity and Reference Frames | 伽利略相对性与参考系
at rest or moves with constant velocity unless acted upon by a net external force (Newton' e space. The Galilean transformation links coordinates between two inertial frames S and S' at a constant relative velocity v along the x-axis. If at t=0 the origins coincide, then x' = x – vt, t' = t, and velocities add: u' = u – v. This implies that time is absolute and simultaneity is universal across all frames.
在牛顿力学中，惯性系是指物体在不受外力时保持静止或匀速直线运动的参考系（牛顿第一定律）。所有惯性系相对绝对空间加速度为零。伽利略变换连接以恒定相对速度v沿x轴运动的两个惯性系S和S'。若t=0时原点重合，则x' = x – vt，t' = t，速度相加：u' = u – v。这意味时间是绝对的，同时性是普适的。
2. The Michelson-Morley Experiment | 迈克尔逊-莫雷实验
In the late 19th century, physicists believed light propagated through a medium called the luminiferous ether. The Michelson-Morley experiment (1887) was designed to detect Earth's motion through the ether by measuring differences in light speed along perpendicular arms of an interferometer. If ether existed, the speed of light would vary with direction, producing a fringe shift as the apparatus rotated. No shift was observed, giving a null result. This suggested that the speed of light is independent of the motion of the source or observer, challenging ether theory and paving the way for special relativity.
19世纪末，物理学家认为光通过一种称为“以太”的介质传播。迈克尔逊-莫雷实验（1887年）旨在通过干涉仪两条垂直臂测量光速差异来探测地球相对于以太的运动。如果以太存在，光速会随方向变化，旋转装置时会产生条纹移动。但实验未观察到移动，得到零结果。这表明光速与光源和观察者的运动无关，动摇了以太理论，为狭义相对论铺平了道路。
3. Einstein's Postulates of Special Relativity | 爱因斯坦狭义相对论的基本假设
Einstein based special relativity on two postulates: (1) The laws of physics are the same in all inertial frames (principle of relativity). (2) The speed of light in a vacuum, c, is the same for all inertial observers, regardless of the motion of the source or observer. These postulates rejected the notion of absolute time and absolute space, requiring a new transformation to replace Galilean transformations.
爱因斯坦基于两个基本假设建立狭义相对论：（1）物理定律在所有惯性系中相同（相对性原理）。（2）真空中的光速c对所有惯性观察者都相同，与光源或观察者的运动无关。这两个假设否定了绝对时间和绝对空间，需要用新的变换代替伽利略变换。
4. Relativity of Simultaneity | 同时性的相对性
A direct consequence of the constancy of c is the relativity of simultaneity: two events that are simultaneous in one inertial frame are not necessarily simultaneous in another frame moving relative to the first. For example, consider a train moving at high speed. An observer on the platform sees lightning strike both ends of the train simultaneously, marking the rails at equal distances. An observer on the train, however, moving towards the light from the front, receives that flash earlier, concluding the front strike occurred first. Thus, simultaneity depends on the observer's state of motion.
光速不变的一个直接推论是同时性的相对性：在一个惯性系中同时发生的两个事件，在相对运动的另一个惯性系中未必同时。例如，考虑一列高速行驶的火车。站台上的观察者看到两道闪电同时击中火车两端，并在铁轨上留下等距离的标记。而火车上的观察者朝向来自前方的光运动，会先接收到前端的闪光，因而判断前端先被击中。因此，同时性取决于观察者的运动状态。
5. Time Dilation | 时间膨胀
clock will measure a longer interval Δt = γ Δt₀. The Lorentz factor γ is given by γ = 1 / √1 – v²/c² and is always greater than 1. This means moving clocks tick more slowly; it is a real effect confirmed by extended lifetimes of muons created in the upper atmosphere.
Δt = γ Δt₀, γ = 1 / √1 – v²/c²
时间膨胀指出，运动的时钟相对于静止时钟走得慢。若与钟相对静止的观察者测得固有时间间隔Δt₀（如该系中两次滴答之间的时间），以速度v相对该钟运动的观察者将测得更长的时间间隔Δt = γ Δt₀。洛伦兹因子γ = 1 / √1 – v²/c²始终大于1。这意味着运动时钟变慢；这是一个已被证实为真实现象，高能宇宙线μ子寿命的延长就是证据。
6. Length Contraction | 长度收缩
is only along the direction of motion; dimensions perpendicular to the velocity remain unchanged. Like time dilation, this effect is reciprocal: each observer sees the other's lengths contracted.
L = L₀ / γ
在物体运动参考系中测得的长度沿相对运动方向收缩。若物体有固有长度L₀（在其静止系中测量的长度），以速度v平行于该长度运动的观察者将测得较短的长度L = L₀/γ。长度收缩仅发生在运动方向上；垂直于速度方向的维度保持不变。与时间膨胀一样，这个效应是相互的：每个观察者都看到对方的长度收缩了。
7. Lorentz Transformations | 洛伦兹变换
consistent with the postulates of relativity. For two frames in standard configuration (S' moving at velocity v along the positive x-axis relative to S, origins coincide at t = t' = 0), the coordinates relate as follows. The Lorentz factor is γ = 1/√1 – v²/c² << c), γ ≈ 1 and the Lorentz transformation reduces to the Galilean transformation.
x' = γ (x – vt)
t' = γ (t – vx / c²)
y' = y, z' = z
洛伦兹变换取代伽利略变换，以符合相对论的基本假设。对于标准位形的两个参考系（S'相对S以速度v沿正x轴运动，t=t'=0时原点重合），坐标关系如上所示，其中γ = 1/√1 – v²/c²。逆变换只需将v替换为-v。在低速条件下（v << c），γ ≈ 1，洛伦兹变换退化为伽利略变换。
8. Relativistic Momentum | 相对论动量
omentum would require infinite force. The relativistic momentum reduces to the classical expression p = m v when v << c.
p = γ m₀ v
为保证动量在所有惯性系中守恒，必须修正动量的定义。相对论动量为 p = γ m₀ v，其中m₀为粒子的静质量。当v趋近于c时，γ无限增大，因此p趋于无穷大。这意味着有质量粒子无法达到光速，因为无限大的动量需要无限大的力。低速时，相对论动量回到经典表达式p = m v。
9. Relativistic Energy and Mass-Energy Equivalence | 相对论能量与质能等价
Einstein' tional to its rest mass. The total energy of a moving particle is E = γ m₀ c² = m₀ c² + (γ – 1) m₀ c², where the term (γ – tion of relativistic energy and momentum allows particle creation and annihilation, as observed in accelerators.
E₀ = m₀ c²
E = γ m₀ c² = m₀ c² + (γ – 1) m₀ c²
爱因斯坦著名的质能等价关系为E₀ = m₀ c²，即静止粒子含有与其静质量成正比的能量。运动粒子的总能量为E = γ m₀ c² = m₀ c² + (γ – 1)m₀ c²，其中(γ – 1)m₀ c²是相对论动能。当v << c时，该式回到经典动能½ m₀ v²。该
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