📚 A.5 Galilean and Special Relativity | 伽利略与狭义相对论
Relativity is one of the most profound frameworks in physics, reshaping our understanding of space, time, and motion. This topic, A.5 in the IB Physics syllabus, explores the transition from classical Galilean relativity—which works perfectly for everyday speeds—to Einstein's special relativity, which governs the behaviour of objects moving close to the speed of light. We will examine the principles, transformations, and remarkable consequences such as time dilation, length contraction, and mass–energy equivalence, all supported by experimental evidence.
相对论是物理学中最深刻的框架之一，重塑了我们对空间、时间和运动的理解。本专题 A.5 是 IB 物理教学大纲的一部分，探讨从适用于日常速度的经典伽利略相对论，到爱因斯坦狭义相对论的过渡，后者支配着以接近光速运动的物体的行为。我们将研究相对论的基本原理、变换以及其惊人的推论，如时间膨胀、长度收缩和质能等价，所有内容均有实验证据支持。
1. Frames of Reference | 参考系
A frame of reference is a set of coordinate axes and a clock used to measure positions and times of events. An inertial frame is one that moves at constant velocity—no acceleration—and in which Newton's first law holds: an object with no net force remains at rest or in uniform motion. Observers in different inertial frames see the same laws of physics, but they may measure different values for position, velocity, and even time.
参考系是一组坐标轴和一个时钟，用于测量事件的位置和时间。惯性系是以恒定速度运动（无加速度）的参考系，其中牛顿第一定律成立：不受净力作用的物体保持静止或匀速直线运动。不同惯性系中的观察者会看到相同的物理定律，但他们可能测量到不同的位置、速度甚至时间值。
In IB Physics, we often compare two inertial frames S and S' where S' moves at constant speed v along the common x‑axis relative to S. The choice of coordinates is crucial for transforming quantities from one frame to another.
在 IB 物理中，我们经常比较两个惯性系 S 和 S'，其中 S' 相对于 S 以恒定速率 v 沿共同的 x 轴运动。坐标系的选择对于将物理量从一个参考系变换到另一个参考系至关重要。
2. Galilean Relativity and Transformations | 伽利略相对性与伽利略变换
he Galilean transformations assume that time is absolute and the same for all observers: t' = t. For two frames S and S' in standard configuration (S' moves at velocity v along the +x direction of S), the position coordinates transform as x' = x - vt, y' = y, z' = z.
在爱因斯坦之前，公认的相对运动描述是由伽利略给出的。伽利略变换假设时间是绝对的，对所有观察者都相同：t' = t。对于标准配置下的两个参考系 S 和 S'（S' 沿 S 的 +x 方向以速度 v 运动），位置坐标变换为 x' = x - vt，y' = y，z' = z。
imply under Galilean relativity. If an object has velocity u in frame S, its velocity in S' is u' nce: if you walk forward at 1 m/s inside a train moving at 10 m/s, your speed relative to the ground is 11 m/s.
在伽利略相对论中，速度简单地相加。如果一个物体在 S 系中的速度为 u，则在 S' 系中的速度为 u' = u - v。这与日常经验相符：如果你在一列以 10 m/s 运动的火车上以 1 m/s 向前走，你相对于地面的速度就是 11 m/s。
However, Galilean relativity fails when speeds approach the speed of light c, because it does not preserve the constancy of c predicted by Maxwell’s equations.
然而，当速度接近光速 c 时，伽利略相对论就失效了，因为它不能保持麦克斯韦方程组所预言的光速不变性。
3. The Principle of Relativity | 相对性原理
Both Galilean and special relativity share a common foundation: the principle of relativity. It states that the laws of physics are the same in all inertial frames of reference. There is no privileged or 'absolute' rest frame; no experiment performed within a smoothly moving laboratory can reveal its constant velocity.
伽利略相对论和狭义相对论共享一个共同的基础：相对性原理。它指出，物理定律在所有惯性参考系中都是相同的。不存在优越的或“绝对”的静止参考系；在一个平稳运动的实验室内进行的任何实验都无法揭示其恒定速度。
For mechanics, Galilean transformations preserve the form of Newton's laws, satisfying the principle. For electromagnetism, however, they fail, which set the stage for Einstein's insight.
对于力学，伽利略变换保持了牛顿定律的形式，从而满足相对性原理。但对于电磁学，伽利略变换却失效了，这为爱因斯坦的洞察铺平了道路。
4. The Conflict with Electromagnetism | 与电磁学的冲突
motion of the source. Under Galilean velocity addition, an observer moving towards a light source would measure a speed greater than c. Experiments, most famously the Michelson– at the speed of light is indeed the same in all directions and for all inertial observers.
麦克斯韦方程组预言，真空中的光以固定速度 c ≈ 3.00 × 10⁸ m/s 传播，与光源的运动无关。根据伽利略速度叠加，一个朝向光源运动的观察者将会测得大于 c 的速度。以迈克尔逊-莫雷实验为代表的著名实验并未找到光以太的证据，而是表明光速在所有方向上且对所有惯性观察者确实都是相同的。
This contradiction meant either the principle of relativity or the Galilean transformation had to be revised. Einstein chose to keep the principle and modify our notions of space and time.
这一矛盾意味着要么相对性原理，要么伽利略变换必须修改。爱因斯坦选择保留相对性原理，并修正我们的空间和时间观念。
5. Einstein’s Postulates of Special Relativity | 爱因斯坦的狭义相对论基本假设
In 1905, Einstein proposed two postulates that form the basis of special relativity:
1905 年，爱因斯坦提出了两条构成狭义相对论基础的基本假设：
- The laws of physics are the same in all inertial frames of reference (the principle of relativity).
- 物理定律在所有惯性参考系中都相同（相对性原理）。
- nertial observers, regardless of the motion of the source or observer: c = 3.00 × 10⁸ m/s.
- 真空中的光速对所有惯性观察者都相同，与光源或观察者的运动无关：c = 3.00 × 10⁸ m/s。
From these simple but radical statements, the entire theory of special relativity follows, replacing Galilean transformations with Lorentz transformations and leading to counterintuitive yet experimentally verified phenomena.
从这些简单但激进的陈述出发，整个狭义相对论由此导出，用洛伦兹变换取代了伽利略变换，并引出了反直觉但已被实验证实的现象。
6. Time Dilation | 时间膨胀
One of the most striking predictions of special relativity is that moving clocks run slow. If an observer at rest in S' measures a time interval Δt₀ between two events occurring at the same location in that frame (the proper time), an observer in S measures a longer interval Δt given by:
狭义相对论最引人注目的预测之一是，运动的时钟走得更慢。如果在 S' 系中静止的观察者测量发生在该系同一地点的两个事件之间的时间间隔 Δt₀（原时），则 S 系中的观察者会测得一个更长的时间间隔 Δt，其关系为：
Δt = γ Δt₀ = Δt₀ / √1 - v²/c²
where the Lorentz factor γ = 1 / √1 - v²/c². Since v < c, γ > 1 and Δt > Δt₀. This effect is negligible at everyday speeds but becomes dramatic as v approaches c. Time dilation has been confirmed by observing extended lifetimes of high‑speed muons from cosmic rays and by precise atomic clocks flown on aircraft.
其中洛伦兹因子 γ = 1 / √1 - v²/c²。由于 v < c，γ > 1 且 Δt > Δt₀。这一效应在日常速度下可以忽略不计，但当 v 接近 c 时会变得非常显著。时间膨胀已通过观测来自宇宙射线的高速 μ 子生命期的延长以及飞机上搭载的精密原子钟得到证实。
7. Length Contraction | 长度收缩
Objects in motion are measured to be shorter along the direction of motion. The proper length L₀ is the length of an object measured in the frame where it is at rest. In a frame where the object moves at speed v, its length L is given by:
运动的物体在其运动方向上被测得缩短。原长 L₀ 是在物体静止的参考系中测得的长度。在物体以速度 v 运动的参考系中，其长度 L 由下式给出：
L = L₀ / γ = L₀ √1 - v²/c²
Length contraction only applies to the dimension parallel to the relative velocity; perpendicular dimensions remain unchanged. This is a physical effect, not an optical illusion, and is necessary to maintain consistency with the postulates and with measurements in different frames.
长度收缩仅适用于平行于相对速度的维度；垂直维度保持不变。这是一种物理效应，而非光学错觉，并且对于与基本假设以及不同参考系中的测量保持一致是必需的。
8. The Lorentz Transformations | 洛伦兹变换
To replace the Galilean transformations and satisfy the constancy of the speed of light, the Lorentz transformations relate coordinates (x, t) in S to (x', t') in S' (where S' moves at velocity v along +x):
为了取代伽利略变换并满足光速不变性，洛伦兹变换将 S 系中的坐标 (x, t) 与 S' 系中的坐标 (x', t') 联系起来（S' 以速度 v 沿 +x 方向运动）：
x' = γ (x - vt)
t' = γ (t - vx / c²)
The inverse transformations are obtained by swapping v with – reduce to Galilean transformations. The Lorentz transformations also show that simultaneity is relative: two events that are simultaneous in S are not necessarily simultaneous in S'.
通过将 v 替换为 –v 可得到逆变换。在低速情况下（v << c），γ ≈ 1，这些变换退化为伽利略变换。洛伦兹变换还表明，同时性是相对的：在 S 系中同时发生的两个事件，在 S' 系中不一定同时。
9. Relativistic Velocity Addition | 相对论速度相加
The Galilean rule u = u' es the correct velocity addition formula for collinear motion. If an object has velocity u' in S' and S' moves at speed v relative to S, then the velocity u measured in S is:
伽利略速度叠加规则 u = u' + v 对于接近光速的速度失效。狭义相对论给出了共线运动的正确速度叠加公式。如果一个物体在 S' 中的速度为 u'，且 S' 相对于 S 以速度 v 运动，那么在 S 中测得的速度 u 为：
u = (u' + v) / (1 + u'v/c²)
This formula ensures that if u' n u = c regardless of v, consistent with the second postulate. For everyday speeds where u'v << c², the denominator is approximately 1 and the classical result is recovered.
该公式确保如果 u' = c，那么不论 v 是多少，u 都等于 c，这与第二条基本假设一致。在日常速度下，u'v << c²，分母约等于 1，经典结果得以恢复。
10. Relativistic Momentum | 相对论动量
To conserve momentum in all inertial frames, the definition of momentum must be modified. The relativistic momentum of a particle with rest mass m₀ and velocity v is:
为了使动量在所有惯性系中都守恒，必须修改动量的定义。一个静止质量为 m₀、速度为 v 的粒子的相对论动量为：
p = γ m₀ v = m₀ v / √1 - v²/c²
nd, meaning an infinite force would be needed to accelerate a massive object to the speed of light.
在低速下，γ ≈ 1 且 p ≈ m₀ v，恢复到牛顿动量。当 v → c 时，γ → ∞ 且动量无限增大，这意味着需要无限大的力才能将具有质量的物体加速到光速。
11. Relativistic Energy and Mass–Energy Equivalence | 相对论能量与质能等价
Einstein's famous equation arises naturally from the new dynamics. The total energy E of a particle is the sum of its rest energy and kinetic energy:
爱因斯坦著名的方程式自然地源于新的动力学。一个粒子的总能量 E 是其静止能量与动能之和：
E = γ m₀ c²
- m₀ c² = (γ - 1) m₀ c². A useful relation connecting energy, momentum, and rest mass is:
静止能量为 E₀ = m₀ c²，这意味着质量本身是能量的一种形式。动能为 K = E - m₀ c² = (γ - 1) m₀ c²。一个连接能量、动量和静止质量的有用关系是：
E² = (m₀ c²)² + (p c)²
xplains nuclear reactions where a tiny loss in mass releases enormous energy, as observed in the Sun and nuclear power.
对于像光子这样的零静止质量粒子，m₀ = 0，可得 E = p c。这一框架解释了核反应中微小的质量损失释放出巨大能量的现象，正如在太阳和核能中所观察到的那样。
12. Experimental Evidence and Applications | 实验证据与应用
ions. Muons created in the upper atmosphere reach the Earth’s surface because time dilation extends their lifetime. The Global Positioning System requires relativistic corrections to both satellite clocks and signal timing to achieve metre‑ ≈ 7000, and their increased momentum and energy perfectly match relativistic predictions.
狭义相对论不仅是一种理论构建；它在粒子加速器、GPS 技术和宇宙射线观测中每天都在得到验证。高层大气中产生的 μ 子能够到达地球表面，是因为时间膨胀延长了它们的寿命。全球定位系统需要对卫星时钟和信号时间进行相对论修正，才能达到米级精度。在大型强子对撞机中，质子被加速到 γ ≈ 7000 的能量，其增加的动量和能量与相对论预言完美吻合。
The equivalence of mass and energy is also demonstrated in pair production and annihilation, and in the mass defect of atomic nuclei. These confirmations give us confidence that the universe at high speeds truly obeys the rules of special relativity.
质能等价性也在粒子对产生和湮灭以及原子核的质量亏损中得到了证明。这些验证使我们确信，高速运动的宇宙确实遵循狭义相对论的规律。
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