IB Physics: Galileo’s Relativity & Special Relativity | 伽利略相对性与狭义相对论考点精讲

📚 IB Physics: Galileo’s Relativity & Special Relativity | 伽利略相对性与狭义相对论考点精讲

Relativity is one of the most elegant and counter-intuitive topics in IB Physics. This guide breaks down Galileo’s principle of relativity, Einstein’s two postulates, time dilation, length contraction and energy-mass equivalence into exam-ready, step-by-step knowledge.

相对论是 IB 物理中最优雅也是最反直觉的考点之一。本精讲将伽利略相对性原理、爱因斯坦两大假设、时间膨胀、长度收缩和质能方程拆解为可直接应试的步步要点，帮助你在考试中准确拿分。

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1. Frames of Reference | 参考系

A frame of reference is a coordinate system from which motion is observed and measured. An inertial frame is one where Newton’s first law holds: an object at rest stays at rest, and an object in motion continues in uniform motion unless acted on by a net force. Inertial frames move at constant velocity with respect to each other.

参考系是观察和测量运动所依据的坐标系。惯性参考系是牛顿第一定律成立的参考系：静止物体保持静止，运动物体保持匀速直线运动，除非受到合外力作用。惯性参考系之间以恒定速度相对运动。

• In physics problems, the Earth’s surface is usually treated as an inertial frame.

  在物理问题中，地球表面通常被近似视为惯性参考系。

• An accelerating car, a spinning merry-go-round, or a free-falling elevator are non-inertial frames.

  加速行驶的汽车、旋转的转盘或自由下落的电梯都是非惯性参考系。

• IB exam questions often ask you to identify whether a given observer is inertial or not.

  IB 考试常要求你判断给定观察者是否为惯性系。

Key idea: Inertial frames differ only by constant relative velocity, not by acceleration.

核心概念：惯性参考系之间只相差恒定相对速度，而非加速度。

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2. Galileo’s Principle of Relativity | 伽利略相对性原理

Galileo argued that the laws of mechanics are identical in all inertial frames. No mechanical experiment performed inside a uniformly moving laboratory can reveal whether the laboratory is at rest or moving at constant velocity. You cannot “feel” constant velocity — only acceleration.

伽利略认为，力学定律在所有惯性参考系中完全相同。在匀速运动的实验室内部进行的任何力学实验，都无法揭示该实验室是静止还是做匀速运动。你无法”感受”恒定速度——只能感受加速度。

Imagine you are on a smoothly flying airplane. Dropping a ball vertically, it lands at your feet — exactly as it would on the ground. The ball’s horizontal motion (inherited from the plane) does not affect the vertical drop. This is Galilean relativity in action.

想象你在平稳飞行的飞机上。垂直释放一个球，它会落在你的脚边——和在地面上完全一样。球的水平运动（来自飞机）不影响垂直下落。这就是伽利略相对性的实际体现。

The laws of mechanics are invariant across all inertial frames.

力学定律在所有惯性参考系中保持不变。

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3. Galilean Transformations | 伽利略变换

The Galilean transformation relates the coordinates of an event as measured in two inertial frames. Consider frame S’ moving at velocity v along the x-axis relative to frame S, with origins coinciding at t = 0:

伽利略变换给出同一事件在两个惯性参考系中的坐标关系。设参考系 S’ 相对于 S 沿 x 轴以速度 v 运动，且在 t = 0 时两原点重合：

x’ = x − vt   y’ = y   z’ = z   t’ = t

Velocities add in Galilean relativity. If a train moves at v and a passenger walks forward at speed u relative to the train, the passenger’s speed relative to the ground is simply u + v.

在伽利略相对论中，速度直接相加。若火车以 v 运动，乘客相对火车以 u 向前走，则乘客相对地面的速度就是 u + v。

Velocity addition:  u = u’ + v

速度叠加公式：  u = u’ + v

However, this simple addition predicts that light speed should change with the observer’s motion — a prediction that experiments have decisively contradicted. This contradiction forced Einstein to rethink the nature of space and time.

然而，这种简单相加预言光速应随观察者运动而变化——这一预言被实验坚决否定。这一矛盾迫使爱因斯坦重新思考时间和空间的本质。

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4. The Michelson-Morley Experiment | 迈克尔逊-莫雷实验

In 1887, Michelson and Morley attempted to detect the “aether” — the hypothetical medium through which light was thought to travel. Using an interferometer, they compared the speed of light in two perpendicular directions as the Earth moved through the supposed aether.

1887 年，迈克尔逊和莫雷试图探测”以太”——当时被认为承载光传播的假想介质。他们利用干涉仪比较地球穿过假设以太时，光在相互垂直两个方向上的传播速度。

The result was null: the speed of light was identical in all directions and at all times of year, regardless of the Earth’s motion. There was no detectable aether, and the speed of light did not obey Galilean velocity addition.

结果是零结果：光速在所有方向、所有季节都完全相同，与地球运动无关。以太探测不到，光速也不服从伽利略速度叠加。

• If aether existed, the fringe pattern would shift as the apparatus rotated.

  如果以太存在，旋转仪器时干涉条纹应当移动。

• No fringe shift was observed — light speed is constant.

  实际未观察到条纹移动——光速恒定。

This provided strong experimental evidence that the speed of light is the same for all inertial observers, setting the stage for Einstein’s theory.

这为”光速对所有惯性观察者相同”提供了强有力的实验证据，为爱因斯坦的理论铺平了道路。

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5. Einstein’s Two Postulates | 爱因斯坦两大假设

In 1905, Einstein proposed special relativity based on two fundamental postulates:

1905 年，爱因斯坦基于两条基本假设提出了狭义相对论：

Postulate 1 (Principle of Relativity): The laws of physics are identical in all inertial frames of reference. This extends Galileo’s principle from mechanics to all of physics, including electromagnetism.

第一假设（相对性原理）：所有惯性参考系中物理定律完全相同。这把伽利略原理从力学推广到全部物理定律，包括电磁学。

Postulate 2 (Constancy of the Speed of Light): The speed of light in a vacuum, c, is the same in all inertial frames — regardless of the motion of the source or the observer.

第二假设（光速不变原理）：真空中的光速 c 在所有惯性参考系中都相同——无论光源或观察者如何运动。

c = 3.00 × 10⁸ m/s  in all inertial frames

c = 3.00 × 10⁸ m/s  在所有惯性系中恒定

These two postulates lead to startling consequences: time and space are not absolute — they depend on the observer’s state of motion. The key parameter is the Lorentz factor γ.

这两条假设引出惊人结论：时间和空间并非绝对——它们取决于观察者的运动状态。核心参数是洛伦兹因子 γ。

γ = 1 / √(1 − v²/c²)

γ = 1 / √(1 − v²/c²)（洛伦兹因子）

When v is much smaller than c, γ ≈ 1, and relativistic effects are negligible. As v → c, γ → ∞, meaning infinite energy would be required to reach light speed.

当 v 远小于 c 时，γ ≈ 1，相对论效应可忽略。当 v → c 时，γ → ∞，意味着要达到光速需要无穷大的能量。

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6. Time Dilation | 时间膨胀

Time dilation states that a moving clock runs slower when observed from a stationary frame. If a time interval Δt₀ is measured in the frame where the clock is at rest (called the proper time), an observer moving relative to that clock measures a longer interval Δt:

时间膨胀指运动时钟在静止观察者看来走得变慢。若时间间隔 Δt₀ 在时钟静止的参考系中测得（称为固有时间），相对时钟运动的观察者测得的间隔 Δt 更长：

Δt = γ Δt₀  (where Δt₀ is the proper time)

Δt = γ Δt₀  （其中 Δt₀ 为固有时间）

The twin paradox is a classic IB exam scenario. One twin travels at high speed to a distant star and returns; the travelling twin ages less than the twin who stayed on Earth. The resolution: the travelling twin’s frame is non-inertial because they must accelerate to turn around, so the symmetry is broken.

双生子佯谬是 IB 考试经典情景。一对双胞胎中一人高速往返遥远恒星，旅行者比地球上的双胞胎老得慢。解释：旅行者的参考系是非惯性系，因为必须加速才能掉头，对称性被打破。

• Proper time Δt₀ is always the shortest time between two events.

  固有时间 Δt₀ 总是两个事件之间的最短时间间隔。

• Muons created in the upper atmosphere reach Earth’s surface because time dilation extends their lifetime.

  高层大气中产生的 μ 子能到达地面，正是因为时间膨胀延长了它们的寿命。

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7. Length Contraction | 长度收缩

Length contraction: an object moving relative to an observer is measured to be shorter along the direction of motion. If L₀ is the proper length (length measured in the object’s rest frame), then the moving observer measures:

长度收缩：物体相对观察者运动时，沿运动方向测得长度变短。若 L₀ 是固有长度（在物体静止参考系中测得的长度），则运动观察者测得：

L = L₀ / γ  (contraction only along the direction of motion)

L = L₀ / γ  （仅沿运动方向收缩）

Important details for exam success:

应试要点：

• Perpendicular dimensions are unaffected — a moving cube still has the same height and width, only its depth shrinks.

  垂直方向不受影响——运动立方体的高和宽不变，只有深度收缩。

• Proper length L₀ is measured in the frame where the object is at rest.

  固有长度 L₀ 在物体静止的参考系中测得。

• Contraction is reciprocal: each observer sees the other’s measuring rods shortened.

  收缩是相互的：每个观察者都看到对方的量尺缩短。

Example: A spaceship of proper length 100 m travels at v = 0.6c. The Lorentz factor is γ = 1/√(1 − 0.36) = 1.25. An Earth observer measures the ship’s length as L = 100/1.25 = 80 m.

例：固有长度 100 m 的飞船以 v = 0.6c 飞行。洛伦兹因子 γ = 1/√(1 − 0.36) = 1.25。地球观察者测得飞船长度为 L = 100/1.25 = 80 m。

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8. Lorentz Transformations | 洛伦兹变换

The Lorentz transformations replace the Galilean transformations to correctly relate spacetime coordinates between inertial frames. For frame S’ moving at speed v relative to S along x:

洛伦兹变换取代伽利略变换，正确关联惯性参考系之间的时空坐标。对沿 x 方向相对 S 以速度 v 运动的 S’ 系：

x’ = γ(x − vt)

t’ = γ(t − vx/c²)

Note that time and space coordinates are intertwined: an event that is simultaneous in one frame is not necessarily simultaneous in another. This destroys the Newtonian idea of absolute simultaneity.

注意时间和空间坐标相互纠缠：在一个参考系中同时的事件，在另一个参考系中不一定同时。这摧毁了牛顿的绝对同时性观念。

Velocity addition under special relativity is no longer simple addition but:

狭义相对论中的速度叠加不再是简单相加，而是：

u = (u’ + v) / (1 + u’v/c²)

u = (u’ + v) / (1 + u’v/c²)（相对论速度叠加）

This formula guarantees that the resultant speed never exceeds c — even if u’ = 0.9c and v = 0.9c, the combined speed is only about 0.994c.

该公式保证合速度永不超过 c——即使 u’ = 0.9c 且 v = 0.9c，合速度也只有约 0.994c。

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9. Relativistic Momentum | 相对论动量

In special relativity, classical momentum p = mv must be modified because velocity addition is no longer linear. The relativistic momentum is:

在狭义相对论中，经典动量 p = mv 需要修正，因为速度叠加不再是线性的。相对论动量为：

p = γmv

p = γmv（相对论动量）

Conservation of momentum still holds in all inertial frames, but only when using the relativistic form. As v → c, γ → ∞, so the momentum grows without bound — another way to see why material objects cannot reach light speed.

动量守恒在所有惯性参考系中仍然成立，但必须使用相对论形式。当 v → c 时，γ → ∞，动量无限增大——这再次说明实物无法达到光速。

In IB problems, you may be asked to calculate γ and then the momentum of a particle at a given speed. Memorise the formula and always state which frame your measurement refers to.

IB 题目中，你可能需要先计算 γ，再求给定速度下粒子的动量。记住公式，并始终说明你的测量对应哪个参考系。

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10. Mass-Energy Equivalence | 质能等价

Einstein’s most famous result states that mass and energy are equivalent. The total energy of a particle of mass m moving at speed v is:

爱因斯坦最著名的结论是质量和能量等价。质量为 m、速度为 v 的粒子的总能量为：

E = γmc² = mc² + K

E = γmc² = mc² + K（总能量）

Here, E₀ = mc² is the rest energy, and K = (γ − 1)mc² is the relativistic kinetic energy. When v = 0, γ = 1, so E = mc².

其中 E₀ = mc² 是静止能量，K = (γ − 1)mc² 是相对论动能。当 v = 0 时 γ = 1，因此 E = mc²。

For low speeds (v ≪ c), K approaches the classical value ½mv², which you can show by a binomial expansion of γ. This is an elegant check the exam board loves to explore conceptually.

低速时（v ≪ c），K 趋近经典值 ½mv²，可用二项式展开 γ 证明。这是考试局偏爱考查的概念性联系。

• Mass-energy equivalence explains nuclear fission and fusion: a small mass defect releases enormous energy.

  质能等价解释核裂变与核聚变：极小的质量亏损释放巨大能量。

• In particle physics, E = mc² is used to calculate the energy required to create new particles.

  粒子物理中，E = mc² 用于计算创生新粒子所需的能量。

• Energy and momentum are related by E² = (pc)² + (mc²)² — a key formula for massless particles like photons.

  能量和动量的关系为 E² = (pc)² + (mc²)² ——这是光子等无质量粒子的关键公式。

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11. Common Exam Pitfalls | 常见考点误区

Many IB students lose marks on relativity due to a few recurring mistakes. Avoid these:

许多 IB 学生在相对论题目上丢分，主要是反复踩中以下误区：

Mistake 1: Confusing proper time with measured time. Proper time Δt₀ is always measured in the frame where the clock is at rest. The dilated time Δt = γΔt₀ is measured in a frame where the clock moves.

误区一：混淆固有时间与测得时间。固有时间 Δt₀ 始终在时钟静止的参考系中测量；膨胀时间 Δt = γΔt₀ 在时钟运动的参考系中测量。

Mistake 2: Applying length contraction to the wrong direction. Only lengths parallel to the motion contract; perpendicular lengths do not.

误区二：对错误方向应用长度收缩。只有平行于运动方向的长度收缩；垂直方向不变。

Mistake 3: Forgetting that γ is always ≥ 1. If you compute γ < 1, you have inverted the formula or misapplied the condition.

误区三：忘记 γ 始终 ≥ 1。如果算出 γ < 1，说明公式颠倒或条件用错。

Mistake 4: Using u = u’ + v when speeds approach c. Always use the relativistic velocity addition formula in such cases.

误区四：速度接近 c 时仍用 u = u’ + v。此时必须使用相对论速度叠加公式。

Mistake 5: Saying “mass increases with speed”. Modern IB convention prefers relativistic momentum γmv and total energy γmc², avoiding the concept of “relativistic mass”. Use γ explicitly instead.

误区五：说”质量随速度增大”。现代 IB 惯例使用相对论动量 γmv 和总能量 γmc²，回避”相对论质量”概念。请明确用 γ 表达。

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12. Summary & Key Formulas | 总结与核心公式

For a quick revision session, here is the complete set of IB-relativity formulas you must master:

快速复习时，以下是 IB 相对论必须掌握的完整公式清单：

Concept | 概念	Formula | 公式	Note | 注意
Lorentz factor | 洛伦兹因子	γ = 1/√(1 − v²/c²)	γ ≥ 1 always | 恒有 γ ≥ 1
Time dilation | 时间膨胀	Δt = γΔt₀	Δt₀ = proper time | 固有时间
Length contraction | 长度收缩	L = L₀/γ	Only along motion | 仅沿运动方向
Momentum | 动量	p = γmv	Conserved in all frames | 所有系中守恒
Total energy | 总能量	E = γmc²	Includes rest energy | 含静止能量
Rest energy | 静止能量	E₀ = mc²	When v = 0 | 当 v = 0
Energy-momentum | 能量-动量	E² = (pc)² + (mc²)²	Useful for photons | 对光子尤其有用
Velocity addition | 速度叠加	u = (u′ + v)/(1 + u′v/c²)	Never exceeds c | 永不超过 c

When tackling IB relativity questions, always follow this strategy: identify the frame of reference, determine which quantity is the “proper” one (measured in the rest frame), compute γ, and then apply the correct transformation. State your assumptions and check whether the velocity is an appreciable fraction of c.

解答 IB 相对论题目时应遵循以下策略：确定参考系，判断哪个量是”固有”量（在静止参考系中测量），计算 γ，然后应用正确的变换。说明你的假设，并检查速度是否为 c 的显著比例。

Mastering relativity is not just about memorising formulas — it is about understanding which observer measures which quantity. With the framework above, you are now equipped to handle any IB special relativity question with confidence.

掌握相对论不只是背公式——关键在于理解哪个观察者测量哪个量。有了以上框架，你已能自信应对任何 IB 狭义相对论考题。

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