📚 IB Physics Option H: Relativity Student Cards | IB物理选项H：相对论学生卡片
These student cards cover the core ideas in IB Physics Option H: Relativity. Relativity asks how measurements of time, length, mass and energy change when objects move close to the speed of light, and how gravity is described by curved spacetime. Use these bilingual notes to revise key definitions, equations and exam traps.
这套学生卡片涵盖 IB 物理选项 H：相对论的核心内容。相对论研究当物体接近光速时，时间、长度、质量和能量的测量如何变化，以及如何用弯曲时空描述引力。使用这些中英对照笔记来复习关键定义、方程和考试陷阱。
1. Frames of Reference | 参考系
A frame of reference is a coordinate system used to measure positions, times and motions. An inertial frame is one in which Newton’s first law holds: a free object moves at constant velocity in a straight line. A non-inertial frame accelerates or rotates relative to an inertial frame, so fictitious forces appear.
参考系是用于测量位置、时间和运动的坐标系统。惯性系是指牛顿第一定律成立的参考系：自由物体保持匀速直线运动。非惯性系相对于惯性系加速或旋转，因此会出现虚拟力。
In relativity, all inertial frames are equally valid for describing physical laws. There is no absolute rest frame. Identifying the correct frame is the first step in solving time dilation and length contraction problems.
在相对论中，所有惯性系对描述物理定律都是等价的。不存在绝对的静止参考系。正确识别参考系是解决时间膨胀和长度收缩问题的第一步。
2. Galilean Relativity and Velocity Addition | 伽利略相对性与速度合成
Before Einstein, Galilean relativity described relative motion at low speeds. If a frame S’ moves at speed v along the x-axis relative to frame S, then positions transform as x’ = x − vt, and velocities add simply as u’ = u − v. Time is assumed to be the same in both frames.
在爱因斯坦之前，伽利略相对性描述低速下的相对运动。如果参考系 S’ 相对于参考系 S 以速度 v 沿 x 轴运动，则位置变换为 x’ = x − vt，速度简单相加为 u’ = u − v。该理论假设两个参考系中的时间相同。
x’ = x − vt, u’ = u − v
This classical model works well for everyday speeds but fails when speeds approach c. It predicts that light speed would depend on the observer, which contradicts experiment.
这个经典模型在日常生活速度下效果很好，但当速度接近 c 时失效。它预言光速会依赖于观察者，这与实验相矛盾。
3. Postulates of Special Relativity | 狭义相对论的两条基本假设
Einstein’s special relativity rests on two postulates. First, the laws of physics are the same in all inertial reference frames. Second, the speed of light in a vacuum is constant at c = 3.00 × 10⁸ m/s for all observers, regardless of the motion of the source or observer.
爱因斯坦的狭义相对论建立在两条基本假设之上。第一，物理定律在所有惯性参考系中相同。第二，真空中的光速对所有观察者都是恒定值 c = 3.00 × 10⁸ m/s，与光源或观察者的运动无关。
- No experiment can detect absolute uniform motion.
- No information or massive object can reach or exceed c in vacuum.
- 任何实验都无法检测绝对的匀速运动。
- 任何信息或具有质量的物体都无法在真空中达到或超过光速 c。
These postulates force us to abandon absolute time and absolute simultaneity. Events that are simultaneous in one frame may not be simultaneous in another moving frame.
这两条假设迫使我们放弃绝对时间和绝对同时性。在一个参考系中同时发生的事件，在另一个运动参考系中可能不同时发生。
4. Time Dilation | 时间膨胀
Proper time Δt₀ is the time interval measured by an observer at rest relative to the event clock. A moving observer measures a longer time interval Δt. This is time dilation.
原时 Δt₀ 是相对于事件时钟静止的观察者测得的时间间隔。运动观察者测得的时间间隔 Δt 更长。这就是时间膨胀。
Δt = γΔt₀ = Δt₀ / √(1 − v²/c²)
Here γ is the Lorentz factor. For example, muons produced in the upper atmosphere reach the ground because their internal decay clock runs slow relative to Earth, increasing their lifetime in the Earth frame.
其中 γ 是洛伦兹因子。例如，在上层大气产生的 μ 子能到达地面，因为它的内部衰变时钟相对地球变慢，在地球参考系中寿命增加。
Always ask: which observer measures the proper time? It is the observer present at both events in the same place.
始终要问：哪个观察者测量原时？原时是在同一位置经历两个事件的观察者所测得的时间。
5. Length Contraction | 长度收缩
Proper length L₀ is the length measured by an observer at rest relative to the object. A moving observer measures a shorter length L along the direction of motion. Length contraction occurs only along the direction of relative motion, not perpendicular to it.
原长 L₀ 是相对于物体静止的观察者测得的长度。运动观察者沿运动方向测得的长度 L 更短。长度收缩只发生在相对运动方向上，垂直于运动方向的长度不变。
L = L₀ / γ = L₀√(1 − v²/c²)
Example: a spaceship travelling at 0.8c has γ = 5/3, so its measured length in the Earth frame is 3/5 of its rest length. The spaceship itself sees the Earth-frame distance to its destination as contracted by the same factor.
例如：一艘以 0.8c 飞行的宇宙飞船，其 γ = 5/3，因此在地球参考系中测得的长度是其静止长度的 3/5。飞船自身看到的到目的地的地球参考系距离也以相同比例收缩。
6. Lorentz Transformations | 洛伦兹变换
The Lorentz transformations replace Galilean transformations at high speed. For relative motion along the x-axis, space and time coordinates transform as follows.
洛伦兹变换在高速下取代伽利略变换。对于沿 x 轴的相对运动，空间和时间坐标变换如下。
x’ = γ(x − vt), t’ = γ(t − vx/c²)
In these equations, γ = 1 / √(1 − v²/c²). The transformation mixes space and time, which explains why simultaneity, length and time are relative. At low speeds, γ ≈ 1 and the Lorentz transformations reduce to the Galilean forms.
在这些方程中，γ = 1 / √(1 − v²/c²)。变换混合了空间和时间，这解释了为什么同时性、长度和时间都是相对的。在低速下，γ ≈ 1，洛伦兹变换退化为伽利略变换。
Exam questions often ask you to use the invariant interval: (Δs)² = (cΔt)² − (Δx)². This quantity is the same in all inertial frames.
考试题经常要求使用不变间隔：(Δs)² = (cΔt)² − (Δx)²。这个量在所有惯性系中相同。
7. Relativistic Momentum and Energy | 相对论动量和能量
At relativistic speeds, momentum no longer equals mv. It must be multiplied by the Lorentz factor. Total energy also includes the Lorentz factor.
在相对论速度下，动量不再等于 mv。它必须乘以洛伦兹因子。总能量也包含洛伦兹因子。
p = γmv, E = γmc²
As v approaches c, γ increases without bound, so an infinite amount of energy would be needed to accelerate a massive object to c. This is why c is a limiting speed.
当 v 接近 c 时，γ 无限增大，因此将一个有质量的物体加速到 c 需要无限大的能量。这就是为什么 c 是一个极限速度。
For a particle at rest, v = 0, so γ = 1 and the rest energy is E₀ = mc². The kinetic energy is E_k = E − E₀ = (γ − 1)mc².
对于静止粒子，v = 0，因此 γ = 1，静能为 E₀ = mc²。动能为 E_k = E − E₀ = (γ − 1)mc²。
8. Mass-Energy Equivalence | 质能等价
Einstein’s famous relation E₀ = mc² states that mass and energy are interchangeable. Even a small amount of mass can release enormous energy, as seen in nuclear fission, fusion and particle-antiparticle annihilation.
爱因斯坦著名的关系式 E₀ = mc² 表明质量与能量可以相互转换。即使少量质量也能释放巨大能量，这在核裂变、核聚变和正反粒子湮灭中可以看到。
E₀ = mc², E² = (pc)² + (m₀c²)²
The second relation connects total energy, momentum and rest mass. For a massless particle such as a photon, m₀ = 0, so E = pc. The momentum of a photon is p = h/λ, where h is Planck’s constant and λ is wavelength.
第二个关系式联系总能量、动量和静质量。对于无质量粒子如光子，m₀ = 0，因此 E = pc。光子的动量为 p = h/λ，其中 h 是普朗克常数，λ 是波长。
In nuclear reactions, the mass defect Δm is converted into energy ΔE = Δm × c². Always use SI units: mass in kg, c = 3.00 × 10⁸ m/s, energy in J.
在核反应中，质量亏损 Δm 转化为能量 ΔE = Δm × c²。始终使用国际单位：质量用 kg，c = 3.00 × 10⁸ m/s，能量用 J。
9. General Relativity: The Equivalence Principle | 广义相对论：等效原理
General relativity extends relativity to accelerated frames and gravity. The equivalence principle states that a uniform gravitational field is locally indistinguishable from a uniformly accelerating reference frame.
广义相对论将相对论扩展到加速参考系和引力。等效原理指出，均匀引力场在局部与匀加速参考系无法区分。
This leads to the idea that gravity is not a force in the Newtonian sense, but a curvature of spacetime caused by mass and energy. Light follows geodesics in curved spacetime, so a light beam bends near a massive object.
这引出一个观点：引力不是牛顿意义上的力，而是质量和能量引起的时空弯曲。光在弯曲时空中沿测地线传播，因此光束在巨大质量附近会发生偏折。
Key exam applications include gravitational lensing, gravitational redshift and the precession of planetary orbits.
关键的考试应用包括引力透镜、引力红移和行星轨道进动。
10. Gravitational Time Dilation and Black Holes | 引力时间膨胀与黑洞
Clocks in stronger gravitational fields run slower. The gravitational time dilation formula for a non-rotating spherical mass M is below, where t₀ is proper time far from the mass and t is time at radius r.
在较强引力场中的时钟走得更慢。对于非旋转球对称质量 M，引力时间膨胀公式如下，其中 t₀ 是远离质量的固有时间，t 是在半径 r 处的时间。
t = t₀ / √(1 − 2GM/(rc²))
A black hole occurs when the escape velocity reaches c. The Schwarzschild radius is the boundary of the region from which light cannot escape.
当逃逸速度达到 c 时就会形成黑洞。史瓦西半径是光无法逃逸的区域边界。
Rₛ = 2GM / c²
For the Sun, Rₛ ≈ 3 km. For Earth, Rₛ ≈ 9 mm. These small values show how intense gravity must be to trap light.
对于太阳，Rₛ ≈ 3 km。对于地球，Rₛ ≈ 9 mm。这些很小的数值表明，要束缚光线，引力必须非常强。
11. Experimental Evidence | 实验证据
Relativity is supported by many experiments. One famous example is muon decay: muons produced at high altitude reach the Earth’s surface in far greater numbers than classical physics predicts because their lifetime is dilated in the Earth frame.
相对论得到许多实验的支持。一个著名例子是 μ 子衰变：在高空产生的 μ 子到达地球表面的数量远多于经典物理预测，因为它们在地球参考系中的寿命被延长了。
Atomic clocks flown in aircraft show time dilation compared with clocks on the ground. Global Positioning System satellites must correct for both special and general relativistic time shifts to maintain accuracy.
飞机上飞行的原子钟显示出相对于地面时钟的时间膨胀。全球定位系统卫星必须同时修正狭义和广义相对论的时间偏移，才能保持精度。
Gravitational lensing by galaxies and the 1919 solar eclipse observation of starlight bending confirmed general relativity. Gravitational waves detected by LIGO also verify Einstein’s theory.
星系产生的引力透镜效应以及 1919 年日食观测到的星光偏折证实了广义相对论。LIGO 探测到的引力波也验证了爱因斯坦的理论。
12. Exam Tips and Common Pitfalls | 考试技巧与常见误区
Always identify the two reference frames before writing equations. For time dilation, decide which observer measures proper time: the one at rest with respect to the event clock, not necessarily the one moving faster in the question’s story.
在写方程之前，始终先确定两个参考系。对于时间膨胀，判断哪个观察者测量原时：原时是与事件时钟相对静止的观察者，而不一定是题目叙述中运动更快的那个。
Never use classical velocity addition at high speed. Use the relativistic velocity transformation u’ = (u − v) / (1 − uv/c²). Do not mix proper length with proper time carelessly.
切勿在高速下使用经典速度相加。应使用相对论速度变换 u’ = (u − v) / (1 − uv/c²)。不要粗心混淆原长和原时。
Convert everything to SI units before adding energies or momenta. Remember that c is huge, so small mass defects produce very large energies. In written answers, define γ and show your substitutions.
在能量和动量相加之前，先统一换算为国际单位。记住 c 的数值很大，因此很小的质量亏损会产生非常大的能量。在书面答案中，定义 γ 并展示代入过程。
Many losing marks come from using a classical formula at v = 0.5c or above. Check whether γ is significantly greater than 1 before simplifying. Practice past paper problems on muons, spaceships and mesons.
许多失分来自在 v = 0.5c 或更高时使用经典公式。在简化之前，检查 γ 是否明显大于 1。练习往年真题中关于 μ 子、宇宙飞船和介子的问题。
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