Hubble's Law and the Expanding Universe | 哈勃定律与宇宙膨胀

📚 Hubble’s Law and the Expanding Universe | 哈勃定律与宇宙膨胀

One of the most profound discoveries in modern cosmology is that the Universe is not static — it is expanding. Hubble’s Law provides the quantitative relationship between the recession velocity of distant galaxies and their distance from Earth, forming the observational foundation of the Big Bang theory.

现代宇宙学中最深刻的发现之一，就是宇宙并非静止不动——它正在膨胀。哈勃定律给出了遥远星系的退行速度与它们到地球距离之间的定量关系，成为大爆炸理论的观测基础。

In this revision guide, we break down everything you need to know about Hubble’s Law for your A-Level Physics exams: the formula, the meaning of the Hubble constant, how redshift supports expansion, and how to estimate the age of the Universe.

在本复习指南中，我们将系统梳理 A-Level 物理考试中关于哈勃定律的全部考点：公式、哈勃常数的含义、红移如何支持宇宙膨胀，以及如何估算宇宙年龄。

1. The Cosmological Principle | 宇宙学原理

Before we can interpret Hubble’s observations, we must assume a foundational idea: the cosmological principle. It states that on sufficiently large scales, the Universe is homogeneous (the same in all places) and isotropic (the same in all directions).

在解读哈勃的观测之前，我们必须先接受一个基础观念：宇宙学原理。它指出在足够大的尺度上，宇宙是均匀的（所有位置都相同）且各向同性的（所有方向都相同）。

There is nothing special about our location in the Universe. Every galaxy sees a similar pattern — distant galaxies moving away from it. The expansion has no single centre; it happens everywhere.

我们在宇宙中的位置没有任何特殊性。每个星系都看到相似的图景——远处的星系正在远离它。膨胀没有单一的中心，它发生在所有地方。

2. Doppler Redshift | 多普勒红移

Galaxy spectra show absorption lines at wavelengths longer than their laboratory values. This shift toward longer wavelengths is called redshift. For nearby galaxies, this is well described by the Doppler effect formula:

星系光谱中的吸收线波长比实验室值更长。这种向长波方向的移动被称为红移。对于较近的星系，多普勒效应公式可以很好地描述这一现象：

z = Δλ / λ = v / c

where z is the redshift, Δλ is the wavelength shift, λ is the rest wavelength, v is the recession velocity, and c is the speed of light (3.0 × 10⁸ m s⁻¹).

其中 z 是红移量，Δλ 是波长变化量，λ 是静止波长，v 是退行速度，c 是光速（3.0 × 10⁸ m s⁻¹）。

As the source moves away from the observer, the waves are stretched, increasing the observed wavelength. This is analogous to the pitch of an ambulance siren dropping as it moves away from you.

当光源远离观测者时，波被拉长，观测到的波长增大。这类似于救护车远去时警笛音调变低的现象。

3. Hubble’s Law | 哈勃定律

In 1929, Edwin Hubble discovered a remarkable correlation: the recession velocity v of a galaxy is directly proportional to its distance d from us.

1929 年，埃德温·哈勃发现了一个惊人的关联：星系的退行速度 v 与它到我们的距离 d 成正比。

v = H₀ × d

where H₀ is the Hubble constant. This is arguably the single most important equation in observational cosmology.

其中 H₀ 是哈勃常数。这可以说是观测宇宙学中最重要的一个方程。

The V-shape of the equation means that a galaxy twice as far away recedes twice as fast. However, this does NOT mean we are at the centre — the same pattern is seen from any galaxy.

这个方程意味着距离加倍，退行速度也加倍。但这并不表示我们位于宇宙中心——从任何星系看都是同样的规律。

4. The Hubble Constant H₀ | 哈勃常数 H₀

The Hubble constant is the gradient of the v–d graph. Its accepted modern value, from the Planck satellite measurements, is approximately:

哈勃常数是 v–d 图像的斜率。根据普朗克卫星的现代测量，其值约为：

H₀ ≈ 2.2 × 10⁻¹⁸ s⁻¹

Equivalently, it is often quoted in units of km s⁻¹ Mpc⁻¹ (kilometres per second per megaparsec):

通常也以 km s⁻¹ Mpc⁻¹（千米每秒每百万秒差距）为单位给出：

H₀ ≈ 68 km s⁻¹ Mpc⁻¹

Note the conversion: 1 Mpc = 3.26 × 10⁶ light-years = 3.1 × 10²² m. The units look strange because velocity and distance are different physical quantities — the constant simply sets the scale of the expansion rate.

注意单位换算：1 Mpc = 3.26 × 10⁶ 光年 = 3.1 × 10²² m。这个单位看起来很特别，因为速度和距离是不同物理量——该常数只是设定了膨胀速率的标度。

5. Measuring H₀ | 测量哈勃常数

To measure H₀, we need two things for each galaxy: its distance d and its recession velocity v. The velocity is obtained from the redshift of its absorption lines using v = zc. Distances are more challenging.

测量 H₀ 需要每个星系的两个数据：距离 d 和退行速度 v。速度可通过吸收线红移使用 v = zc 获得。距离则更富挑战性。

Astronomers use a ‘distance ladder’. Cepheid variable stars serve as standard candles: their pulsation period is directly related to their luminosity. By comparing a Cepheid’s known luminosity with its apparent brightness, its distance can be found.

天文学家使用”距离阶梯”。造父变星作为标准烛光：其脉动周期与光度直接相关。通过比较造父变星已知的光度和视亮度，可以求出距离。

For even more distant galaxies, Type Ia supernovae are used as standard candles, as they peak at a known intrinsic brightness. These extend measurements to billions of parsecs.

对于更遥远的星系，Ia 型超新星被用作标准烛光，因为它们的峰值亮度是已知的本征亮度。这可将测量延伸到数十亿秒差距之外。

6. Age of the Universe | 宇宙年龄的估算

Hubble’s Law leads to a remarkably simple estimate of the age of the Universe. If we run the expansion backwards in time, all matter was located at a single point. The time that has elapsed since then is approximately:

哈勃定律引出一个非常简洁的宇宙年龄估算。如果我们将膨胀在时间上倒推，所有物质曾位于同一个点。从那时起至今所经过的时间约为：

t ≈ 1 / H₀

To see why: a galaxy at distance d recedes at speed v = H₀d. The time taken to reach that separation from zero is t = d/v = d/(H₀d) = 1/H₀.

原因如下：距离为 d 的星系以速度 v = H₀d 退行。从零距离分开到当前距离所需时间为 t = d/v = d/(H₀d) = 1/H₀。

With H₀ = 2.2 × 10⁻¹⁸ s⁻¹, we find t = 4.5 × 10¹⁷ s ≈ 1.4 × 10¹⁰ years (14 billion years). This agrees remarkably well with independent age estimates from stellar evolution and CMB measurements.

代入 H₀ = 2.2 × 10⁻¹⁸ s⁻¹，得 t = 4.5 × 10¹⁷ s ≈ 1.4 × 10¹⁰ 年（140 亿年）。这与从恒星演化和 CMB 测量得到独立年龄估计惊人地一致。

7. Cosmological Redshift — Beyond Simple Doppler | 宇宙学红移——超越简单多普勒效应

For very distant galaxies, the redshift is caused not just by the Doppler effect at emission, but by the expansion of space itself. As photons travel toward us, the space they pass through expands, stretching their wavelength. This is called cosmological redshift.

对于极遥远的星系，红移不仅由发射时的多普勒效应引起，更是空间本身膨胀的结果。光子在向我们传播的过程中，途经的空间在不断膨胀，从而拉长了它们的波长。这被称为宇宙学红移。

This distinction matters for large redshifts. When z > 1, the simple formula v = zc breaks down and you must use the relativistic Doppler formula:

这种区别对大红移系统很关键。当 z > 1 时，简单公式 v = zc 不再适用，必需使用相对论性多普勒公式：

z = √((1 + v/c) / (1 − v/c)) − 1

At A-Level, you are usually expected to use the non-relativistic approximation v = zc for z ≤ 0.1, and be aware that the full relativistic treatment applies for larger values.

在 A-Level 考试中，通常要求对 z ≤ 0.1 的星系使用非相对论近似 v = zc，并了解更大红移值需要使用完整的相对论公式。

8. The Expanding Universe and the Big Bang | 膨胀宇宙与大爆炸

The observation that all galaxies recede from each other implies that the Universe began from an extremely hot, dense state — the Big Bang. Evidence supporting this includes:

所有星系彼此远离的观测暗示着宇宙始于一个极热、极密的状态——大爆炸。支持这一理论的证据包括：

The Cosmic Microwave Background (CMB) — a faint afterglow of the early Universe, at 2.7 K, detected in all directions.

宇宙微波背景辐射（CMB）——早期宇宙的余晖，温度为 2.7 K，在所有方向均可探测到。
The abundance of light elements (hydrogen, helium, lithium) matches predictions from Big Bang nucleosynthesis.

轻元素（氢、氦、锂）的丰度与大爆炸核合成预测一致。
The redshift–distance relationship itself, which cannot be explained by a static Universe.

红移–距离关系本身即是证据，静态宇宙无法解释这一现象。

9. Dark Energy and Accelerated Expansion | 暗能量与加速膨胀

Surprising observations of distant Type Ia supernovae (1998 Nobel Prize in Physics) revealed that the expansion of the Universe is NOT slowing down due to gravity — it is accelerating. This discovery was completely unexpected.

对遥远 Ia 型超新星的观测（1998 年诺贝尔物理学奖）揭示了一个惊人事实：宇宙膨胀并没有因引力而减速——而是在加速。这一发现完全出乎意料。

The cause is labelled ‘dark energy’, a mysterious form of energy permeating space. In the standard ΛCDM model, dark energy accounts for about 68% of the total energy content of the Universe.

其原因被名为”暗能量”，一种弥漫于空间的未知能量形式。在标准 ΛCDM 模型中，暗能量约占宇宙总能量的 68%。

This acceleration means the age estimate t = 1/H₀ is an approximation. The true age of the Universe (13.8 billion years) is slightly larger than 1/H₀ because expansion was faster in the distant past.

这种加速意味着 t = 1/H₀ 的年龄估算只是一个近似值。宇宙的真实年龄（138 亿年）略大于 1/H₀，因为远古时期膨胀更快。

10. Worked Example | 典型例题解析

Question: A distant galaxy shows a hydrogen absorption line at 680 nm, while its laboratory wavelength is 656 nm. Calculate (a) the redshift z, (b) the recession velocity v, and (c) the distance to the galaxy using H₀ = 68 km s⁻¹ Mpc⁻¹.

题目：某遥远星系的氢吸收线波长测得为 680 nm，而实验室静止波长为 656 nm。求：(a) 红移 z；(b) 退行速度 v；(c) 使用 H₀ = 68 km s⁻¹ Mpc⁻¹ 计算该星系的距离。

(a) z = Δλ/λ = (680 − 656)/656 = 24/656 = 0.0366

(b) Using v = zc: v = 0.0366 × 3.0 × 10⁸ = 1.10 × 10⁷ m s⁻¹ = 1.1 × 10⁴ km s⁻¹

(c) From v = H₀d: d = v/H₀ = 1.1 × 10⁴ / 68 = 162 Mpc ≈ 1.6 × 10² Mpc

Answer check: An expansion age t = 1/H₀ ≈ 1.4 × 10¹⁰ years. A galaxy 162 Mpc away at 1.1 × 10⁴ km s⁻¹ would indeed have taken about 14 billion years to get there.

答案核对：膨胀年龄 t = 1/H₀ ≈ 1.4 × 10¹⁰ 年。距离 162 Mpc、速度 1.1 × 10⁴ km s⁻¹ 的星系正好需要大约 140 亿年才到达这个位置。

11. Common Exam Pitfalls | 常见考试易错点

Avoid these mistakes to secure full marks in exam questions:

在考试中避免以下错误以确保获得满分：

Confusing redshift z with wavelength λ. z is dimensionless; λ is measured in metres or nanometres.

将红移 z 与波长 λ 混淆。z 是无量纲的，而 λ 以米或纳米为单位。
Forgetting unit conversion: always convert km s⁻¹ Mpc⁻¹ to s⁻¹ before using t = 1/H₀. 1 Mpc = 3.1 × 10¹⁹ km, so H₀ = 68 km s⁻¹ Mpc⁻¹ = 68 / (3.1 × 10¹⁹) = 2.2 × 10⁻¹⁸ s⁻¹.

忘记单位换算：在计算 t = 1/H₀ 前，务必先将 km s⁻¹ Mpc⁻¹ 转换为 s⁻¹。1 Mpc = 3.1 × 10¹⁹ km，故 H₀ = 68 km s⁻¹ Mpc⁻¹ = 68 / (3.1 × 10¹⁹) = 2.2 × 10⁻¹⁸ s⁻¹。
Misinterpreting the Universe’s expansion as galaxies moving through space. It is space itself that expands — galaxies are at rest with respect to their local space.

误解宇宙膨胀为星系在空间中运动。实际上是空间本身在膨胀——星系在它们所在的局部空间中保持静止。
Applying v = zc for extremely distant galaxies (z > 1). Use the relativistic formula or state the approximation limit.

对于极遥远星系（z > 1）使用 v = zc。此时应使用相对论公式或说明近似条件。

12. Summary — Key Equations | 总结——核心公式

Memorise this formula sheet before your exam:

考试前记住以下公式清单：

Hubble’s Law	v = H₀ × d
Redshift	z = Δλ / λ
Non-relativistic velocity	v = zc (valid z ≤ 0.1)
Age of Universe (approximate)	t = 1 / H₀
Unit conversion	H₀ = 68 km s⁻¹ Mpc⁻¹ = 2.2 × 10⁻¹⁸ s⁻¹

TutorHao tip: In calculations, if you see ‘estimate the age of the Universe’, immediately write down t = 1/H₀ and convert all units to seconds first. Most marks are awarded for correct unit handling.

TutorHao 提示：当题目要求”估算宇宙年龄”时，立即写出 t = 1/H₀ 并先将所有单位转换为秒。大多数分数在于正确的单位处理。

Understand the physics: Hubble’s Law is not just a formula to plug numbers into — it is the observational key to the birth, evolution, and fate of our Universe. Master it and you unlock the deepest connection between A-Level physics and modern cosmology.

理解其物理本质：哈勃定律并非只是套用数字的公式——它是揭示宇宙诞生、演化和最终命运的关键观测证据。掌握它，你就打通了 A-Level 物理与现代宇宙学之间最深刻的一环。

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Hubble’s Law and the Expanding Universe | 哈勃定律与宇宙膨胀