Topic 6

Relativity

Albert Einstein's twin pillars: special relativity rewrites space and time into spacetime, while general relativity describes gravity not as a force but as the curvature of that spacetime itself — the motion of planets, the bending of light, and the expansion of the universe.

The two postulates & inertial frames
Lorentz transforms, time dilation & length contraction
Spacetime & the invariant interval
The light-clock thought experiment
General relativity: gravity as curvature
Cosmology & black holes

1. The Two Postulates

Inertial reference frames are those moving at constant velocity relative to each other. Before Einstein, physicists believed there was an absolute rest frame (the "aether"). Einstein's special relativity (1905) rests on two bold postulates:

Postulate 1 (Principle of Relativity) — The laws of physics are the same in every inertial frame. No inertial experiment can tell you whether you're "really" moving.

Postulate 2 (Invariance of $c$) — The speed of light in vacuum, $c = 299{,}792{,}458$ m/s, is the same for all observers, regardless of the motion of the source or the observer.

The second postulate is the shock. In everyday life, speeds add: if you walk 3 mph forward on a train going 60 mph, someone on the platform sees you at 63 mph. But light doesn't work that way. If you shine a flashlight forward on a train going 0.9$c$, both you and the platform watcher measure the light beam at exactly $c$ — not $1.9c$. Space and time must bend to keep this true.

The futility of catching up to light. Nothing with mass can reach $c$. As an object's speed approaches $c$, its relativistic factor $\gamma = 1/\sqrt{1-v^2/c^2}$ blows up to infinity, requiring infinite energy to accelerate further. Photons (massless) travel at $c$; everything else travels below it.

2. Time, Length & the Lorentz Transform

To keep $c$ constant across frames, space and time mix together. The Lorentz factor $\gamma$ governs all relativistic effects:

$$\gamma = \frac{1}{\sqrt{1 - v^2/c^2}}$$

$$\Delta t' = \gamma\,\Delta t\qquad\text{(time dilation)}\qquad L' = \frac{L_0}{\gamma}\qquad\text{(length contraction)}$$

Time dilation: a clock in motion ticks slower by factor $\gamma$. Length contraction: objects shrink along their direction of motion by $\gamma$. These are not illusions — they're real, measurable effects. The GPS satellites orbiting Earth experience both: their clocks run fast by 45 μs/day due to weaker gravity (general relativity), but slow by 7 μs/day due to their speed (special relativity). Without relativistic corrections, GPS would drift by kilometers within hours.

$$\begin{pmatrix} ct' \\ x' \end{pmatrix} = \begin{pmatrix} \gamma & -\gamma\beta \\ -\gamma\beta & \gamma \end{pmatrix}\begin{pmatrix} ct \\ x \end{pmatrix},\qquad \beta = v/c$$

3. The Light-Clock Thought Experiment

The cleanest way to see time dilation: imagine a clock made of two mirrors with a photon bouncing between them. In the clock's rest frame, the photon travels a vertical path of length $L$ each tick. But to a watcher moving past, the photon traces a longer diagonal path. Since light travels at the same speed $c$ in both frames, the moving observer sees the photon travel further — and since speed is the same, the ticks must take longer:

Light clock and time dilation. On the left, a clock at rest with a photon bouncing between mirrors. On the right, the same clock as seen by a fast-moving observer: the photon now travels a longer diagonal path at the same speed $c$, so each tick takes longer by the factor $\gamma$. Adjust velocity to see how the effect grows as $v$ approaches $c$.

$$\Delta t' = \gamma\,\Delta t = \frac{\Delta t}{\sqrt{1-v^2/c^2}}$$

This is the only relativity thought-experiment I know of that has a ready-made simulation. The derivation is pure geometry: a right triangle with legs $L$ (half the vertical bounce) and $v\,\Delta t'/2$ (horizontal drift), hypotenuse $c\,\Delta t'/2$, giving $\gamma$ by Pythagoras.

4. Spacetime & the Invariant Interval

Hermann Minkowski (1908) realized Einstein's equations reveal a deeper truth: space and time aren't separate. They fuse into a four-dimensional spacetime. Events are labeled by $(ct, x, y, z)$. The Lorentz transforms are rotations in spacetime, mixing the time and space coordinates just as a ordinary rotation mixes $x$ and $y$ axes.

$$\boxed{ds^2 = -c^2 dt^2 + dx^2 + dy^2 + dz^2}$$

This is the invariant interval $ds^2$: every observer, regardless of motion, agrees on its value. It's the spacetime analogue of the Pythagorean distance. The minus sign on the time component (the "mostly-plus" metric) is what makes timelike and spacelike intervals fundamentally different, and sets a cosmic speed limit.

A spacetime diagram: vertical axis is time ($ct$), horizontal is space ($x$). Straight lines are object worldlines. The dashed red 45° lines are light cones — nothing inside can escape to outside.

5. General Relativity: Gravity as Curvature

In 1915, Einstein extended relativity to accelerating frames and gravity. The key insight: gravity isn't a force — it's the curvature of spacetime caused by mass and energy. Objects in free fall aren't being "pulled"; they're following the straightest possible paths through curved geometry. The Einstein field equations relate curvature to the matter that creates it:

$$\boxed{G_{\mu\nu} = \frac{8\pi G}{c^4}\,T_{\mu\nu}}$$

The left side ($G_{\mu\nu}$, the Einstein tensor) describes how spacetime curves. The right side ($T_{\mu\nu}$, the energy-momentum tensor) describes where mass, energy, and momentum are. They are equal — curved spacetime tells matter how to move; matter tells spacetime how to curve.

Equivalence principle. There is no local experiment that can distinguish free fall in a gravitational field from floating in empty space. This is why an astronaut in orbit feels weightless — they and the spacecraft are both following the same straightest path through curved spacetime.

$$r_s = \frac{2GM}{c^2}\qquad\text{(Schwarzschild radius / event horizon)}$$

Testable predictions. General relativity made three verified predictions that Newton couldn't explain: (1) the bending of starlight by the Sun (1919 Eddington expedition — the famous "eclipse" result), (2) the anomalous perihelion advance of Mercury (43 arcseconds/century unexplained by Newtonian gravity), and (3) gravitational redshift (light loses energy climbing out of a gravity well).

6. Cosmology & Black Holes

Einstein's equations applied to the universe as a whole give the Friedmann equations. In 1929, Hubble confirmed what these equations predicted: the universe is expanding. Run the clock backward, and everything was denser — the Big Bang.

When enough mass concentrates in one place, spacetime curves so severely that not even light can escape — a black hole. The boundary is the event horizon at the Schwarzschild radius $r_s = 2GM/c^2$. Inside, all future paths lead to the singularity.

Hubble's law: more distant galaxies recede faster, evidence that the universe began in a hot, dense state.

Key Takeaways

The speed of light $c$ is constant for all observers, which forces space and time to mix — leading to time dilation and length contraction.

Space and time are unified into 4D spacetime; the interval $ds^2$ is invariant, and Lorentz transforms are rotations in this geometry.

Gravity is not a force but the curvature of spacetime — Einstein's equation $G_{\mu\nu} = (8\pi G/c^4)T_{\mu\nu}$ links matter to geometry.

General relativity's predictions — light bending, Mercury's orbit, gravitational redshift — are confirmed to high precision.

Cosmological solutions gave us the Big Bang and the expanding universe; extreme curvature produces black holes and event horizons.

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