Modern Physics · Chapter 19

Special Relativity

In 1905, Einstein showed that the constancy of the speed of light — confirmed by every experiment — forces a complete revision of our concepts of space and time.

Prerequisites

Classical mechanics (Ch. 1–7) \cdot Electromagnetism (Ch. 13–16) motivates why c must be const

ant

Learning Goals

State Einstein's two postulates and explain why the second one forces a revision of space and time.
$Calculate the Lorentz factor \gamma and use it to compute time dilation and length contraction$ .
Solve problems involving relativistic energy, rest-mass energy, and kinetic energy.
Apply the relativistic velocity addition formula and verify that light speed is frame-independent.
Resolve the twin paradox by identifying the non-inertial frame that breaks the symmetry.

19.1 The Two Postulates

Special relativity rests on two postulates, both of which are confirmed to extraordinary precision by experiment:

Definition 19.1 — Einstein's Two Postulates (1905)

I. Principle of Relativity: The laws of physics are identical in all inertial reference frames. No experiment performed entirely within a closed system can detect uniform motion.II. Constancy of Light Speed:

The speed of light in vacuum is c = 2.998\times10^{8} m/s in all inertial frames, regardless of th

motion of the source or observer.

The second postulate is the radical one. In Newtonian mechanics, velocities add: a ball thrown forward at 20 m/s from a train moving at 30 m/s moves at 50 m/s relative to the ground. But if a flashlight on the train emits light, Postulate II requires the light to travel at c relative to both the train and the ground — simultaneously. This is impossible in Newtonian mechanics, and it forces the revision of space and time.

19.2 Time Dilation

Consider a "light clock": a photon bouncing between two mirrors separated by distance d. In its rest frame, one tick takes Δt₀ = 2d/c. Now observe the same clock from a frame where it moves sideways at speed v. The photon must travel a longer diagonal path. By Postulate II, it still travels at c, so the tick takes longer.

\Deltat = \gamma \Deltat_{0} \qquad where \qquad \gamma = 1/\sqrt(1 - v^{2}/c^{2})

(19.1)

The factor γ ≥ 1 is the Lorentz factor. Time dilation: a moving clock runs slow relative to a stationary observer. The "proper time" Δt₀ is the time measured in the clock's own rest frame. This is not an illusion or a measuring error — it is a real physical effect. Muons produced in the upper atmosphere at 0.998c live long enough to reach the ground only because of time dilation.

Figure 19.1. Minkowski spacetime diagram. The gray axes are the rest frame S; the blue line is the ct'-axis (worldline of the moving

frame's origin); the red line is the x'-axis (the moving frame's simultaneity plane). As \beta

n coordinate time — time dilation made visible.

19.3 Length Contraction

The same geometry that dilates time also contracts length. An object of proper length L₀ (measured in its rest frame) appears contracted along the direction of motion when observed from a frame where it moves at speed v:

L = L_{0}/\gamma = L_{0} \sqrt(1 - v^{2}/c^{2})

(19.2)

Lengths perpendicular to the motion are unchanged. Length contraction and time dilation are two sides of the same coin — both follow from the Lorentz transformation, and both are required for the speed of light to be the same in all frames.

Example 19.1 — Muon Survival

$Muons are produced at h = 10 km altitude moving at v = 0.998c. Their mean lifetime at rest$ $is \tau_{0} = 2.2 \mus. How many mean lifetimes does it take them to reach the ground in the lab f$ ame? In the muon's frame?

γ:

\gamma = 1/\sqrt(1 - 0.998^{2}) = 1/\sqrt(1 - 0.996) = 1/\sqrt0.004 = 1/0.0632 \approx 15.8

Lab frame travel time:

\Deltat = h/v = 10000/(0.998 \times 3\times10^{8}) = 3.34\times10^{-5} s = 33.4 \mus

Number of lifetimes (lab):

33.4 \mus / 2.2 \mus \approx 15.2 — classically, almost none would survive

Muon frame:

\Deltat_{0} = \Deltat/\gamma = 33.4/15.8 = 2.11 \mus — less than one lifetime. Most survive. ✓

Equivalently:

In muon frame, the atmosphere is length-contracted: L = 10000/15.8 = 633 m. Short trip

19.4 Relativistic Energy and Momentum

The Lorentz transformation forces revisions to momentum and energy. The relativistic momentum is p = γmv, and the total relativistic energy is:

E = \gammamc^{2}

(19.3)

At rest (v = 0, γ = 1), this gives Einstein's most famous result: E₀ = mc². Mass is a form of energy. The kinetic energy is K = (γ−1)mc². The fundamental energy-momentum relation holds in all frames:

E^{2} = (pc)^{2} + (mc^{2})^{2}

(19.4)

For a photon (m = 0): E = pc, so E = hf = hc/λ. For a particle at rest: E = mc². These are special cases of the same equation.

Example 19.2 — Kinetic Energy at High Speed

$An electron (m = 9.11\times10^{-31} kg) is accelerated to v = 0.99c. Find its kinetic energy in Me$ $V. (1 MeV = 1.602\times10^{-13} J, mc^{2} = 0.511 MeV$

γ:

\gamma = 1/\sqrt(1 - 0.99^{2}) = 1/\sqrt(0.0199) = 1/0.141 = 7.09

K = (\gamma-1)mc^{2} = (7.09 - 1) \times 0.511 MeV = 6.09 \times 0.511 =

3.11 MeV

Compare Newtonian:

K_{classical} = \frac{1}{2}mv^{2} = \frac{1}{2}(0.511)(0.99)^{2} = 0.250 MeV — a factor of 12 too small

19.5 Relativistic Velocity Addition

If frame S moves at v relative to the lab, and an object moves at u in frame S (along the same direction), its velocity in the lab is:

u_{lab} = (u + v) / (1 + uv/c^{2})

(19.5)

For u, v ≪ c, the denominator ≈ 1 and we recover the Galilean result. But if u = c: u_lab = (c + v)/(1 + v/c) = c(1+v/c)/(1+v/c) = c. Light always travels at c, regardless of the source's motion — Postulate II is built into the algebra.

Definition 19.2 — Common Traps

Time dilation is symmetric between inertial observers: the asymmetry in twin-style problems comes from changing frames.
Length contraction is along motion only: transverse dimensions are unchanged.
Velocity addition is not Galilean: no massive object can be boosted past c.
Rest energy is real energy: $E = mc^{2} is part of the total relativistic energy budget$

Exercises — 19.1–19.5 Special Relativity

Calculate the Lorentz factor \gamma for a particle moving at v = 0.99c

Straightforward

A clock moves at 0.866c (\gamma = 2). The lab measures a time interval of 1 \mus. What does the m

oving clock read?

μs

Straightforward

An electron is accelerated to 0.99c (\gamma = 7.09). Find its relativistic kinetic energy in Me

MeV

Intermediate

4.A rocket moves at 0.9c relative to Earth and fires a missile forward at 0.9c relative to the rocket. What is the missile's speed relative to Earth? Why doesn't simple addition give 1.8c?

Intermediate

5.Twin A travels at 0.8c to a star 4 light-years away and returns. Twin B stays on Earth. Who is younger when they reunite, and by how much? Explain why this is not a paradox.

Challenging

Key Takeaways

Two postulates: physics is the same in all inertial frames; c is constant in all frames.
$Time dilation: \Deltat = \gamma\Deltat_{0} — a moving clock runs slow by factor \gamma = 1/\sqrt(1-v^{2}/c^{2}$ .
$Length contraction: L = L_{0}/\gamma — moving objects are shortened along their motion$ .
$Relativistic energy: E = \gammamc^{2}, so E_{0} = mc^{2} at rest — mass and energy are equivalent$ .
$Energy-momentum: E^{2} = (pc)^{2} + (mc^{2})^{2} — for photons (m=0): E = pc$ .
$Relativistic velocity addition prevents anything from exceeding c: u = (u'+v)/(1+u'v/c^{2}$ .