Part 3: Special Relativity

Einstein's Two Postulates

In June 1905, Albert Einstein published "On the Electrodynamics of Moving Bodies" — a paper that would redefine our understanding of space and time. Its foundation rests on just two postulates, both seemingly innocuous, yet together producing revolutionary consequences.

First Postulate: The Principle of Relativity

This means there's no "absolute rest" — a physics experiment performed on a smoothly moving train gives identical results to one on the ground. You cannot determine your absolute velocity through space — only your velocity relative to other objects.

Second Postulate: Constancy of the Speed of Light

This is the radical departure from Newton. Whether you run toward a light beam or away from it, you measure exactly the same speed c. This seems impossible if you're thinking in Newtonian terms — but it's experimentally verified to extraordinary precision.

flowchart TD
    A["Postulate 1:
Laws same in all
inertial frames"] --> D["Relativity of
Simultaneity"]
    B["Postulate 2:
Speed of light
is constant"] --> D
    D --> E["Time Dilation"]
    D --> F["Length Contraction"]
    E --> G["Lorentz
Transformations"]
    F --> G
    G --> H["Relativistic
Momentum"]
    G --> I["Mass-Energy
Equivalence"]
    H --> J["Nothing exceeds c"]
    I --> K["E = mc²"]
                            

Relativity of Simultaneity

The first shocking consequence: events that are simultaneous in one frame may not be simultaneous in another. This is not an illusion or measurement error — it's a fundamental property of spacetime.

Time Dilation

Moving clocks run slower. This isn't a mechanical effect — time itself passes more slowly for a moving observer. This is one of relativity's most startling and experimentally verified predictions.

The Light Clock Derivation

Consider a "light clock" — a photon bouncing vertically between two mirrors separated by distance d. For the stationary clock, one tick takes:

Now observe this clock from a frame where it moves horizontally at speed v. The photon travels a diagonal path (longer distance), but still at speed c. By the Pythagorean theorem:

The Time Dilation Formula

Solving for \Delta t:

Since \gamma > 1 for any v > 0, the moving clock always takes longer per tick — it runs slow. The proper time \Delta t_0 (measured by the clock at rest) is always the shortest.

import numpy as np
import matplotlib.pyplot as plt

# Time dilation visualization
c = 1  # natural units
speeds = np.linspace(0, 0.99, 500)
gamma = 1 / np.sqrt(1 - speeds**2)

# For 1 year of proper time, how much time passes for stationary observer?
proper_time = 1  # year
dilated_time = gamma * proper_time

# Print key values
print("Time Dilation: 1 year of proper time")
print("=" * 45)
print(f"{'Speed (v/c)':<12} {'γ':<10} {'Stationary time':<18}")
print("-" * 45)
for v in [0.1, 0.5, 0.8, 0.9, 0.95, 0.99, 0.999]:
    g = 1 / np.sqrt(1 - v**2)
    print(f"{v:<12.3f} {g:<10.4f} {g*proper_time:<18.4f} years")

Length Contraction

Objects are shorter in the direction of motion — as measured by an observer they pass. The contracted length is:

where L_0 is the proper length (measured in the object's rest frame). Since \gamma > 1, the measured length L is always less than L_0.

import numpy as np

# Length contraction examples
c = 3e8  # m/s
L0 = 100  # proper length in meters (e.g., a spacecraft)

print("Length Contraction of a 100-meter spacecraft")
print("=" * 50)
print(f"{'Speed (v/c)':<12} {'γ':<10} {'Observed Length (m)':<20} {'% Contracted':<12}")
print("-" * 50)

for beta in [0.1, 0.3, 0.5, 0.7, 0.8, 0.9, 0.95, 0.99]:
    gamma = 1 / np.sqrt(1 - beta**2)
    L = L0 / gamma
    contraction = (1 - L/L0) * 100
    print(f"{beta:<12.2f} {gamma:<10.4f} {L:<20.2f} {contraction:<12.1f}%")

# Muon example
print("\n\nReal Example: Cosmic Ray Muons")
print("=" * 50)
muon_lifetime = 2.2e-6  # seconds (proper time)
v_muon = 0.998 * c  # typical cosmic ray muon speed
gamma_muon = 1 / np.sqrt(1 - (0.998)**2)

# Classical prediction: distance muon can travel
d_classical = v_muon * muon_lifetime
# Relativistic prediction: dilated lifetime
d_relativistic = v_muon * (gamma_muon * muon_lifetime)

print(f"Muon proper lifetime: {muon_lifetime*1e6:.1f} μs")
print(f"Muon speed: 0.998c")
print(f"Lorentz factor: γ = {gamma_muon:.1f}")
print(f"Classical travel distance: {d_classical:.0f} m = {d_classical/1000:.2f} km")
print(f"Relativistic travel distance: {d_relativistic:.0f} m = {d_relativistic/1000:.1f} km")
print(f"Muons created at ~15 km altitude DO reach Earth's surface!")

Lorentz Transformations

The Lorentz transformations replace the Galilean transformations, correctly relating coordinates between two inertial frames in relative motion at velocity v along the x-axis:

The inverse transformations (solving for unprimed from primed) simply replace v with -v:

Mass-Energy Equivalence: E = mc²

The most famous equation in physics states that mass and energy are interchangeable:

This is the rest energy — the energy contained in an object simply by virtue of having mass. Since c^2 \approx 9 \times 10^{16} m²/s², even a tiny amount of mass contains enormous energy.

import numpy as np

# E = mc² calculations — energy content of everyday objects
c = 3e8  # m/s
c_squared = c**2

print("E = mc² — Energy Content of Everyday Objects")
print("=" * 55)
print(f"c² = {c_squared:.2e} m²/s² = {c_squared:.2e} J/kg")
print()

objects = [
    ("Grain of sand", 0.001e-3),      # 1 mg
    ("Penny (US)", 2.5e-3),            # 2.5 g
    ("Apple", 0.2),                     # 200 g
    ("Human (70 kg)", 70),
    ("Car (1500 kg)", 1500),
]

for name, mass in objects:
    energy_joules = mass * c_squared
    energy_kilotons = energy_joules / 4.184e12  # 1 kiloton TNT = 4.184e12 J
    print(f"{name:<20} mass={mass:.3e} kg → E = {energy_joules:.2e} J ({energy_kilotons:.1f} kilotons TNT)")

print(f"\nComparison: Hiroshima bomb ≈ 15 kilotons TNT")
print(f"A single kilogram fully converted = {1*c_squared/4.184e12:.0f} kilotons = {1*c_squared/4.184e15:.1f} megatons")

Relativistic Momentum & Energy

The total energy of a particle includes both rest energy and kinetic energy:

The relativistic momentum is:

These combine into the energy-momentum relation:

For massless particles (photons): E = pc. For particles at rest: E = mc^2.

The Twin Paradox

Perhaps the most famous "paradox" in relativity: one twin stays on Earth while the other makes a high-speed round trip to a distant star. When the traveler returns, they've aged less than the stay-at-home twin.

import numpy as np

# Twin Paradox calculation
c = 1  # natural units (c = 1 light-year/year)
distance = 4.37  # light-years to Alpha Centauri
v = 0.9  # speed as fraction of c

# Earth frame calculations
travel_time_one_way = distance / v  # years
total_earth_time = 2 * travel_time_one_way

# Traveler frame calculations
gamma = 1 / np.sqrt(1 - v**2)
total_traveler_time = total_earth_time / gamma

# Age difference
age_difference = total_earth_time - total_traveler_time

print("Twin Paradox: Trip to Alpha Centauri at 0.9c")
print("=" * 50)
print(f"Distance: {distance} light-years")
print(f"Speed: {v}c")
print(f"Lorentz factor γ: {gamma:.4f}")
print(f"\nEarth twin (stays home):")
print(f"  Time elapsed: {total_earth_time:.2f} years")
print(f"\nTraveling twin:")
print(f"  Time elapsed: {total_traveler_time:.2f} years")
print(f"\nAge difference: {age_difference:.2f} years")
print(f"The traveler is {age_difference:.2f} years YOUNGER!")

# Multiple speeds comparison
print(f"\n{'Speed':<8} {'Earth Time':<13} {'Traveler Time':<15} {'Age Gap':<10}")
print("-" * 46)
for v in [0.5, 0.7, 0.9, 0.95, 0.99, 0.999]:
    t_earth = 2 * distance / v
    g = 1 / np.sqrt(1 - v**2)
    t_traveler = t_earth / g
    gap = t_earth - t_traveler
    print(f"{v:<8.3f} {t_earth:<13.2f} {t_traveler:<15.2f} {gap:<10.2f} years")

Practice Exercises

Conclusion & Next Steps

Special relativity emerges from two simple postulates and reshapes our understanding of reality:

• Time dilation: moving clocks run slow by factor \gamma
• Length contraction: moving objects are shorter by factor 1/\gamma
• Simultaneity is relative: events can be simultaneous in one frame but not another
• Mass-energy equivalence: E = mc^2 — mass IS energy
• Speed limit: nothing with mass can reach or exceed c

But special relativity only handles inertial (non-accelerating) frames. To properly unify these effects into a geometric framework, we need Minkowski spacetime.