Part 2: Crisis of Classical Physics

The Problem: When Classical Physics Broke Down

By the late 19th century, physicists were confident they had nearly completed the picture of the universe. Newton's mechanics explained motion from falling apples to orbiting planets. Maxwell's electromagnetism unified electricity, magnetism, and light. Thermodynamics governed heat and energy. The universe seemed like a clockwork machine, and physics was almost "done."

Then everything fell apart.

The Assumptions of Classical Physics

Classical physics rested on several assumptions that seemed obviously true:

1. Absolute time — time flows at the same rate for everyone, everywhere
2. Absolute space — there exists a fixed "stage" on which physics plays out
3. Galilean velocity addition — velocities add simply: if you walk at 5 km/h on a train moving at 100 km/h, you move at 105 km/h relative to the ground
4. Instantaneous action — forces (like gravity) act instantaneously across any distance

Every one of these assumptions would be overthrown by Einstein's relativity.

Maxwell's Puzzle: The Speed of Light Problem

Maxwell's equations give the speed of electromagnetic waves as:

But relative to what? In Newtonian physics, every speed is relative to something. Sound travels at 343 m/s relative to air. Water waves move relative to water. So light must travel at c relative to... something. Physicists called this hypothetical medium the luminiferous aether.

flowchart TD
    A["Earth moves through
stationary aether"] --> B["Light speed should
vary with direction"]
    B --> C["Moving with aether:
c - v (slower)"]
    B --> D["Moving against aether:
c + v (faster)"]
    B --> E["Perpendicular to aether:
√(c² - v²)"]
    C --> F["Michelson-Morley
should detect this"]
    D --> F
    E --> F
    F --> G["Result: NO
difference detected!"]
                            

The Michelson-Morley Experiment (1887)

The Michelson-Morley experiment is often called "the most famous failed experiment in physics." It was designed to detect Earth's motion through the aether — and its shocking null result changed everything.

The Aether Hypothesis

The reasoning seemed bulletproof:

• Light is a wave (demonstrated by interference and diffraction)
• Waves need a medium to propagate through
• Therefore, space must be filled with a medium — the "luminiferous aether"
• Earth orbits the Sun at ~30 km/s, so it must move through the aether
• This motion should create an "aether wind" that affects light speed

The expected effect was small — about 1 part in 100 million — but Albert Michelson's interferometer was sensitive enough to detect it.

The Interferometer Design

Michelson's brilliant design split a beam of light into two perpendicular paths, bounced them off mirrors, and recombined them. If one path took longer (due to the aether wind), the recombined beams would show interference fringes — a shift in the pattern.

import numpy as np

# Michelson-Morley Experiment Calculations
c = 3e8  # speed of light (m/s)
v = 3e4  # Earth's orbital speed (m/s) = 30 km/s
L = 11  # arm length in meters (Michelson's setup)
wavelength = 550e-9  # yellow light wavelength (m)

# Time for light to travel parallel to motion (round trip)
t_parallel = (2 * L / c) * (1 / (1 - (v/c)**2))

# Time for light to travel perpendicular to motion (round trip)
t_perpendicular = (2 * L / c) * (1 / np.sqrt(1 - (v/c)**2))

# Time difference
delta_t = t_parallel - t_perpendicular

# Expected fringe shift
fringe_shift = delta_t * c / wavelength

print("Michelson-Morley Experiment Analysis")
print("=" * 45)
print(f"Speed of light: {c:.2e} m/s")
print(f"Earth orbital speed: {v:.2e} m/s")
print(f"v/c ratio: {v/c:.2e}")
print(f"Arm length: {L} m")
print(f"Light wavelength: {wavelength*1e9:.0f} nm")
print(f"\nTime (parallel path): {t_parallel:.15e} s")
print(f"Time (perpendicular): {t_perpendicular:.15e} s")
print(f"Time difference: {delta_t:.3e} s")
print(f"\nExpected fringe shift: {fringe_shift:.2f} fringes")
print(f"Actual observed shift: ~0.00 fringes")
print(f"\nConclusion: NO aether wind detected!")

The Null Result and Its Implications

The experiment was repeated many times — at different times of day, different seasons (when Earth's velocity direction changes), at different altitudes. The result was always the same: no detectable motion through the aether.

This left only a few possibilities:

1. Earth happens to be at rest relative to the aether (rejected — Earth orbits the Sun)
2. The aether is somehow dragged along with Earth (contradicted by stellar aberration)
3. Objects contract in their direction of motion (Lorentz-FitzGerald hypothesis)
4. There is no aether, and the speed of light is truly constant for all observers (Einstein's solution)

Failure of Galilean Relativity

The Velocity Addition Problem

Galilean velocity addition says that if you're on a train moving at speed u and throw a ball forward at speed v', the ball's speed relative to the ground is:

This works perfectly for everyday speeds. But now consider light:

The correct relativistic velocity addition (derived in Part 3) is:

Notice that when both u and v' are much smaller than c, the denominator is approximately 1, and we recover the Galilean formula. But when speeds approach c, the denominator prevents the sum from ever exceeding c.

import numpy as np

# Compare Galilean vs Relativistic velocity addition
c = 1  # natural units

def galilean_addition(u, v_prime):
    """Classical velocity addition"""
    return u + v_prime

def relativistic_addition(u, v_prime):
    """Einstein's velocity addition formula"""
    return (u + v_prime) / (1 + u * v_prime / c**2)

print("Velocity Addition: Galilean vs Relativistic")
print("=" * 55)
print(f"{'Ship speed':<12} {'Ball speed':<12} {'Galilean':<12} {'Relativistic':<12}")
print("-" * 55)

cases = [
    (0.01, 0.01, "Everyday speeds"),
    (0.5, 0.5, "Half light speed"),
    (0.9, 0.9, "Near light speed"),
    (0.99, 0.99, "Ultra-relativistic"),
    (0.9, 1.0, "Ship + light beam"),
    (0.99, 1.0, "Fast ship + light"),
]

for u, v_prime, label in cases:
    gal = galilean_addition(u, v_prime)
    rel = relativistic_addition(u, v_prime)
    print(f"{u:<12.2f} {v_prime:<12.2f} {gal:<12.2f} {rel:<12.4f}  ← {label}")

print(f"\nKey insight: Relativistic addition NEVER exceeds c = {c}")
print(f"Even 0.99c + 0.99c = {relativistic_addition(0.99, 0.99):.6f}c (not 1.98c!)")

The Lorentz-FitzGerald Contraction

Before Einstein, Hendrik Lorentz and George FitzGerald independently proposed that objects physically contract in their direction of motion through the aether. The contraction factor was:

This could explain the Michelson-Morley null result — if the interferometer arm parallel to Earth's motion shortened by just the right amount, the travel times would equalize. But this was an ad hoc hypothesis — a mathematical patch without physical understanding.

Einstein's genius was showing that this contraction isn't a mysterious physical effect — it's a consequence of the geometry of spacetime. Length contraction, time dilation, and the relativity of simultaneity all emerge naturally from two simple postulates (Part 3).

Other Cracks in Classical Physics

The speed-of-light problem wasn't the only crisis. Around 1900, classical physics failed spectacularly in several other domains, leading to the quantum revolution alongside the relativity revolution.

The Blackbody Radiation Catastrophe

Classical physics predicted that a heated object should radiate infinite energy at short wavelengths — the "ultraviolet catastrophe." This was clearly absurd. In 1900, Max Planck resolved this by proposing that energy comes in discrete packets (quanta):

where h = 6.626 \times 10^{-34} J·s is Planck's constant and \nu is frequency.

The Photoelectric Effect

When light hits a metal surface, electrons are ejected. Classical wave theory predicted that brighter light should always eject faster electrons. Instead, the kinetic energy of ejected electrons depends only on light frequency, not intensity. Einstein explained this in 1905 using Planck's quantum idea — the same year he published special relativity.

where \phi is the work function (minimum energy to free an electron).

Practice Exercises

Conclusion & Next Steps

By 1905, the foundations of classical physics had crumbled:

• The Michelson-Morley experiment showed no aether wind — the speed of light is invariant
• Galilean velocity addition fails at high speeds — you can't simply add velocities near c
• Maxwell's equations demanded a constant speed of light, incompatible with Newtonian mechanics
• The Lorentz-FitzGerald contraction was a mathematical band-aid, not a true explanation

The stage was set for a revolution. What was needed was not another patch, but a complete reimagining of space, time, and motion. That reimagining came from Albert Einstein's two deceptively simple postulates — and their extraordinary consequences.