Einstein's 1905 theory rests on just two postulates: (1) The laws of physics are the same in all inertial
reference frames. (2) The speed of light in vacuum (c = 299,792,458 m/s) is the same for all observers.
From these two simple statements, revolutionary consequences follow - time slows down, lengths contract,
and mass becomes energy.
Why c is special: It's not
just the speed of light - it's the speed of causality itself. Nothing carrying information can
exceed c. Even gravity propagates at c.
2 Time Dilation
A moving clock ticks slower relative to a stationary observer. This isn't an illusion - it's a real
physical effect confirmed by muon decay experiments and GPS satellite corrections.
Δt' = γΔt = Δt / √(1 − v²/c²)
Δt = proper time (in the moving frame)Δt' = dilated time
(observed from stationary frame)γ = Lorentz factor (always ≥ 1)
Worked Example 1
Muon Decay - Time Dilation in Action
Problem: Muons created in the upper atmosphere travel at 0.998c.
Their rest-frame half-life is 1.56 μs. Without time dilation, how far would they travel? With time
dilation, how far?
Without relativity
d = v × t = 0.998 × 3×10⁸ × 1.56×10⁻⁶
d = 467 m (wouldn't reach ground from ~15 km)
With time dilation
γ = 1/√(1 − 0.998²) = 1/√(0.004) = 15.8
t' = γ × 1.56 μs = 24.7 μs
d' = 0.998c × 24.7 μs
d' = 7,390 m ≈ 7.4 km (reaches ground!)
Answer: Time dilation extends the muon's observed lifetime from
1.56 μs to 24.7 μs, allowing it to travel 7.4 km instead of 467 m. We detect muons at sea level - direct
proof of time dilation.
3 Length Contraction
Objects moving at relativistic speeds appear shorter along the direction of motion to a stationary
observer.
L = L₀ / γ = L₀√(1 − v²/c²)
L₀ = proper length (measured in the object's rest
frame)L = contracted length (measured by moving observer)
Worked Example 2
Length Contraction - Spacecraft
Problem: A 100 m spacecraft travels at 0.8c. How long does it
appear to a stationary observer?
Apply contraction
γ = 1/√(1−0.64) = 1/0.6 = 1.667
L = 100/1.667
L = 60 m (40% shorter!)
Answer: 60 m. The ship is physically 100 m in its own frame but
measures 60 m to an outside observer. Width and height are unaffected - only the direction of motion
contracts.
4 Mass-Energy Equivalence
Einstein's most famous equation reveals that mass and energy are interchangeable.
E = mc²
E = energy (J), m = mass (kg)c² = 9 × 10¹⁶
m²/s²1 kg of mass = 9 × 10¹⁶ J = 21.5 megatons TNT
Worked Example 3
Energy from Mass - Nuclear Fusion
Problem: In the Sun, 4 hydrogen nuclei fuse into 1 helium
nucleus. The mass difference is 0.0287 atomic mass units (u). How much energy is released per fusion? (1
u = 1.661 × 10⁻²⁷ kg)
E = Δmc²
Δm = 0.0287 × 1.661×10⁻²⁷ = 4.77 × 10⁻²⁹ kg
E = 4.77×10⁻²⁹ × (3×10⁸)² = 4.77×10⁻²⁹ × 9×10¹⁶
E = 4.29 × 10⁻¹² J = 26.7 MeV
Answer: 26.7 MeV per fusion reaction. The Sun performs ~3.8 ×
10³⁸ fusions per second, converting 4 million tonnes of mass to energy every second - and has enough
hydrogen for another 5 billion years.
5 General Relativity - Gravity as Geometry
General relativity (1915) extends special relativity to include gravity. Mass and energy curve spacetime,
and objects follow the straightest possible paths (geodesics) through curved spacetime - this is what we
experience as gravity.
A massive object like the Sun warps the fabric of spacetime around it, much like a bowling ball placed on a stretched rubber sheet creates a depression. Planets orbit the Sun not because they are being pulled by a force acting at a distance, but because they follow the natural curves of the warped spacetime around the Sun. Light itself bends around massive objects - an effect called gravitational lensing - and has been observed and measured precisely. GPS satellites must correct for both special relativistic time dilation (clocks run slow at high speed) and general relativistic time dilation (clocks run fast at higher altitude) - a net gain of ~38 microseconds per day that must be corrected to maintain metre-level precision.
Predictions confirmed:
Gravitational lensing, gravitational redshift, frame-dragging, gravitational time dilation (GPS
satellites lose 45 μs/day without correction), and gravitational waves (detected by LIGO in
2015).
6 Relativistic Energy & Momentum
At relativistic speeds, the classical formulas for kinetic energy and momentum break down. Einstein showed that mass and energy are equivalent - two sides of the same coin - through the most famous equation in science: E = mc². A tiny mass converts to an enormous amount of energy, since c² = 9 × 10¹⁶ m²/s² is immense. One gram of matter, fully converted, releases as much energy as ~21 kilotons of TNT.
E² = (pc)² + (mc²)²
Total energy: E = γmc²Relativistic momentum: p =
γmvKinetic energy: KE = (γ−1)mc²
γ stays near 1 at everyday speeds but rises sharply above ~0.8c, approaching infinity as v → c.
v = 0.1c
γ = 1.005, KE ≈ ½mv² (classical OK)
v = 0.5c
γ = 1.155, 15.5% heavier
v = 0.9c
γ = 2.294, mass more than doubles
v = 0.99c
γ = 7.089, 7× rest mass
v = 0.999c
γ = 22.37
Worked Example 4
Relativistic Kinetic Energy
Problem: A proton (m = 1.673 × 10⁻²⁷ kg) travels at 0.9c.
Compare its relativistic KE to the classical (½mv²) value.
Classical vs relativistic
Classical: KE = ½mv² = ½(1.673×10⁻²⁷)(0.9×3×10⁸)²
= 6.09 × 10⁻¹¹ J
Relativistic: γ = 1/√(1−0.81) = 2.294
KE = (γ−1)mc² = 1.294 × 1.673×10⁻²⁷ × 9×10¹⁶
KE_rel = 1.95 × 10⁻¹⁰ J (3.2× the classical value!)
Answer: Relativistic KE = 1.95 × 10⁻¹⁰ J vs classical 6.09 ×
10⁻¹¹ J. Classical physics underestimates by 3.2×. At the LHC, protons reach γ ≈ 7,000 - classical
mechanics is hopelessly wrong at such speeds.