Special Relativity: Time Dilation, Length Contraction, E = mc²

By Engr. Henry Akpan

When Common Sense Breaks Down

In everyday life, time ticks at the same rate for everyone, a metre stick is a metre long whoever measures it, and speeds simply add. In 1905, Albert Einstein showed that all three ideas are wrong at speeds approaching that of light. Moving clocks run slow, moving objects shrink, and nothing can travel faster than light. These predictions sound strange, but they have been confirmed countless times, and they are built into the GPS system in every smartphone. This tutorial introduces special relativity as taught in PHY 101/102 and modern physics courses: frames of reference, Einstein's two postulates, the Lorentz factor, time dilation, length contraction, the relativistic addition of velocities, mass–energy equivalence and relativistic energy, with worked examples including muons and GPS satellites.

Frames of Reference

A frame of reference is a viewpoint from which positions and times are measured, such as the ground, a moving train or a spacecraft. An inertial frame is one moving at constant velocity, neither accelerating nor rotating. In an inertial frame, Newton's first law holds: an object with no force on it stays at rest or keeps moving in a straight line at a constant speed.

Galileo realised that no experiment in mechanics can tell you whether you are at rest or moving steadily. In a smooth train with the blinds down, a dropped ball falls straight down just as it would on the platform. Special relativity extends this idea to all of physics, including light.

Einstein's Two Postulates

1. The principle of relativity: the laws of physics are the same in every inertial frame of reference.
2. The constancy of the speed of light: the speed of light in a vacuum, c ≈ 3.00 × 10⁸ m/s, is the same for every observer, whatever the motion of the source or the observer.

The second postulate defies common sense. If you chase a beam of light at half the speed of light, you still measure it passing you at c, not ½c. The Michelson–Morley experiment (1887) had already failed to detect any change in the speed of light caused by Earth's motion. Einstein accepted this result and worked out its consequences: if the speed of light is the same for everyone, then measurements of time and length must differ between observers.

The Lorentz Factor

All the effects of special relativity involve the Lorentz factor γ (gamma):

γ = 1 ÷ √(1 − v²/c²)

• At everyday speeds, v/c is tiny, so γ is almost exactly 1, and relativistic effects are negligible. For a jet airliner at 250 m/s, γ differs from 1 by less than one part in a trillion.
• At v = 0.6c, γ = 1 ÷ √(1 − 0.36) = 1 ÷ 0.8 = 1.25.
• At v = 0.8c, γ = 1 ÷ √(1 − 0.64) = 1 ÷ 0.6 ≈ 1.67.
• At v = 0.995c, γ ≈ 10.
• As v approaches c, γ grows without limit.

Time Dilation: Moving Clocks Run Slow

The time between two events measured by a clock that is present at both (moving with them) is the proper time t₀. An observer who sees that clock moving at speed v measures a longer time between the same events:

t = γt₀

Why? Imagine a "light clock" in a moving spaceship: a pulse of light bouncing between two mirrors, one above the other. To the astronaut, the light goes straight up and down. To an observer on Earth, the ship moves sideways while the light travels, so the light follows a longer, diagonal zigzag path. Since the light travels at the same speed c for both observers, each tick must take longer as seen from Earth.

Worked example 1

A spacecraft travels at 0.8c. Its astronaut measures 3.0 hours between two meals. How much time passes between the meals according to mission control on Earth?

γ ≈ 1.67, so t = 1.67 × 3.0 ≈ 5.0 hours.

Evidence: muons from the upper atmosphere

Muons are unstable particles produced when cosmic rays strike the upper atmosphere, about 10 km up. At rest, they have a mean lifetime of only about 2.2 μs.

• Moving at 0.995c, they would take about 33.5 μs to reach the ground, more than 15 mean lifetimes. Without relativity, only about 1 muon in 4 million should survive the journey.
• With γ ≈ 10, their lifetime as measured from the ground is about 22 μs. About 22% should survive, which is what is actually detected.

Muons reach Earth's surface in large numbers, exactly as time dilation predicts.

Length Contraction

An object's length measured in the frame where it is at rest is its proper length L₀. An observer who sees it moving at speed v measures it to be shorter along the direction of motion:

L = L₀ ÷ γ

Lengths perpendicular to the motion are unchanged.

Worked example 2

A spaceship is 100 m long at rest. How long is it as measured by an observer it passes at 0.6c?

L = 100 ÷ 1.25 = 80 m.

The muon example looks different from the muon's own point of view. In its frame, its lifetime is only 2.2 μs, but the 10 km of atmosphere rushing past it is contracted to about 10 ÷ 10 = 1 km, short enough to cross in time. Both views give the same prediction for the number of muons detected.

Adding Velocities

In everyday physics, if a train moves at v and a passenger throws a ball forward at u relative to the train, the ball's speed relative to the ground is u + v. This cannot be exactly right, because it would let speeds exceed c. The relativistic formula is:

w = (u + v) ÷ (1 + uv/c²)

Worked example 3

A spaceship moving at 0.6c relative to Earth fires a probe forwards at 0.6c relative to the ship. How fast does the probe move relative to Earth?

w = (0.6c + 0.6c) ÷ (1 + 0.36) = 1.2c ÷ 1.36 ≈ 0.88c, not 1.2c.

If either speed is c itself, the formula gives exactly c, in agreement with the second postulate. And at everyday speeds, uv/c² is negligible, so the formula reduces to w = u + v.

Mass–Energy Equivalence

Einstein's most famous result is that mass is a form of energy. An object of mass m at rest has a rest energy:

E₀ = mc²

For example, the rest energy of an electron is 9.11 × 10⁻³¹ × (3.00 × 10⁸)² ≈ 8.2 × 10⁻¹⁴ J, or 0.511 MeV. The mass defect of nuclei, which powers nuclear reactors and the Sun, is a direct application of this equation, as covered in the tutorial on nuclear energy.

Total energy and kinetic energy

A moving object has total energy E = γmc², so its kinetic energy is:

KE = (γ − 1)mc²

At low speeds, this reduces to the familiar ½mv². As v approaches c, γ and therefore the energy needed grow without limit. That is why no object with mass can reach the speed of light: it would need infinite energy.

Worked example 4

Find the kinetic energy of an electron moving at 0.8c, in MeV.

KE = (1.667 − 1) × 0.511 MeV ≈ 0.34 MeV.

The classical formula ½mv² would give ½ × 0.511 MeV × 0.64 ≈ 0.16 MeV, which is less than half the true value. At high speeds, the classical formula is badly wrong.

The momentum of a moving object also becomes p = γmv, which is why particle accelerators need ever-stronger magnets to bend the paths of particles as they approach the speed of light.

Relativity in Everyday Technology

GPS satellites orbit at about 3.9 km/s and carry extremely precise atomic clocks. Because of their speed, special relativity makes their clocks run slow by about 7 microseconds a day. General relativity, which describes gravity, makes them run fast by about 45 microseconds a day, because gravity is weaker at their height. The net effect is about 38 microseconds a day. That sounds tiny, but light travels about 11 km in that time, so without correcting for relativity, GPS positions would drift by kilometres every day.

Common Mistakes with Special Relativity

• Swapping the proper and dilated times. The proper time is measured by the clock moving with the events, and it is always the shortest.
• Multiplying by γ for length contraction. Moving lengths are shorter, so divide: L = L₀/γ.
• Adding velocities simply at speeds close to c.
• Using KE = ½mv² at relativistic speeds. Use (γ − 1)mc².
• Thinking the effects are an illusion caused by the time light takes to travel. They are real differences in measured time and length, confirmed by experiment.

Special Relativity: Practice Questions

1. State Einstein's two postulates of special relativity.
2. Find γ for a speed of 0.5c.
3. A clock on a spaceship moving at 0.6c ticks for 10 s. How long is this interval as measured from Earth?
4. A rod is 2.0 m long at rest. Find its length when it moves along its length at 0.8c.
5. Two particles approach each other, each at 0.5c relative to the laboratory. Find the speed of one relative to the other.
6. Calculate the rest energy of a proton (mass 1.67 × 10⁻²⁷ kg), in joules.

Answers: 1. The laws of physics are the same in all inertial frames, and the speed of light in a vacuum is the same for all observers. 2. 1 ÷ √0.75 ≈ 1.15. 3. 1.25 × 10 = 12.5 s. 4. 2.0 ÷ 1.667 = 1.2 m. 5. (0.5c + 0.5c) ÷ (1 + 0.25) = 0.8c. 6. 1.67 × 10⁻²⁷ × (3.00 × 10⁸)² ≈ 1.5 × 10⁻¹⁰ J (about 938 MeV).

Conclusion

Special relativity rests on two postulates: the laws of physics are the same in every inertial frame, and the speed of light is the same for every observer. From these, with the Lorentz factor γ = 1/√(1 − v²/c²), come time dilation (t = γt₀), length contraction (L = L₀/γ) and the velocity-addition formula, which never gives a speed above c. Mass is a form of energy (E₀ = mc²), and kinetic energy is (γ − 1)mc², so no massive object can reach the speed of light. Muon lifetimes and GPS satellites confirm these predictions every day.