Time Dilation 

• If two identical, synchronised clocks are placed 

side by side they will read the same time for as 

long as they both remain side by side. 

• However if one of the clocks is moving, it will show 

that less time has elapsed (i.e. time has slowed 

down) than the stationary clock. 

• This effect is called time dilation. 

• It has been confirmed by experiment. 

• Two highly accurate atomic clocks were 

synchronised. One was placed in a jet aircraft and 

the other in a laboratory on Earth. After the jet 

returned (having travelled at 1000 km/hr), the 

travelling clock was a tiny fraction of a second 

behind the Earth clock. 

Proof – A Thought Experiment: 

• Within a truck moving with velocity v, a light pulse is transmitted, reflected by a 

mirror and received! 

• Truck's frame of reference i.e. observed within the truck, the light pulse travels a 

distance of 2L in time t0. 

• Earth's frame of reference i.e. observed outside the truck, the light pulse travels 

2x. 

• Light travels a longer distance but at the same speed ⇒ longer time taken t ≥ t0. 

t0 = 2L/c 

 ⇒

 L = ct0/2 

t = 2x/c

 ⇒

 x = ct/2 

• Using x2 = L2 + (vt/2) 2 sub for t and t0 

• t - relativistic time i.e. measured in a frame of reference moving relative to object. 

• t0 - rest time i.e. measured in the same frame as (at rest with) the object. 

• Relativity factor:

As v ≤ c, t ≥ t0. 

• Time dilation is significant at relativistic speeds i.e. v 

→ c, γ → ∞ i.e. t → ∞. 

Actual Experiment - Muon Decay Experiment 

• Muons are produced in the upper layers of the Earth’s atmosphere. 

• They travel close to the speed of light (99% c). 

• But they have very short life-times i.e. a half-life of 2 

μs. 

• Classically, very few muons would be expected to reach the Earth’s surface; 

most would have decayed by that time. 

• But with time dilation, the half-life of muons increase (time slows down!). 

• Measure count rates of muons at top and bottom of a mountain. 

• For an observer on the ground, a higher than expected count rate is measured – 

fewer have decayed because of the longer half-life. 

Worked Example 

• Muons travel at 0.99c. 

• Muons should take 

t = s/v = 6000 / (0.99 × 3 × 108) = 20 

μs  to reach Earth’s surface. 

• They have a half-life of 2 

μs. 

• So (20 

μs / 2 μs =) 10 half-lives should occur in this time. 

• Expect 1/210 muons to reach Earth’s surface. 

• If 1000 muons are detected in the upper layers of 

the Earth’s atmosphere then  expect: 

1 muon to be detected at the Earth’s 

surface. 

• With time dilation, the half-life increases to 

t = 

γ t0 = × 2 μs = 7.09 × 2 μs = 14 μs 

• So (20 

μs / 14 μs =) 1.4 half-lives occur. 

• Observe 1/21.4 muons to reach Earth’s surface. 

380 muons are detected at the Earth’s 

surface. 

• A higher than expected count rate is detected!