Time Dilation

 to dilate is to become larger

 A moving clock ticks more slowly than a clock at rest

Actual difference of elapsed time between two events as measured

by observers moving relative to each other

Time where clock is at rest

Time where clock is relative to the observer. Proper
moving relative to the time
observer

t0
t
1 v2 / c2

Clocks moving relative to an observer are measured by that observer to

run more slowly, as compared to the clock at rest.

 This effect arises neither from technical aspects of the clocks nor from the
fact that signals need time to propagate, but from the nature of space-
time itself.
PH1001/20
 Time Dilation
The figure shows the laboratory
clock in operation.

The time interval between ticks is

the proper time t0 and

The time needed for the light pulse

to travel between the mirrors at the
speed of light c is t0/2.

Hence t0/2 = L0/c

PH1001/21
 Time Dilation
The figure shows the
moving clock with its
mirrors perpendicular to
the direction of motion
relative to the ground.
The time interval
between ticks is t.

Because the clock is moving, the light pulse, as seen from the
ground, follows a zigzag path.

On its way from the lower mirror to the upper one in the time t/2,
the pulse travels a horizontal distance of v(t/2) and a total
distance of c(t/2).
PH1001/22
 Time Dilation
Since L0 is the vertical
distance between the
mirrors,

As 2L0/c is the time interval t0

between ticks on the clock on
the ground,

t0 = time interval on clock at rest relative to an observer

=proper time
t = time interval on clock in motion relative to an observer
v = speed of relative motion
c = speed of light PH1001/23
 every observer finds that clocks in motion relative to him tick
more slowly than clocks at rest relative to him
Experimental verification
• Time Dilation and Muon Decay
Muon Decay
 Cosmic rays enter the upper
atmosphere and interact with
particles in the upper atmosphere
creating  mesons (pions), decay
into other particles called muons
 Obey radioactive law:

 ( 0.693t t1 / 2 )
N  N 0e
N: no. of muons at t
No: no. of muons at t=0
Half life: t1/2 =1.5 x 10-6 sec
PH1001/24
 v=0
 0 = 2.2 s v = 0.995 c
  = 22 s = 10 0

The observer and the muon are now in the same frame of reference,
and in this frame the muon’s lifetime is only 2.2 µs.

To the observer, the muon can travel only 0.66 km before decaying.

The only way to account for the arrival of the muon at ground level is
if the distance it travels, from the point of view of an observer in the
moving frame, is shortened by virtue of its motion.

The principle of relativity tells us the extent of the shortening—it must

be by the same factor of “k” that the muon lifetime is extended from
the point of view of a stationary observer.

PH1001/25
Example
1. What is the lifetime of a muon travelling at 0.60 c (1.8 X 108 m/s) if its rest
lifetime is 2.2 s?
Ans. t0 2.2 10 6 s
t t  2.8 10 6 s
1 v2 / c2 1  (0.6c) 2 / c 2

2. How long will a 100 year trip (as observed from earth) seem to the astronaut
who is travelling at 0.995 c?
t0
Ans. t 10 year
1 v2 / c2

3. A particle travels at 1.90×108 m/s and lives 2.10×10−8 s when at rest relative to an
observer. How long does the particle live as viewed in the laboratory?
Ans:
t0 2.10 10 8 s
t   2.7110 8 s
1 v2 / c2
1
1.9 108 s 2

3.0 108 s 2

PH1001/26
4. A spacecraft is moving relative to the earth. An observer on the earth finds that,
between 1 P.M. and 2 P.M. according to her clock, 3601 s elapse on the spacecraft’s
clock. What is the spacecraft’s speed relative to the earth?
Here to = 3600 s is the proper time interval on the earth and t = 3601 s is the
time interval in the moving frame as measured from the earth. Then

Today’s spacecraft are much slower than this. For instance, the
highest speed of the Apollo 11 spacecraft that went to the moon
was only 10,840 m/s, and its clocks differed from those on the
earth by less than one part in 109.
Most of the experiments that have confirmed time dilation made
use of unstable nuclei and elementary particles which readily
attain speeds not far from that of light.
APPOLLO 11 PH1001/27
 Length Contraction Faster means shorter

Length where observer is Length where observer is

moving relative to the at rest relative to the
length being measured. length being measured.

L  L0 1  v 2 / c 2
The length of an object is measured to be shorter when it is
moving relative to the observer than when it is at rest.
Observers from earth would see a
spaceship shorten in the length of travel

(a)

(b) Only shortens in the direction of travel

PH1001/28