Special Relativity: Time Dilation and Length Contraction

SPH4U

Einstein’s Postulates

• All laws of physics are valid in all inertial (non-accelerated) reference frames.

Einstein’s Postulates

• Light travels at c = 3.0 x 108 m/s relative to all reference frames.

Simultaneity

This implies that events that are simultaneous (occurring at the same time) for one observer are not simultaneous for an observer in a different reference frame.

C will see the lightning flashes at A and B at the same time.

D will see the flash at B happen before the flash at A.

Measurement

And that observers in different reference frames measure time and length differently.

One observer bounces a laser off a mirror on the ceiling.

The other sees the light travelling a longer path in a longer time.

Time DilationBy the Pythagorean Theorem:

ct ct’

vt’

Time DilationBy the Pythagorean Theorem:

ct ct’

vt’

2

222

22

222

222

22

222222

222

1'

''

''

''

''

c

vtt

tc

vtt

ttc

vt

tctvtc

ctvtct

Time Dilation

(for an event that occurs in the non-prime reference frame)

ct ct’

vt’

2

22

2

1

'1'

cv

ttor

c

vtt

Time Dilation

(for an event that occurs in the non-prime reference frame)

ct ct’

vt’

2

22

2

1

'1'

cv

ttor

c

vtt

dilation = expansion

Minutephysics

http://www.youtube.com/watch?v=ajhFNcUTJI0

Gamma

The factor is often written as :2

2

1

1

cv

ttort

t

''

Note that is dimensionless (it has no units).

Length Contraction

Similarly, it can be shown that lengths parallel to the direction of motion are contracted:

2

22

2

1

'1'

cv

LLor

c

vLL

'' LLorL

L

Muons

Muons created in the upper atmosphere have a lifetime of 2.2 s in their reference frame but reach the Earth’s surface because they see the distance to the surface as contracted and/or we see the lifetime as dilated.

Example

What is the apparent lifetime of a muon travelling at 0.99c?

Example

What is the apparent lifetime of a muon travelling at 0.99c?

09.7)99.0(

1

1

1

1

2

2

2

2

cc

cv

Example

What is the apparent lifetime of a muon travelling at 0.99c?

09.7)99.0(

1

1

1

1

2

2

2

2

cc

cv

st

st

tt

16'

)2.2)(09.7('

'

More Practice

Textbook Questions• p. 573 #2, 3• p. 576 #6, 7, 8

Recommended: “Old School” Relativity

http://www.scivee.tv/node/2415