FOURTH LECTUREConsequences of Lorentz

Transformation

Length Contraction

22

11

1 cv

tvxx

22

22

1 cv

tvxx

2222

12012

11 cv

L

cv

xxLxx

2210 cvLL

Length Contraction

Bob’s reference frame:

The distance measured by the spacecraft is shorter

Sally’s reference frame:

Sally

Bob

0

0

LLv

t t

The relative speed v is the same for both observers:

22

0

/1 cv

tt

220 /1 cvLL

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Length contraction only occurs in the directionof motion—lengths in the perpendicular directions do not change.

V = 0 v = 0.87c v=0.995c v=.999c v=c

Time Dilation

22

21

11 cv

c

vxt

t

22

22

21 cv

c

vxt

t

220

22

0

22

12012

1

1

1

cvttor

cv

tt

cv

ttLtt

PROPER FRAME

The inertial frame of reference in which the observed body is at rest is called the proper frame.

PROPER LENGTH

The length of a rod as measured in the inertial frame in which it is at rest is called the PROPER LENGTH, the relation between the proper length L0 and the apparent or non-proper length L is as follows

2210 cvLL

Proper Time

The time interval recorded by a clock fixed with respect to the observed event is called the Proper Time ,the relation between the proper time t0 and the apparent or non-proper time t is as follows

220 1 cvtt

Time DilationOne consequence: Time Changes

Equipment needed: a light clock and a fast space ship.

Time DilationIn Bob’s reference frame the time between A & B is Δt0

Sally on eart

h

Bob

Beginning Event A

Ending Event B

c

Dt

20

D

Δt0

Bob

Time DilationIn Sally’s reference frame the time between A & B is Δt

Bob

A BSally on eart

h

22 2 22 2 2

2

v ts D L D

Length of path for the light ray:

c

st

2and

Δt

Time Dilation

22

0

/1 cv

tt

Δt0 = the time between A & B measured by Bob

Δt = the time between A & B measured by Sally

v = the speed of one observer relative to the other

Time Dilation = Moving clocks slow down

If Δt0 = 1s, v = .999 c then:s 500

999.1

s 12

t

Time Dilation

• Bob’s watch always displays his proper time

• Sally’s watch always displays her proper time

How do we define time?

The flow of time each observer experiences is measured by their watch – we call this the proper time

• If they are moving relative to each other they will not agree

Time DilationA Real Life Example: Lifetime of muonsMuon’s rest lifetime = 2.2x10-6 secondsMany muons in the upper atmosphere (or in the laboratory) travel at high speed. If v = 0.999 c. What will be its average lifetime as seen by an observer at rest?

s 101.1999.1

s 102.2

/1

3

2

6

22

0

cv

tt

Experimental Verification of Time Dilation

M – meson Decay: Time dilation has been verified in experiments on nuclear particle , called m-mesons. Fast moving m-mesons , are created in the cosmic rays at a height of about 10 kilometers from the surface of the earth and reach the earth in large numbers. Theses m-meson have a typical speed of 2.994x108 m/s , which is 0.998 of the speed of light c. A m-meson is found to have an average life – time of 2x10-6 s after which it decays into an electron. Obviously, a m-meson in its life-time can travel a distance of only 2.994x108 m/s x 2x10-6 s≈ 600m or 0.6 km .

HOW DO m-MESONS TRAVEL A DISTANCE OF 10 Km TO REACH THE EARTH ?

Rossi and Hall in 1941 attributed this result to the time dilation effect. The m-mesons has a life –time t0≈ 2x10-6 s in its own frame of reference , in observer`s frame of reference on the earth, however , the life time is lengthened owing to the relative motion, to the value t given by

st

cvtt56

26220

1017.363.0102

)998.0(11021/

In , a meson whose speed is 0.998c ( ) can travel a distance

Hence, despite their brief life-time it is possible for the m-mesons to reach the ground from the large altitudes at which they are actually formed . More recently , the dilation caused by the thermal vibration of the nuclei in certain crystals has also been verified .

A similar experiment was done with pions by Ayres in 1971, the proper life time measured for point at rest is known to be 26 ms

s51017.3

sm /10994.2 8

kmm 5.995001017.310994.2 58

What will be the apparent length of a meter stick measured by an observer at rest when the stick is moving along its velocity equal

Solution

c2

3

mL

cvLL

5.04

1

4

311

1 220