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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
:// . . / / / / - - -300.http www pbs org wgbh nova einstein rela car q html
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