Consequences of Lorentz Transformation

Bob’s reference frame: The distance measured by the spacecraft is shorter Sally’s reference frame: Sally Bob The relative speed v is the same for both observers:

Length contraction only occurs in the direction of motion— lengths in the perpendicular directions do not change. V = 0 v = 0.87c v=0.995c v=.999c v=c

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

 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 L 0 and the apparent or non- proper length L is as follows 

 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 t 0 and the apparent or non-proper time t is as follows

One consequence: Time Changes Equipment needed: a light clock and a fast space ship.

In Bob’s reference frame the time between A & B is Δ t 0 Sally on earth Bob Beginning Event A Ending Event B Δt0Δt0

Bob In Sally’s reference frame the time between A & B is Δ t Bob AB Sally on earth Length of path for the light ray: and ΔtΔt

Δt 0 = 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 Δt 0 = 1s, v =.999 c then:

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

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

  – meson Decay: Time dilation has been verified in experiments on nuclear particle, called  -mesons. Fast moving  -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 2.994x10 8 m/s, which is of the speed of light c. A  -meson is found to have an average life – time of 2x10 -6 s after which it decays into an electron. Obviously, a  -meson in its life-time can travel a distance of only 2.994x10 8 m/s x 2x10 -6 s≈ 600m or 0.6 km.

 Rossi and Hall in 1941 attributed this result to the time dilation effect. The  -mesons has a life –time t 0 ≈ 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

 In, a meson whose speed is 0.998c ( ) can travel a distance  Hence, despite their brief life-time it is possible for the  -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  s

 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