FOURTH LECTURE Consequences of Lorentz Transformation Length Contraction

FOURTH LECTURE Consequences of Lorentz Transformation

Length Contraction

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

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Length contraction only occurs in the direction of motion—lengths in the perpendicular directions do not change. V = 0 v = 0. 87 c v=0. 995 c v=. 999 c v=c

Time Dilation

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

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 t 0 and the apparent or nonproper time t is as follows

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

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

Time Dilation In Sally’s reference frame the time between A & B is Δt Bob A Sally on earth Δt B Length of path for the light ray: and

Time Dilation 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 If Time Dilation = Moving clocks slow down Δt 0 = 1 s, v =. 999 c then:

Time Dilation How do we define time? The flow of time each observer experiences is measured by their watch – we call this the proper time • Sally’s watch always displays her proper time • Bob’s watch always displays his proper time • If they are moving relative to each other they will not agree

Time Dilation A Real Life Example: Lifetime of muons Muon’s rest lifetime = 2. 2 x 10 -6 seconds Many 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?

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 mmeson have a typical speed of 2. 994 x 108 m/s , which is 0. 998 of the speed of light c. A mmeson is found to have an average life – time of 2 x 10 -6 s after which it decays into an electron. Obviously, a m-meson in its life-time can travel a distance of only 2. 994 x 108 m/s x 2 x 10 -6 s≈ 600 m 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 t 0≈ 2 x 10 -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 0. 998 c ( � 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 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 � , a meson whose speed is ) can travel a distance

� � 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