## Length Contraction

An object at rest in an inertial frame S is continually changing its time coordinates. It is described by an infinite succession of space-time events. A point particle at rest in an inertial frame S is described in space-time by a line made up of all the point events that have the same values of (x, y, z) in S. The two lines parallel to the t-axis, and represent x=x_{1} and x=x_{2}, could be the world lines of the ends of the body whose length in S is always measured as l_{0}.

l_{0}=x_{2}-x_{1}

## Derivation

Suppose now that we want to define a procedure for finding the length of the body by means of measurements made in some other frame S’. Measuring the positions of the two ends of the body at the same time t’ as judged in S’, any line representing t’=constant will intersect the world lines of the ends of the body at two points in the x-t diagram. The length l of the object as measured in S’ is given by

l=x_{2}’-x_{1}’

Now as measured in S these same two-point events have x coordinates equal to x_{1} and x_{2}, respectively independent of the time as measured in S. Using the Lorentz transformation we have

so that the measured length of the moving rod, x_{2}-x_{1} Is contracted by a factor γ from its rest length, x_{2}’-x_{1}’.

So what is length is measured to be greatest its rest relative to the observer so then it must be the velocity V relative to the observer its measured length is contracted in the direction of this motion by the factor

Whereas its dimensions are perpendicular to the direction of its motion and are unaffected.

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## Time dilation

A Clock is measured to go at its fastest rate when it is at rest related to the observer. When it moves with velocity relative to the observer, its rate is measured to have been slowed down by a factor

Consider a clock to be at rest at the position x’ in the S’ frame. It may simplify matters to picture the hand of this clock going around and to let unit time be the time it takes the hand of the clock to go around once. Hence, the events we observe (the two successive coincidences of the hand of the clock with a given marker on the face of the clock) span the time interval t’ to t’+1 in the primed coordinates. The S frame observer records these events as occurring at times

The clock in the S’ frame is at a fixed position x’, but the times t_{1 }and t_{2} are read from two different clocks in the S frame, namely the stationary S clock that happens to be coincident with the moving clock at the beginning of the interval and the stationary S clock coincident with the moving clock at the end of the interval.

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These clocks are synchronized, however, so that the time interval they record for the event is simply

Clearly, if, instead of unit time, the S’ clock recorded a time interval t_{2}’-t_{1}’, the S clock would have recorded the corresponding interval

Hence, unit time measured on the S’ clock is recorded as a longer time on the S clocks. From the point of view of observer S, the moving S’ clock appears slowed down, that is, it appears to run at a rate that is slow by a factor

This is known as time dilation. This result applies to all S’ clocks observed from S, for the location x’ in our proof was arbitrary.

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