Slide show Notes_04.ppt

Another effect emerging from the Einstein’s postulates:Length contraction.We now turn the “light-clock” sideways. Suppose that the frame O in which the clock rests moves in the horizontal direction with speed u .

Will the frequency of the flashes be different than the frequency for theclock in upright position?No, of course not! The distance the light signal travels does not change,and the speed of light is the same in all directions!

Length contraction, continued:Suppose there is another Identical clock in frame O’that moves with in the horizontal direction with speed u relative to the Oframe.The figure shows how theobserver in the O frame sees the situation in O’:(a) When the bulb is at A1 ,it flashes, and the light pulse starts traveling toward the mirror at B1 ..It reaches the mirror aftertime Δt1 . However, in theperiod the light travels, theentire clock shifts to the right by uΔt1 . When the signal reaches the mirror,it is already at B2 .

Note: here the clocklength is “L”, not “L0”because we don’tknow whether the observer in O seesthe same clock length.

Length contraction, continued:(b) The signal is now reflectedand travels back toward the bulb.but while it travels, the bulb keepsmoving. So when the signal reaches the bulb after Δt2 , the bulb is already shifted to the rightby uΔt2 from the position at which it was at the moment of reflection.

As follows from the whole scenariodepicted in the figure, on its way toward the mirror the light signal traveled a distance:

The same figure as in the preceding slide, repeated:

1tuL And on the way back, a distance:

2tuL Because for the observer in O the speed of light has always a constant valueof c, these distances can also be written, respectively, as: 21 and tctc

Length contraction, continued:

So, we get two equations which we can solve for : and 21 tt

2222

021tot

222

111

/112

11:p"return tri" ssignal' theof

time totalobtain the which weFrom

and

cucL

ucucucL

ucucLttt

ucLttctuL

ucLttctuL

Length contraction, calculations continued:

tott , let’s stress it again, is the “tick period” registered by the observer in the O system – so the clock moves relative to him/her.

We have also shown (in the preceding Notes_03.ppt presentation) thatif a time period Δt0 elapses between two events in the O’ frame, than the observer in the O frame watching the same events registers a longer elapse of time between them, equal to:

220

/1 cu

tt

If Δt0 is the period between the two “ticks” for the observer in O’, thenthe left-side Δt is the same as the Δt tot we have calculated in the preceding slide. So, we can equate the two expressions:

220

22/1/1

12

cu

tcuc

L

Length contraction, calculations continued (2):

We can also use the expression for Δt0 (the “tick” period for the observertraveling together with the clock) from the preceding slide show:

cLt 0

02

220

22/1

2/1

12cuc

Lcuc

L

Putting everything together:

Which reduces to a compact expression:

022

0 /1 LLcuLL Meaning

that:

Objects in a frame moving relative to the one we are in appear shorter than they really are.