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 the clock 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 O frame. The figure shows how the observer in the O frame sees the situation in O’: (a) When the bulb is at A 1, it flashes, and the light pulse starts traveling toward the mirror at B 1.. It reaches the mirror after time Δt 1. However, in the period the light travels, the entire clock shifts to the right by uΔt 1. When the signal reaches the mirror, it is already at B 2. Note: here the clock length is “L”, not “L 0 ” because we don’t know whether the observer in O sees the same clock length.

Length contraction, continued: (b) The signal is now reflected and travels back toward the bulb. but while it travels, the bulb keeps moving. So when the signal reaches the bulb after Δt 2, the bulb is already shifted to the right by uΔt 2 from the position at which it was at the moment of reflection. As follows from the whole scenario depicted in the figure, on its way toward the mirror the light signal traveled a distance: The same figure as in the preceding slide, repeated: And on the way back, a distance: Because for the observer in O the speed of light has always a constant value of c, these distances can also be written, respectively, as:

Length contraction, continued: So, we get two equations which we can solve for

Length contraction, calculations continued:, 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) that if a time period Δt 0 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: If Δt 0 is the period between the two “ticks” for the observer in O’, then the left-side Δt is the same as the Δt tot we have calculated in the preceding slide. So, we can equate the two expressions:

Length contraction, calculations continued (2): We can also use the expression for Δt 0 (the “tick” period for the observer traveling together with the clock) from the preceding slide show: Putting everything together: Which reduces to a compact expression: Meaning that: Objects in a frame moving relative to the one we are in appear shorter than they really are.