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22.1 Length Contraction
Discussion
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@Sander Bouwhuis. No, length contraction is not a side effect of time asynchrony. Consider we know the length of a train when at rest, and we know its velocity when in motion, so we can calculate its contracted length. We place two clocks along the track so that the distance between them is equal to the contracted length of the train, and we synchronize those clocks. As the train passes, the clock furthest in the direction of the train records the time when the front of the train reaches it, while the other clock records the time when the back of the train reaches it. Both clocks will record the same time. Since those clocks are synchronized, the length of the train must be contracted when in motion relative to the clocks.
Of course, those at rest relative to the train will say that the clocks are in motion and out of sync. So it’s all relative.
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I’ve seen arguments against the validity of length contraction as a horizontal light clock, should actually tick at different rate than a vertical clock due to the contracted distance. You can’t have 2 different readings of time from the same source. So is it possible to perform an experiment of Brian’s concept here for clarification or has an experiment ever been conducted?
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While it’s true that a shorter horizontal clock would cycle faster than a vertical clock if they were both at rest, a horizontal clock in motion would cycle slower than a vertical clock in motion (according to an observer at rest if the clocks had the same velocity and direction) if it weren’t for length contraction. This may seem counter-intuitive at first, but consider that the light in a moving horizontal clock must not only cover the distance between both sides of the clock, but must also cover the additional distance created by the movement of the clock. If the light must travel 3/2 times the rest-length of its clock in one direction to catch up to the wall moving away from it, but can only travel 1/6 the rest-length of the clock when it reverses before running into the other wall, then the light travels an additional 2/3 the rest-length of the clock each cycle.
Consider a vertical clock and a horizontal clock each with length L = 1/2 feet when at rest. When moving, these clocks have velocity v = 4/5 ft/ns (or 4/5c). The equation for determining the cycle time of a light clock at rest (whether vertical or horizontal) is t = 2L/c. This yields a value of 1 ns for both clocks. For a vertical clock in motion, a factor of γ (5/3 in this case) must be applied, t = 2Lγ/c, yielding 5/3 ns according to an observer at rest. The equation for determining the cycle time of a moving horizontal clock (which accounts for length contraction) is L/γ/(c-v) + L/γ/(c+v), which yields 5/3 ns, just like the moving vertical clock. However, if we remove the factor of γ from our equation, L/(c-v) + L/(c+v), we get 25/9 ns, an additional 1+1/9 ns over the vertical clock. So the factor of γ, the length contraction, is necessary to keep the clocks in sync.
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I’ve seen arguments against the validity of length contraction as a horizontal light clock, should actually tick at different rate than a vertical clock due to the contracted distance. You can’t have 2 different readings of time from the same source. So is it possible to perform an experiment of Brian’s concept here for clarification or has an experiment ever been conducted?
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From George’s perspective Gracie seconds are longer so Gracies meters are longer and if Gracie’s meters are longer fewer meters will fit in a particular physical distance (in this case the train) so Gracie will measure the train to be shorter. Again the metric is changing (how much physical distance is actually in a metre) due to the definition a meter: light covers a constant amount of distance in a given time. If the time unit changes (in this case a second becomes longer) light will cover more physical distance in that time unit and so the metre becomes longer. With a longer metre you will measure less metres for any given physical distance. The physical length hasn’t change just the metric for measuring the length has.
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// Roelof said: “From George’s perspective Gracie seconds are longer so Gracies meters are longer” //
From George’s perspective Gracie seconds are longer than George’s clock seconds, that’s true, but that not make Gracie meters longer. Gracies meters are shorter from George perpective. Both measure the same time dilation and length contraction in the other frame of reference.
// Roelof said:”f the time unit changes (in this case a second becomes longer) light will cover more physical distance in that time unit and so the metre becomes longer” //
You are giving a strange definition of a meter, the distance traveled by light in Geroge reference frame each second of Gracies reference frame, that has no sense.
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