Twin paradox and length contraction

[resaypi]

We know that the twin paradox is not a paradox since one of the twins accelerate and time passes more slowly for him relative to his twin on earth. My question is why can't we apply the same reasoning for length contraction? If one of the twin A measures the lengths of sticks in the reference frame of B, why shouldn't B measure the length of sticks of A to be longer? My approach is natural motion is to remain unaccelerated, and acceleration causes the shortening of the sticks.

 

[sylas]

We know that the twin paradox is not a paradox since one of the twins accelerate and time passes more slowly for him relative to his twin on earth. My question is why can't we apply the same reasoning for length contraction? If one of the twin A measures the lengths of sticks in the reference frame of B, why shouldn't B measure the length of sticks of A to be longer? My approach is natural motion is to remain unaccelerated, and acceleration causes the shortening of the sticks.

No; length contraction and time dilation are only a consequence of relative velocities of two observers.

Length contraction and time dilation are, in a sense, two aspects of the same thing; the Lorentz transformations. The speed of light remains constant; therefore the factor by which time dilates is the same as the factor by with length contracts. The factor is called the "gamma factor".

Consider the twins. One moves from Earth to a distant star at 60% light speed, and returns.

For the twin who stayed at home the distance between Earth and the star is 6 light years. The traveling twin therefore takes 10 years to get there and ten years to return. Their clock is dilated by the gamma factor, which is (1-0.6²)^-0.5 = 1.25, and so they age by 20 / 1.25 = 16 years.

For the traveling twin, the distance between the Earth and the star is 6/1.25 = 4.8 light years. The star moves (in their frame) at 60% light speed, and therefore takes 4.8/0.6 = 8 years to move from its starting position to their location. (Note that in the traveling twin's frame(s), it is actually the star and the Earth which are moving.) It also takes another 8 years to move back out again, for 16 years to total. Same result.

The "paradox" arises when one makes the mistake of simply applying factors to the stay-at-home twin, as if the traveling twin is in a single reference frame. You can't do that. (Add in edit: I put quotes around paradox because it isn't a paradox at all, as Filip points out; merely a common error in reasoning.) The time dilation and length contraction actually fall out as consequences of the Lorentz transformations, and these depend on velocity only. Not acceleration.

Cheers -- sylas

 

[Filip Larsen]

Time dilation is, according to the clock hypothesis, only dependent on the relative speed, not acceleration and the twin paradox is not a paradox because the situation, even without acceleration, is not symmetric between the two twins as can verified by analyzing the relativistic intervals of a simplified "experiment".

 

[resaypi]

then what determines which clock should move slower?

from this picture the point of return (acceleration) causes time to dilate by the loss of simultanety. so acceleration should affect time dilation.

 

[sylas]

then what determines which clock should move slower?

The rate at which a clock ticks is frame dependent. Which clock ticks slower depends exclusively on the frame in which they are being compared. The clock that has a higher velocity has the slower tick rate. And therefore, since velocity depends on the frame, so too does the ticking rate.

There is no such thing as an absolute comparison of the clocks. A clock is not altered by acceleration, nor even by velocity. Its the same clock, which ticks at different rate depending on which frame is used to define the rate.

Cheers -- sylas

 

[resaypi]

One more question is, why is simultaneity lost in the twin paradox