# length contraction question

we know from eintein's theory of relativity that lets say, a ruler is travelling to a speed if light, then we can say that the ruler (from our view as observers) has shorten. but why, lets say we have a 15 km runway, and we let an electron run through it, in electron's perspective, the length of runway is now only around, very short lets say in centimeters. why is that so? i can grasp the idea if we can see actual objects flying, but when i think of the path as being shorten, i dont get the idea why its shorten also. thanks

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Questions that include the word "why" are always difficult to answer, because you end up having to to say that's just the way the universe is. However I can offer you one way to look at this, though it doesn't tell you why the universe is that way.

Amongst the things physicists look for are invarients. That is, quantities that are constant. You often find that if you understand the invarients they tell you a lot about the system you're looking at. In the case of special relativity there is an invarient called the line element, and denoted by $ds$. If you move a distance $dx$, $dy$ and $dz$ in space, and $dt$ in time, then the line element is given by:

$$ds^2 = -c^2dt^2 + dx^2 + dy^2 + dz^2$$

The significance of $ds$ is that all observers in all frames will agree on the value of $ds$ i.e. it is an invarient.

The fact that $ds$ is invarient is pretty much all you need to know about special relativity. It even encodes the fact that the speed of light is constant. The weird effects like time dilation and length contraction arise from the fact that in the equation of $ds^2$ the term in $dt^2$ has a negative sign while all the spatial terms have a positive sign.

I guess your next question is why $ds$ is given by the above equation and why it's an invarient, but those questions I can't answer. That's just the way the universe is!

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In the frame of the electron, the electron is stationary and the runway is rushing past it. So whether a runway is rushing past the electron or a ruler is rushing past the electron, both are length contracted in the direction of motion.

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