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As an object moves faster, its time runs slower by factor X. I get why this is the case, but I'm not sure why distance is decreased by factor X as well.

Questions:

1) Why does a moving object's travel distance decrease by factor X as its time decreases by factor X? The explanation I've heard is that because of v = d/t, since the moving object's speed is constant from different reference frames, its distance must decrease as time decreases. But I thought motion was relative?

2) Why does the object's physical length, from the reference frame of an observer, decrease by factor X? Even if the object travels a smaller distance from part 1), I'm completely lost on how that would affect its length.

Also, I'm aware of the unification between space and time in General Relativity, but I'm learning about Special Relativity and would prefer an explanation in that realm.

Inertial Ignorance
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3 Answers3

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A core tenet of special relativity is that the spacetime interval between two events:

$$ \Delta s^2=(c\Delta t)^2-\Delta\vec x^2 $$

is a Lorentz invariant (i.e. it's the same if measured in any two reference frames).

Since you know that $\Delta t^2$ varies between two frames due to time dilation, it follows that $\Delta\vec x^2$ must also vary in order to keep $\Delta s^2$ invariant. This is length contraction.

Chris
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You say:

I get why time dilates

But in what sense, do you "get"? If you know that time dilation is a result the transformation laws between inertial frames being Lorentz transformations and not Galilean, then length contraction is another result from the same transformation equations. Time dilation doesn't cause length contraction, though they are related.

PhyEnthusiast
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This is another "deep space" scenario but a little different. You're in a space station when a ship going 0.5c, or 150 million meters/sec, approaches. You know the ship has a length of 150 meters and it has a central master clock with large numeric displays on the sides, one at the nose and one at the tail. Because you have extremely sharp eyes you're able to read the clock's time on each display as the ship flashes by. You expect to read a time difference of exactly one microsecond, but, because the ship's clock is running slower than yours, you don't see 1 microsec but 0.866 microsecond as the time difference. And going 150 million m/sec, that makes the apparent length 130 meters instead of 150.

Pete Carter
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