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Suppose there are two inertial frames of reference ##S## and ##S'## with coordinates ##(x,ct)## and ##(x',ct')## such that ##S'## is moving relative to ##S## with velocity ##v##. Suppose ##v>0##, that implies ##\gamma >1##.

We know that a Lorentz boost is given by:

$$

x' = \gamma (x -vt), \qquad t' = \gamma (t - \frac{v}{c^2}x ) \qquad (1)

$$

And the inverse transforms are given by

$$

x = \gamma (x' +vt'), \qquad t = \gamma (t' + \frac{v}{c^2}x' ). \qquad (2)

$$

Suppose ##|x_A - x_B| = L## and ##|x'_A - x'_B| = L'## is the length of a rod in the ##S## and ##S'## frame of reference respectively, then we know that there is a length contraction:

$$

L' = |x'_A - x'_B| \stackrel{(1)}{=} |\gamma (x_A - vt) - \gamma(x_B - vt)| = \gamma L \qquad (3)

$$

However using equation (2) a paradox arises:

$$

L = |x_A - x_B| \stackrel{(2)}{=} |\gamma (x'_A +vt') - \gamma (x'_B +vt')| = \gamma |x'_A - x'_B| = \gamma L' \qquad (4)

$$

That means

$$

L' \stackrel{(3)}{=} \gamma L \stackrel{(4)}{=} \gamma \gamma L' \stackrel{(3)}{=}\gamma^3 L \iff L = \gamma^3 L \implies 1 = \gamma^3 \implies \gamma = 1,

$$

a contradiction.

What is going wrong? I really cannot find it.

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# I Length contraction paradox

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