- #1
Jonsson
- 79
- 0
Hello there,
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.
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.