Four Vectors

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A reference frame S' is moving at 0.6_c_ in the z direction with respect to a stationary frame S. An event occurs in S' with the coordinates (x', y', z', ct')=(1, 0, 2, 3). What are the coordinates (x, y, z, ct) of the event with respect to the stationary frame S?

(1, 0, 4.75, 5.25)

CORRECT

(1, 0, 5.25, 4.75)

(1, 0, 2.64, 3.3)

(1, 0, 3.3, 2.64)

(1, 0, 2, 3)

Explanation

To apply a Lorentz transformation, we need gamma and beta:

$\beta=\frac{v}{c}=\frac{3}{5}$

$\gamma=\frac{1}{\sqrt{1-\frac{v^2}{c^2}}}=\frac{5}{4}$

Then, apply the Lorentz transformation:

$z=\gamma(z'+\beta ct')=\frac{5}{4}(2+\frac{3}{5}\cdot 3)=4.75$

$ct=\gamma(ct'+\beta z')=\frac{5}{4}(3+\frac{3}{5}\cdot 2)=5.25$

A container of water is carried on a space ship traveling at 60% the speed of light relative to the Earth. From the perspective of an observer on Earth, what is the speed of the light in the water?

$n_{water}=1.33$