Solution

Lorentz transformation (spaceship S' → Earth S)

The spaceship moves in the +z direction with velocity $v = \beta c$

$z = \gamma (z' + vt'), \qquad t = \gamma \left(t' + \frac{v}{c^2}z'\right), \qquad \gamma = \frac{1}{\sqrt{1-\beta^2}}$

The inverse transformations (spaceship from Earth) are:

$z' = \gamma (z - vt), \qquad t' = \gamma \left(t - \frac{v}{c^2}z\right)$

We start with (a) Longitudinal motion ($z' = A \sin \omega t'$)

In the spaceship frame, the body oscillates about the origin (z' = 0). From Earth, the center of oscillation moves with velocity$\beta c$, so the coordinate z of the body will be:

$z = \underbrace{\beta c t}_{\text{center}} + \Delta z$

where$\Delta z$ is the position relative to the center. To find the relationship between z' and t' observed from Earth, we use the inverse transformations:

$z' = \gamma(z - \beta c t), \qquad t' = \gamma\left(t - \frac{\beta}{c}z\right)$

Substituting into the equation of motion $z' = A \sin \omega t'$
we obtain:

$\gamma(z - \beta c t) = A \sin\left[\omega \gamma \left(t - \frac{\beta}{c}z\right)\right]$

Solving approximately gives:

$\boxed{z' = \frac{A}{\gamma} \sin\!\left(\frac{\omega t'}{\gamma}\right)\left(1 + \frac{\beta z'}{\omega c}\right)}$

We have in this case

The amplitude is reduced by a factor of $\gamma$ (length contraction).
The frequency is reduced by $\gamma$ (time dilation).
The phase depends on position.

b) Transverse motion

The motion is perpendicular to the direction of relative motion. Transverse coordinates do not contract (y = y'), and the time t' is uniformly dilated:

$y = y', \qquad t' = \gamma\left(t - \frac{\beta}{c}z\right)$

However, since the motion is transverse, we can directly use$y = A \sin \omega t'$
and substitute t' as a function of t:

$\boxed{y' = A \sin\!\left(\frac{\omega t'}{\gamma}\right)}$

In this case

The amplitude A does not change (there is no contraction in the transverse direction).
The frequency is reduced by $\gamma$ (time dilation).
The oscillation is perfectly harmonic in S', and upon passing to S an additional dependence appears because t' varies with z if the motion is not purely transverse. Nevertheless, for a fixed point in S, the observed frequency is $\omega/\gamma$