(III) (a) In reference frame S, a particle has momentum p→ = pₓî along the positive x axis. Show that in frame S’, which moves with speed v as in Fig. 36–12, the momentum has components

p'ₓ = pₓ - (vE/c²) / √(1 - v²/c²)

p'ᵧ = pᵧ

p'_𝓏 = p_𝓏

E' = E - pₓv / √(1 - v²/c²)

(These transformation equations hold, actually, for any direction of p→ , as long as the motion of S' is along the x axis.) (b) Show that pₓ , pᵧ, p_𝓏, E/c transform according to the Lorentz transformation in the same way as x, y, z, ct.

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Verified step by step guidance

1. Identify the given quantities and the frame of reference. In frame S, the particle has momentum along the x-axis, represented as \( \vec{p} = p_x \hat{i} \). Frame S' is moving with a velocity v relative to S along the x-axis.
2. Use the Lorentz transformation for momentum and energy to find the components in the moving frame S'. The Lorentz transformation equations for momentum components are \( p'_x = \gamma (p_x - vE/c^2) \), \( p'_y = p_y \), and \( p'_z = p_z \), where \( \gamma = 1/\sqrt{1-v^2/c^2} \) is the Lorentz factor.
3. Apply the Lorentz transformation for energy, which is given by \( E' = \gamma (E - p_x v) \). This equation accounts for the change in energy due to the relative motion along the x-axis.
4. Verify that the transformed momentum components \( p'_x \), \( p'_y \), and \( p'_z \) and the transformed energy \( E' \) satisfy the relativistic energy-momentum relation \( E'^2 = (p'_x^2 + p'_y^2 + p'_z^2)c^2 + (m_0c^2)^2 \), where \( m_0 \) is the rest mass of the particle.
5. For part (b), demonstrate that the components \( p_x \), \( p_y \), \( p_z \), and \( E/c \) transform according to the Lorentz transformation similarly to the spacetime coordinates \( x \), \( y \), \( z \), and \( ct \). This can be shown by comparing the transformation equations for momentum and energy with those for the spacetime coordinates.