The Relativity of Velocities — Question 5

Question 5

A transverse photon remains lightlike after aberration

A photon is emitted along +y in a spacecraft frame: $u_x\prime=0$, $u_y\prime=c$. The spacecraft moves at $v=0.60c$ along x. For $S\prime$ moving at +v along x, use $u_x=(u_x\prime+v)/(1+vu_x\prime/c^2)$ and $u_y=u_y\prime/[\gamma_v(1+vu_x\prime/c^2)]$, with $\gamma_v=(1-v^2/c^2)^{-1/2}$.

1. Find its laboratory components and speed.

2. Find its laboratory propagation angle from +x.

Question 5 - Solution

A transverse photon remains lightlike after aberration

(a) Method and calculation

$$\boxed{u_x=0.60c},\qquad\boxed{u_y=c/1.25=0.80c}.$$$$|\mathbf u|=c\sqrt{.60^2+.80^2}=\boxed{c}.$$

(b) Interpretation and further calculation

$$\tan\theta=u_y/u_x=4/3.$$$$\boxed{\theta=53.1301^\circ}.$$

Physical check

The boost changes the propagation direction while preserving the magnitude of a vacuum light velocity.