The Relativity of Velocities — Question 7

Question 7

The velocity transformation preserves the entire subluminal disk

Let a particle in $S\prime$ have two-dimensional velocity $(u_x\prime,u_y\prime)$ with $u\prime^2=u_x\prime^2+u_y\prime^2<c^2$. 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. Simplify $1-(u_x^2+u_y^2)/c^2$ into a factored expression.

2. Use the expression to prove the speed bound and then examine the lightlike limit.

Question 7 - Solution

The velocity transformation preserves the entire subluminal disk

(a) Method and calculation

$$D=1+vu_x\prime/c^2,\quad1-u^2/c^2=\frac{D^2-(u_x\prime+v)^2/c^2-(1-v^2/c^2)u_y\prime^2/c^2}{D^2}.$$$$\boxed{1-\frac{u^2}{c^2}=\frac{(1-v^2/c^2)(1-u\prime^2/c^2)}{(1+vu_x\prime/c^2)^2}}.$$

(b) Interpretation and further calculation

$$|v|<c,\ u\prime<c\quad\Rightarrow\quad\boxed{u<c}.$$$$u\prime=c\quad\Rightarrow\quad\boxed{u=c}.$$

Physical check

The denominator is nonzero in the physical domain. The transformation cannot turn a massive-particle velocity into a lightlike or superluminal one.