The Lorentz Transformation — Question 1

Question 1

A full event transform includes unchanged transverse coordinates

An event in S has $(ct,x,y,z)=(1000,600,200,-100)\,\mathrm m$. The frame $S\prime$ moves at $0.60c$ along x. Use $ct\prime=\gamma(ct-\beta x)$, $x\prime=\gamma(x-\beta ct)$, $y\prime=y$, $z\prime=z$, where $\beta=v/c$ and $\gamma=(1-\beta^2)^{-1/2}$.

1. Find all four transformed coordinates.

2. Check the inverse map for ct and x and interpret the resulting $x\prime$.

Question 1 - Solution

A full event transform includes unchanged transverse coordinates

(a) Method and calculation

$$\gamma=1.25,\quad ct\prime=1.25(1000-0.60\cdot600)=\boxed{800\,\mathrm m}.$$$$\boxed{x\prime=0,\quad y\prime=200\,\mathrm m,\quad z\prime=-100\,\mathrm m}.$$

(b) Interpretation and further calculation

$$ct=1.25(800+0.60\cdot0)=1000\,\mathrm m.$$$$x=1.25(0+0.60\cdot800)=600\,\mathrm m.$$

Physical check

The event lies in the moving frame plane x-prime equals zero, but its nonzero transverse coordinates mean it is not at the spatial origin.