The Lorentz Transformation — Question 8

Question 8

Null coordinates turn a boost into reciprocal scale changes

Define $U=ct-x$ and $V=ct+x$. 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. Derive the transformations of U and V.

2. For $\beta=0.60$, find both scale factors and show the interval product is unchanged.

Question 8 - Solution

Null coordinates turn a boost into reciprocal scale changes

(a) Method and calculation

$$U\prime=ct\prime-x\prime=\gamma(1+\beta)U.$$$$V\prime=ct\prime+x\prime=\gamma(1-\beta)V.$$

(b) Interpretation and further calculation

$$\boxed{U\prime=2U},\qquad\boxed{V\prime=V/2}.$$$$\boxed{U\prime V\prime=UV=c^2t^2-x^2}.$$

Physical check

One null coordinate expands by precisely the factor that contracts the other.