Past exam of the symmetry and relativity course of the University of Oxford / 2019 / 1 / d / i / Solution

For the standard passive Lorentz boost, write $\beta =v/c$ andA proper orthochronous Lorentz transformation preserves the sign of the time component of a nonzero future causal vector and preserves its Minkowski norm. ThereforeThe upper-right panel of part c uses $x^{\prime }$ horizontally and $c{t}^{\prime }$ vertically. Its null rays still have slopes $\pm 1$. The grey dashed old axes satisfy $x^{\prime }=-\beta c{t}^{\prime }$ for the old time axis and $c{t}^{\prime }=-\beta x^{\prime }$ for the old spatial axis; changing the coordinate grid has not changed which events are reachable.