= Solution
For the standard passive <Lorentz boost>, write $\beta=v/c$ and
$$
ct'=\gamma(ct-\beta x),\qquad x'=\gamma(x-\beta ct).
$$
A <proper orthochronous Lorentz transformation> preserves the sign of the time component of a nonzero future <causal vector> and preserves its <Minkowski norm>. Therefore
$$
\boxed{J^+(0)=\{(t',x'):ct'\ge|x'|\}.}
$$
The upper-right panel of part <1/c> uses $x'$ horizontally and $ct'$ vertically. Its null rays still have slopes $\pm1$. The grey dashed old axes satisfy $x'=-\beta ct'$ for the old time axis and $ct'=-\beta x'$ for the old spatial axis; changing the coordinate grid has not changed which events are reachable.
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