Solution
= Solution
Let $t^a$ exhibit the <stable causality> of $g$, so $g-t_at_b$ has no <closed timelike curves>. Set
$$
r^a=\frac{t^a}{\sqrt2},\qquad g'_{ab}=g_{ab}-r_ar_b=g_{ab}-\frac12t_at_b.
$$
The vector $r^a$ remains timelike for $g'$. Widening the cones of $g'$ once more using $r$ gives
$$
g'_{ab}-r_ar_b=g_{ab}-t_at_b,
$$
which is causal by assumption. Hence
$$
\boxed{g-r r\ \text{is itself stably causal}}.
$$