Solution
= Solution
Parametrize the spacelike geodesic by unit length. With $J=r^2\dot\theta=b$, its norm equation gives
$$
\dot r^2=r^2-b^2.
$$
The total angular change is therefore
$$
|\Delta\theta|=2\int_b^\infty
\frac{b\,dr}{r^2\sqrt{r^2-b^2}}.
$$
Putting $r=b\sec u$ evaluates the half-integral as $1/b$, and hence
$$
\boxed{|\Delta\theta|=\frac2b}.
$$