= Solution
In the parent's rest frame, momentum conservation makes the massive daughter and photon momenta equal and opposite, of magnitude $p$. The <relativistic two-body decay> formulas give
$$
p=\frac{M^2-m^2}{2M}c,\qquad
E_R=\frac{M^2+m^2}{2M}c^2.
$$
Since $v=pc^2/E_R$ and $r=m/M$,
$$
\boxed{v=cf(r),\qquad f(r)=\frac{1-r^2}{1+r^2}}.
$$
The daughter's <Lorentz factor> is
$$
\gamma=\frac{E_R}{mc^2}=\frac{1+r^2}{2r}.
$$
If it decays after <proper time> $\tau$, <time dilation> gives $t=\gamma\tau$, while uniform motion gives $x=v\gamma\tau$. Therefore the event coordinates in the parent frame are
$$
\boxed{(ct,x)=
\left(\frac{1+r^2}{2r}c\tau,\,
\frac{1-r^2}{2r}c\tau\right)}.
$$
Back to article page
