Solution
= Solution
Because $\tau$ is <proper time>, the <four-velocity> has normalization
$$
g_{\mu\nu}\dot x^\mu\dot x^\nu=-1.
$$
At $t=0$, the initial data therefore give
$$
-\dot t(0)^2+v_0^2=-1.
$$
Both $t$ and $\tau$ are future oriented, so the positive root must be chosen:
$$
\boxed{\dot t(0)=\sqrt{1+v_0^2}.}
$$