I hope this is not too silly a question: We often see
$$\frac{dt}{d\tau}=\gamma=\frac{1}{\sqrt{1-v^2}},$$ taking $c=1$.
Problem: I don't understand why...
In the Minkowski metric, using the $(-+++)$ signature and taking $c=1$, $$ds^2=-dt^2+d\vec x^2\\ d\tau^2=-ds^2\\ \implies d\tau^2=dt^2-d\vec x^2\\ \implies 1=\left(\frac{dt}{d\tau}\right)^2-v^2\\ \implies \frac{dt}{d\tau}=\sqrt{1+v^2}\neq \frac{1}{\sqrt{1-v^2}}$$
What has gone wrong? with my reasoning?