4. Given point $P$ on the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ with eccentricity $\sqrt{2}$, $F_{1}, F_{2}$ are the two foci of the hyperbola, and $\overrightarrow{P F_{1}} \cdot \overrightarrow{P F_{2}}=0$, then the ratio of the inradius $r$ to the circumradius $R$ of $\triangle P F_{1} F_{2}$ is
荅事 $\frac{\sqrt{6}}{2}-1$.

Answer $\frac{\sqrt{6}}{2}-1$.