4. As shown in the figure, $P$ is a point on the diagonal $B D_{1}$ of the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, and the plane $A P C \parallel$ plane $D A_{1} C_{1}$. If $A A_{1}=2 A D$, then the tangent value of the dihedral angle $P-A B-C$ is $\qquad$ .
Let $E, F$ be the centers of the upper and lower bases, respectively, then $D F / / E B_{1}, \triangle P E B \sim \triangle P B_{1} D_{1} \Rightarrow \frac{P B}{P D_{1}}=\frac{E B}{B_{1} D_{1}}=\frac{1}{2}$, thus $P B=\frac{1}{3} D_{1} B$. Draw $P G \perp B D$ at $G$, draw $G H \perp A B$ at $H$, and connect $P H$. Then $\angle P H G$ is the plane angle of the dihedral angle $P-A B-C$. By $P B=\frac{1}{3} D_{1} B \Rightarrow P G=\frac{1}{3} A A_{1}, G H=\frac{1}{3} A D \Rightarrow \tan \angle P H G=2$.