4. In the rectangular prism $A B C D-A_{1} B_{1} C_{1} D_{1}$, $A B=$ $2 \sqrt{2}, B C=A A_{1}=2, P$ is a moving point on the diagonal $A C_{1}$, $Q$ is a moving point on the base $A B C D$ (points $P$ and $Q$ can coincide). Then the minimum value of $B_{1} P+P Q$ is $\qquad$
4.3.

Fold $\triangle A B_{1} C_{1}$ to be coplanar with $\triangle A C C_{1}$, and by the shortest distance from a point to a plane being the perpendicular segment, we get the result.