4. For the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1$, the left focus is $F_{1}$, the vertices are $A_{1}$ and $A_{2}$, and $P$ is any point on the right branch of the hyperbola. Then the two circles with diameters $P F_{1}$ and $A_{1} A_{2}$ must ( ).
(A) intersect
(B) be internally tangent
(C) be externally tangent
(D) be separate
4.B.

Let the other focus of the hyperbola be $F_{2}$, and the midpoint of segment $P F_{1}$ be $C$. In $\triangle F_{1} F_{2} P$, $C$ is the midpoint of $P F_{1}$, and $O$ is the midpoint of $F_{1} F_{2}$. Therefore, we have
$$
O C=\frac{1}{2}\left|P F_{2}\right|=\frac{1}{2}\left(\left|P F_{1}\right|-\left|A_{1} A_{2}\right|\right) .
$$

Hence, the two circles with diameters $P F_{1}$ and $A_{1} A_{2}$ must be internally tangent.