4. As shown in Figure 2, a tangent line is drawn from the left focus $F$ of the hyperbola $\frac{x^{2}}{a^{2}}-\frac{y^{2}}{b^{2}}=1(a>0, b>0)$ to the circle $x^{2}+y^{2}=a^{2}$. The point of tangency is $T$, and the extension of $F T$ intersects the right branch of the hyperbola at point $P$. If the midpoint of segment $P F$ is $M$, and $O$ is the origin, then the relationship between $|O M| - |M T|$ and $b-a$ is ( ).
(A) $|O M|-|M T|>b-a$
(B) $|O M|-|M T|<b-a$
(C) $|O M|-|M T|=b-a$
(D) Uncertain
4.C.

Let the right focus of the hyperbola be $\boldsymbol{F}^{\prime}$, then
$$
|P F|=\left|P F^{2}\right|+2 a \text{. }
$$

Since $|P F|=2|M F|,\left|P F^{\prime}\right|=2|O M|$, we have
$$
|M F|=|O M|+a \text{. }
$$

In the right triangle $\triangle O T F$, since
$$
|T F|=\sqrt{|O F|^{2}-|O T|^{2}}=b,
$$

we get $|M F|=|M T|+|T F|=|M T|+b$.
Thus, $|M T|+b=|O M|+a$, which means
$$
|O M|-|M T|=b-a
$$