As you all know, the 4-velocity vector components in the momentarily comoving reference frame (MCRF) is given by $$\vec U=(1,0,0,0)\ .$$ On the other hand, the 4-acceleration vector is given by $$\vec A=\cfrac{d\vec U}{d\tau}\ ,$$so the components of $\vec A$ in MCRF should be equal to $(0,0,0,0)$. But the textbook said $$\vec A = (0,a^1,a^2,a^3)\ .$$I know the components are given in that way is because of the orthogonality of $\vec U$ and $\vec A$, but i don’t know why it is not $(0,0,0,0)$.
By mathematics, it should be $(0,0,0,0)$, so what is wrong about the mathematics?
What is wrong is, as usual, not mathematics but your understanding of it. The key word here is “momentarily”. The $4$-velocity takes the form $(1,0,0,0)$ at one particular time with respect to one particular inertial frame chosen for that purpose. Unless the body is moving with uniform velocity, the $4$-velocity will no longer take the form $(1,0,0,0)$ in that frame at other times – though it will again be possible to choose a different inertial frame in which it will again take that form. Thus, there is not one momentarily comoving reference frame, but a different one for each instance, and the fact that the $4$-velocity takes the same form when expressed in these frames at these times does not prevent the $4$-velocity from changing over time and thus having a non-zero derivative.