While following a derivation for asymptotic solutions to a scalar field in Schwarzschild spacetime, it became necessary to approximate the interior null co-ordinates as a function of the exterior null co-ordinates, where exterior refers to free space and interior refers to inside some massive ball where gravitational potential exists and affects the field. The ball's boundary radius $R_0$ is greater than its Schwarzschild radius and so there is no event horizon and the reason why the two regions exist.
The null co-ordinates in the exterior region are $$u = t - r^* + R_0^*\ ,\ v = t + r^* - R_0^*$$ where the notation "$*$" means that $d^* = d + 2M \text{ln}\bigg|\dfrac{d}{2M} - 1\bigg|$
For the purpose of finding the exterior asymptotic solutions, the exact structure of inside the ball is not required, and the interior null co-ordinates can be kept general $U$, $V$. One can define the transformation functions; $$\alpha(u) = U\ ,\ \beta(V) = v$$
A common origin is intoduced without loss of generality at $t = 0, r = R_0$. If the boundary of the ball begins to collapse, such that it has a trajectory $R(t)$, the metrics of each region, $ds^2 = Adudv$, $ds^2 = BdUdV$ must match at this boundary for all $t$, where $A, B$ are conformal factors that may be functions of $r, t$.
Apparently, this all the information one needs to conclude; $$\alpha^{\prime}(u) = \frac{dU}{du} = (1 - \dot{R})A\bigg(\bigg[\sqrt{AB(1 - \dot{R}^2) + \dot{R}^2}\bigg] - \dot{R}\bigg)^{-1}$$
I cannot derive this calculation. I suspect I am lacking some theory or method. The closest I can get is by considering the Jacobian transformation, which at the boundary is; $$A = B \bigg(\alpha^{\prime}(u) \dfrac{1}{\beta^{\prime}(V)} - \dfrac{\partial U}{\partial v}\dfrac{\partial V}{\partial u}\bigg)$$
One of the biggest difficulties I think is trying to introduce a time derivative of the boundary, I have no idea how that would manifest itslef.
Any insights, or directions would be most appreciated. I understand there's quite a bit of physics but really that just provides context, I am solely looking for a way to derive the $\alpha^{\prime}(u)$ equation. Thanks