Solution
= Solution
The <Lorentz transformation> is
$$
\boxed{\frac{\partial\bar x^{\bar\alpha}}{\partial x^\mu}
=\begin{pmatrix}
\gamma&-\gamma v&0&0\\
-\gamma v&\gamma&0&0\\
0&0&1&0\\
0&0&0&1
\end{pmatrix}.}
$$
It is a boost along the $x$-axis from the frame in which the black hole moves to its rest frame. Direct matrix multiplication shows that replacing $v$ by $-v$ gives its inverse.