The Schwarzschild metric in isotropic coordinates $\bar{x}^{\bar{\alpha}}=(\bar{t}, \bar{x}, \bar{y}, \bar{z}), \bar{\alpha}=0, \ldots, 3$, is given by:

$$d s^{2}=\bar{g}_{\bar{\alpha} \bar{\beta}} d \bar{x}^{\bar{\alpha}} d \bar{x}^{\bar{\beta}}=-\frac{(1-A)^{2}}{(1+A)^{2}} d \bar{t}^{2}+(1+A)^{4}\left(d \bar{x}^{2}+d \bar{y}^{2}+d \bar{z}^{2}\right)$$

where

$$A=\frac{m}{2 \bar{r}}, \quad \bar{r}=\sqrt{\bar{x}^{2}+\bar{y}^{2}+\bar{z}^{2}}$$

and $m$ is the mass of the black hole.

(a) Let $x^{\mu}=(t, x, y, z), \mu=0, \ldots, 3$, denote a coordinate system related to $\bar{x}^{\bar{\alpha}}$ by

$$\bar{t}=\gamma(t-v x), \quad \bar{x}=\gamma(x-v t), \quad \bar{y}=y, \quad \bar{z}=z,$$

where $\gamma=1 / \sqrt{1-v^{2}}$ and $-1<v<1$. Write down the transformation matrix $\partial \bar{x}^{\bar{\alpha}} / \partial x^{\mu}$, briefly explain its physical meaning and show that the inverse transformation is of the same form, but with $v \rightarrow-v$.

(b) Using the coordinate transformation matrix of part (a), or otherwise, show that the components $g_{\mu \nu}$ of the metric in coordinates $x^{\mu}$ are given by

$$d s^{2}=g_{\mu \nu} d x^{\mu} d x^{\nu}=f(A)\left(-d t^{2}+d x^{2}+d y^{2}+d z^{2}\right)+\gamma^{2} g(A)(d t-v d x)^{2}$$

where $f$ and $g$ are functions of $A$ that you should determine. You should also express $A$ in terms of the coordinates $(t, x, y, z)$.

(c) Consider the limit $v \rightarrow 1$ with $p=m \gamma$ held constant. Show that for points $x \neq t$ the function $A \rightarrow 0$, while $\gamma^{2} A$ tends to a finite value, which you should determine. Hence determine the metric components $g_{\mu \nu}$ at points $x \neq t$ in this limit.