Coordinate transformations in general relativity

Question

Let's assume a non-rotating point mass with mass $M$. A non-massive object travels with constant velocity $\mathbf{v}_t$, with respect to the point mass, in the vicinity of the point mass. A non-massive observer, with constant velocity $\mathbf{v}_o\neq\mathbf{v}_t$, with respect to the point mass, is observing the target.

Without the point mass special-relativistic Lorentz transformations can be applied to perform a coordinate transformation. The question is how the coordinate transformation looks like in general-relativistic case, i.e. by considering the effect of the point mass?

In principle, the transformation should contain the Lorentz transformation as a limiting case for $M\rightarrow 0$.

Usually, the Schwarzschild metric is cited for a point mass potential $${\displaystyle \mathrm {d} s^{2}=-\left(1-{\frac {2M}{r}}\right)\mathrm {d} t^{2}+{\frac {1}{1-{\frac {2M}{r}}}}\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\;\mathrm {d} \phi ^{2}},$$

which for $M\rightarrow 0$ gives

$${\displaystyle \mathrm {d} s^{2}=-\mathrm {d} t^{2}+\mathrm {d} r^{2}+r^{2}\mathrm {d} \theta ^{2}+r^{2}\sin ^{2}(\theta )\;\mathrm {d} \phi ^{2}},$$

i.e. the classical special relativistic metric in spherical coordinates. But how to derive the transformation from the Schwarzschild metric?

Answer 1

The Schwarzschild metric $G$ is:

$$G=\left[ \begin {array}{cccc} -1+2\,{\frac {M}{r}}&0&0&0 \\ 0& \left( 1-2\,{\frac {M}{r}} \right) ^{-1}&0&0 \\ 0&0&{r}^{2}&0\\ 0&0&0&{r}^{2} \left( \sin \left( \theta \right) \right) ^{2}\end {array} \right] $$

we first transformed the metric $G$ to $\eta$

$$G'=T_1\,G\,T_1=\eta= \left[ \begin {array}{cccc} {\frac {1}{\sqrt {1-2\,{\frac {M}{r}}}}}&0 &0&0\\ 0&{\frac {1}{\sqrt { \left( 1-2\,{\frac {M}{r }} \right) ^{-1}}}}&0&0\\ 0&0&{r}^{-1}&0 \\ 0&0&0&{\frac {1}{r\sin \left( \theta \right) }} \end {array} \right] \left[ \begin {array}{cccc} -1+2\,{\frac {M}{r}}&0&0&0 \\ 0& \left( 1-2\,{\frac {M}{r}} \right) ^{-1}&0&0 \\ 0&0&{r}^{2}&0\\ 0&0&0&{r}^{2} \left( \sin \left( \theta \right) \right) ^{2}\end {array} \right] \left[ \begin {array}{cccc} {\frac {1}{\sqrt {1-2\,{\frac {M}{r}}}}}&0 &0&0\\ 0&{\frac {1}{\sqrt { \left( 1-2\,{\frac {M}{r }} \right) ^{-1}}}}&0&0\\ 0&0&{r}^{-1}&0 \\ 0&0&0&{\frac {1}{r\sin \left( \theta \right) }} \end {array} \right] = \left[ \begin {array}{cccc} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0\\ 0&0&0&1\end {array} \right] $$

Then we transformed $G'=\eta$ to spherical coordinates

$$T_2\,\eta\,T_2= \left[ \begin {array}{cccc} 1&0&0&0\\ 0&1&0&0 \\ 0&0&r&0\\ 0&0&0&r\sin \left( \theta \right) \end {array} \right] \left[ \begin {array}{cccc} -1&0&0&0\\ 0&1&0&0 \\ 0&0&1&0\\ 0&0&0&1\end {array} \right] \left[ \begin {array}{cccc} 1&0&0&0\\ 0&1&0&0 \\ 0&0&r&0\\ 0&0&0&r\sin \left( \theta \right) \end {array} \right]= \left[ \begin {array}{cccc} -1&0&0&0\\ 0&1&0&0 \\ 0&0&{r}^{2}&0\\ 0&0&0&{r}^{2} \left( \sin \left( \theta \right) \right) ^{2}\end {array} \right] $$

so the transformation matrix to bring the Schwarzschild metric $G$ to a spherical coordinates (metric $G_s\quad$) is:

$$T=T_2\,T_1=\left[ \begin {array}{cccc} {\frac {1}{\sqrt {1-2\,{\frac {M}{r}}}}}&0 &0&0\\ 0&{\frac {1}{\sqrt { \left( 1-2\,{\frac {M}{r }} \right) ^{-1}}}}&0&0\\ 0&0&1&0 \\ 0&0&0&1\end {array} \right] $$

$$T\,G\,T=G_s$$