homework and exercises - Finding the appropriate coordinate transformation given two metrics

Given the two-dimensional metric

$$ds^2=-r^2dt^2+dr^2$$ How can I find a coordinate transformation such that this metric reduces to the two-dimensional Minkowski metric?

I know that $g_{\mu\nu}=\begin{pmatrix}-r^2&0\\0&1\end{pmatrix}$ (this metric) and $\eta_{\mu\nu}=\begin{pmatrix}-1&0\\0&1\end{pmatrix}$ (Minkowski). Obviously, the matrix transformation is $\begin{pmatrix}1/r^2&0\\0&1\end{pmatrix}g_{\mu\nu}=\eta_{\mu\nu}$, but how is that related to the coordinate transformation itself?

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EDIT: would the following transformation be acceptable?

$$r'=r\cosh t$$ $$t'=r\sinh t$$ Such that: $dr'=\cosh t\ dr+r\sinh t\ dt,\quad dt'=\sinh t\ dr+r\cosh t\ dt$

And: $ds'^2=-dt'^2+dr'^2=-r^2dt^2+dr^2=ds^2$

Where we have: $ds'^2=\eta_{\mu\nu}dx^{\mu}dx^{\nu}$ as requested.

Is that correct? Also, is there a formal way of "deriving" the proper change of coordinates (since mine is more of an educated guess)?

Answer

If you were to Wick rotate $t \rightarrow i \theta$, the metric would be $ds^2 = dr^2 + r^2 d\theta^2$, which is just flat space in polar coordinates. The standard cartesian coordinates can be obtained by $x=r\cos\theta$, $y=r\sin\theta$. The same procedure works in the original Lorentzian signature metric, but with hyperbolic trig functions instead of sines and cosines.

By the way, this is two-dimensional Rindler space, which is just a patch of two-dimensional Minkowski space: http://en.wikipedia.org/wiki/Rindler_coordinates.