Sometimes more than one coordinate system can usefully describe the same spacetime geometry. This is true in particular for the Schwarzschild geometry surrounding a spherically symmetric mass \(M\). The usual Schwarzschild metric is

with the same \(d t^{2}\) term, while the other terms contain a new radial coordinate \(\bar{r}\) instead of \(r\), and where \(u=u(\bar{r})\).

(a) Find \(\bar{r}\) in terms of \(r\) and \(\mathcal{M}\), choosing a constant of integration so that \(\bar{r} \rightarrow r\) as \(r \rightarrow \infty\).

(b) What is an advantage of using the isotropic metric?

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ds=-(1-2M/r)cdi + (1 - 2M/r) dr + rds? where M = GM/c and d = (do + sin Od). The so-called "isotropic" metric, describing exactly the same spacetime, has the form ds = (12M/r)c di + e (dr + F d),