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I know that an einstein rosen bridge is derived by a coordinate transformation on the schwarzschild metric, but I can't find much on it online, could someone please show how to change the metric into a "wormhole" one and explain it a bit?

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The usual Schwarzchild metric is given by

$$ ds^{2}=(1-2 M / r) d t^{2}-\frac{d r^{2}}{1-2 M / r}-r^{2} d \Omega^{2}. $$ We can make a coordinate transformation of the form $u^2 = r - 2M$.

This gives $1-2M/r = \frac{u^2}{u^2 + 2M}$ and $dr = 2u du$ such that the line element becomes

$$ ds^{2}=\frac{u^2}{u^2 + 2M} d t^{2}-4(u^2 + 2M)du^2-(u^2 + 2M)^2 d \Omega^{2}. $$ This is the line element for the Einstein-Rosen bridge. This coordinate transformation gives the spacetime two distinct asymptotically flat regions, one defined by $u \rightarrow \infty$ and the other by $u \rightarrow -\infty$. Where a singularity once was, at $r = 2M$ (which was only a coordinate singularity anyway), we now see that there is a 'bridge' (now located at $u = 0$) connecting each asymptotically flat region $u = \pm \infty$. This is the so-called wormhole.

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  • $\begingroup$ Thank you for your help. One thing that I'm confused about is if we set r= 2m, then u=0, making no time pass. Is this correct? $\endgroup$ Jul 28, 2021 at 17:00
  • $\begingroup$ It seems like it might be, if only because the Schwarzchild black hole is entirely hypothetical: Astrophysical black holes (for which there's much observational evidence) are described by the Kerr or (if electrically charged) Kerr-Newman metrics, reportedly much more complex. $\endgroup$
    – Edouard
    Jul 28, 2021 at 19:13
  • $\begingroup$ ah i see. I wonder if you could or if some source can derive the wormhole mathematics from a kerr metric, for a spinning black hole I guess. And I also had some trouble understanding the -infinity to positive infinity, could you show how they found that from the line element? $\endgroup$ Jul 28, 2021 at 19:36

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