Deriving Einstein-Rosen Bridge

Question

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?

Answer 1

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.

Answer 2

$u=0$ is on the event horizon of the Schwartzschild black hole, where $r=2M$. By defining $u^2$ to equal $r-2M$, the u coordinate does not cover the region inside the black hole $r$ less than $2M$. Both positive and negative $u$ are outside, being a kind of reflection off the event horizon. I say the interpretation of this as a wormhole linking two different regions of spacetime is a complete mathematical artifice and also untrue, because the metric does not describe a spacetime. The reason is, $du$ has the dimensions of root(length) so is not a spatial coordinate. You can't for example define velocity $c$ using such a coordinate. So I do not ascribe any physical significance to this metric.