# Deriving Einstein-Rosen Bridge

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?

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.

• 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? Commented Jul 28, 2021 at 17:00
• 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. Commented Jul 28, 2021 at 19:13
• 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? Commented Jul 28, 2021 at 19:36
• Nice answer (+1) I can see the metric but cannot see a "bridge" in it... Could you state the definition of a "ER-bridge" in terms of the geodesic equations, please? If the geodesic velocity becomes very high between 2 separate points in the 4D volume, is this what's defined as a "wormhole" between 2 points? Commented Aug 9 at 6:24

$$u=0$$ 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.

• (+1) Can one define a wormhole in plain terms using the geodesic equations, please? When is something a "wormhole" and when is something not a "wormhole"? Commented Aug 9 at 6:29
• There's nothing wrong with the metric in the earlier answer. It covers regions I and III of the Kruskal-Szekeres diagram, while the usual static metric covers regions I and II. If you don't like the units of $u$, you can take $u^2=2M(r-2M)$ instead. Commented Aug 9 at 17:11