# What is unstable about the wormhole in the Reissner-Nordström solution?

In the conformal diagram for the Reissner-Nordström solution (included below), the inner horizon of the black hole leads to a wormhole which has a singularity, but it is possible to exit it through a white hole in another universe. I've read several sources that call this wormhole "unstable", but without any justification for this fact. What makes it unstable?

• Read this post May 19, 2015 at 0:50
• @Cicero Thanks, but I'm interested in how the instability follows from the conformal diagram. The post you linked invokes a thermodynamic argument, and does not address the instability of the wormhole specifically.
– Drew
May 19, 2015 at 1:04
• Can you provide a citation of where this specific wormhole solution is considered unstable (so I can provide a more relavant response). Thanks May 19, 2015 at 1:20
• Recipe for a wormhole: 1) borrow a second universe from your friendly neighbor... Oops... not every mathematical solution that arises from equations of motion of a physical theory actually has to have a realization in nature. Thermodynamics goes a long way to explain why. Regarding your question in detail I am not sure that one can actually get around that argument because even a black hole is unstable, also for thermodynamic reasons, so no matter what the stability of the white hole would be, the entire construct has to be unstable one way, or another. May 19, 2015 at 1:38

I can give you only a partial account because this is an area I'm only partially familar with. If you're feeling brave there is a rigorous analysis of the physics involved in this paper.

The Reissner-Nordström geometry is perfectly stable as long as no extra matter is present, so in the conformal diagram you've included there is no instability. The trouble is that matter falling in from our universe meets matter falling in from the other universe at the inner horizon and produces an effect called mass inflation. This produces a contribution to the stress-energy tensor that makes the Reissner-Nordström geometry unstable.

For completeness I should note that this is all interesting but physically irrelevant, because the Reissner-Nordström geometry describes an eternal charged black hole and no such object can exist in the real universe. The black hole must have a finite age due to the Big Bang and cannot exist forever due to Hawking evaporation, so no wormholes and parallel universes actually exist. It was a sad day when I realised this was the case.