Cauchy Horizon

When I was a kid, I read pop-science books about black holes that said that a rotating black hole has a second event horizon inside the main one. As a physics student, I found out that it’s actually called the Cauchy horizon, but I’m not really sure what its properties are. Wikipedia says it’s “the boundary of a Cauchy problem”, but doesn’t tell me what that is. I’ve heard that one of the properties of a Cauchy horizon is that someone passing through it sees time speed up to infinity in the outside universe. Does anyone have a better definition of a Cauchy horizon?

Take your pick.

This site has a pretty good explanation of black hole interiors. In particular, I’ll direct your attention to the diagram in the section titled “Charged or Rotating Singularities”. The inner horizon, aka the Cauchy horizon, are the lines in pink.

So what’s a Cauchy horizon? Essentially, it’s the boundary of the region of spacetime beyond which we have no way of predicting what will happen without specifying “boundary conditions at the singularity.” (“Solving the Cauchy problem” is mathematical physics speak for "solving the time-evolution of the system given some set of initial conditions.)

A much simpler, but completely analogous, situation to think about is a wave travelling on a string of finite length. Unless we specify what’s happening at the boundary of the string, we will only be able to predict with certainty what’s going to happen for a given amount of time at any point along the string; we don’t know, without specifying the boundary conditions, whether or not someone came along and jiggled the end of the string after we set up our experiment and “let it run.” If you look at an analogous spacetime diagram for the “wave on a string problem”, the diagram would look like an infinitely long (in the time direction) rectangle of finite width (in the space direction); the Cauchy horizon in this problem would be the lines corresponding to the path of a pulse that some malicious person could have sent in from the end of the string at the exact same time we specified the initial data. (Apologies if I’m not making this clear — my kingdom for a chalkboard!)

Why doesn’t this problem crop up for a regular Schwarzchild (non-rotating) black hole? In that case, the singularity is spacelike, and in fact corresponds to the “end of time” for observers in the black hole. So regardless of what happens at the singularity, it won’t affect events inside of the black hole. The Kerr (rotating) black hole, OTOH, has a “timelike” singularity; in other words, the singularity stretches forward in time, and so there could, conceivably, be time-dependent stuff happening at the singularity that would affect future events beyond the Cauchy horizon.

Hope that made sense — it’s late at night here. Feel free to post further questions and I (or Chronos) will be sure to check back in & answer them.