I Event and Cauchy horizons for a charged black hole

[Afonso Campos]

Consider the Reissner-Nordstrom metric for a black hole:

$$ds^{2} = - f(r)dt^{2} + \frac{dr^{2}}{f(r)} + r^{2}d\Omega_{2}^{2},$$

where

$$f(r) = 1-\frac{2M}{r}+\frac{Q^{2}}{r^{2}}.$$

We can write

$$f(r) = \frac{1}{r^{2}}(r-r_{+})(r-r_{-}), \qquad r_{\pm} = M \pm \sqrt{M^{2}-Q^{2}}.$$

Then $r_{+}$ is called the event horizon and $r_{-}$ is called the Cauchy horizon.

Why is $r_{+}$ called the event horizon and why is $r_{-}$ called the Cauchy horizon?

 

[PeterDonis]

Why is $r_{+}$ called the event horizon and why is $r_{-}$ called the Cauchy horizon?

Because that's what they are.

https://en.wikipedia.org/wiki/Event_horizon

https://en.wikipedia.org/wiki/Cauchy_horizon

 

[Paul Colby]

Things don't look like they end well for $|Q| > M$?

 

[PeterDonis]

Things don't look like they end well for |Q|>M|Q| > M?

In that case there is no event horizon or Cauchy horizon, just a naked singularity. Most physicists appear to consider that case as not being physically reasonable.

 

[Afonso Campos]

Most physicists appear to consider that case as not being physically reasonable.

There are some reasonable examples of naked singularities though: https://arxiv.org/abs/1006.5960

 

[PeterDonis]

There are some reasonable examples of naked singularities

"Reasonable" does not mean "I have a mathematical model". "Reasonable" means "I have reason to believe this mathematical model describes something that exists in our actual universe". The paper you cite gives no reason to believe that.