- #1
Afonso Campos
- 29
- 0
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?
$$ds^{2} = - f(r)dt^{2} + \frac{dr^{2}}{f(r)} + r^{2}d\Omega_{2}^{2},$$
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?