# Event horizon of static black holes

I am interested in finding event horizon of static space times of the following forms : $$$$ds^2=-f(r)dt^2+\frac{1}{g(r)}dr^2+r^2d\Omega^2,$$$$ where we have $$f(r)\neq g(r)$$. According to the Carroll's book :

1-determinig the point at which r=constant hypersurfaces become null is easy; $$\partial_{\mu}r$$ is a one form normal to such hypersurfaces, with norm $$$$g^{\mu\nu}\partial_{\mu}r\partial_{\nu}r=g^{rr},$$$$ we are looking for the place where the norm of our one vanishes $$$$g^{rr}(r_H)=0,$$$$

so from this definition, $$g(r_H)=0$$ is the location of event horizon.

2-But $$K^{\mu}=\delta^{\mu0}$$ is a time like Killing vector which becomes null $$g_{\mu\nu}K^{\mu}K^{\nu}=g_{tt}(r_{H'})=0$$ for some hypersurface which is located at $$r=r_{H'}$$.

It seems to me that 1&2 are contradictory according to Carroll's book which claims

A-Every event horizon $$\Sigma$$ in a stationary, asymptotically flat space time is a Killing horizon for some Killing vector field $$\xi^{\mu}$$.

B-If the space time is static, $$\xi^{\mu}$$ will be the Killing vector field $$\partial_t^{\mu}$$ representing time translations at infinity.

if $$f(r_{H'})\neq g(r_H)$$, the above conditions are not satisfied.

3-By looking at the null geodesics $$$$\frac{dr}{dt}=\sqrt{f(r)g(r)},$$$$ it seems to me that the event horizon should be the outer radios for which one of the functions $$f(r)$$ or $$g(r)$$ is equal to zero.

which one is the definition of event horizon? I am interested in an explicit calculation and not the general explanations as event horizon is the hypersurface for which spacetime is divided to two separately causally disconnected regions.

I found some answers to the similar questions without explicit calculations :

How to derive the Schwarzschild radius?

What is the radius of the event horizon?

Does condition $g_{00}(r_0)=0$ define the event horizon on $r_0$?

• In any case i have encountered $g(r)$ will eventually end up being something like $f(r)*h(r)$ with $h(r)$ being an analytic non zero function everywhere. Commented Aug 23, 2023 at 8:09

If you use arbitrary metrics and coordinates the event horizon is at the contravariant $$g^{\rm rr}=0$$. In static Schwarzschild spacetime this gives the same result as the covariant $$g_{\rm tt}=0$$, otherwise the latter gives the static limit which can also happen at the ergosphere if there is one.