What is the role of an event horizon in Hawking Radiation?

Question

I got to know that the particle anti-particle pair idea of Hawking Radiation isn’t completely accurate. Hence I went on to learn what the phenomenon actually is, but got stuck on how the horizon contributes to scattering and everything. Please help.

Answer 1

Let us consider particle creation by a static gravitational field. In such a field the energy of a particle would be $E=-\xi_\mu p^{\mu}$ (metric signature is "mostly plus"), where $p^\mu$ is the 4-momentum of a particle and $\xi^\mu$ is a Killing vector field. Momentum $p^\mu$ is a future-directed timelike (null) vector for massive (massless) particles, and so the energy of a particle is positive in the regions where Killing vector field is timelike and future-directed.

Let us assume that two particles are created. If both of them are in the region where $\xi^\mu$ is timelike & future directed, then the total energy of the pair would be positive and such process would be forbidden by the energy conservation. (Likewise, creation of just a single particle is also forbidden by the same law.) But, if one of the particles is created inside a region where $\xi^\mu$ is spacelike it becomes possible for the total energy of the pair to be zero and so such process would be allowed (subject to other conservation laws). This means that static gravitational field can create particles only if there is a spacetime region where the Killing vector field $\xi^\mu$ is spacelike, while at asymptotic infinity it must be timelike and future directed. The boundary of such a region would be the Killing horizon, where $\xi^2=0$. But for a static spacetime, a Killing horizon is also necessarily an event horizon.

So, energy conservation allows particle creation by a static spacetime only if it contains a black hole. This argument also generalizes on a more general case of stationary spacetimes.