This type of metric is called a static spacetime.
when $A$ and $B$ are functions only of $r$ and not of $t$. Michael Seifert points out that the analysis I've given above is only applicable when the metric is time independant i.e.
The left side $dr/dt$ is the coordinate velocity of the light ray, so the location of the event horizon is where this is zero:įor example, the Schwarzschild metric has $A(t)$ and $B(t)$ equal to:Īnyone interested in this area may want to look at Anonymous' previous question on the subject as my answer to that provides more detail on what the metric tells us. For a radial trajectory $d\theta = d\phi = 0$, and the metric becomes:Īnd we know that for light $ds = 0$ so we get the equation: So we can just look for the radius at which the coordinate velocity of the radial light rays is zero. In this case the spherical symmetry makes the problem easy because radial light rays will be normal to the horizon. So what we need to do is look at the trajectories followed by light rays and find where they become trapped.
The event horizon of an asymptotically-flat spacetime is the boundary between those events from which a future-pointing null geodesic can reach future null infinity and those events from which no such geodesic exists.Ī null geodesic is the path followed by a light ray, so the horizon marks the surface at which light just cannot escape to infinity. Let's start with what we mean by a horizon:
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