# How does one find the Vaidya black hole event horizon?

As for a definition, there are quite precise ones for what an event horizon is. One can define it as the boundary of the causal past of future null infinity, i.e., $$\mathcal{H}=\partial J^-(\mathscr{I}^+).$$

Well, the causal past of an event $$J^-(\mathfrak{e})$$ is defined as the set of all events $$q$$ such that there is one future-directed causal geodesic $$\gamma : [a,b]\to M$$ with $$\gamma(a)=q$$ and $$\gamma(b)=\mathfrak{e}$$. The causal past of a set is the union of the causal pasts of every points in the set.

So we would need to find all causal geodesics which "end up" in $$\mathscr{I}^+$$. This seems bad already, but then we need to somehow take the boundary of that set.

So in practice, in order to precisely "find the horizon" in coordinates, this seems cumbersome.

For example, I've read that for this Vaidya metric

$$ds^2=-\left(1-\dfrac{2m\theta(v)}{r}\right)dv^2+2dvdr+r^2(d\theta^2+\sin^2\theta d\phi^2),$$

the horizon starts at $$r = 0$$ when $$v = -4m$$, then satisfies $$r=v/2 +2m$$ and finally remains at $$r = 2m$$. So in coordinates the horizon seems to be defined here as

$$\mathcal{H}=\{(v,r,\theta,\phi) : r = 0,v=-4m \quad \text{or} \quad r = v/2+2m, v\in [-4m,0],\quad \text{or} \quad r = 2m, v\in [0,+\infty)\}$$

It is totally not obvious how this kind of thing follows from the abstract definition.

So the question is:

How that abstract definition is used in practice to conclude that the event horizon of the Vaidya black hole as above is given by the above relations?

• Calculations of the event horizon for the Vaidya spacetime could be seen in this SageManifolds notebook. – A.V.S. Nov 12 '18 at 19:06