The Causal Diagram of the Black Hole

A week ago, I explained the idea of causal diagrams, or Penrose-Carter diagrams, and we discussed the diagram for the static black hole metric shown below.

As I pointed out, though a solution of Einstein's field equations, this diagram does not actually describe a situation we find in reality. The black hole shown in this diagram is accompanied by a white hole, and both have existed since forever, and will continue to exist, unchanging, until eternity. Today, I thus want to discuss the metric for a realistic black hole, a black hole formed from collapse of matter. I will also briefly touch on the evaporation but, as you know if you've been around for a while, the exact way the evaporation proceeds, in particular the final stage, is still under debate.

To obtain the causal diagram of the black hole, recall that Einstein's field equations are local and the black hole solution is a vacuum solution. Yes, that is right. This means that in General Relativity empty space is not necessarily flat. (Flat meaning the curvature tensor vanishes identically. Empty space however has a property called "Ricci-flatness.") If we want to describe collapsing matter, we thus know that outside of that matter the previously found solution, depicted above, still holds. So, what we do is drawing into the diagram the surface of the collapsing matter, and keep the part that is outside that matter. This is shown below.



Now the blue shaded part is the one that no longer correctly describes the black hole that forms from collapse and has to be discarded. This means in particular that the white hole as well as the second asymptotically flat regions are both gone and do not exist in real world situations (addressing a concern that Andrew brought up in the previous post).

What we do then is to attach an interior solution that does not describe vacuum. In some simplified cases this can be done explicitly. For example if the collapsing density is homogeneous (which would be a piece of a FRW-metric), or if it is null dust (described by the Vaidya-metric). Then, one can calculate the interior solution and use a matching condition to join both parts together. For our purposes however, we don't have to bother with the details since we just want to capture the causal structure. For what the causal structure is concerned, the inside solution is rather dull. There is nothing specific going on. The radius just shrinks until it falls below the Schwarzschild radius associated to its total mass. Then the horizon forms, and the matter collapses to a singular point. This is shown in the diagram below.



Note that there is no particular meaning to curves that are exactly horizontal or vertical, we are thus free to deform them, which has been done to make the r=0 curve vertical. This is fine as long as we make sure that the null curves on 45° angles remain the same, and thus spacelike remains spacelike and timelike remains timelike.

As pointed out in the previous post, the use of radial coordinates means that ingoing curves look as if they are reflected at r=0 when they actually go through. The lightray marked v₀ in the above figure is the last light ray that just manages to escape the forming horizon. It is in this background, not the static background, that Hawking did his calculation which showed that black holes do emit radiation.

Knowing the black hole, once formed, emits radiation of course brings up the next question: how do we incorporate the evaporation into the diagram? One can add the evaporation of the black hole by using another non-vacuum patch that describes outgoing radiation which leads to a decreasing of the mass. The Schwarzschild-radius of the black hole then gets closer to the singularity until both, the horizon and the singularity, vanish in the endpoint of evaporation. In this process, the event horizon remains lightlike. What changes for the observer at scri minus is the mass associated to the black hole. When the black hole is completely evaporated, we are left with a spacetime filled with very dilute radiation. This spacetime is to good precision flat and described by another piece of Minkowski metric. If you patch the pieces together you get the diagram below.

If you followed me so far, then we are now in an excellent shape to discuss the black hole information loss problem, which can basically be read off the causal diagram, and the possible solutions Lee and I classified in our recent paper. Let me know in the comments if you're interested in another post on that.