What Is the Connection Between Black Hole Temperature and Event Horizons?

[Nabeshin]

We know the temperature of a BH is related to the area of it's event horizon. So we can imagine if we are in a very cold universe with a very nice detector measuring the temperature of a black hole and being able to infer it's radius. Great.

Now, I want to explain the difference between apparent and event horizons just so the issue is clear. The apparent horizon marks the boundary between outgoing and ingoing null rays, that is, inside the apparent horizon even "outgoing directed" rays are headed towards the singularity. The event horizon, on the other hand, marks the boundary between which signals cannot propagate to infinity. The two are coincident for static spacetimes, but the apparent horizon will always lay inside the event horizon. The point is that the event horizon somehow "knows" about matter which is going to fall into the hole. I.e if I have a collapsing star the event horizon grows steadily in anticipation of matter collapsing at the singularity, it "knows" that although an outgoing light ray might initially travel outwards, because of the infalling mass it will ultimately be pulled back into the hole. Okay, great. Since the EH is just a mathematical boundary it makes sense that it can have such properties. Hopefully you see where this is going.

Now, an observer monitoring the black hole could prepare some system which has a 50/50 chance of falling into the hole after a given time. However, the observe should see the temperature of the hole increase in accordance to the EH area increasing even before he may have otherwise deduced what the outcome of the (presumable) quantum event was. Is this an issue?

I haven't yet thought through this terribly carefully but on its face it seems interesting. Can anyone see any logical consequences of this which might be problematic? Of course, the setup of the problem may be wrong. I'm not sure about Hawking's solution, but perhaps it only applies to a static spacetime, in which case the whole area argument would be bust.

 

[Driftwood1]

...quantum uncertainty also dictates that the event horizon cannot be fixed precisely in time and space

This is why Hawking radiation is allowed to be emmitted - another way fo looking at is that there is quantum uncertainty in the speed of light when the position of a particle is known to be at the evenet horizon boundary and so this uncertainty allows one of the particle pairs to escape.

50/50 chance needs to be defined in a practical sense