# Firewall solution for black hole paradox and creation of black event horizon

AMPS paper [1] proved that black hole complementarity explanation of blackhole paradox requires there is a "firewall": high-energy wall which destroys everything entering event horizon.

However, I'm confused how "firewall" explains the following setup:

• lets assume that are two entangled particles Bob and Alice
• black hole event horizon is created in space between these two particles
• Bob is now behind black hole event horizon while Alice is outside
• Now black-hole evaporates and you have again Alice entangled with more than two particles

So how "firewall" solves this?

UPDATE: Here is my thought experiment how to create black hole by horizon appearing - not growing. Lets assume that is absolutely nothing around Bob. Then you can create light rays pointing to Bob from far far away. If you have enough of these light rays, you will create event horizon around Bob before light rays even reach Bob. So Bob will not be swallowed by growing event horizon and event horizon will just appear around Bob.

You can't just create event horizons of finite area out of thin air like that.

Let me explain with a very idealized thought experiment: imagine you have a thin spherically symmetric shell of dust particles of radius $R$ and total mass $M$. Say that initially $R > r_s$ (where $r_s = 2M$ in natural units). Say that the particles are initially at rest so that they will inevitably collapse under their mutual gravitational attraction, that is, the radius of the shell will decrease with time. Eventually the radius will become smaller than $r_s$ and the dust particles will be gobbled up by the event horizon. You are left with something that to the outside observer is indistinguishable from a Schwarzschild black hole.

Now place yourself inside the shell. You calculated that at an instant $t_0$ the dust shell will reach the Schwarzschild radius, so you might want to try to escape before an event horizon forms and you are irrevocably trapped.

Because the shell is spherically symmetric, the metric inside is flat, so you can just use special relativity to figure this out. If you are at the center, because you can't exceed the speed of light, you need to give yourself at least $r_s/c$ time before collapse in order to be able to escape. If you wait too long, welp, that's too bad: you're trapped, just as you would be if you had waited until the dust shell reached its Schwarzschild radius.

This implies that a retroactive event horizon has formed whose radius grows linearly with time until it reaches $r_s$. So Bob has actually been swallowed by a growing event horizon. If you believe in firewalls, you must believe this horizon has a firewall, too.

• I have updated my question: Here is my thought experiment how to create black hole by horizon appearing - not growing. Lets assume that is absolutely nothing around Bob. Then you can create light rays pointing to Bob from far far away. If you have enough of these light rays, you will create event horizon around Bob before light rays even reach Bob. So Bob will not be swallowed by growing event horizon. – user2196351 Mar 28 '15 at 0:21
• @user2196351 That is exactly the same situation, except that you have replaced the dust shell by light rays. Conceptually, it's the same thing: the horizon will still have to grow because an event horizon is determined by the entire future history of the spacetime. – Leandro M. Mar 28 '15 at 0:44
• Sorry - I'm still confused :( So in "non-firewall" world the event horizon will form around Bob and Bob will not be aware of it and still be entangled with Alice outside event horizon. That is causing paradox because Hawking radiation is in a pure state. What will happen to Bob in "firewall" version of events? – user2196351 Mar 28 '15 at 1:06
• An event horizon can never "form around" a region of spacetime. It has to grow to encompass that region. You can see that it must by imagining trying to escape from that region, as I did in my answer: because you can't travel faster than the speed of light, if you can't escape before an event horizon "forms", then you were yourself trapped by an event horizon. Bob will have to cross an event horizon and therefore, a firewall (if firewalls exist at all). – Leandro M. Mar 28 '15 at 1:12
• I got it now. This "firewall" version of definitely weird because it assumes that something "knows" around Bob that he cannot escape - even if that dust shell is approaching at speed light (that is reason I used light rays). Am I right? – user2196351 Mar 28 '15 at 1:38