How does blackhole event horizon grow?

[mathfeel]

From a far away observer, thing falling into a BH takes infinite time to cross the horizon. At the same time, the horizon radius is proportional to BH's mass. But if we never really see any energy fall into a BH, how did it acquire a horizon in the first place as seem from outside? i.e. What would a far away observer see when he watches a massive star collapsing into a BH.

Similar question is, suppose there is a BH of mass M there. Then debris of mass M also falls into it. Since we never see them fall through the EH, does the EH grow from 2M to 4M?

 

[Pengwuino]

The typical solutions you see such as the Schwarzschild solution are time-independent solutions. They do not take into account the back-reaction of the metric to particles being introduced. You assume you simply have some "test particle" and you determine the various properties of the metric (for example, this idea that particles never actually reach the event horizon). If you were to introduce something the size of the black hole, your assumptions of time-independence and spherical symmetry go out the window and those metrics make no sense.

 

[mathfeel]

The typical solutions you see such as the Schwarzschild solution are time-independent solutions. They do not take into account the back-reaction of the metric to particles being introduced. You assume you simply have some "test particle" and you determine the various properties of the metric (for example, this idea that particles never actually reach the event horizon). If you were to introduce something the size of the black hole, your assumptions of time-independence and spherical symmetry go out the window and those metrics make no sense.

Fine. But we can imagine the hypothetical situation in which a spherical shell of small masses falls radially, uniformly in all direction into the BH. The shell's mass is small so that we can neglect any gravitation interaction within it. One can then imagine continuous, discrete "collapse" of such shells. The question is, would the event horizon ever grow after enough masses has gotten closed enough (as seen from outside) as to swallow some masses that was stuck (again, as seen from outside) right outside the horizon?

 

[Pengwuino]

I'm sure it would, although that is beyond what I know. What I do know, or have heard at least, is that models have been created under those exact kinda stipulations, but are highly unrealistic.

 

[George Jones]

It is useful to look at the Vaidya metric. See Figure 5.7 on page 134 (pdf page 150) of Eric Poisson's notes,

http://www.physics.uoguelph.ca/poisson/research/agr.pdf,

which evolved into the excellent book, A Relativist's Toolkit: The Mathematics of black hole Mechanics. Radiation falls into a black hole from v1 to v2, but the left diagram of Figure 5.7 shows that the event horizon starts to grow before the first radiation crosses the event horizon.

It is a remarkable property of the event horizon that the entire history of the spacetime before its position can be determined

Here, "history" means past and future.

 

[grav-universe]

The event horizon forms wherever the gravitational potential is - V ≥ c^2 / 2. Before the mass has condensed enough, an actual event horizon doesn't exist, although the gravitational time dilation can still be found externally in terms of the Schwarzschild radius for the mass, which becomes sqrt(1 - 2 G M / (r c^2)) = sqrt(1 - 2 V / c^2) = sqrt(1 - R/r), where R is the Schwarzschild radius. Externally V = - G M / r while internally the potential is greater, depending upon the density profile of the body. For instance, for a uniformly dense body, it is V{internal} = G M (r^2 - 3 r_body^2) / (2 r_body^3).

According to the equivalence principle, local observers that are stationary within the gravitational field are accelerating like accelerating observers on a platform (ignoring minor gradients), while freefallers are inertial. This is because any experiments they perform locally will have the same results as their counterparts. A freefaller falling from infinity will measure a local speed of a stationary observer of v = sqrt(2 G M / r) = c sqrt(R/r) as the freefaller passes. Since SR is valid locally, the time dilation is just the kinetic component with the relative speed, so becomes sqrt(1 - (v/c)^2) = sqrt(1 - R/r). As the freefaller continues to accelerate past the surface of the body, the relative speed continues to increase, so a greater time dilation is measured until the freefaller reaches the center where the relative speed is greatest, so the time dilation is greatest at the center, which relates directly back to the gravitational potential and the time dilation that the distant observer will measure.

With the greatest potential (or smallest since it is negative) at the center of a body, that is where the event horizon will first form when the infall of the body produces a potential there of - V = c^2 / 2. As the infall continues, the potential at the center increases past that and the center will now lie within the event horizon while the event horizon itself is now further out, wherever - V = c^2 / 2. So it starts at the center and grows outward.

If the density profile of a body were right at the brink of producing an event horizon but has not yet done so, then introducing another mass will still cause the potential at the center to pass the boundary condition first since it is already greatest there, and the event horizon will still grow from the center outward even if the second body is nowhere near the center. The event horizon of an infalling body will continue to grow until it has consumed all of its mass, although if no more mass is introduced, the surface of the body will never quite cross the event horizon in finite time according to a distant observer due to the gravitational time dilation involved.

 

[Naty1]

"But if we never really see any energy fall into a BH, how did it acquire a horizon in the first place as seem from outside? i.e. What would a far away observer see when he watches a massive star collapsing into a BH."

The simple answer: stuff (mass, energy/radiation) falls in before the horizon forms.

When a black hole initially forms, whatever mass/energy is involved "disappears" behind an event horizon. [There are numerous "horizons" inlcuding cosmic and Unruh type which offer their own insights. For example the Unruh effect (concerning an accelerating observer) requires Hawking radiation.]

But what an observer sees is "relative"...analogous to time dilation and length contraction in special relativity

George Jones above describes an "absolute" horizon" which

...begins to form at a collapsing star's center well before the stars surface shrinks through the critical circumference . The absolute horizon is just a point when created, but expands smoothly and emerges through the star's surface precisly when the surface shrinks through the critical circumference. It then stops expanding, and thereinafter coincides with the suddenly created apparent horizon. ...The apparent horiozn is created suddenly (instanteously) where the star's surface shrinks through the critical circumference...

from BLACK HOLES AND TIME WARPS, BY Kip Thorne

Wikipedia has an ever so slightly different description for how the apparent and absolute horizon coincide:
http://en.wikipedia.org/wiki/Apparent_horizon

I copied the following from another thread which provides some mathematical insight (I think this was posted by Dalespam...oops or Jesse M??)

For example, in the curved spacetime around a nonrotating black hole, if you use Schwarzschild coordinates the coordinate velocity of light decreases as you approach the horizon and actually reaches zero on it, whereas if you use Kruskal-Szekeres coordinates in the same spacetime, the coordinate speed of light is the same everywhere (see this page# for a discussion of different coordinate systems that can be used in a spacetime containing a nonrotating black hole).

[# link doesn't work]


First of all, it helps to understand some of the weaknesses of Schwarzschild coordinates which are "fixed" by Kruskal-Szekeres coordinates. The first is that in Schwarzschild coordinates it takes an infinite coordinate time for anything to cross the horizon, even though physically it only takes a finite proper time for a falling object to cross the horizon. The second is that inside the horizon, Schwarzschild coordinates reverse the role of time and space--the radial coordinate in Schwarzschild coordinates is physically spacelike outside the horizon but timelike inside, while the time coordinate in Schwarzschild coordinates is physically timelike outside the horizon but spacelike inside. In Kruskal-Szekeres coordinates, in contrast, objects crossing the horizon will cross it in a finite coordinate time, and the Kruskal-Szekeres time coordinate is always timelike while its radial coordinate is always spacelike. And light rays in Kruskal-Szekeres coordinates always look like straight diagonal lines at 45 degree angles, while the timelike worldlines of massive objects always have a slope that's closer to vertical than 45 degrees.

 

[Naty1]

Some good additional insights here:
https://www.physicsforums.com/showthread.php?t=488051&highlight=event+horizon