# Can a Black Hole really grow?

1. Mar 14, 2012

### Gio83

Hi everyone,

I'm not aware if a similar question has already been posed, so if this is the case I apologize and I beg you to redirect me to the relevant discussion.

In the context of General Relativity, the Schwarzschild solution in the frame of reference of an observer at fixed distance from the origin (call him A) possesses the well known event horizon (EH).
Such EH is a boundary of the spacetime for observer A.
Another observer B, freely falling towards the origin, will not be aware of any boundary because in its reference frame there is no horizon (EH is in fact a "removable singularity" of the metric).
Returning to the point of view of A, he observes B falling towards the EH, registering his clock speed (with the aid of light flashes every second, as usual) on his way down and he notes that B's clock speed is slowing down with a rate that indicates that it will take an infinite amount of time for B to reach the radius of EH.

Long story short: from the point of view of A, will B ever cross the horizon?

And: given the relation that links the radius of the EH with the mass beyond it, how can a black hole grow (from the point of view of a distant fixed observer) if everything will fall behind the EH only for tââ ?

Best regards

2. Mar 14, 2012

### Staff: Mentor

Note that the Schwarzschild metric is a static metric, it really cannot be used to determine how the event horizon grows.

If by "point of view of A" you mean "Schwarzschild coordinates" then the answer is no. That is a limitation of Schwarzschild coordinates, not a limitation of the physics.

3. Mar 14, 2012

### julian

Actually it begs the question - how does an event horizon ever form in the first place cus of similar reasoning....the answer is that Hawking found out that a collapsing star will give out a burst of radiation and then fall below its own event horizon within a finite time as measurede by an outsider's obsevation (then he found out that the resulting black hole kept giving out radiation and this is how he discovered Hawking radiation)...so I'm guessing that if you are matter that can be converted into radiation then you can make the blackhole grow and outsiders can see you make it grow if you take into account QFT on curved spacetime...

4. Mar 14, 2012

### julian

I only found out the other day that it was Penrose who taught the world you need more than one coordinate patch to understand solutions of Einstein'sequations and that Einstein died never knowing what the Schwarzschild solution meant.

5. Mar 14, 2012

### George Jones

Staff Emeritus

6. Mar 14, 2012

### yuiop

Short story long:

Imagine an observer falls into a black hole. Lets say it takes him 10 seconds to arrive at the EH and 20 seconds to arrive at the singularity as measured by his own clock. As measured by a clock far away from any massive objects, an infinite amount of time will have passed before 10 seconds registers on the in-falling observer's clock. In that time the Universe will have expanded and the background temperature of the universe will be near absolute zero degrees Kelvin. Black holes will have evaporated before that 10 seconds passes on the in-falling observer's clock and in this analysis the Black hole will be gone before the observer arrives at the EH. The problem with this analysis is that it uses the static solution and ignores the mass of the in-falling observer. What actually happens is that the mass of the in-falling observer contributes to the total mass of the black hole gravitationally and EH expands and moves outwards to meet the the faller. In this way mass can arrive at and pass through the EH in finite time, even from an external coordinate time point of view.

7. Mar 14, 2012

### Staff: Mentor

This part of your analysis is not correct even if the infalling observer's mass is negligible compared to the BH mass, and even if we add BH evaporation to the mix (the OP, as far as I can tell, did not mean to do so, he was asking about the case where the BH is "eternal" and never evaporates). For an evaporating BH of any significant size, there is still plenty of "room", so to speak, for infalling observers to cross the horizon and be destroyed in the r = 0 singularity before the BH evaporates, even if no significant mass is ever added to the BH after its initial formation. Light signals from the events of those observers crossing the original EH of the BH will eventually escape to infinity, and will be visible to the faraway external observer at the same time that he sees the last flash of light from the BH's evaporation; so if the BH evaporates it's no longer true that "the BH will have evaporated before that 10 seconds passes on the infalling observer's clock", because the light from that "10 seconds passed" event will appear to the faraway observer along with the light from the final BH evaporation. But that won't save any infalling observers; they'll still have fallen in and been destroyed, so they would certainly not agree that the BH was "gone" before they arrived at the EH.

The part about the EH expanding and moving outward to meet the faller is correct (with the minor proviso that to make it precise, we have to stipulate that the infaller's radial extension is much smaller than the radius of the EH, so that he basically crosses the EH "instantaneously" instead of taking a finite time to do so).

But the rest is not correct; even if the infalling object adds a non-negligible amount of mass to the BH, the event of the infaller arriving at the new EH ("new" meaning it includes the effect of the infaller's mass being added to the BH) will still occur at a time "t = infinity" according to an external observer, assuming the BH is "eternal" and never evaporates (more precisely, the time it takes light from the infaller to reach the faraway external observer diverges to infinity as the infaller approaches the *new* EH). (If the BH eventually evaporates, see above; the additional mass of the infaller doesn't change that part.)

8. Mar 15, 2012

### Gio83

I was certainly not referring to a possible limitation of General Relativity. I'm aware that issues that are present in some circumstances can be resolved by enforcing the diffeomorphism invariance.

You are right, if I'm to describe a dynamical situation I cannot appeal to a static metric

So, let's choose the Vaidya metric: I see the fact that "the event horizon starts to grow before the first radiation crosses the event horizon" as a causality issue (is effectively a consequence that precedes the cause). This is due to the fact that EH is defined as a global object, while it would be preferable to deal with locally defined objects.
In this sense it occurs to me that maybe it's better to use the concept of trapping horizon (TH), a local definition of horizon which prevents "causal accidents" in non-stationary cases.

But also considering the TH and its causal nature, does not affect what the "external and static" observer will see: from its point of view the object will never cross the infinite red-shift surface.

I repeat: I'm not claiming that something is wrong here, because I know perfectly that these situations are a consequence of the rules of the game. I'm just amused by the conclusions that sometimes one is lead to draw (sometimes - I admit - erroneously).

9. Mar 15, 2012

### Staff: Mentor

No, it isn't; but the usual way in which this is described does make it seem that way. Here's a better way to describe it: there are events which lie *outside* the original horizon radius and are at times *before* the infalling shell of radiation reaches the new horizon radius, which still are unable to send light signals that escape outward to infinity. This is because light signals emitted from such events do not make it far enough outward in a short enough time to be outside the new horizon radius before the infalling shell of radiation gets there. In other words, the light signals from these events start moving outward, but then get trapped by the infalling shell of radiation and are forced back inward again (or, in the marginal case, end up right at the new horizon radius and stay there forever). No reversed causality anywhere.

Using the TH description, light signals emitted from the events I describe above start out outside the local TH, so they move outward, but when the infalling shell of radiation reaches the new EH radius, the TH jumps outward past the light signals and traps them. That's the problem with the TH: it can "jump" discontinuously outward, whereas the EH is a continuous null surface.

10. Mar 15, 2012

### yuiop

If you look at the interior Sharwzchild solution and consider an idealised sphere of gas with unifrom density that has a radius of 9/8 of the Schwarzschild radius you will see that an event horizon forms initially at the centre. As the sphere of gas continues to collapse and the density increases, the event horizon (defined here as the radius where gravitational time dilation is infinite) moves outwards so that gradually the event horizon approaches the surface of the sphere of gas at the Schwarzschild radius.

11. Mar 15, 2012

### yuiop

I do not agree. If the black hole is eternal and static (no mass added and no evaporation) and the infaller is massless, then an infinite amount of time will have to pass before those 10 seconds registers on the infallers clock and the infaller never arrives at the EH. If the infaller has significant mass and if the black hole is gaining more energy from background radiation than it is losing from Hawking evaporation (normal for most black holes) then the EH will expand and engulf the infaller. If the infaller is massless and black hole is evaporating and losing mass, the radius of the EH will keep shrinking and the infaller will never arrive at the receding EH and there will be nothing left of the BH when the infaller eventually arrives at the centre of where the BH used to be. In this dynamic case I agree that more than 10 seconds will have passed on the infallers clock and an external observer might agree that the infaller has passed where the EH used to be, but no one sees the infaller pass the current EH. In short nothing can cross a static or receding EH but an expanding EH can engulf a falling object.

I think we are in some sort of agreement on this part, although I do not recall claiming anything about what an external observer "sees" in my first post.

12. Mar 15, 2012

### Staff: Mentor

yuiop, I'm not sure if we are disagreeing or not, because I can't tell whether you intend your statements to be taken as true for *all* observers, or only for observers that always stay far away from the BH.

If you mean "an infinite amount of time according to the exterior Schwarzschild time coordinate", then yes, that part is true. But the statement that "the infaller never arrives at the EH" has to be similarly qualified. To the infaller himself, he certainly does arrive at the EH, after 10 seconds have elapsed on his clock, and falls through it.

This is true, in the sense that, if the infaller would have taken 10 seconds by his own clock to reach the EH in the above case, he will take *less* than 10 seconds by his own clock to reach the new EH including the additional mass. But the light from the event where the infaller crosses the new (expanded) EH is trapped at the new EH; it never makes it outward, so it is never seen by a faraway observer. (This assumes that the Bh never evaporates.)

If you are trying to describe this scenario from the viewpoint of the faraway observer, then this is more or less OK, although I would not phrase it this way, for the same reason I would not phrase the description of the original scenario (BH doesn't gain mass or evaporate) the way you did. These statements are more or less true from the viewpoint of the faraway observer, but they are *not* true from the infaller's viewpoint; as I said before, even if the BH is evaporating, the infaller can still cross the EH, reach the singularity, and be destroyed before the BH evaporates, in a finite time by the infaller's clock. So from the infaller's viewpoint, it is *not* true that "the infaller will never arrive at the receding EH", etc.

Before the infaller reaches the "receding" EH, yes. But he will still reach it in a finite time by his own clock; that finite time will just be greater than 10 seconds.

(The only caveat to this is that if the evaporating BH is small enough before the infaller starts falling in in the first place, it might evaporate completely before he can reach it; but in that case the infaller sees the same thing that the faraway observer sees; he *stops* falling in as soon as the radiation from the final evaporation of the BH passes him on its way outward. But the original BH would have to be quite small for this to happen; for a BH of any significant size, such as a stellar-mass hole, there is plenty of room, as I said before, for infalling observers to cross the EH and be destroyed in the singularity before the BH evaporates.)

No *faraway observer* sees this. But the infaller *does*.