# B Can a black hole event horizon grow at the speed of light?

1. Apr 26, 2016

### Gigel

Assume a spherical black hole that is eating matter from its surroundings. Then its Schwarzschild radius will increase with a speed proportional to the mass flux that enters the black hole. The question is: is this speed limited by the speed of light in vacuum c?

If the event horizon is a purely geometrical notion, then I'd say it can grow at any speed. But if the event horizon were holding some quantum fluctuations with attached energy, which move with the event horizon, then the horizon would be a physical object with mass and its speed would be limited by c.

It would also be interesting whether a speed of growth larger than c could be observed, i.e. if it could exist relative to a stationary or to a falling referential.

Btw, in order to obtain a speed of growth equal to c, a rough classical estimate gives a mass flux of about 100 000 Suns / second. Not unachievable for a supermassive black hole, yet this is far higher than the (average) mass flux of a quasar.

2. Apr 26, 2016

### Staff: Mentor

It is not correct to refer to the rate of increase of the hole's Schwarzschild radius as a "speed". The hole's horizon is not a thing that has a location in space. It's an outgoing null surface, i.e., a surface made up of outgoing light rays.

The rest of your post is based on this mistaken interpretation.

3. Apr 26, 2016

### PAllen

In a sense, a BH event horizon is always moving at the speed of light, whether matter is falling in or not. More accurately, a trapping surface is the surface formed by ougoing null geodesics (light paths) that make no further 'progress'. In the classic, stable, BH, an event horizon is a trapping surface. As a result, all free fall observers 'observe' the event horizon passing at exactly c. More, any observer sees the event horizon pass at c - the same as outgoing light.

What you are describing is a coordinate dependent quantity which can be made pretty much whatever you want.

4. Apr 26, 2016

### Simon Bridge

That may need more clarification for pretty much anyone who has not covered a general relativity course.

It is very common to describe a black hole as gobbling up mass and getting bigger.

OK then, if there is an observer for which mass of the black hole changes with time (usually their coordinate time or something), then the Swarzchilde radius also changes with time and the rate of that change, in the observer's frame, would have the units of speed (whatever it is correct to call it). Would it be possible to find a frame where this "speed" is greater than light? ... (The usual unspoken assumption is the observer is far away from the BH and stationary wrt it's center of mass, so it's probably good to address this situation too.)

Now the reason I'm asking is that this is something where I am personally at a bit of a loss to describe this properly and I'd like to be able to.
I've got about as far as PAllen's description...

5. Apr 26, 2016

### PAllen

The simplest, classical, exact model of a growing BH is the ingoing vaidya metric (https://books.google.com/books?id=b...jAH#v=onepage&q=ingoing vaidya metric&f=false). So far as I am aware, there is no particular limit on m(v), thus no limit to the coordinate dependent rate of expansion of the trapping surface of an ingoing Vaidya metric.

6. Apr 26, 2016

### Staff: Mentor

A possibly more concrete way of trying to model this would be a scenario like the following: I am far away from a BH, but I have a fleet of small space beacons deployed along the radial line between my ship and the hole, at a range of altitudes from just below my ship to just above the hole's horizon. All of the beacons have attached rockets that enable them to hover at their assigned altitudes. Each beacon continually emits light signals radially outward, and I use the signals to monitor the beacons.

Now I see stuff starting to fall past me and on into the hole. (For simplicity, assume it's ingoing null radiation, so we can say we are using the ingoing Vaidya metric.) As it falls in, the hole's horizon starts moving outward, and I start losing contact with beacons--first the one closest to the original horizon, then ones further out, etc., etc. I can use the time by my clock when I receive the last signal from each beacon, combined with its assigned altitude and the known properties of the spacetime geometry, to calculate a "speed" at which the horizon is moving outward.

How fast can this speed be? It doesn't look at first like there is any upper limit on the rate m(v) at which energy falls into the hole, which would mean there would be no upper limit on the speed calculated as above either. But I see a potential issue with that. If m(v) can be arbitrarily large, that means the energy flux coming in along a particular ingoing null geodesic (corresponding to a particular value of the null coordinate v) can also be arbitrarily large. But where is all this energy coming from? If the energy flux coming in can be arbitrarily large, then it would seem that the energy outside my radius has to be allowed to be arbitrarily large as well--but that's not consistent with me being outside the hole's horizon!

So I think there might be some kind of upper limit on m(v) imposed by the assumption that the hole starts out with a particular mass. But I haven't taken this line of thought any further yet (for example, to try to come up with a relationship between the upper limit on m(v) and the mass of the hole, the radius at which my ship is hovering, etc.)

7. Apr 27, 2016

### Ibix

For @Gigel - noting that this is a B thread - the answer seems to be that many common concepts like speed don't apply without very precise re-definition (hence the experiment @PeterDonis is talking about). And arguably the concept of speed can't be applied to the event horizon because it's not really a thing in many senses.

It's complicated, in short, and any answer is likely to come with a list of caveats and nit-pickingly precise definitions as long as your arm.

8. Apr 27, 2016

### PAllen

Interesting argument, but I'm not sure it places any limit on 'some reasonable definition' of horizon grown being able to be > c, at least for a while. The energy density of the radiation in the ingoing Vaidya metric is proportional to M(v),v/r2 (comma = partial derivative notation). You can choose M(v) such that its rate of change per some time measure is arbitrarily large at some time, but such that the M(v),v is decreasing as a function of v. Note that for a given time slice (in a reasonable foliation), dv/dr approaches unity for large r. This means that with such an M(v) function (with partial by v decreasing as v increases), the integral of energy density over a whole exterior slice can be finite.

9. Apr 27, 2016

### Staff: Mentor

The question isn't whether the integral of energy density over the exterior of a spacelike slice is finite; the question is whether it satisfies, heuristically, $2M(r) < r$ for all $r$, where $r$ is the areal radius $\sqrt{A / 4 \pi}$, $A$ being the area of the 2-sphere containing a given event (the usual coordinates for the Vaidya metric are set up so the coordinate $r$ is equal to the areal radius, but the areal radius itself is an invariant), and $M(r)$ is the integral of the energy density over a spacelike slice out to areal radius $r$. If that condition is not satisfied, there will be a trapped surface somewhere in the "exterior" region. I don't know what limits this places on the function $M(v)$ or its derivative; it might be that it still makes no difference when the above condition is taken into account. But I think it's worth checking.

10. Apr 27, 2016

### PAllen

I believe it is even more complicated than that. Your 'inside horizon' criterion is based on asymptotically flat spacetime that is vacuum beyond some radius of interest (or where a large, isolated spacetime volume can be approximated this way). With no vacuum anywhere, I am not sure this criterion is even applicable. Actual analyses of Vaidya that I've read discuss only a trapping surface at r=2M(v), irrespective of M(v). I would think they would mention the possibility of other trapping surfaces, if present.

11. Apr 27, 2016

### PAllen

I found a good reference giving the real math on this:

http://www.blau.itp.unibe.ch/newlecturesGR.pdf

At pages 709 - 710, the key results are given. There is a true event horizon if M(v) approaches a finite limit as v->infinity (and this true horizon is NOT coincident with the r=2M(v) trapping surface), and the limit from above as v goes to zero, of M(v)/v is greater than 1/16. To me this 'almost proves' that there is no problem with (in some appropriate sense) ftl growth of the trapping surface in the 'early' phase.

Last edited: Apr 27, 2016
12. Apr 27, 2016

### Staff: Mentor

I don't think it has to be vacuum outside some $r$, but I would agree that $M(r)$ has to approach a finite limit as $r \rightarrow \infty$. See further comments below.

I think this is equivalent, at least heuristically, to requiring $M(r)$ to approach a finite limit as $r \rightarrow \infty$ in any spacelike hypersurface.

To put it another way, I think it shows that it is possible for the "growth speed" of a trapped surface to be arbitrarily large, but only for a finite period of time (where "time" here can either mean a finite range of the $v$ coordinate or a finite range of proper time for any observer "hovering" far away from the horizon and observing light signals from beacons as I described in an earlier post).

13. Apr 27, 2016

### PAllen

In the same reference I linked in #11, in a later section on Vaidya, they show the limit of M(v) as v->infinity is the ADM mass of an ingoing Vaidya solution. This concurs with Peter's observation that for a physically meaningful solution, you want this limit to be finite.

Last edited: Apr 27, 2016
14. Apr 27, 2016

### Simon Bridge

BTW @Gigel : the above pretty much means you asked a hard and interesting question. Well done.

15. Apr 28, 2016

### Gigel

Well, thanks to the people above the discussion took an interesting turn. I didn't know about the Vaidya metric for a dynamic BH; I am not very familiar with General Relativity though. What I was thinking about had more to do with the Hawking radiation and possible quantum correlations between emitted particles, that may have given the event horizon a mass and thus a physical (not just a purely geometrical) nature. But even so, it turns there is a limit proportional to the BH mass (at least as v -> 0). This is even more interesting than a constant universal limit.

Also, what I meant initially by speed of growth would be more correctly called rate of increase of the Schwarzschild radius.

I tried to see if the prospect of a rate of increase equal to c is realistic. Factoring in the Eddington limit (i.e. photon pressure on infalling matter), it turns out that in order to obtain a mass flux of 100 000 Suns/second, the BH would have a mass larger than (roughly) 10^51 kg, which is about 1% of the normal matter in the visible Universe. The Eddington limit may be circumvented, say a mini-BH eating elementary particles. Otherwise, it seems improbable for a natural BH to grow at dR/dt ~ c.

16. Apr 28, 2016

### PAllen

The trapping surface discussed by Peter and I is the direct analog of the Schwarzschild radius - it is the same formula with 'current BH' mass input. However, in a growing BH it is a light trapping surface, but it is not the location of the true horizon until the BH settles to its final mass (then they would coincide again).

Theorists (rather than phenomonologists) are not very interested in the Eddington limit. We just imagine bombarding a BH with neutrons (for example), and ask if there is any limit to how fast the SC radius can grow. We concluded there is no limit whatsoever to speed of growth for a short period of time.

17. Apr 28, 2016

### Staff: Mentor

I don't know where you're getting this from, but it's not correct. The event horizon is a globally defined null surface; it doesn't have a "mass", regardless of whether or not Hawking radiation is being emitted. A trapping surface (a surface where, locally, outgoing light just fails to move outward) is a locally defined surface that can be null, spacelike, or even timelike, and again has no "mass", regardless of whether or not Hawking radiation is being emitted. So none of these things have any limit on their "speed" from anything like "having a mass" or "being a physical object" or anything of that sort.

Photon pressure from what photons? A black hole is not a star. If you're thinking of Hawking radiation exerting radiation pressure, such pressure will be negligible by many, many orders of magnitude for a BH of stellar mass or larger.

18. Apr 28, 2016

### PAllen

The Eddington limit is applied to accreting BH (as well as many other bodies). It reflects the tendency of the radiation released by accreting matter to blow away other accreting matter.

19. Apr 28, 2016

### Staff: Mentor

I agree that this will happen in some cases, but we're talking about the general case, which would have to include things like pure ingoing radiation with no matter and no radiation going outward; after all, that's what the Vaidya metric describes. In cases like that, there is no Eddington limit; so I don't think the Eddington limit can be used to derive a general limit on the derivative of $m(v)$ and hence on the "horizon speed".

20. Apr 28, 2016

### PAllen

Of course I agree. However, in the real world, rather than theory, I expect that BH growth rates are rather slow except for mergers (possibly with neutron stars). Even with a BH star interaction, I suspect that the star would be torn apart and absorbed 'slowly'.