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

• B

## Main Question or Discussion Point

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.

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PeterDonis
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Then its Schwarzschild radius will increase with a speed proportional to the mass flux that enters the black hole.
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.

PAllen
2019 Award
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.

Jeff Rosenbury
Simon Bridge
Homework Helper
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...

PAllen
2019 Award
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...
The simplest, classical, exact model of a growing BH is the ingoing vaidya metric (https://books.google.com/books?id=bk2XEgz_ML4C&pg=PA173&lpg=PA173&dq=ingoing+vaidya+metric&source=bl&ots=d486QSlY7D&sig=RJWkKduaHKo-haRdbBJ4zyksJEE&hl=en&sa=X&ved=0ahUKEwiPjJ2htq3MAhWEVD4KHSZgA8YQ6AEIRjAH#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.

PeterDonis
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2019 Award
the Swarzchilde radius also changes with time and the rate of that change, in the observer's frame, would have the units of speed
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.
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.)

Simon Bridge
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.

PAllen
2019 Award
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.)
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.

PeterDonis
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2019 Award
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.
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.

PAllen
2019 Award
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.
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.

PAllen
2019 Award
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.

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PeterDonis
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Your 'inside horizon' criterion is based on asymptotically flat spacetime that is vacuum beyond some radius of interest
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.

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 then 1/16.
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 me this 'almost proves' that there is no problem with (in some appropriate sense) ftl growth of the trapping surface in the 'early' phase.
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).

PAllen
2019 Award
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.

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

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.

PAllen
2019 Award
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.
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.

PeterDonis
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2019 Award
Hawking radiation and possible quantum correlations between emitted particles, that may have given the event horizon a mass
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.

Factoring in the Eddington limit (i.e. photon pressure on infalling matter)
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.

PAllen
2019 Award
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.

PeterDonis
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2019 Award
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.
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".

PAllen
2019 Award
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".
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'.

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...
I think some confusion is in using units of speed for something that is not speed. I could rig up a series of lights such that they blink in sequence (we've all seen this effect), and that gives the illusion of a pulsing light traveling. I can rig it over vast distances and pre-synchronize the sequence to faster than c. That is, the next light bulb turns on at an interval before the light from the previous bulb gets there. I could describe the system as having a sequence that covers distance at faster than c. Distance the sequence progresses divided by the time intervals.

The light bulbs are not truly a thing moving, but a series of pre-arranged independent events. The propagation of the BH event horizon seems a similar thing that is convenient to use distance and time to describe the expansion, but is not a "thing" that moves.

Another example would be two hypothetical moving lines in a plane that move towards each other. Say they are moving at below c. But the point of intersection can "move" at above c, if they are nearly parallel. The point of intersection is not really a "thing" though. It can be described as a thing. And I might imagine the construction of some very large scissors in space. At the end, they are virtually a vertical shear, and the path of the "cut" travels faster than the blades.

I don't know if any of those situations is at all illuminating, but the first thing I thought of was that there is not truly a speed involved as in the first response:

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.

Simon Bridge
Homework Helper
So OK then - it's "not really a speed" ... how does that address the question differently from above?
The locus of points at some time is still different from the locus at an earlier time ...

Note: prev posts allowed geometry like the point of intersection between two lines to have a speed which is not restricted to sub-light.
Then we discussed the observer issues.

So OK then - it's "not really a speed" ... how does that address the question differently from above?
The locus of points at some time is still different from the locus at an earlier time ...

Note: prev posts allowed geometry like the point of intersection between two lines to have a speed which is not restricted to sub-light.
Then we discussed the observer issues.
I think that c as an upper limit on the movement of things (light is at c in a vacuum, and matter is sub-c) does not apply to a non-thing, such as the Schwarzschild radius location. It seems that the possibility of expansion across space at faster than light "rates" is unlikely, but as I read this sequence of posts, there is not an upper limit to locational change of that boundary.

I would again use the analogy of intersections. A guillotine that is used as a paper cutter could have a shallow angle. The boundary between the cut and uncut paper could move at faster than light, but the blade of that guillotine could not. The sense I am getting from the prior posts is that the boundary is not a thing. So the ordinary limit of c does not apply.

It also does seem that the light-speed limit on matter involved, and on the speed with which gravity can move need consideration. Perhaps it is theoretically possible for the boundary to calculation-ally expand at faster than light, but in my own confusion on this, it seems like the speed of light would apply to the rate at which the new mass of the black hole could expand the boundary. Since we have no instantaneous gravitational field, wouldn't the expansion outward from the center of mass be limited by the light-speed limit on the force of gravity? The situation being proposed is weird because I'm not sure where the new mass going into the black hole is coming from. But this trapped light, now is a CHANGE in the mass of the black hole. And a CHANGE in mass should propagate as a gravity wave, at the speed of light (again correct this ... I cannot claim to fully understand the recent gravity wave confirmation).

I find myself saying two contradictory things. The mathematical boundary from a changing mass could move faster than c. And the changing mass can only have effects that move outward from the center of mass at c. At least one of them must be wrong. Perhaps it is my preference for treating it as a center of mass.

PeterDonis
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The sense I am getting from the prior posts is that the boundary is not a thing. So the ordinary limit of c does not apply.
Yes, that's correct.

It also does seem that the light-speed limit on matter involved, and on the speed with which gravity can move need consideration.
This applies to whatever it is that is falling into the black hole. It does not apply to the event horizon itself. Note that the "speed" at which the event horizon can appear to move outward (in appropriate coordinates) is not the same thing as "the speed of gravity". When matter or energy falls into a black hole, by the time the event horizon has moved outward, observers far away have already observed the gravitational effects of the new, larger mass of the hole. Those gravitational effects are not propagating "outward" from the new location of the horizon. They propagate from the matter or energy that falls in, and there is plenty of time for them to do so at the speed of light.

I'm not sure where the new mass going into the black hole is coming from.
It's coming from the matter or energy that falls into the hole to increase its mass.

It's coming from the matter or energy that falls into the hole to increase its mass.
I was more wondering if it mattered if it all came from one direction. Or was asymmetric some other way. If you are on one side of the black hole and this large light energy comes into it from the other side, It seems like the speed of light limit would matter. If it was radially symmetric inputs, then the "center of mass" was the same at the moment the energy entered the black hole.

That seems a rather farfetched hypothetical ... but I was thinking of that asymmetric input, and whether that required a speed of gravity consideration.