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

[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.

 

[PeterDonis]

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]

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.

 

[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...

 

[PAllen]

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=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.

 

[PeterDonis]

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.)

 

[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]

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]

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]

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]

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.

 

[PeterDonis]

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]

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.

 

[Simon Bridge]

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

 

[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.

 

[PAllen]

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]

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]

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]

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]

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'.

 

[votingmachine]

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]

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.

 

[votingmachine]

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]

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.

 

[votingmachine]

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.

 

[PeterDonis]

I was more wondering if it mattered if it all came from one direction.

That will change some of the details, but it won't change anything I said.

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.

Matter for what? It will affect when you observe the effects of the increased mass of the hole, compared to when someone on the same side as the energy coming in will observe them, yes. But it won't affect the fact that the gravitational effects of the increased mass propagate at the speed of light, or the fact that the "speed" of the event horizon has nothing to do with the speed of propagation of those gravitational effects. They're simply two different things.

 

[votingmachine]

Matter for what? It will affect when you observe the effects of the increased mass of the hole, compared to when someone on the same side as the energy coming in will observe them, yes. But it won't affect the fact that the gravitational effects of the increased mass propagate at the speed of light, or the fact that the "speed" of the event horizon has nothing to do with the speed of propagation of those gravitational effects. They're simply two different things.

I'll have to think about this ... the event horizon is the location far enough from the black hole where light emitted can escape. Inside that, light cannot escape. That location depends on the black hole mass.

Say a hypothetical black hole had an event horizon as a sphere with a radius of a light year. And you are on one side of it, outside of that 1 light year radius of course, and a whole lot of light enters the event horizon 2 light years away from you, on the other side of the black hole. And say you had the ability to measure the event horizon simultaneously, it seemed that it would take two years for the new mass to show up as increased gravity in the black hole, on your side.

I am perplexed by:

It will affect when you observe the effects of the increased mass of the hole, compared to when someone on the same side as the energy coming in will observe them

and

the event horizon has nothing to do with the speed of propagation of those gravitational effects. They're simply two different things.

I thought they would be the same thing. That observing the effects of the increased mass WAS observing the event horizon change.

 

[PeterDonis]

I'll have to think about this ... the event horizon is the location far enough from the black hole where light emitted can escape.

This is how it is often described in ordinary language, but this description is misleading. The EH is not really a "location". It is an outgoing null surface--a surface made up of outgoing light rays, i.e,. null curves. A "location", if you dig into what that word actually implies, is something that can only be modeled as a timelike curve, not a null curve.

In other words, the EH is not a "place", just as a light ray in general is not a "place". (This can be viewed as another way of saying that light always moves at $c$ in any inertial frame--it can never be at rest. Whereas a "place" is something that can be at rest.)

Say a hypothetical black hole had an event horizon as a sphere with a radius of a light year. And you are on one side of it, outside of that 1 light year radius of course, and a whole lot of light enters the event horizon 2 light years away from you, on the other side of the black hole. And say you had the ability to measure the event horizon simultaneously, it seemed that it would take two years for the new mass to show up as increased gravity in the black hole, on your side.

This doesn't actually make sense. First, the "radius" attributed to the horizon is not a physical radius. It does not correspond to a "distance" from the horizon to the singularity, or anything of that sort; that idea doesn't even make sense; it simply doesn't apply to the actual spacetime geometry at and inside the horizon. If the singularity is anywhere relative to the horizon, it is to the future of the horizon--i.e., the singularity is best thought of as a moment of time, not a place in space. And the EH, as above, is best thought of as an outgoing light ray, so it is neither a moment in time nor a place in space. So trying to think about the EH having a "radius", "expanding outward", etc., as you are doing in the quote above, is trying to apply concepts to something they simply don't apply to.

Second, you can't "measure the horizon" from the outside, because light signals, or anything else you could use to make such a measurement, can't get to you from at or inside the horizon. So there's no way to compare the time when you observe the effects of the increased mass of the hole due to the light falling in on the other side, with any time at which you "measure the horizon to grow"--because no such measurement is possible. See further comments below on that.

I thought they would be the same thing. That observing the effects of the increased mass WAS observing the event horizon change.

No, it isn't. You can't actually observe the horizon change directly anyway, because no light signals can escape from the horizon. The "speed" that was described earlier in this thread is a calculated, coordinate-dependent number; it is not an actual observable.

 

[Gigel]

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.

c upper limit does not apply to purely geometrical objects, only to matter. A geometrical object (say the projection of a rotating lighthouse signal at a large enough distance) can have an apparent speed higher than c as it follows the trajectory of its projection, but it cannot transmit energy or information from one point to the other of its trajectory. It can do that only from its source to the point of projection, and then the speed is at most c. That was my contention: whether the event horizon of a black hole is a purely geometrical entity, or it has some mass and physical nature.

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.

What you are describing here is a form of synthesized wave, i.e. it is formed in place instead of propagating. Such a wave can have any apparent speed, but since it doesn't propagate it cannot transmit information at speeds higher than c. Actually something on this line was thought by some guys. Just a link: http://arxiv.org/abs/physics/0405062v1 Their idea was to synthesize a circular electromagnetic wave moving at speeds higher than c (apparent speed) by electrically polarizing a circular dielectric at different places on its circumference. According to them, the polarized dielectric behaves like an antenna that produces an analog of the Cherenkov effect.

 

[George K]

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.

I'm not a Physicist and surely I'm not qualified to give an appropriate answer. However, a Black Hole is an astronomical, massive object and the event horizon is the limit (i.e. the physical surface) of this object. (In fact we can calculate the mass inside the event horizon -i.e. the mass of the Black Hole- from the radius of the event horizon.) So, by thinking that any physical (massive) object cannot be increased in size at a speed larger than c, I guess that this should be, also, valid for Black Hole.

 

[votingmachine]

I'm not a Physicist and surely I'm not qualified to give an appropriate answer. However, a Black Hole is an astronomical, massive object and the event horizon is the limit (i.e. the physical surface) of this object. (In fact we can calculate the mass inside the event horizon -i.e. the mass of the Black Hole- from the radius of the event horizon.) So, by thinking that any physical (massive) object cannot be increased in size at a speed larger than c, I guess that this should be, also, valid for Black Hole.

Increased in size is not a thing moving. It can be described in terms of rate, with units that are the same as velocity, but it isn't velocity. If you are building a rock wall, and add a 1 meter rock, the wall increased in size INSTANTLY by that meter. That happens no matter how slowly and carefully you place that rock. You could even build an entire kilometer of wall, starting at one end and finishing at the other, at faster than light rate of construction (ignore all the PRACTICAL impossibilities). Because the wall is not moving when we speak of its rate of completion in kilometers per second. You can even use construction rates for things like a tunnel. The Chunnel tunnel is 50.5 km long, and took years to complete. The tunnel was completed at a slow rate, of kilometers per year. In that case, the "speed" is the increase in length of the absence of the rock.

There are practical considerations that are limited by light speed. But any construction process that moves in segments, say laying a railroad track ... the addition of a new piece is an instantaneous extension in length. I can build a lego tower at FTL rates of increasing size, every time I add a brick.

 

[votingmachine]

This is how it is often described in ordinary language, but this description is misleading. The EH is not really a "location". It is an outgoing null surface--a surface made up of outgoing light rays, i.e,. null curves. A "location", if you dig into what that word actually implies, is something that can only be modeled as a timelike curve, not a null curve.

...

Second, you can't "measure the horizon" from the outside, because light signals, or anything else you could use to make such a measurement, can't get to you from at or inside the horizon. So there's no way to compare the time when you observe the effects of the increased mass of the hole due to the light falling in on the other side, with any time at which you "measure the horizon to grow"--because no such measurement is possible. See further comments below on that.

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No, it isn't. You can't actually observe the horizon change directly anyway, because no light signals can escape from the horizon. The "speed" that was described earlier in this thread is a calculated, coordinate-dependent number; it is not an actual observable.

I see the practical impossibility. I ended up deleting a section of the post to ignore the many parts that were impossible. I can't conceive of a practical way to determine the difference, because of the time issue. But even though the "location" is an unmeasurable, the outside universe has space that approaches (asymptotic hyperbolic curve is my understanding)the event horizon. And if you COULD measure that (even retrospectively with observations in the far future), and piece together an accurate understanding of where the event horizon was and how it grew, it seems that the boundary of the unobservable would follow the properties of the mass of the black hole, and that an asymmetric mass change would generate an asymmetric event horizon.

As a related question, what about the event horizons of the two black holes colliding (detected by LIGO)?

The simulation shows asymmetry. And I would have assumed that the asymmetry was spatially changed by the gravity waves, just as the LIGO detector was spatially changed.

I will include that I do recognize that this is completely outside of the realm of measurable physics. Black holes just are not amenable to exact measurement science. Computer simulations are just cool video's. And the hypothetical asymmetric growth of a black hole can probably never be measured or observed with any satisfactory accuracy. If I've strayed too far from the guidelines for physics questions ... just say so and I will drop it. I think the answer to the OP question has been made, and any secondary question about a hypothetical asymmetric mass change in a black hole is far afield from that. I am questioning if a response time limit of c applies, based on a naive understanding that mass changes will change the gravitational force with a light-speed "transmission" limit.

There is no reason to speak of the locational displacement of event horizons as a "thing", for which the speed of light limit applies. Even if it was the case that growth was limited to light speed by some other physics that matter, there would be no objection to the concept of a non-thing with rate measured in km per sec having a larger than c number.

 

[PeterDonis]

a Black Hole is an astronomical, massive object and the event horizon is the limit (i.e. the physical surface) of this object. (In fact we can calculate the mass inside the event horizon -i.e. the mass of the Black Hole- from the radius of the event horizon.)

This is misstated. The event horizon is not a "physical surface" of anything. An idealized black hole is a vacuum solution of the Einstein Field Equation; there is no stress-energy anywhere. A more realistic solution describing a hole formed from the collapse of a massive object like a star has a region of nonzero stress-energy occupied by the collapsing object, but that region occupies very little of the interior region (i.e., the region of spacetime inside the event horizon), and anyone falling into the hole well after the collapse will encounter only vacuum at and well inside the horizon.

Also, the mass of the hole is not "inside the event horizon". It is really a global property of the spacetime, not a localized property of anything.

 

[George K]

Increased in size is not a thing moving. It can be described in terms of rate, with units that are the same as velocity, but it isn't velocity. If you are building a rock wall, and add a 1 meter rock, the wall increased in size INSTANTLY by that meter. That happens no matter how slowly and carefully you place that rock. You could even build an entire kilometer of wall, starting at one end and finishing at the other, at faster than light rate of construction (ignore all the PRACTICAL impossibilities). Because the wall is not moving when we speak of its rate of completion in kilometers per second. You can even use construction rates for things like a tunnel. The Chunnel tunnel is 50.5 km long, and took years to complete. The tunnel was completed at a slow rate, of kilometers per year. In that case, the "speed" is the increase in length of the absence of the rock.

There are practical considerations that are limited by light speed. But any construction process that moves in segments, say laying a railroad track ... the addition of a new piece is an instantaneous extension in length. I can build a lego tower at FTL rates of increasing size, every time I add a brick.

Yes, that was a good example and I completely understood your argument. However, I'm not sure that this is valid for the EH of a BH. I mean the EH is (approximately) spherical and this means that its increment must be symmetrical in every direction. It's not like you are connecting LEGOS in series. The falling mass-energy must "re-arrange" the size of the EH as a whole, i.e. the "new" increased EH must have -again- a symmetric, spherical shape. How fast this "re-arrangement" of the EH surface can be done (in order to get -again- a symmetrical increment of this surface)?
As an example, let's consider the merge of the two massive BHs that was recently observed by LIGO (via the gravitational waves). The new larger BH (the result of the merging) didn't happen in zero time. There was a violent bouncing of the whole system and it took some time to settle down and a new (spherical) EV to be formed.

 

[PeterDonis]

an asymmetric mass change would generate an asymmetric event horizon.

It would initially generate an asymmetry in the horizon; but the asymmetry would propagate away as gravitational waves, and the final state of the horizon would be symmetrical.

The simulation shows asymmetry. And I would have assumed that the asymmetry was spatially changed by the gravity waves, just as the LIGO detector was spatially changed.

Yes, the gravitational waves detected by LIGO included waves that propagated away the asymmetry in the horizon of the new hole formed by the merger. (This is the "ringdown" phase of the process.) The final state of the horizon of the new hole is symmetrical, as above.

Black holes just are not amenable to exact measurement science.

Sure they are. LIGO measured properties of black holes.

the hypothetical asymmetric growth of a black hole can probably never be measured or observed with any satisfactory accuracy.

Sure it can; LIGO measured it. See above.

I am questioning if a response time limit of c applies, based on a naive understanding that mass changes will change the gravitational force with a light-speed "transmission" limit.

In GR, gravity is not a force, it's spacetime curvature. Changes in spacetime curvature propagate at the speed of light. That is what gravitational waves are: changes in spacetime curvature propagating at the speed of light.

There is no reason to speak of the locational displacement of event horizons as a "thing", for which the speed of light limit applies.

This is correct. It is correct because the "locational displacement" of the event horizon is not a change in spacetime curvature that has to propagate at the speed of light. It is just a number we can calculate (and the number can be different depending on the coordinates we choose) that has no physical effects. The actual changes in spacetime curvature associated with a black hole growing in mass do propagate at the speed of light, as above.