RELATIVITY QUESTIONS! (and other common queries)

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Wolfkeeper » Tue Dec 15, 2015 3:54 am UTC

In my view, when a true event horizon forms at any miniscule point, QM happens and the Hawking radiation immediately tears it apart. Very small black holes are insanely unstable. In other words, the parts fly apart until the density to form an event horizon is no longer quite there; but the gravity is still enormous; so it forms the same shape, and from outside you wouldn't see any major difference. I call this the 'snowplow'.

So on the one hand you have gravity pulling everything in tighter, and on the other, Hawking radiation stopping the event horizon from ever quite forming.

If you bring a mass nearby from outside, it triggers the snowplow again and the apparent event horizon moves out towards it and the density goes down until it just isn't quite a true event horizon again.

Can I prove this? No. But it seems to be consistent with all known physics so far as I can tell. And it's a plausible model to reason with.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby gmalivuk » Tue Dec 15, 2015 4:17 am UTC

But how does the inner event horizon "know" it's now inside a larger black hole? You're positing some new unknown physics that would cause that to happen (contrary to every theorem that says a spherically symmetric shell has flat space inside it), and then calling your picture consistent with all known physics.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Wolfkeeper » Tue Dec 15, 2015 4:40 am UTC

In the absence of a successful theory of quantum gravity that's a difficult question. Perhaps it's because the speed of light is special in physics. The horizon is notable in that the time-like curve is exceeding the speed of light relative to the rest mass of the black hole.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby gmalivuk » Tue Dec 15, 2015 4:47 am UTC

And yet you're positing that matter inside the new horizon moves outward under those circumstances, faster than light in its own frame?

That is not consistent with all known physics.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Wolfkeeper » Tue Dec 15, 2015 4:50 am UTC

No, at, or slower than the speed of light on average.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby gmalivuk » Tue Dec 15, 2015 4:54 am UTC

There is no slower-than-light path that moves radially outward through an event horizon, though.

And again, what kind of signal comes from the outer horizon to "tell" the inner horizon that it's time to "snowplow" all that matter up to a higher radius? I don't need a mature theory of quantum gravity to say that's also not particularly consistent with known physics.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Eebster the Great » Tue Dec 15, 2015 5:32 am UTC

Wolfkeeper wrote:Can I prove this? No. But it seems to be consistent with all known physics so far as I can tell. And it's a plausible model to reason with.

The thing is, you can't say that something is "consistent with all known physics" until you know all known physics, and you can't even tell if it's a plausible model until you know some known physics.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby doogly » Tue Dec 15, 2015 1:25 pm UTC

Yeah, it would be nice to know which QFT and GR books, at least, you have worked through. Gives us somewhere to base the conversation.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Wolfkeeper » Tue Dec 15, 2015 5:12 pm UTC

gmalivuk wrote:There is no slower-than-light path that moves radially outward through an event horizon, though.

QM infamously isn't constrained by things like having to be slower than light at all times; virtual particles and Bell's inequality are a couple of the more famous manifestations of that.
And again, what kind of signal comes from the outer horizon to "tell" the inner horizon that it's time to "snowplow" all that matter up to a higher radius? I don't need a mature theory of quantum gravity to say that's also not particularly consistent with known physics.

QM routinely violates conservation laws, provided the books balance in the end. The information paradox would appear to violate one of those conservation laws, and hence QM will act to arrange the wavefunctions of the masses involved to prevent the information being destroyed which would stop the event horizon from forming.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby gmalivuk » Tue Dec 15, 2015 5:29 pm UTC

"QM sometimes kinda-sorta breaks this rules" is not a valid justification for "therefore these rules are broken in exactly this way at this time", just like "Earth is not flat" is not a justification for "therefore Earth is a cylinder".

You still haven't even hinted at what actual mechanism might make the inner Hawking radiation suddenly explode outward.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Hypnosifl » Tue Dec 15, 2015 10:13 pm UTC

Wolfkeeper wrote:In the absence of a successful theory of quantum gravity that's a difficult question. Perhaps it's because the speed of light is special in physics. The horizon is notable in that the time-like curve is exceeding the speed of light relative to the rest mass of the black hole.

In the infinitesimal neighborhood of any point on a time-like worldline, you can define a set of "local inertial frame" (see here for a discussion of how this relates to the "equivalence principle"), and in all of those local inertial inertial frames, the object moving along that worldline will be moving slower than light, even if the object is on or inside the event horizon. I'm not sure what you mean when you define the speed "relative to the rest mass of the black hole", if you're talking about some large-scale coordinate system that includes the singularity (or at least points arbitrarily close to it) along with points on a worldline far from the singularity, then no such large-scale coordinate system can qualify as "inertial", and in any spacetime there are an infinite number of different non-inertial coordinate systems you can choose. The idea that "light always travels at c, massive objects always move slower than c" does not generally hold in non-inertial frames--even in the flat spacetime of special relativity, you're free to define non-inertial frames like Rindler coordinates where the speed of light isn't constant, and a massive object can move faster than c (or faster than the speed of a light ray at another point in the coordinate system). In addition, in the case of a non-rotating black hole it's actually possible to design a non-inertial coordinate system where all radially-moving light rays do move at a constant coordinate speed, and all massive objects moving on timelike worldlines in a radial direction move slower than that speed--see [url=http://en.wikipedia.org/wiki/Kruskal–Szekeres_coordinates]Kruskal-Szekeres coordinates[/url]. So, the conclusion from all this is that there is no coordinate-independent physical sense in which anything "weird" happens with the speed of light or timelike worldlines in the vicinity (exterior or interior) of a black hole apart from the singularity itself, according to general relativity.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Eebster the Great » Wed Dec 16, 2015 5:13 am UTC

Wolfkeeper wrote:
gmalivuk wrote:There is no slower-than-light path that moves radially outward through an event horizon, though.

QM infamously isn't constrained by things like having to be slower than light at all times; virtual particles and Bell's inequality are a couple of the more famous manifestations of that.

What specifically is moving faster than light in these examples?

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby doogly » Wed Dec 16, 2015 12:28 pm UTC

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Carlington » Wed Dec 16, 2015 1:23 pm UTC

Nicias, thanks for the explanation before. That makes sense, of course once you start to accelerate some funky stuff is going to go on with regards to "your" frame.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Wolfkeeper » Wed Dec 16, 2015 5:24 pm UTC

gmalivuk wrote:"QM sometimes kinda-sorta breaks this rules" is not a valid justification for "therefore these rules are broken in exactly this way at this time", just like "Earth is not flat" is not a justification for "therefore Earth is a cylinder".

You still haven't even hinted at what actual mechanism might make the inner Hawking radiation suddenly explode outward.

I already have, twice, but you didn't get it, either time.

With Hawking radiation, we're talking about something, that is, in a very real sense, stronger than gravity, particularly at the very, very small scale.

Black holes have a lifetime that is dependent on their mass. The lifespan is proportional to the mass cubed. So very tiny blackholes are incredibly unstable.

So take a mass distribution which is critically above that of a black hole. Now add one proton. The proton would seed the blackhole, the black hole would form around the proton first. So the only thing in the blackhole is the proton.

I calculate that the lifetime is 3.9e-97 seconds; ridiculously small. Meanwhile the Planck time is 5.39e-44.

So the black hole is instantly destroyed, and replaced by Hawking radiation. Note that most of the Hawking radiation is found outside where the black hole was. Crucially the diameter is now bigger and it is no longer a blackhole.

Now that might trigger further blackholes, and further explosions. But these explosions can be expected to be spacelike, not timelike- because we're far, far below the Planck time. So as you pile mass into the 'blackhole', by any means at all, it keeps expanding, but a stable event horizon isn't ever made.

So am I saying that GR theorists are idiots for ever postulating macroscopic event horizons? No way, absolutely not. Under standard GR assumptions, event horizons certainly form. If you take an infinitely smooth mass distribution, and squeeze it, then an event horizon forms of indefinite size and at faster than the speed of light.

But they fail to form when mass is quantised and Hawking radiation can be produced; in other words when you allow for the effects of QM within the GR framework. With quantised mass, i.e. fundamental particles, their presence sharply curves spacetime and seeds the creation of these tiny, tiny event horizons, but they're always far too small to be stable and the mass density automagically lowers until the infinitesimal event horizon disappears.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby doogly » Wed Dec 16, 2015 5:57 pm UTC

Wolfkeeper wrote: The proton would seed the blackhole, the black hole would form around the proton first. So the only thing in the blackhole is the proton.

I'm not sure where to land on whether this is false or meaningless, but I think I can go with both pretty comfortably.

Could you address the earlier request for the base literature you've studied? I do think it would help to ground things, I'd rather point at things than just berate and complain at you.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby sevenperforce » Wed Dec 16, 2015 6:42 pm UTC

RE: Wolfkeeper...

So what you appear to be suggesting (if I understand it right) is that no persistent event horizon is ever formed, but rather that black holes form and evaporate (perhaps via quantum tunneling) instantly at some infinitesimally small mass, with the radiation from their evaporation instantly rushing outward. However, since this radiation formerly had the density of a black hole and has just tunneled out of its former event horizon, it will collapse back into a black hole as soon as it encounters any other mass or energy. Due to its infinitesimal mass, however, it will again evaporate outward, only to collapse each time it encounters additional mass-energy. As the emission of evaporative radiation will not necessarily comprise only a single photon, but may include multiple photons in different directions, it will separate into multiple infinitesimally small black holes.

What we would have, then, is a loosely-bound spherically-symmetric shell of radiation "packets" expanding outward against their collective gravity, arrested at their collective Schwarzschild radius but able to individually quantum-tunnel out.

I don't know enough about GR to know whether there are any immediate and problematic issues with this view. It does make sense, though, in an odd sort of way (which usually is a bad sign, haha, but oh well), and might be able to be mathematically extended to the point that it could explain some additional aspects of black holes. In this model, a black hole (perhaps "black cloud" or "black shell" is a better term?) would initiate at the center of the mass distribution and expand outward, growing in radius and in number as the "packets" encounter more and more mass-energy. However, the warping of spacetime at the event horizon would affect the evaporative tunneling probability as a function of frequency (perhaps via the expanding geometry and decreasing curvature of the dynamic surface), so that the distribution of escaping evaporative radiation becomes cooler and cooler as the overall black hole becomes larger and larger. If the escaping packets do not encounter sufficient mass-energy to continue re-collapsing, the black hole will evaporate entirely. At around 1e22 kg, the temperature of the escaping radiation becomes equal to the temperature of the CMBR and an equilibrium is reached, where the "black cloud" comprises a spherical shell of infinitesimal black holes which are continually evaporating and re-collapsing.

I guess my question to Wolfkeeper would be this: if that's really how it works, then what predictions can we make based on this model about the behavior of black holes that differs from current theory?

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby gmalivuk » Wed Dec 16, 2015 6:55 pm UTC

Wolfkeeper wrote:I already have, twice, but you didn't get it, either time.
No, what you have done is spew nonsense with no basis in known physics, and you've done it far more than twice.

So take a mass distribution which is critically above that of a black hole. Now add one proton. The proton would seed the blackhole, the black hole would form around the proton first. So the only thing in the blackhole is the proton.
What do you mean by this? Where are you adding the proton? Where would it seed the black hole? The only way the only thing in a black hole is a proton is if it's a proton-massed black hole, in which case I don't get the point of starting with an almost-dense-enough mass distribution.

I still think a better thought experiment is the infalling shell, because then you can keep it spherically symmetric instead of imagining a particle that either shows up from nowhere or falls in at one particular point on the surface.

Suppose we have mass M distributed somehow in a sphere within or just above radius R=2M, and another mass M falling in as a thin shell of dust. No part of the configuration is above the critical density for a black hole with radius 2R=4M until the entire shell has fallen past 2R. When only half the shell is past 2R, then within that radius there's only enough mass for a black hole of radius 1.5R, so you can't yet posit that your Hawking snowplow has started to push the original mass distribution outward. No single part of the infalling mass can "seed" a black hole because there's no first particle that brings anything above the critical density. Rather, if anything it's the last particle past 2R that accomplishes this.

At this point, a ship hovering at 1.5R sees what? The M below the ship is still emitting its modest Hawking radiation, and no sphere of mass above 1.5R has any effect on the spacetime seen by the ship at and below its own height. So you're positing that through some quantum handwavium, the outer shell sends an invisible signal (which is not based on any curvature of spacetime) into the lower black hole, which tells its Hawking radiation to suddenly heat up and explode itself and the ship out to 2R?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby cyanyoshi » Wed Dec 16, 2015 7:14 pm UTC

About Hawking radiation, is that something unique to black holes, or does it form around basically anything with mass? And if anything can emit Hawking radiation, then how would the composition of the thing with mass change over time?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby doogly » Wed Dec 16, 2015 7:58 pm UTC

It isn't really something that comes out of a thing (which is the mental picture wolfie has, and the source of much wrongness.) It something someone notices when they notice a horizon. Apparent horizons as well as event horizons will get this.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Yakk » Wed Dec 16, 2015 8:46 pm UTC

The infalling shell thing -- is there an "inside out" horizon there, above the things *inside* the shell?

I'm trying to picture it, and man, that geometry gets strange. The stuff inside the dust-shell is in locally flat space-time, but it is in a sense on top of an infinitely tall "plateau" corresponding to the event horizon "around" them?

Insofar as you cross the event horizon at the speed of light (in a sense) (the event horizon in a locally flat and stationary remapping of the environment crosses that environment at the speed of light, right?), the infalling event horizon crosses locally flat and stationary reimaging of the person inside the shell at ... what, the speed of light? Faster? That wouldn't make sense.

Usually when you fall through an event horizon, your "arrow of time" rotates towards the center of the black hole in a continuous manner. With an infalling shell, it seems stranger.

So, what does the person sitting inside the infalling shell experience as it collapses inward?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Hypnosifl » Wed Dec 16, 2015 9:11 pm UTC

Wolfkeeper wrote:With Hawking radiation, we're talking about something, that is, in a very real sense, stronger than gravity, particularly at the very, very small scale.

Why do you think it's stronger than gravity? Do you think something is "escaping" the event horizon in Hawking radiation? If so, I recommend reading this article which clears up some misconceptions about Hawking radiation that arise from rough layman's explanations of how it works, including the incorrect idea that it is produced right at the location of the event horizon.
Wolfkeeper wrote:So take a mass distribution which is critically above that of a black hole. Now add one proton. The proton would seed the blackhole, the black hole would form around the proton first. So the only thing in the blackhole is the proton.

Why do you think it would form "around the proton first" as opposed to forming around the entire mass distribution? One thing to be aware of is the fact that when physicists talk about an event horizon they are usually talking about something called the "absolute horizon" (though there is another notion called the 'apparent horizon') and this absolute horizon is defined in a non-local way. Where the absolute horizon is located at any given moment is defined to depend on the entire future history of the region (the horizon marks the boundary between spacetime points where light emitted from those points can eventually escape to infinity, vs. points where it all light emitted from such a point will eventually hit the singularity), it isn't something that you can directly detect the presence of in a local sense. There's a good discussion of this in Kip Thorne's book Black Holes and Time Warps, check out this page and this one on the google books preview. So if you drop a proton onto a mass that just needs that extra proton to form a black hole, I'm pretty sure that what happens is that the event horizon starts to expand outward from the center of the mass before the proton reaches it, and reaches the radius of the mass at the same moment the proton hits it. That's similar to the diagram in Eddington-Finkelstein coordinates on this page, showing a spherical mass of collapsing radius (the worldline of the surface is the white line) and also showing how the absolute event horizon (the pinkish-red line) expands outward from the center, meeting the surface of the mass at the exact moment the radius reaches the Schwarzschild radius for that mass, after which the horizon ceases to expand.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby sevenperforce » Wed Dec 16, 2015 9:50 pm UTC

Hypnosifl wrote:Why do you think it would form "around the proton first" as opposed to forming around the entire mass distribution? One thing to be aware of is the fact that when physicists talk about an event horizon they are usually talking about something called the "absolute horizon" (though there is another notion called the 'apparent horizon') and this absolute horizon is defined in a non-local way. Where the absolute horizon is located at any given moment is defined to depend on the entire future history of the region (the horizon marks the boundary between spacetime points where light emitted from those points can eventually escape to infinity, vs. points where it all light emitted from such a point will eventually hit the singularity), it isn't something that you can directly detect the presence of in a local sense.

So if you drop a proton onto a mass that just needs that extra proton to form a black hole, I'm pretty sure that what happens is that the event horizon starts to expand outward from the center of the mass before the proton reaches it, and reaches the radius of the mass at the same moment the proton hits it. That's similar to the diagram in Eddington-Finkelstein coordinates on this page, showing a spherical mass of collapsing radius (the worldline of the surface is the white line) and also showing how the absolute event horizon (the pinkish-red line) expands outward from the center, meeting the surface of the mass at the exact moment the radius reaches the Schwarzschild radius for that mass, after which the horizon ceases to expand.

I'm not so sure that the concept of a "mass that just needs that extra proton to form a black hole" is meaningful. There is no critical mass for a black hole, only a critical density, and that critical density is reached under some pretty specific conditions. Moreover, even if that case were possible, what about the speed of the proton? If the mass has a nonzero radius, then there is some velocity just below the speed of light at which the proton will impact the surface before the information associated with its impending arrival will be able to reach the center of the mass distribution. In fact, given that this hypothetical mass distribution is already pretty much equal to the critical density, the proton will be moving just shy of the speed of light no matter what.

gmalivuk wrote:I still think a better thought experiment is the infalling shell, because then you can keep it spherically symmetric instead of imagining a particle that either shows up from nowhere or falls in at one particular point on the surface.

Suppose we have mass M distributed somehow in a sphere within or just above radius R=2M, and another mass M falling in as a thin shell of dust. No part of the configuration is above the critical density for a black hole with radius 2R=4M until the entire shell has fallen past 2R. When only half the shell is past 2R, then within that radius there's only enough mass for a black hole of radius 1.5R, so you can't yet posit that your Hawking snowplow has started to push the original mass distribution outward. No single part of the infalling mass can "seed" a black hole because there's no first particle that brings anything above the critical density. Rather, if anything it's the last particle past 2R that accomplishes this.

Because I'm interested (and I freely admit that I may be making extremely critical errors right from the outset, so feel free to smack me down if necessary)...

As I understand it, a black hole condition exists where the density of matter at some point in space is great enough that it falls within its own gravitational radius. While it makes sense that it is the last infalling particle which accomplishes this, I'm not sure about how we can properly characterize the point in space where the event horizon initiates.

Let's take the case of a spherically symmetric mass distribution which is slightly lower than the critical density for a black hole. One could imagine that this takes the form of the most massive neutron star possible. Then, rather than a thin shell of dust falling in uniformly, let us suppose that a single photon falls onto the surface of the neutron star.

The force exerted by this final photon's mass-energy within the gravitational field of the neutron star creates a pressure wave which travels inward toward the core of the neutron star. In a rather extreme analogue of the straw that broke the camel's back, this increase in pressure overcomes the quantum degeneracy pressure of the neutron soup at the core, and the core begins to collapse. The collapse will propagate inward, away from the impact point of the photon's pressure wave and toward the center of the core.

At some time tc, the critical density of a black hole is exceeded at the very core, and a black hole of infinitesimal mass is created. It should be noted that at tc, the neutron star is not yet a black hole. It is still intact, with a surface outside its gravitational radius. It is only the infinitesimally small volume at the center of the core which has momentarily exceeded the critical density.

By the shell theorem, this black hole is unaffected by any of the mass "above" its event horizon. An infinitesimally small black hole presents an infinitesimally small tunneling length to the outside of its event horizon, and so evaporation via quantum tunneling is instantaneous. The radiation expands outward from the event horizon with a density infinitesimally lower than the critical density. Thus, each particle it encounters must also collapse into an infinitesimally small black hole, causing a sort of chain reaction of collapse and evaporation moving outward in a shell.

I suppose that the evaporative radiation will technically be a wavefunction with some specific probability distribution, but it shouldn't do any harm to conceptualize it as a single photon moving outward along a random vector. Initially, all such vectors intersect the infalling distribution of matter, spawning more infinitesimally small black holes, but as the population of black holes increases, more and more of the possible vectors will intersect other black holes and be reabsorbed. This causes the growth rate to slow in proportion to the overall radius of the shell. Although the tunneling distance remains the same, the curvature of space at that tunneling distance increases exponentially as the cumulative mass of the black hole shell rises, and so the temperature of escaping radiation becomes cooler and cooler.

This process would continue until no further particles are encountered, at which point the radiation begins to escape in earnest without recollapsing and the black hole evaporates into what we would term Hawking radiation, or until the amount of incoming radiation meets or exceeds the amount of escaping radiation.

Seems like it fits to me...but again, I'm not well-versed in these things.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Hypnosifl » Wed Dec 16, 2015 10:11 pm UTC

sevenperforce wrote:I'm not so sure that the concept of a "mass that just needs that extra proton to form a black hole" is meaningful. There is no critical mass for a black hole, only a critical density, and that critical density is reached under some pretty specific conditions.

A black hole does not have a characteristic density--larger black holes have lower densities than small ones, and supermassive black holes can have densities lower than that of water. But for any given mass, there is a critical radius, the Schwarzschild radius, such that if the mass is confined to a sphere smaller than that radius a black hole will form. So, just imagine a sphere of mass M whose radius R is just slightly larger than the Schwarzschild radius for that mass, but then if a proton's mass m is added to it, the Schwarzschild radius for mass (M+m) is greater than or equal to R. In terms of the equations, this just means choosing an R such that 2GM/c^2 < R ≤ 2G(M+m)/c^2
sevenperforce wrote:Moreover, even if that case were possible, what about the speed of the proton? If the mass has a nonzero radius, then there is some velocity just below the speed of light at which the proton will impact the surface before the information associated with its impending arrival will be able to reach the center of the mass distribution. In fact, given that this hypothetical mass distribution is already pretty much equal to the critical density, the proton will be moving just shy of the speed of light no matter what.

But that was my point about the absolute horizon behaving non-locally--information about the impending arrival doesn't have to be able to reach the center in order for the absolute horizon to begin expanding outward from the center, it can expand in "anticipation" of events that are outside the future light cone of the center at the moment the horizon begins to expand. This doesn't violate relativity since the absolute horizon is not a real physical entity you can measure locally, it's just an abstract partition of spacetime into two regions (one region such that light emitted from any point in the region is doomed to eventually hit the singularity, perhaps arbitrarily far in the future, and the second region such that light emitted from any point in the region can, if it's emitted at the right angle, continue on forever without hitting the singularity).

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Copper Bezel » Thu Dec 17, 2015 9:18 am UTC

Hypnosifl wrote:A black hole does not have a characteristic density--larger black holes have lower densities than small ones, and supermassive black holes can have densities lower than that of water. But for any given mass, there is a critical radius, the Schwarzschild radius, such that if the mass is confined to a sphere smaller than that radius a black hole will form. So, just imagine a sphere of mass M whose radius R is just slightly larger than the Schwarzschild radius for that mass, but then if a proton's mass m is added to it, the Schwarzschild radius for mass (M+m) is greater than or equal to R. In terms of the equations, this just means choosing an R such that 2GM/c^2 < R ≤ 2G(M+m)/c^2

Since the radius just goes linearly with mass, so that the required density goes inversely with the square of the mass, it seems important to note that if a particular clump of matter in a degenerate star is just on the edge of becoming a black hole, then a slightly larger spherical shell including it already has, and that goes all the way up until you run out of star. The idea that tiny black holes should ever realistically form at all is entirely Wolfkeeper's.

Edit: To be clear, I don't think I'm saying anything that Hypnosifi isn't, and if I am, they can correct me - just pulling this out for emphasis.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby sevenperforce » Thu Dec 17, 2015 9:53 pm UTC

Hypnosifl wrote:A black hole does not have a characteristic density--larger black holes have lower densities than small ones, and supermassive black holes can have densities lower than that of water. But for any given mass, there is a critical radius, the Schwarzschild radius, such that if the mass is confined to a sphere smaller than that radius a black hole will form.

Sure, black holes don't have a characteristic density...but they have a critical density which is a function of mass or of volume. For any given mass or for any given volume, we can talk about the critical density where the object will be a black hole.

Copper Bezel wrote:Since the radius just goes linearly with mass, so that the required density goes inversely with the square of the mass, it seems important to note that if a particular clump of matter in a degenerate star is just on the edge of becoming a black hole, then a slightly larger spherical shell including it already has, and that goes all the way up until you run out of star.

If that's the case, then it makes things a great deal simpler and yet a great deal more complicated.

The issue I'm seeing is whether the density gradient goes up quickly enough to make a difference. If the Schwarzschild radius goes up linearly with mass but the mass goes up linearly with volume, then obviously this will hold true...but that's a constant-density case. Density will be some function of the radius, so if the density function goes up with the inverse cube of the radius, or if it's an exponential relationship, things get messy.

One presently-unsolved problem in astrophysics is the issue of what occupies the space between neutron stars, which have been observed up to 2.01 M but could theoretically go up to 3 M, and the lowest-mass stellar black holes, which start around 5 M. Possible solutions include quark-degenerate stars and various other ideas. However, if neutron stars themselves could contain black holes of very low mass which produce a Hawking radiation pressure sufficient to prevent further collapse, this would be a very interesting solution indeed.

The density of neutron stars is supposed to range between 3.7e17 and 5.9e17 kg/m3 (1.86e-4 to 2.98e-4 M/km3). At this density, a neutron star would need to have a mass of 5.6-7.1 M in order to fall within its own Schwarzschild radius and become a black hole; this corresponds to a Schwarzschild radius of between 16.5 and 20.9 km.

The density at the center of a neutron star, however, could go up as high as 8e17 kg/m3 for neutron-degenerate matter, or 1e23 kg/m3 for hypothetical preon-degenerate matter (corresponding to 4.02e-4 and 5.0e2 M/km3, respectively). The former density corresponds to a radius of 14.19 km for a black hole, but matter at the latter density would need to reach a radius of only 40.12 meters before becoming a black hole of approximately 14.21 Jupiter masses.

Thus, if we imagine a 3 M neutron star with an average density of 5.9e17 kg/m3, an outer core density of 8e17 kg/m3, and an inner preon core density of 1e23 kg/m3, we get something like this:

smns.png
smns.png (6.73 KiB) Viewed 7583 times

So although the Schwarzschild radius of the entire object is well inside the outer core, the Schwarzschild radius of the core is only slightly smaller, and the Schwarzschild radius of the inner core is just barely smaller than the inner core itself. A collapse of the inner core would thus result in it forming a black hole before the rest of the star constituted a black hole; that is, you would end up with a small black hole at the center and a neutron-star shell around it.

The Eddington limit for a black hole's accretion disk is reached when the pressure from radiation released by infalling matter exceeds the gravitational pull of the black hole. Similarly, it is possible to have an Eddington limit for Hawking radiation, where the radiation pressure is so great that the black hole cannot grow further.

If the core-collapse of the degenerate-matter core of a neutron star (perhaps during a neutron star collision) happened in such a way that density grew exponentially as radius decreased, then it would be possible to have an arbitrarily small black hole form at the center and immediately begin evaporating in Hawking radiation, which would produce enough radiation pressure to stabilize the rest of the core and prevent further collapse.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Copper Bezel » Thu Dec 17, 2015 10:11 pm UTC

Wait, how does that scenario connect with the gap between masses of neutron stars and black holes? Are you saying this scenario is a partial collapse, where some of the things we've observed as two solar-mass neutron stars are more massive and closer to that threshold of five, but have these collapsed cores you're positing?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby sevenperforce » Thu Dec 17, 2015 10:34 pm UTC

No, that wouldn't affect the observable mass. It could, however, explain why the gap exists. We understand neutron degeneracy pressure up to point, but we haven't been able to develop an equation of state that tells us how the core is expected to behave itself. However, if the density of a neutron star core increases exponentially with decreasing radius, then the formation of a microscopic black hole could serve to arrest collapse during formation, affecting the final outcome and resulting in that gap.

Neutron degeneracy pressure is believed to "top out" at 1.6e35 Pa. By my calculations, the Hawking radiation at the event horizon of a 6e10 kg black hole would approximately match this pressure, allowing a neutron-degenerate shell to "float" on top of the Hawking radiation at the event horizon of the black hole. A 6e10 kg black hole would have a radius of 8.92e-17 meters. Coincidentally (or perhaps not?), this is the approximate range of the weak force...precisely where we would expect neutron degeneracy to break down.

One could propose that during a core-collapse supernova, the very center exceeds neutron degeneracy pressure and a black hole is formed with an event horizon radius on the order of the weak force. Depending on the size of the progenitor star, this takes place sooner or later in the core-collapse timeline, and for stars above a certain threshold, neutron-degenerate matter "piles up" on the surface of the event horizon's radiation pressure too quickly, resulting in total collapse into a black hole of at least 5 solar masses. For stars below that threshold, the "pile-up" on the surface of the event horizon stops and results in the rebound stage of the supernova, blowing off everything but around 2-3 solar masses.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Hypnosifl » Thu Dec 17, 2015 10:59 pm UTC

sevenperforce wrote:Neutron degeneracy pressure is believed to "top out" at 1.6e35 Pa. By my calculations, the Hawking radiation at the event horizon of a 6e10 kg black hole would approximately match this pressure, allowing a neutron-degenerate shell to "float" on top of the Hawking radiation at the event horizon of the black hole. A 6e10 kg black hole would have a radius of 8.92e-17 meters. Coincidentally (or perhaps not?), this is the approximate range of the weak force...precisely where we would expect neutron degeneracy to break down.

But would this be a stable equilibrium, or an unstable one? At equilibrium, the rate that the black hole is losing mass due to Hawking radiation is balanced out by the rate it gains mass from nucleons (or reflected Hawking radiation, or radiation from other sources) falling into it. But presumably there will be small statistical fluctuations away from the equilibrium. What happens when a fluctuation causes the black hole's mass to be a little larger than the equilibrium mass, so that the Hawking radiation is slightly weaker? Would there be any feedback effect causing the rate of nucleons falling in to become slightly smaller here too, or would they fall in at a greater rate due to fewer being deflected by Hawking radiation? The latter would seem to suggest the departure from equilibrium would be unstable, and the mass of the black hole would continue to increase from such a fluctuation, until it had consumed the whole neutron star.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby sevenperforce » Fri Dec 18, 2015 1:37 pm UTC

I'm reasonably sure it would be a stable equilibrium. The equilibrium, if it exists at all, would be a radiative balance between the radiation pressure flowing out of the black hole and the reflection or absorption/emission of radiation from the inner surface of neutron-degenerate plasma. There aren't any free neutrons at the center of a neutron star, so there wouldn't be any "deflection" going on; it's just light coming out and light going back in. That's the only way for an equilibrium to actually be maintained at all; otherwise it would just be slowly consuming the neutron star from the inside out.

If a small fluctuation caused an increase in the black hole's mass, its radiative output would decrease, reducing the radiation pressure on the inner surface of neutron-degenerate plasma. But because the inner surface would be absorbing less energy, it would radiate less energy, reducing the radiation flux back into the black hole and returning it to its equilibrium mass.

There are different types of core collapse supernovae, all with different outcomes depending on the metallicity and mass of the progenitor star. Depending on these variables, it may be that a larger star ends up creating a smaller black hole (due to a more rapid collapse), producing a more energetic Hawking radiation front which repulses the infalling neutron-degenerate plasma at a larger radius.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Yakk » Fri Dec 18, 2015 2:51 pm UTC

At the energy density in question, isn't thinking of mass-energy having a particular particle form a bit off?

We have a ridiculous pressure and an energy density: won't it be a soup of conserved quantities rather than particles?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby doogly » Fri Dec 18, 2015 2:55 pm UTC

I've been trying very hard to get some of our interlocutors to address quantum field theory. Good luck.
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby sevenperforce » Fri Dec 18, 2015 3:55 pm UTC

Yakk wrote:At the energy density in question, isn't thinking of mass-energy having a particular particle form a bit off?

We have a ridiculous pressure and an energy density: won't it be a soup of conserved quantities rather than particles?

Indeed. Though this will not in any sense prevent the formation of a black hole.

In an iron-core-collapse supernova, the core grows to a mass of 1.4 M, surpassing the Chandrasekhar limit and triggering core collapse. The core collapses not only under its own gravity, but under the tremendous weight of the entire rest of the star. How fast it collapses will depend primarily on the total mass of the star.

The neutron-degenerate analogue to the Chandrasekhar limit is the Tolman-Oppenheimer-Volkoff limit, which is not known exactly but is estimated to be between 1.5 and 3.0 M. That's odd, because we have observed neutron stars all the way up to 2.01 M. The core-collapse process which forms neutron stars is exponentially more energetic than the corresponding process to form white dwarves; if the limit is on the order of two solar masses, then the cataclysmic collapse of that much matter would be expected to far exceed the limits of neutron degeneracy pressure.

We're talking about a collapse with speeds on the order of 20-40% of c. It's like the compressive strength of a material; if your measurements indicate that a brick building won't be able to rise to more than fifty feet without the bottom bricks crumbling, then you can be quite certain that a fifty-foot-high stack of bricks is going to crumble if you're dropping it from a mile high. One would thus imagine that the only way to get a neutron star to 2.01 M would be by gradual accretion...but gradual accretion is unlikely in the midst of a supernova.

It is therefore suspected by astrophysicists that the inner core of a neutron star comprises something more exotic, like a quark-gluon plasma, which serves to arrest collapse and give the neutron star a stable radius to allow the supernova to "bounce" off of. However, if the collapse of the center of the iron core takes place at arbitrarily high speeds, an equivalent "bounce" could result from the formation of a microscopic black hole which pushes back the neutron-degenerate plasma (or even a more dense quark-gluon plasma) with Hawking radiation pressure.

Paradoxically, a heavier star would produce a smaller black hole due to the more rapid collapse achieving the necessary density at a smaller volume. But the smaller black hole would produce a larger "bounce" radius, thus making the "core" larger and enabling the formation of a larger neutron star. For even heavier stars, the "core" would be large enough that the neutron star mass plateaus and more mass is blasted off in the supernova. At some even greater progenitor mass, the equilibrium is unstable (perhaps due to the very large size of the "core") and so what would have formed a neutron star collapses into the black hole with a minimum mass of 5 M. In the extreme case, the initial black hole is so small that its radiation pressure pushes everything away and it evaporates without leaving any compact remnant at all.

Seems solid...and I think I could make a few predictions once the math is worked out.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Hypnosifl » Fri Dec 18, 2015 6:13 pm UTC

sevenperforce wrote:I'm reasonably sure it would be a stable equilibrium. The equilibrium, if it exists at all, would be a radiative balance between the radiation pressure flowing out of the black hole and the reflection or absorption/emission of radiation from the inner surface of neutron-degenerate plasma. There aren't any free neutrons at the center of a neutron star, so there wouldn't be any "deflection" going on; it's just light coming out and light going back in. That's the only way for an equilibrium to actually be maintained at all; otherwise it would just be slowly consuming the neutron star from the inside out.

Doesn't the neutron-degenerate plasma carry mass/energy which can potentially cross the event horizon, and isn't it capable of absorbing energy like any other medium? Even if some of the plasma was entering the black hole, an equilibrium could still be maintained if not all the Hawking radiation was being reflected back as well, but instead some was being absorbed by the surrounding plasma.

The main reason I'm skeptical of the equilibrium being a stable one is just that if it was, I can't imagine all the physicists who have analyzed neutron stars and quark stars in the past could have just missed what seems like a reasonably straightforward possibility...

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby sevenperforce » Fri Dec 18, 2015 8:24 pm UTC

Hypnosifl wrote:
sevenperforce wrote:I'm reasonably sure it would be a stable equilibrium. The equilibrium, if it exists at all, would be a radiative balance between the radiation pressure flowing out of the black hole and the reflection or absorption/emission of radiation from the inner surface of neutron-degenerate plasma. There aren't any free neutrons at the center of a neutron star, so there wouldn't be any "deflection" going on; it's just light coming out and light going back in. That's the only way for an equilibrium to actually be maintained at all; otherwise it would just be slowly consuming the neutron star from the inside out.

Doesn't the neutron-degenerate plasma carry mass/energy which can potentially cross the event horizon, and isn't it capable of absorbing energy like any other medium? Even if some of the plasma was entering the black hole, an equilibrium could still be maintained if not all the Hawking radiation was being reflected back as well, but instead some was being absorbed by the surrounding plasma.

We don't really have any decent equation of state for matter at this kind of density. So we don't have any way of knowing how the neutron-degenerate plasma will behave. Maybe it's absorbing more than it's emitting, but given that we're dealing with a very abrupt event horizon very close by, it could just as easily be behaving as a perfectly-reflective inside-out blackbody.

I'm expecting that in most cases, the mass of the black hole would be less than 6e10 kg, so that the radiation pressure reaches 1.6e35 Pa at some distance above the event horizon. For example, if the black hole forms with a mass of just 3e10 kg, then the event horizon has a radius of 4.46e-17 meters and a surface radiation pressure of 2.56e36 Pa. This would drop to the same 1.6e35 Pa at a greater radius of 1.26e-16 meters, meaning that there would be a gap of 3.68e-17 between the event horizon and the inner surface of neutron-degenerate plasma. Unless I miss my guess, that gap is going to prevent the plasma from crossing the event horizon.

The main reason I'm skeptical of the equilibrium being a stable one is just that if it was, I can't imagine all the physicists who have analyzed neutron stars and quark stars in the past could have just missed what seems like a reasonably straightforward possibility...

Well, I'm glad that someone with what is clearly a decently broad grasp of astrophysics and relatively thinks that this is a "reasonably straightforward" conjecture!

It may be that there is some blindingly obvious foil, like, "Oh, didn't you know that the quark-gluon plasma we generated ten years ago at CERN has a volumetric binding energy seven orders of magnitude greater than neutron-degenerate plasma despite being only half as dense?" Or maybe there isn't. I don't know.

A core-collapse supernova progenitor has three main variable attributes: mass, angular momentum, and metallicity. Since angular momentum can be arbitrarily low, we can ignore it for our purposes, leaving us with only mass M and metallicity m as our variables. There is therefore some function ρ(M,m) which determines the peak density which a collapsing iron-core supergiant can produce at the center of its formerly-iron core. If ρ(M,m) is greater than 2.02e58 kg/m3 for any M,m then a black hole with a mass of less than 6e10 kg will form. While such a density is obviously many many orders of magnitude greater than even purely-hypothetical preon stars, it is still 38 orders of magnitude lower than the Planck density.

It should be noted that one of the predictions of loop quantum gravity is that black holes may contain Planck stars, with a singularity density on the order of 1e96 kg/m3, which solves a few of the current problems with black holes. That's somewhat similar to the idea of the (non-loop-quantum-gravity) ultramicroscopic black hole cloud that I suggested a while back, where the black hole event horizon is formed of infinitesimal-mass black holes which evaporate with such high-energy Hawking radiation that the radiation itself is infinitesimally smaller than its own Schwarzschild radius and thus re-collapses as soon as it encounters any additional energy, even if it's just the CMBR.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby tomandlu » Wed Dec 30, 2015 9:12 am UTC

A related (ish) question - if we assume that the universe will eventually reach a state of maximum entropy, how does a black hole ever stop being a black hole and become part of the general noise?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Eebster the Great » Wed Dec 30, 2015 9:30 am UTC

tomandlu wrote:A related (ish) question - if we assume that the universe will eventually reach a state of maximum entropy, how does a black hole ever stop being a black hole and become part of the general noise?

Black holes can form and evaporate many times before the heat death of the universe.

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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby tomandlu » Wed Dec 30, 2015 10:01 am UTC

Eebster the Great wrote:
tomandlu wrote:A related (ish) question - if we assume that the universe will eventually reach a state of maximum entropy, how does a black hole ever stop being a black hole and become part of the general noise?

Black holes can form and evaporate many times before the heat death of the universe.


Thanks - hadn't really taken that in, but a quick google and a read of Wikipedia's article on Hawking Radiation has clarified the matter (sic).
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby Neil_Boekend » Wed Dec 30, 2015 2:16 pm UTC

Can a black hole be so massive that it is in thermal equilibrium with the cosmic background radiation? Would an even heavier black hole gain mass from the background radiation?
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Re: RELATIVITY QUESTIONS! (and other common queries)

Postby thoughtfully » Wed Dec 30, 2015 2:25 pm UTC

That's the case for pretty much any black hole these days. The CMBR is 2.76K, but the emission from a stellar mass BH is way under that (*), and the huge ones at the centers of galaxies are even less.

(*)The Hawking Temperature for a minimal stellar mass BH (3 solar masses) is 2.06e-8 K.

The mass of a BH with the temp of the CMBR is around the mass of the Moon, IIRC.
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