When will neutronization occur in the black hole formation process?

[bcrelling]

We have a large mass, and we increase it slowly- dropping in one atom at a time. Will a black hole form suddenly, or will it gradually become blacker with the addition if each atom?

I assume that a mass marginally below the threshold must at least partially have the properties of a black hole?

Thanks

 

[Matterwave]

Creation of a black hole is quite catastrophic. It is not a gradual process. Degeneracy pressure can hold up the mass until it can't, at which point the mass basically crumples and catastrophically collapses into a black hole. Before the mass collapses, it can be quite dense, like a neutron star, but it is not dense enough to capture light and so is not "black", but it will become dense enough to redshift light emerging from the surface, so it does get redder.

 

[Doug Huffman]

Your 'marginally' embraces the Sorites paradox, less one is it still marginally, less two, et cetera. I believe that your hypothetical would cause a catastrophe.

LOL quite a coincidence of both using catastrophe.

 

[PeterDonis]

We have a large mass, and we increase it slowly- dropping in one atom at a time. Will a black hole form suddenly, or will it gradually become blacker with the addition if each atom?

As Matterwave said, the process of forming a black hole is not gradual; there are abrupt transitions in the state of the matter. If we assume that no nuclear, chemical, or other reactions occur, and there are no thermal fluctuations (for example, we drop iron atoms at absolute zero onto a mass composed, at least at low masses, of other iron atoms at absolute zero), there are still at least three abrupt transitions we will observe, and a fourth that also qualifies as a transition even if it isn't necessarily abrupt:

(0) The transition from an object whose mass is too low for its own self-gravity to be significant, like a rock, to an object whose mass is large enough for self-gravity to be a significant factor in its structure, like a planet. This transition may not be abrupt; there are examples in nature of objects at various points along this spectrum (from rocks to asteroids to "dwarf planets" to planets).

(1) The transition from normal matter, made of atoms and held up against gravity by ordinary inter-atomic forces, to white dwarf matter, made up of electrons and nuclei not organized into atoms and held up against gravity by electron degeneracy pressure. This is an abrupt transition; there is no stable sequence of intermediate states in between ordinary matter and white dwarfs.

(2) The transition from white dwarf matter to neutron star matter, which is made of neutrons (electrons and protons are collapsed into neutrons during the process of neutron star formation) and held up against gravity by neutron degeneracy pressure. This is also an abrupt transition.

(3) The transition from neutron star matter to a black hole, when the mass of a neutron star exceeds the maximum possible mass at which the star can hold itself up against gravity. The basic reason there is such a maximum mass is that, as the mass increases, the neutrons get squeezed closer together and become relativistic, and relativistically degenerate matter has a lower adiabatic index (the exponent in the equation of state relating pressure and density) than non-relativistically degenerate matter. So as the density continues to go up as the mass increases, the pressure can no longer increase fast enough to keep up, and a point is reached where the star is no longer stable and collapses. This is an abrupt transition.

I assume that a mass marginally below the threshold must at least partially have the properties of a black hole?

A neutron star near the maximum possible mass (and therefore the minimum possible size) will have some gravitational redshift, but it will be fairly small by black hole standards; there is a large gap between the properties of such an object and the properties of a black hole with a mass just above the maximum possible neutron star mass.

 

[tionis]

One thing I don't understand is: Is it the imploding mass that causes a black hole to form, or is it the huge spacetime curvature formed around the imploding mass that causes the BH to form? In other words, if there were no spacetime for the mass to curve, would a BH actually ever form?

 

[PeterDonis]

if there were no spacetime for the mass to curve

Then we wouldn't be discussing General Relativity, we'd be discussing some other theory. Do you have one? If not, your question doesn't really have any point, because if we don't have a theory to constrain our speculations, we can say anything we want.

 

[bcrowell]

The OP asked about a hypothetical process, but the replies talk about gravitational collapse of a star. Gravitational collapse of a star doesn't have to be the way a black hole forms. For example, there could be primordial black holes that were formed shortly after the big bang, long before stars formed.

We have a large mass, and we increase it slowly- dropping in one atom at a time. Will a black hole form suddenly, or will it gradually become blacker with the addition if each atom?

I assume that a mass marginally below the threshold must at least partially have the properties of a black hole?

There is no threshold, no minimum mass for a black hole. General relativity allows black holes to exist with any mass whatsoever. However, the known pathways to formation of a black hole from a dying star are pathways that only work if the star is fairly massive.

Hypothetically, one could have a black hole of any tiny size, and then any matter that you trickled into it would simply increase its mass from there.

 

[tionis]

Then we wouldn't be discussing General Relativity, we'd be discussing some other theory. Do you have one? If not, your question doesn't really have any point, because if we don't have a theory to constrain our speculations, we can say anything we want.

The first part of my question is within the context of GR:

One thing I don't understand is: Is it the imploding mass that causes a black hole to form, or is it the huge spacetime curvature formed around the imploding mass that causes the BH to form?

The second part:

In other words, if there were no spacetime for the mass to curve, would a BH actually ever form?

.. is speculation on my part based on incomplete or misunderstood knowledge of the first part of the question, which is why I asked in the first place. Do you have an answer for the first part of my question?

 

[PeterDonis]

Do you have an answer for the first part of my question?

The black hole is the spacetime curvature that is formed by the imploding mass.

 

[PAllen]

To add to bcrowell's point, there is no minimum size to the Oppenheimer-Snyder collapse. In lay terms, that means if you posit matter that is incapable of pressure, then any amount of it will eventually (smoothly) collapse to a BH. General relativity does not include any theory of matter. Classical use of GR typically assumes that matter must locally behave as expected by SR (e.g. the dominant energy condition). However, this places no requirement on pressure, so the perfectly smooth (non-catastrophic) collapse of 1 gram of pressure-less mathematical dust to a BH is allowed in classical GR.

Peter's comments apply to matter as we know it, governed by equations of state based on the standard model of particle physics.

[edit: The closest real world analog I can think of the pressure-less dust is iron filings. Iron is incapable of either fission or fusion. If you add iron to an iron ball slowly enough to dissipate heat (which toward the end will be enormous ), the first catastrophe will be collapse from ordinary matter to neutron star with release of a flood of neutrinos. This would constitute some weak form of supernova (which obviously has never been observed). Then, as you kept adding iron, at somewhere above 3 solar masses, you would get catastrophic collapse to a BH. These are best guesses, as there is no process in the universe that slowly accumulates iron.]

 

[bahamagreen]

The question was about dropping in one atom at a time... and the nature of the beginning of the thing becoming a BH.

It seems to me that the OP might be wondering:

Will the transition to BH happen after the addition of one particular atom in the series of atoms added?
(assuming that each dropping of an atom is spaced apart in time sufficient for its effect to be fully made manifest.)

Or better yet, let the things being dropped in be protons or hydrogen atoms... so the increment is small.

I think the OP is wondering if the pre-BH object is subject to HUP fluctuations and wondering how the addition of a single increment compares to the magnitude of these fluctuations... whether a fluctuation might be sufficient to put the object over the critical mass to become a BH when the BH is in the state where the addition of one more increment of dropped matter would otherwise do so...

Since the size at which an incrementally built BH is known, what orders of incremental mass compare to the fluctuation magnitude?

Could an object that is just short of becoming a BH make the transition through a HUP fluctuation?

Also, as to micro BHs, I don't know if it is thought that there are micro-versions of the above - micro objects just short of the density to become a BH. If so, would their HUP fluctuations with respect to their size be bigger than the large BHs? If so, it seems they would very subject to making the transition, and so quite rare in their pre-BH state.

 

[PAllen]

The question was about dropping in one atom at a time... and the nature of the beginning of the thing becoming a BH.

It seems to me that the OP might be wondering:

Will the transition to BH happen after the addition of one particular atom in the series of atoms added?
(assuming that each dropping of an atom is spaced apart in time sufficient for its effect to be fully made manifest.)

Or better yet, let the things being dropped in be protons or hydrogen atoms... so the increment is small.

I think the OP is wondering if the pre-BH object is subject to HUP fluctuations and wondering how the addition of a single increment compares to the magnitude of these fluctuations... whether a fluctuation might be sufficient to put the object over the critical mass to become a BH when the BH is in the state where the addition of one more increment of dropped matter would otherwise do so...

Since the size at which an incrementally built BH is known, what orders of incremental mass compare to the fluctuation magnitude?

Could an object that is just short of becoming a BH make the transition through a HUP fluctuation?

Also, as to micro BHs, I don't know if it is thought that there are micro-versions of the above - micro objects just short of the density to become a BH. If so, would their HUP fluctuations with respect to their size be bigger than the large BHs? If so, it seems they would very subject to making the transition, and so quite rare in their pre-BH state.

I would say much of this is unknown. Even the minimum mass of BH you would get by incrementally growing a neutron star is unknown (3 to 5 solar masses is an estimate, but that is an error bar of a whole sun). Qualititatively, all agree (including my hypothetical adding iron to a neutron star), that the final collapse to a BH would be sudden.

 

[bahamagreen]

By "sudden", does that mean at the speed of sound through the material of the object, or more like the speed of light around the surface... or does the geometry of the BH make the usual space and time measures confounded by the variations of an observer's position and motion? Maybe the transition must always appear "instantaneous"?

 

[PAllen]

By "sudden", does that mean at the speed of sound through the material of the object, or more like the speed of light around the surface... or does the geometry of the BH make the usual space and time measures confounded by the variations of an observer's position and motion? Maybe the transition must always appear "instantaneous"?

I don't know of any real analysis of a model of incremental BH formation from a neutron star (it would be a lot of work, and since it isn't known to occur, maybe no one has been motivated). Or maybe I've missed it in the literature. So I really can't answer the 'how fast' question with any confidence. It may not make much difference because the speed of sound would be a significant fraction of the speed of light for a neutron star (e.g. > .3 c).

On the other hand, viewed from afar, the process would be slower, but not really that slow - the object would become blacker than CMB filled empty space relatively fast, but much slower than as experienced by an particle of the neutron star.

 

[tionis]

so, are there any spacetime-free solutions for mass/energy?
And if there are, what woud it say about an imploding mass?
Would we get to observe a mass collapse without becoming a black hole?

 

[martinbn]

Just one more comment. If the characteristic feature of a black hole is the event horizon, then you can get a black hole without anything unusual happening, at least for some time. Take a spherical configuration of stars that fall towards the centre. If you choose the masses well, an event horizon will form at the centre and grow outward i.e. you have a black hole. If you are somewhere there you will not see anything unusual. In fact a horizon may be growing and passing through you room right now.

 

[PAllen]

so, are there any spacetime-free solutions for mass/energy?
And if there are, what woud it say about an imploding mass?
Would we get to observe a mass collapse without becoming a black hole?

What do you mean by spacetime free solutions? Since SR and GR are built on spacetime, that seems to be asking what theory would be true if the currently best verified theories were wrong? There is no meaningful way to answer that. It is not like picking matter without electric charge, which is possible. There is just no theory left if you remove spacetime.

 

[PAllen]

Just one more comment. If the characteristic feature of a black hole is the event horizon, then you can get a black hole without anything unusual happening, at least for some time. Take a spherical configuration of stars that fall towards the centre. If you choose the masses well, an event horizon will form at the centre and grow outward i.e. you have a black hole. If you are somewhere there you will not see anything unusual. In fact a horizon may be growing and passing through you room right now.

Yes, I've used this example many times (stars as dust; implements hypothetical smooth Oppenheimer-Snyder type collapse), as a way of talking about what you would see throughout a collapsing volume, and also to discuss how radical a change to classical GR is implied by 'active horizon' hypotheses (e.g. firewalls).

[edit: similarly, if allowed to pose implausible initial conditions, if you had a sufficiently large sparse dust cloud, with a uniform density of e.g. air, but of a size just short its being in its own SC radius, then it would smoothly collapse further, with growth of event horizon from center, without any happening to the dust until well after it was already inside its event horizon.]

 

[PeterDonis]

If you add iron to an iron ball slowly enough to dissipate heat (which toward the end will be enormous ), the first catastrophe will be collapse from ordinary matter to neutron star with release of a flood of neutrinos.

Wouldn't you pass through a white dwarf stage first? At that stage, the iron nuclei would still be separate entities, but the electrons would be degenerate. At the neutron star stage (the next stage), there would no longer be iron nuclei or electrons, just a big blob of neutrons. Ordinary iron ball -> white dwarf -> neutron star is the sequence according to the Harrison-Wakano-Wheeler equation of state (which is basically what we're talking about here--the discussion in Thorne's Black Holes and Time Warps is what I've been basing my comments on).

 

[PAllen]

Wouldn't you pass through a white dwarf stage first? At that stage, the iron nuclei would still be separate entities, but the electrons would be degenerate. At the neutron star stage (the next stage), there would no longer be iron nuclei or electrons, just a big blob of neutrons. Ordinary iron ball -> white dwarf -> neutron star is the sequence according to the Harrison-Wakano-Wheeler equation of state (which is basically what we're talking about here--the discussion in Thorne's Black Holes and Time Warps is what I've been basing my comments on).

More like a black dwarf, with different composition than any that form naturally (none are iron). But, yes, I forgot the stage of electron degeneracy. The only stage that would release much energy is the neutron star formation, when an enormous surge of neutrinos are released.

 

[tionis]

What do you mean by spacetime free solutions? Since SR and GR are built on spacetime, that seems to be asking what theory would be true if the currently best verified theories were wrong? There is no meaningful way to answer that. It is not like picking matter without electric charge, which is possible. There is just no theory left if you remove spacetime.

Since spacetime couples to mass, I thought you could get rid of the spacetime part, just like you can have a matter-free spacetime and still have black holes form. I always thought black holes formed because some intrinsic property of matter, but now it's dawning on me that mass got nothing to do with it, that spacetime is actually responsible for it. For all we know, the collapsing mass is still there shrouded behind the horizon of spacetime. So the vacuum solutions that go to infinity and all that are for the collapsing spacetime. I think I finally got it. The star is still there, it's just that the spacetime around it is behaving as though the mass has gone to

 

[mfb]

Mass is always somewhere in spacetime. Spacetime is the framework for everything.

The star is still there

No, whatever remains has nothing to do with a star.

it is behaving as though the mass has gone to

No, the mass stays the same.

 

[tionis]

mfb -- yes, the hole's mass is the same as that of the collapsing object, but why does spacetime behaves as if the mass is infinite?

 

[PAllen]

mfb -- yes, the hole's mass is the same as that of the collapsing object, but why does spacetime behaves as if the mass is infinite?

It doesn't. It behaves as if the mass is M, the mass while the star was 'normal'. Inside the horizon, you have a curvature singularity, but this does not represent mass.

 

[PeterDonis]

More like a black dwarf, with different composition than any that form naturally (none are iron).

Yes, good point. This also affects the maximum mass limit; for a body composed of iron nuclei, it's around 1.2 solar masses, but if it's mostly hydrogen and helium nuclei, it's 1.44 solar masses (this is the limit that Chadrasekhar derived in the 1930's).

 

[stevebd1]

mfb -- yes, the hole's mass is the same as that of the collapsing object, but why does spacetime behaves as if the mass is infinite?

On the outside of the event horizon (r>2M), spacetime is time-like according to Schwarzschild metric, this means that time (t) is temporal. Inside the event horizon (r>2M) spacetime is space-like which means space (r) is temporal, this means there is no stable radius, matter couldn't support itself at a constant r, regardless of pressure, within space-like spacetime and all matter collapses towards r=0 which is where the notion of the singularity and 'infinite' density comes from.

We have a large mass, and we increase it slowly- dropping in one atom at a time. Will a black hole form suddenly, or will it gradually become blacker with the addition if each atom?

I assume that a mass marginally below the threshold must at least partially have the properties of a black hole?

Thanks

To add to the other posts, according to the Schwarzschild interior metric, the point of no return is 9/4M (or 2.25M), this is the point when the time dilation at the centre of mass reaches zero and as the star collapsed further, the event horizon would begin to move outwards towards the surface as in this diagram (the pink line represents the event horizon, the blue lines the collapsing star).

 

[PeterDonis]

On the outside of the event horizon (r>2M), spacetime is time-like according to Schwarzschild metric, this means that time (t) is temporal. Inside the event horizon (r>2M) spacetime is space-like which means space (r) is temporal

You are not stating this correctly. Spacetime is not "timelike" or "spacelike"; there are always timelike, spacelike, and null vectors everywhere in spacetime. Coordinates like $t$ and $r$ can be timelike or spacelike (or null), but whether or not they are depends on the coordinate chart you choose. There are charts covering Schwarzschild spacetime in which the "time" $t$ is timelike everywhere, and there are charts in which $r$ is spacelike everywhere.

The correct way to state what you are trying to say here is this: Schwarzschild spacetime contains a set of curves along which the spacetime geometry is constant. In standard Schwarzschild coordinates, these are curves of constant $r$, $\theta$, $\phi$. A more technical way of describing these curves is that they are integral curves of a Killing vector field; in standard Schwarzschild coordinates, this vector field is just the coordinate basis vector field $\partial / \partial t$ (technically this doesn't work at the horizon in these coordinates, because they are singular there, but there are ways to finesse that).

The key fact about this family of curves is this: outside the horizon, they are timelike; inside the horizon, they are spacelike; and on the horizon itself, they are null. This is an invariant fact about the spacetime geometry, independent of coordinates. It's unfortunate that many sources aren't careful enough about how they state this.

this means there is no stable radius, matter couldn't support itself at a constant r, regardless of pressure, within space-like spacetime and all matter collapses towards r=0 which is where the notion of the singularity and 'infinite' density comes from.

Just to clarify based on my comments above, since a curve of constant $r$ is spacelike inside the horizon, and since no particle of matter can travel along a spacelike worldline, it is impossible for matter to be static at constant $r$ inside the horizon. This in itself, however, does not tell you that the matter must be collapsing towards $r = 0$; to show that, you need to show that all timelike curves inside the horizon have decreasing $r$ (which is easy to show).

 

[Ken G]

I actually don't think the formation of a neutron star or a black hole involves a catastrophe in the force balance, though it is popular to explain it that way. To me, a catastrophe in the force balance would require that there be no hydrostatic solution, say via a bifurcation, even when we treat the system as not undergoing any thermal changes, i.e., not losing heat to its surroundings and not undergoing internal changes in composition. Creation of a neutron star is often characterized as what you get when you have a white dwarf at zero temperature whose mass exceeds the Chandrasekhar mass-- there is indeed no hydrostatic solution for such an object, so we might characterize its collapse as a catastrophe. However, this is an artifact of the questionable assumptions, because there is no reason to assume you have an object that maintains zero temperature in a real core-collapse scenario. Instead, you have a mass, which can exceed the Chandrasekhar mass, and you have an internal energy density, which because of the history of the object can certainly yield a nonzero temperature. That object can have a hydrostatic solution if the energy density is high enough, and indeed it seems to me it normally will be, because objects like this are generally created by dropping in mass that either already has a high energy content, or falls in from a far enough distance to acquire a high energy content. So I think it would be pretty unexpected to ever arrive at a state that does not have a hydrostatic solution, such that it could be said to collapse as a true catastrophe.

If that is indeed true (and an alternative possibility is that the general relativity of the situation leads to unstable orbits such that hydrostatic equilibrium is indeed impossible), then the core collapse could involve an object that has a hydrostatic solution that is possible, for its given energy content, if that object could just be left alone long enough to find that solution. But the object is not left alone, it is undergoing change that comes either from loss of heat via neutrino losses (perhaps the URCA process), or from photodisintegration of its iron that is changing its internal composition. When the gas is highly relativistic, even if it does have a hydrostatic solution if none of these changes were occurring, the rate at which the hydrostatic structure would need to reconfigure to adapt to these changes is strongly leveraged by the fact that relativistic gas is very soft-- by which I mean simply that large reconfigurations are necessary given even small internal changes in composition or heat content. At some point, these reconfigurations are so leveraged that they take much longer than just the sound crossing time, and meanwhile the thermal timescale is getting shorter and shorter because it could in principle get as short as the light crossing time. When the thermal timescale gets shorter than the force-balance timescale, you get free fall, but not because there exists no instantaneous hydrostatic solution, it is because the hydrostatic solution takes too long to achieve. So the catastrophe is a thermal instability, not a catastrophe in the force balance. The force balance contributes to the problem only by how soft it is-- small thermal changes lead to large reconfigurations which drive further thermal changes, until the thermal physics runs away. Take away the thermal changes and the force balance might have no problem finding a continuously changing configuration as you add mass particle by particle, you would just drive a steady and continuous transition to a neutron star that could be made as slow as you like by adding particles as slowly as you like. The thermal runaway precludes that.

So what I'm saying is, I suspect that if you added mass gradually to a white dwarf, then before you even quite get to the Chandra mass you will start to see the thermal adjustment timescale becoming comparable to the force-balance establishment timescale, and when the two are about equal, you will lose control of the process-- it will take on a life of its own that no longer cares how fast you are adding particles, and will lead to a collapse, not because there is no force balance (as is true above the Chandra mass), but because there is a thermal instability that operates faster than the forces can equilibrate. That's not a formal catastrophe in the sense of a bifurcation in which the force balance is lost, but it is a catastrophe in the sense that it is a thermal runaway.

Of course, in real stars, we don't know how fast the mass is being added-- it might be added abruptly, like in a white-dwarf merger. But in that case, as I said above, there's still no reason to assume the temperature will be zero, so there could still be a force balance that is possible at the higher mass. You'll still have to wait for the heat to get out, so you will still be controlled by the thermal timescale, and the catastrophe will still happen when the thermal timescale gets short enough to rival the force-balance timescale. So I think that will be true whether you add mass slowly or abruptly-- the core collapse is a thermal instability that plays out in situations where there generally would be a hydrostatic solution, but it simply does not have time to set up given the rapidly changing thermal environment. The criterion for the catastrophe is equating the thermal adjustment rate to the relativistically softened force-balance rate, which happens before you get to the Chandra mass and does not matter how fast you add the particles once you get to that critical limit.

 

[PeterDonis]

That object can have a hydrostatic solution if the energy density is high enough

Of course this is true--that's what stars are, and a star can be in hydrostatic equilibrium with a mass much larger than the Chandrasekhar limit (or its analogue for neutron stars). But a star has an internal energy generation mechanism that can balance its rate of heat loss to space so that its temperature remains constant and its structure remains static.

A white dwarf with mass above the Chandrasekhar limit has no such mechanism. It can gradually contract and generate heat from the contraction, but this is not a hydrostatic equilibrium. In principle it could still take a long time to contract to the point where a serious instability set in; but since it can't maintain a static structure while losing energy to space (which it has to since space is effectively at zero temperature), the way a star can, it will inevitably reach a point where it is unstable at its current temperature (which may be close to zero temperature) and collapses.

Of course a white dwarf above the limit could collapse to a neutron star instead of a black hole; but the same logic applies to neutron stars, since there is also a maximum mass limit for those. Any object above that maximum mass limit will eventually collapse to a black hole, though it might take time if the object stars out at high enough temperature for kinetic pressure to be significant in determining its structure (and if its mass is not so large that even the extra kinetic pressure doesn't help).

objects like this are generally created by dropping in mass that either already has a high energy content, or falls in from a far enough distance to acquire a high energy content

I don't think this is really true; supernova explosions are currently thought to be a major source (if not the main source) of stellar mass black holes, and those don't require any external object to fall into the star that goes supernova.

an alternative possibility is that the general relativity of the situation leads to unstable orbits such that hydrostatic equilibrium is indeed impossible

GR does impose such a limit: it says that a spherically symmetric object with a radius less than 9/8 of the Schwarzschild radius for its mass cannot be in hydrostatic equilibrium. AFAIK no white dwarf comes anywhere near this limit, but some neutron stars come fairly close to it. Note that this result does not depend on the temperature of the object.

the core collapse could involve an object that has a hydrostatic solution that is possible, for its given energy content, if that object could just be left alone long enough to find that solution

I agree that this is possible, and that something like a supernova explosion could cause an object to collapse into a black hole even though, strictly speaking, the mass left behind would have enabled a hydrostatic equilibrium as a neutron star (or even possibly a white dwarf, though I think that would be unlikely since a star small enough to reach that mass after the explosion would not go supernova in the first place), if the process had happened slowly.

How common this is is a different question, which we can't really assess in too much detail at this point, since we don't know the exact maximum mass limit for neutron stars, we only have a range (IIRC about 1.5 to 3 solar masses), and we don't have very good mass values for many black hole candidates, so we can't assess what percentage of actual black holes have masses that are below the neutron star maximum mass limit.

 

[PAllen]

Those who do numerical simulations of these things claim that collapse to a neutron star has matter going an appreciable fraction of the speed of light, and that this occurs quickly. For a BH, matter crosses a near horizon 'observer' at near c. In the case of a neutron star, it is debatable 'how catastrophic' it is, because for a neutron star, speed of sound is over .3c, so the collapse is not 'dust like'. However for BH formation, different bits of matter are essentially independent of each other because mutual influence cannot propagate fast enough.