Black holes might not be singularities

[Lord Torath]

From SyFy Wire: Fuzzy Black Holes Could Solve Hawking's Information Paradox (This article is locked behind a member account, but membership is free - I used a dummy email to set mine up).
Here's a link to the research paper from the Turkish Journal of Physics that is cited by the SyFy article, and is not behind a pay/membership wall.

The short version is that the inside of black holes might not be a singularity surrounded by vacuum out to the event horizon, but could be filled with stretched strings and branes.

This avoids the paradox of the information being preserved in the Hawking Radiation somehow needing to jump the vacuum between the singularity and the event horizon.

[Yora]

I've never been completely convinced about the singularity. I've always seen it as a hypothesis at best, or even only a postulate or assumption. Could be correct, but with no observational evidence either for or against it, the idea that stuff in the center takes a different form doesn't seem that ground breaking or a major overhaul of physics to me.
The event horizon is basically as far as my understanding of black holes goes. The physics of what happens with matter and energy as it approaches the center always was beyond me.

[Chronos]

If our models are correct for describing the exterior geometry, and if they can be extrapolated all the way in, then there's a singularity. Most physicists are pretty confident about the former, but we're considerably less so about the latter. I imagine that nearly everyone suspects (and has suspected for a long time) that, once we have a working theory of quantum gravity (we don't yet), it'll probably turn out that the center of a black hole isn't quite a singularity. But since we don't yet have that working theory of quantum gravity, nobody knows for sure.

[Anymage]

Typing "fuzzball" into youtube turns up a couple of interesting videos on the theme.

It's an interesting alternate hypothesis from string theorists, but like most things string theory doesn't get farther than "potentially interesting" until strings themselves can be meaningfully tested.

[Fat Rooster]

I've always found it amusing that people ever believed that a 'singularity' was a real thing. The whole point of a singularity is that it is basically the mathematics telling you that you have made a mistake. Sure, that isn't all that helpful in telling you what the mistake is, but singular points occur in many models, and all it means is that that particular model is not appropriate for that situation, and we already know the GR isn't good enough even at the surface. When faced with the choice between abandoning the powerful statistical laws that pervade the rest of the universe, or admitting we just don't have the theory to understand it, I find it bizarre that anyone would insist that our model is perfect and a point of infinite density must exist.

I've had a thought in my head about that that I've been meaning to run the numbers on, but getting the specific know how would take a lot of time: Hawking radiation appears in a frame of reference static with regards to the black hole, but the only interesting thing about that frame of reference is that it is stable over long timescales. The event horizon is there, but only really has any meaning in this frame of reference. If we have a frame of reference moving towards the centre of a black hole within it's event horizon; then a line exists where events on one side of the line cannot affect things on the other. Everything is still moving inward, but photons from some points will be able to slow their collapse to the point of being able to affect some other point, or they will not. A new event horizon should exist and be distinct in every frame of reference.

Where that would get important is that all of these horizons should be emitting Hawking radiation. Granted, it isn't outwardly important, because it will all collapse back in again, but while it is doing that it is black hole mass that is not at the centre. As you get nearer the centre the effect might get more extreme, to the point that I wonder if collapse could be halted by internal Hawking radiation, avoiding the singularity without resorting to anything exotic.

As I said, I lack the skills to actually run the numbers, so if anyone here has them they are welcome to. If this has been done before and a reference is available that would be cool too!

[BaronOfHell]

From what I understand, it takes an infinite amount of time to pass the event horizon from the perspective of any observer that isn't falling through the event horizon, which is why I wonder why should there even be anything beyond the event horizon?
From what I understand, the physics describing the gravity of a black hole doesn't care if all the mass is evenly distributed within the black hole or all the mass is on the horizon itself.

[halfeye]

From what I understand, it takes an infinite amount of time to pass the event horizon from the perspective of any observer that isn't falling through the event horizon, which is why I wonder why should there even be anything beyond the event horizon?
From what I understand, the physics describing the gravity of a black hole doesn't care if all the mass is evenly distributed within the black hole or all the mass is on the horizon itself.

That's all very well for stellar mass and heavier black holes, but a black hole with the mass of the Moon would have a 0.1 mm event horizon radius. That's not happening without something very, very dense in there.

https://en.wikipedia.org/wiki/Schwarzschild_radius

Moon 7.35×1022 kg 1.09×10−4 m

[Mastikator]

From what I understand, it takes an infinite amount of time to pass the event horizon from the perspective of any observer that isn't falling through the event horizon, which is why I wonder why should there even be anything beyond the event horizon?
From what I understand, the physics describing the gravity of a black hole doesn't care if all the mass is evenly distributed within the black hole or all the mass is on the horizon itself.

The closer you get to the event horizon the longer it takes light emitted/reflected from your body to make it to a far away observer. As you approach the horizon the time it takes reaches infinity. As such you can never see someone cross the event horizon. That is just special relativity.
Another effect of relativity is that as you approach the horizon you also approach the speed of light, the effect of that is that information traveling from one side of your body to the other side will take much more time, so your clock will appear to move slower. And as you cross the even horizon you are moving at the speed of light, time stops for you. From your point of view not only do you fall in, you fall in super duper fast. (assuming you don't die from spaghettification, which is my one of my favorite scientific terms)

These "twin paradoxes" are generally resolved when the twin returns, but in the case of a black hole the faller inner never returns. (hypothetically the information will evaporate from the black hole, but that information will be maximally jumbled and take a virtually infinite amount of time)

[BaronOfHell]

@halfeye
I don't understand how scale changes anything. My thought was that perhaps the particles are all close to the event horizon itself, creating the same level of density, after all compared to the size of a particle, 0.1 mm is huge.

@mastikator
From what I understand there never is a paradox? An example I recall from spec. relativity is that one twin (twin 1) leaves earth on a rocket ship travelling some high percentage of the speed of light while the other (twin 2) stays on Earth.

Because, say, a light beam is emitted from the rocket towards some destination, then from the point of view of twin 1 the light travels away from the rocket at the speed of light, while from the point of view of twin 2 the light beam only travels a percentage of the speed of light faster than the rocket. Yet both twins can agree on when the light beam hits the destination.
The implication is that for twin 1, space and time distorts so e.g. distances becomes shorter, and twin 1 simply experiences a shorter trip. When twin 1 arrives at his destination he will only have aged the same amount of time he experienced the shortened ride to take, while twin 2 saw twin 1 making a much longer travel, hence more time has passed.
Twin 2 is now older than twin 1, as he has experienced more time. Is this example relevant to what you meant by the twin paradox?

In regards to the black hole, I imagine the observer who isn't falling into the black hole sees some guy fall towards the black hole horizon, but he slows down as he closes in on the event horizon, never stopping completely, but moving ever slower, yet never reaching the horizon, like the series of 1/1 + 1/2 + 1/4 + 1/8, getting ever closer to 2, yet never reaching 2. Let's imagine after some years the observer gets tired of watching the guy getting closer and closer to the event horizon and goes home.
From the perspective of the person falling into the black hole, if this person can look back on the observer, he'd see the observer age a few years rapidly, and then disappear quickly as he stops observing to go home.
Like with the example above, for the observer not falling into the black hole a lot of time will pass, while the person falling into the black hole, almost no time will have passed.
Since the time passed from the perspective of any observer not falling into the black hole, like for instance the entire universe, will get closer and closer to infinity, the closer the person falling into the black hole gets to the event horizon, I imagine if the process of falling into a black hole is survivable (which I understand it is for sufficient large black holes, though I suppose if the shrunk while falling into it, it wouldn't be, as one would end up as a bunch of particles), the black hole itself will evaporate due to hawking's radiation the closer one gets to the event horizon from the perspective of the guy falling into a black hole, and ultimately while from the perspective of a traveler falling into a black hole, crossing the event horizon takes no more time than crossing any other distance, the event horizon itself will shrink as you get closer, because unimaginable amounts of time has passed in the universe and the black hole has shrunk due to hawking's radiation.
Hence while the process of falling into a black hole takes next to no time, I imagine the person falling into the black hole will never cross the event horizon and in stead will see the black hole evaporate as the event horizon gets smaller and smaller until the black hole is no more. At this point in time a huge amount of time has passed in the universe, but perhaps only seconds for the traveler falling into the black hole.

This may be entirely incorrect, but from what I understand there are no genuine paradoxes in relativity, hence if the guy falling into the black hole looks like he takes an infinite amount of time to reach the event horizon, then he does so, but black holes aren't infinite to my understanding due to hawking's radiation. So he closes into the black hole until the black hole itself evaporates, and it may takes eons of time from the perspective of an observer not falling into the black hole, and it may be a matter of seconds for the person falling into the black hole.

[Rakaydos]

Random thought- the net gravitational pull on the inside of a uniform sphere is zero. Therefore the center of a black hole (assuming there is no singularity) is flat space, which means that light (hawking radiation) can travel across the interior of a black hole, only to be absorbed by the interior event horizon of the opposite side.

[Lord Torath]

Random thought- the net gravitational pull on the inside of a uniform sphere is zero. Therefore the center of a black hole (assuming there is no singularity) is flat space, which means that light (hawking radiation) can travel across the interior of a black hole, only to be absorbed by the interior event horizon of the opposite side.

I'm pretty sure that's just for shells, not solid spheres. For a solid sphere, the net gravity you feel is for that of the sphere under your feet. If you dig halfway down to the center, you feel the pull of a sphere of half the diameter of the whole.

[Rakaydos]

I'm pretty sure that's just for shells, not solid spheres. For a solid sphere, the net gravity you feel is for that of the sphere under your feet. If you dig halfway down to the center, you feel the pull of a sphere of half the diameter of the whole.

That's because a shell is self supporting. If you presume a non-pressure mechanisim for a black hole to not crush itself into a singularity, then suddently, there IS no pressure on the center of a black hole, where there is no spheres below it.

I checked my intuition at a far more science literate forum I go to, and was linked to this video:

https://youtu.be/351JCOvKcYw?t=680

680 seconds in the above video seems to observe a similar effect to what I predict, though of course quantum bull means that weird things are also happening. It asserts that the interior of the fuzzball black hole is not in spacetime at all.

[Radar]

Random thought- the net gravitational pull on the inside of a uniform sphere is zero. Therefore the center of a black hole (assuming there is no singularity) is flat space, which means that light (hawking radiation) can travel across the interior of a black hole, only to be absorbed by the interior event horizon of the opposite side.

If general relativity is fully correct, it was proven that singularity has to appear. This was done by Penrose in 1965. Pretty much everyone expects the general relativity to fail within a black hole, but there is no way of checking, how things actually work inside an event horizon and we have no other proofs of general relativity failings on which we could base a better theory.

[Chronos]

Quoth Fat Rooster:

Sure, that isn't all that helpful in telling you what the mistake is, but singular points occur in many models, and all it means is that that particular model is not appropriate for that situation, and we already know the GR isn't good enough even at the surface.

We have no reason to believe that there is any problem with general relativity at the event horizon of a reasonable-sized (i.e., significantly larger than the Planck mass) black hole.

At the center, sure, like I said, everyone knows that we need a better theory there; we just don't know what that better theory is. But the event horizon is no problem.

[Lord Torath]

That's because a shell is self supporting. If you presume a non-pressure mechanisim for a black hole to not crush itself into a singularity, then suddently, there IS no pressure on the center of a black hole, where there is no spheres below it.

Gah! That's me misreading your statement. "At the center" is not the same as "a random point inside".

Sorry for the confusion.

Edit: I watched the video. That was really interesting. I know that black hole event horizons do not follow the standard density laws (a black hole with double the diameter of a smaller one would be expected to have eight times the mass, but it doesn't), and I thought that having a black 'shell' might explain that. If the mass was distributed on a shell, then you'd expect a black hole with double the diameter to have four times the mass. Some quick checking revealed I'm totally wrong, though. The relationship between mass and diameter is linear, not square or cubic. Double the mass, double the diameter.

[BaronOfHell]

I may be way off here, but a quick thought, from what I understand, the mass rotating a black hole lies on a disk similar to how a moon or planets would orbit another celestial body, if the majority of the mass is in this disk, is the mass of a black hole then actually spherically distributed?

Also I suppose if the disk gets compressed enough, wouldn't it become closer to being a 1 dimensional object?

[halfeye]

I may be way off here, but a quick thought, from what I understand, the mass rotating a black hole lies on a disk similar to how a moon or planets would orbit another celestial body, if the majority of the mass is in this disk, is the mass of a black hole then actually spherically distributed?

Also I suppose if the disk gets compressed enough, wouldn't it become closer to being a 1 dimensional object?

They worked out that the singularity of a spinning black hole is a ring with an infinitely narrow cross section.

[johannessmid]

@mastikator
From what I understand there never is a paradox? An example I recall from spec. relativity is that one twin (twin 1) leaves earth on a rocket ship travelling some high percentage of the speed of light while the other (twin 2) stays on Earth.

Because, say, a light beam is emitted from the rocket towards some destination, then from the point of view of twin 1 the light travels away from the rocket at the speed of light, while from the point of view of twin 2 the light beam only travels a percentage of the speed of light faster than the rocket. Yet both twins can agree on when the light beam hits the destination.
The implication is that for twin 1, space and time distorts so e.g. distances becomes shorter, and twin 1 simply experiences a shorter trip. When twin 1 arrives at his destination he will only have aged the same amount of time he experienced the shortened ride to take, while twin 2 saw twin 1 making a much longer travel, hence more time has passed.
Twin 2 is now older than twin 1, as he has experienced more time. Is this example relevant to what you meant by the twin paradox?

There is a "paradox" - if you think simply in terms of relativity, you might analyze this by saying "I can consider a reference frame where the twin on earth is not moving and the rocket is moving away; or a frame where the twin on the rocket is not moving and the earth is moving away. These should be equivalent, so each twin experiences the same amount of time by the time they meet up again." But this is not what happens, so therefore "paradox."

In regards to the black hole, I imagine the observer who isn't falling into the black hole sees some guy fall towards the black hole horizon, but he slows down as he closes in on the event horizon, never stopping completely, but moving ever slower, yet never reaching the horizon, like the series of 1/1 + 1/2 + 1/4 + 1/8, getting ever closer to 2, yet never reaching 2. Let's imagine after some years the observer gets tired of watching the guy getting closer and closer to the event horizon and goes home.
From the perspective of the person falling into the black hole, if this person can look back on the observer, he'd see the observer age a few years rapidly, and then disappear quickly as he stops observing to go home.
Like with the example above, for the observer not falling into the black hole a lot of time will pass, while the person falling into the black hole, almost no time will have passed.
Since the time passed from the perspective of any observer not falling into the black hole, like for instance the entire universe, will get closer and closer to infinity, the closer the person falling into the black hole gets to the event horizon, I imagine if the process of falling into a black hole is survivable (which I understand it is for sufficient large black holes, though I suppose if the shrunk while falling into it, it wouldn't be, as one would end up as a bunch of particles), the black hole itself will evaporate due to hawking's radiation the closer one gets to the event horizon from the perspective of the guy falling into a black hole, and ultimately while from the perspective of a traveler falling into a black hole, crossing the event horizon takes no more time than crossing any other distance, the event horizon itself will shrink as you get closer, because unimaginable amounts of time has passed in the universe and the black hole has shrunk due to hawking's radiation.
Hence while the process of falling into a black hole takes next to no time, I imagine the person falling into the black hole will never cross the event horizon and in stead will see the black hole evaporate as the event horizon gets smaller and smaller until the black hole is no more. At this point in time a huge amount of time has passed in the universe, but perhaps only seconds for the traveler falling into the black hole.

This may be entirely incorrect, but from what I understand there are no genuine paradoxes in relativity, hence if the guy falling into the black hole looks like he takes an infinite amount of time to reach the event horizon, then he does so, but black holes aren't infinite to my understanding due to hawking's radiation. So he closes into the black hole until the black hole itself evaporates, and it may takes eons of time from the perspective of an observer not falling into the black hole, and it may be a matter of seconds for the person falling into the black hole.

If you think a bit, you'll know this cannot be true. Mass does fall into blackholes, we see the astrophysical effects. We've observed blackholes merging through gravitational waves. If what you are describing was true no mass could ever enter a blackhole, no blackholes could ever merge. I think you're focusing too much on "relativity" i.e. "the same thing is happening to both reference frames" but that is Special Relativity, and only strictly true for inertial reference frames in flat space.

[Chronos]

Quoth BaronOfHell:

I may be way off here, but a quick thought, from what I understand, the mass rotating a black hole lies on a disk similar to how a moon or planets would orbit another celestial body, if the majority of the mass is in this disk, is the mass of a black hole then actually spherically distributed?

There is often a bunch of ordinary matter in the vicinity of a black hole, and that ordinary matter usually forms a disk. But that's not the black hole itself, just stuff that happens to be near it. A black hole itself can also be rotating, and if so it's not spherically symmetric, but it's still pretty close to a sphere.

Quoth johannessmid

There is a "paradox" - if you think simply in terms of relativity, you might analyze this by saying "I can consider a reference frame where the twin on earth is not moving and the rocket is moving away; or a frame where the twin on the rocket is not moving and the earth is moving away. These should be equivalent, so each twin experiences the same amount of time by the time they meet up again." But this is not what happens, so therefore "paradox."

No, actually, you can use either of those reference frames, and you get the exact same answer. But the key is that there aren't two relevant reference frames: There are three. There's the reference frame where the Earth is at rest, the frame where the twin on the rocket flying away from Earth is at rest, and the frame where the twin on the rocket returning to Earth is at rest. You only get a paradox if you assume those last two reference frames are the same. But they're not.

[Bohandas]

No, actually, you can use either of those reference frames, and you get the exact same answer. But the key is that there aren't two relevant reference frames: There are three. There's the reference frame where the Earth is at rest, the frame where the twin on the rocket flying away from Earth is at rest, and the frame where the twin on the rocket returning to Earth is at rest. You only get a paradox if you assume those last two reference frames are the same. But they're not.

What happens on the case of a universe that's elliptical or that otherwise loops around over large scales?

In regards to the black hole, I imagine the observer who isn't falling into the black hole sees some guy fall towards the black hole horizon, but he slows down as he closes in on the event horizon, never stopping completely, but moving ever slower, yet never reaching the horizon, like the series of 1/1 + 1/2 + 1/4 + 1/8, getting ever closer to 2, yet never reaching 2. Let's imagine after some years the observer gets tired of watching the guy getting closer and closer to the event horizon and goes home.
From the perspective of the person falling into the black hole, if this person can look back on the observer, he'd see the observer age a few years rapidly, and then disappear quickly as he stops observing to go home.
Like with the example above, for the observer not falling into the black hole a lot of time will pass, while the person falling into the black hole, almost no time will have passed.
Since the time passed from the perspective of any observer not falling into the black hole, like for instance the entire universe, will get closer and closer to infinity, the closer the person falling into the black hole gets to the event horizon, I imagine if the process of falling into a black hole is survivable (which I understand it is for sufficient large black holes, though I suppose if the shrunk while falling into it, it wouldn't be, as one would end up as a bunch of particles), the black hole itself will evaporate due to hawking's radiation the closer one gets to the event horizon from the perspective of the guy falling into a black hole, and ultimately while from the perspective of a traveler falling into a black hole, crossing the event horizon takes no more time than crossing any other distance, the event horizon itself will shrink as you get closer, because unimaginable amounts of time has passed in the universe and the black hole has shrunk due to hawking's radiation.
Hence while the process of falling into a black hole takes next to no time, I imagine the person falling into the black hole will never cross the event horizon and in stead will see the black hole evaporate as the event horizon gets smaller and smaller until the black hole is no more. At this point in time a huge amount of time has passed in the universe, but perhaps only seconds for the traveler falling into the black hole.

I think it may be of relevance that infalling matter causes the event horizon to expand, and so as long as there was a steady stream of infalling matter one theoretically would eventually cross the event horizon by this mechanism

[chaincomplex]

I was lurking for D&D stuff as I usually do but after running into this discussion I felt compelled to stop lurking, create an account, and say something, since (not necessarily gravitational) singularities are basically my field. Strictly speaking I work in stochastics and renormalization in math and physics, but I feel comfortable enough to comment about this as a doctoral student.

Chronos is correct on all counts and they're obviously a physicist. From an outsider's perspective it doesn't particularly matter if there is or isn't a physical singularity within the black hole because for everything observable from without is just built into the geometry that has to describe all of observable spacetime. It's not that crazy to imagine that a different model has to describe the singularity itself. Obviously as suggested this additional structure could pop out of a quantized gravity, which we haven't done yet.

So the deal with singularities in physics in general is that it's an artifact of models that appears when then there's not enough geometric data to describe something. Usually there is just enough to suggest that some measurable runs off to zero or infinity but all information is lost otherwise. This isn't just something that happens in general relativity, it happens in classical and quantum physics. Just add thermal or quantum white noise to particle packets and suddenly waves, diffusion, and transport create singularities. What we're seeing is additional spectral and geometric structure preventing everything from running off to infinity in the real world.

So going back to general relativity, turns out you can compute many dynamics on worldlines that go through singularities by renormalizing with respect to known symmetries. This is a geometric structure (well, formally a topological group) that makes it clear that black hole singularities don't have to be unphysical since they can still reproduce "normal" physics. Of course we can't be sure that any of this math is correct, since we can't experiment with the interior of black holes, but what I'm trying to get at with this is that interpreting singularities should be done with a lot of care.

The black hole singularity is a spot where curvature goes to infinity. The little fractal angles of Brownian motion also has infinite curvature by the same measure. I assure you those infinite curvatures are very real, and have implications in practical applications in physics and engineering, but this doesn't stop Brownian motion from being a prosaic, physical thing.

That black hole singularities are infinities in spacetime is of course concerning in some intuitive sense, but we should zoom out a bit and recognize that an ant walking on a Brownian noise-perturbed surface can still do ant things even while literally walking on infinities, and this should temper how we understand the physicality of infinities.

Anyways thanks for coming to my TED Talk.

[Radar]

I was lurking for D&D stuff as I usually do but after running into this discussion I felt compelled to stop lurking, create an account, and say something, since (not necessarily gravitational) singularities are basically my field. Strictly speaking I work in stochastics and renormalization in math and physics, but I feel comfortable enough to comment about this as a doctoral student.

Chronos is correct on all counts and they're obviously a physicist. From an outsider's perspective it doesn't particularly matter if there is or isn't a physical singularity within the black hole because for everything observable from without is just built into the geometry that has to describe all of observable spacetime. It's not that crazy to imagine that a different model has to describe the singularity itself. Obviously as suggested this additional structure could pop out of a quantized gravity, which we haven't done yet.

So the deal with singularities in physics in general is that it's an artifact of models that appears when then there's not enough geometric data to describe something. Usually there is just enough to suggest that some measurable runs off to zero or infinity but all information is lost otherwise. This isn't just something that happens in general relativity, it happens in classical and quantum physics. Just add thermal or quantum white noise to particle packets and suddenly waves, diffusion, and transport create singularities. What we're seeing is additional spectral and geometric structure preventing everything from running off to infinity in the real world.

So going back to general relativity, turns out you can compute many dynamics on worldlines that go through singularities by renormalizing with respect to known symmetries. This is a geometric structure (well, formally a topological group) that makes it clear that black hole singularities don't have to be unphysical since they can still reproduce "normal" physics. Of course we can't be sure that any of this math is correct, since we can't experiment with the interior of black holes, but what I'm trying to get at with this is that interpreting singularities should be done with a lot of care.

The black hole singularity is a spot where curvature goes to infinity. The little fractal angles of Brownian motion also has infinite curvature by the same measure. I assure you those infinite curvatures are very real, and have implications in practical applications in physics and engineering, but this doesn't stop Brownian motion from being a prosaic, physical thing.

That black hole singularities are infinities in spacetime is of course concerning in some intuitive sense, but we should zoom out a bit and recognize that an ant walking on a Brownian noise-perturbed surface can still do ant things even while literally walking on infinities, and this should temper how we understand the physicality of infinities.

Anyways thanks for coming to my TED Talk.

Just something I wanted to mention: the infinite curvature in the Brownian motion comes from the continuum approximation of the stochastic motion, so instead of a discrete series of collisions you treat it as a probability distribution smoothly evolving over time. Makes the calculations way easier, but introduces those kind of artifacts. As long as everyone using such models is aware of the limitations, it is obviously possible to deal with it.

[Chronos]

Quoth Bohandas:

What happens on the case of a universe that's elliptical or that otherwise loops around over large scales?

In such a universe, special relativity doesn't completely apply at large length scales, because the universe on such scales has a preferred reference frame, and so you'd have to do all of your calculations in that preferred frame (which may not even be the reference frame of either of the twins).

And thanks for that contribution, Radar. The singularities themselves are not something that I've ever particularly focused on, certainly not in non-relativity contexts.

[chaincomplex]

Just something I wanted to mention: the infinite curvature in the Brownian motion comes from the continuum approximation of the stochastic motion, so instead of a discrete series of collisions you treat it as a probability distribution smoothly evolving over time. Makes the calculations way easier, but introduces those kind of artifacts. As long as everyone using such models is aware of the limitations, it is obviously possible to deal with it.

Actually I was thinking of rough paths! Aka we get to have our cake and eat it too. We want Brownian motion to have infinitely fractal data *and* we want to talk about its (not necessarily quadratic) variation anyways.

In many cases it suffices to put in a wavelength cutoff to anything that has some kind of Laplacian action on a noisy term. It's an artifact of the model for e.g. the Navier-Stokes with random forcing, lattices with perturbations, etc. We don't have to worry about Brownian motion being fractal because as you zoom in in many cases, the data gets smothered by the lower-order dynamics and we're all good. As you say, just keep in mind that it's a limitation and be on our way.

In other cases, such as KPZ dynamics or a \phi^4 QFT, we need the full Brownian geometry because the deep wave numbers both empirically and theoretically control macroscopic trends. Also in the continuum limit of a lattice path integral in QM the measure on the space of paths becomes (usually some Girsanov transformation of) Brownian, so if we're interested in working with the machinery of this limit for theoretical reasons, we have to care about the regularity structures of infinite Brownian motion.

Not to mention a lot of these tools get adapted for singularities that have nothing at all to do with randomness or geometry. For instance, reflecting boundary conditions for anything continuum with diffusive or transport characters is very singular unless mollified or renormalized. Or even just Schrodinger's with an unbounded self-adjoint operator, having to wade into Schwartz distributions and projection-valued measures is a way to deal with an implicitly singular behavior.

tl;dr We can just treat BM as a continuum approximation of something more discrete (in a lattice or spectral sense) in many cases but not always.

edit: speaking of, Mourrat & Weber 2016 (arXiv:1601.01234) is a really interesting paper studying one such case in depth

[Lord Torath]

I was lurking for D&D stuff as I usually do but after running into this discussion I felt compelled to stop lurking, create an account, and say something, since (not necessarily gravitational) singularities are basically my field.

Welcome to the Playground!
I too lurked for a while before signing up and posting.

*snip*

In other cases, such as KPZ dynamics or a \phi^4 QFT, we need the full Brownian geometry because the deep wave numbers both empirically and theoretically control macroscopic trends. Also in the continuum limit of a lattice path integral in QM the measure on the space of paths becomes (usually some Girsanov transformation of) Brownian, so if we're interested in working with the machinery of this limit for theoretical reasons, we have to care about the regularity structures of infinite Brownian motion.

Not to mention a lot of these tools get adapted for singularities that have nothing at all to do with randomness or geometry. For instance, reflecting boundary conditions for anything continuum with diffusive or transport characters is very singular unless mollified or renormalized. Or even just Schrodinger's with an unbounded self-adjoint operator, having to wade into Schwartz distributions and projection-valued measures is a way to deal with an implicitly singular behavior.

tl;dr We can just treat BM as a continuum approximation of something more discrete (in a lattice or spectral sense) in many cases but not always.

edit: speaking of, Mourrat & Weber 2016 (arXiv:1601.01234) is a really interesting paper studying one such case in depth

I understand most of those words. Brownian motion? No problem.

Can you describe KPZ dynamics and \phi^4 QFT briefly?

[Bohandas]

I imagine QFT probably stands for quantum field theory

[NichG]

KPZ is a differential equation that (among other things) models the growth of surfaces through random aggregation. It gives you rough surfaces, the way that a random walk gives you a rough path...

[chaincomplex]

Both posts above are correct.

The Kardar-Parisi-Zhang equation modulates interface dynamics that have random noise. This interface could be: (1) thermal or quantum chemical interactions at nanoscale; (2) shear flow turbulence-quiescent boundaries in maritime or aerospace; (3) deformations of surfaces of plasma packets. I want to point out that surface behavior implies behavior for the rest of the space (e.g. PDE boundary conditions, Stokes's theorem, etc.), so it's pretty important generically.

The \phi^4 quantum field theory is so named because a power-4 term appears in the action functional for the Feynman path integral you get for a class of theories in quantum electrodynamics (Yang-Mills) and their extensions, especially with regards to the Higgs field (of Higgs boson fame). There are applications of the former outside of theoretical physics in superconductivity and material electronics so it's not just abstract nonsense. Generally all Ginzburg-Landau theories are \phi^4, as well as some important selections from Klein-Gordon theories for relativistic waves.

And what I'm saying in my prior post is that one can't just do discrete approximations to the path integral by naively shunting plasma flow or quantum electrodynamics onto lattices because approximations only work robustly if they converge -- and they don't in the regime of "just the equations please". This is about the point in my research pitch where I start introducing unnecessarily hard-looking math in the hopes of wowing and intimidating the audience.

By the way, turbulence is like a """""baby""""" version of this, i.e. you can't just do Navier-Stokes on a lattice and recover turbulence at all. I put lots of air quotes because it's probably a harder problem mathematically and computationally than everything else I discussed, but there is a strong belief that turbulence can be captured by some spectral methods that dodges having to do hard-looking math; it's just painfully hard "classical" math.

Lastly I will say that while these aspects have been neglected for many years in the math and physics communities, we've recently been seeing recognition that they're important with Hairer getting a Fields Medal and Parisi a Nobel Prize respectively.

[Battleship789]

WElcome to the Playground!
I too lurked for a while before signing up and posting.

I understand most of those words. Brownian motion? No problem.

Can you describe KPZ dynamics and \phi^4 QFT briefly?

As mentioned by Bohandas, QFT is quantum field theory. The \phi^4 qualifier refers to a specific type of field theory involving a scalar field (the ϕ in question, basically a function whose value does not depend on a specific choice of coordinates) which has the typical Klein-Gordon equation of scalar field theories

Spoiler: Klein-Gordon equation explanation

Show

A relativistic generalization of the Schrodinger equation from quantum mechanics, consists of a ϕ² term (the "mass" term) and a term with squared derivatives of ϕ (the "kinetic" term.)

but with one additional term where ϕ is raised to the fourth power. This type of theory is of particular interest because, unlike other scalar field theories, quartic scalar field theories are "renormalizable" in that they can be quantized (in the quantum mechanics sense) in a way that doesn't lead to unmanageable singularities/infinities.

With the right setup the scalar field in question has an associated physical particle (the Higgs boson) and the interaction of the field with other fundamental fields gives the W^+/- and Z bosons of the weak nuclear force their masses (as opposed to the massless bosons of electromagnetism and the strong nuclear force, the photon and gluon, respectively.) The dependence on a fourth power leads to descriptions like "Mexican-hat potential" for the theory since the potential for these theories is of the form V(ϕ) = V₀ - a*ϕ² + b*ϕ⁴, which takes the described shape when rotated around the vertical axis (looking up the phrase yields some nice plots if you have trouble visualizing it.) The fourth power is also interesting due to the introduction of self-interaction terms, meaning a particle which follows the field theory can not only interact with other particles but with itself! (Though not a ϕ⁴ theory, one example of this phenomenon is found with gluons which have interactions where a single gluon can split into 2 or 3 other gluons.)

[chaincomplex]

This type of theory is of particular interest because, unlike other scalar field theories, quartic scalar field theories are "renormalizable" in that they can be quantized (in the quantum mechanics sense) in a way that doesn't lead to unmanageable singularities/infinities.

The real thunk is that physicists have been renormalizing these field theories forever by arbitrarily adding and subtracting infinities, hence why we knew they were renormalizable before we, well, formally renormalized them to begin with. As you might imagine this gives mathematicians migraines.

That's not to say formalizing it contributed nothing -- the techniques were arbitrary and couldn't be generalized, and a lot of data was missing that we could've used for practical stuff. For example, say you had a plasma jet under KPZ. The jet is causing a chemical reaction as it mingles with the atmosphere. How do we find how much reaction is happening? Previously we had to infer this experimentally. Now we have the tools to compute it analytically using Hairer's regularity structures.

Of course in practice we're just inferring it from experiments anyways because teaching higher stochastic calculus to every plasma engineer is a waste of their time and this is too new to be in any multiphysics repository. But I expect that one day these will be black box algorithms that plasma engineers will know to use.