>Physicists have taken this to mean that the contorted space-time fabric of a black hole may be made of atomlike parts, just like a gas.
as far as i see the "singularity" at the center of black hole is just a mathematical artifact of the smoothness of the GR. And while that smoothness is a valid approximation at macro scales, by all the accounts the world isn't that smooth at the micro scales, and similarly to white dwarfs and neutron stars it seems naturally for a black hole core to be some next step of degenerate matter, something like quark-gluon soup.
It's more tricky than that, because inside a black hole things turn really weird.
Of course things might get unpredictable close to the singularity if you really assume it is just a ton of matter squished into a tiny point as seen from the outside, but the overall spacetime geometry inside black holes would be untouched by that. Especially if you consider large, supermassive black holes that even have comfortable tidal forces at the horizon. If you now look at black hole geometries in GR, you'll find that the singularity is not a point in space, it is actually a moment in time. Once you cross the event horizon, that moment becomes part of your future, which means that there is absolutely nothing you could do to escape it. So a more accurate description of a singularity would not say "super dense point in space" - it would literally describe it as "the end of time." As in the opposite of the big bang.
Personally, I’m not convinced that there is an “inside”. From the outside there doesn’t appear to be one — you never see anything cross the horizon, including the substance that originally formed the black hole! Conversely, observers see the black hole evaporating from just above the surface, with all infalling matter recovered in this way. World lines approach the horizon, hug it closely, then leave. They don’t fall “in”.
In my mind scientists filled in a blank with their imaginations, but the blank they are filling in may not even exist as a location. It’s like complex (imaginary) numbers, you can talk meaningfully about solutions to equations: but ordinary +, -, *, / arithmetic “can’t get you there”. A black hole could be like a pinhole stretched out into a larger hole in a stretchy sheet of fabric. From the point of view of an ant walking on the surface the hole is a boundary that can be approached, but not entered, and around it the fabric is highly distorted.
There have been some good papers published recently on related topics. For example, Penrose diagrams as typically drawn use a simplification that black holes extend forever forward and backward in time. This allows infinitesimals to add up over infinity, which is a non-physical sleight of hand. Real black holes form over time, and worldlines don’t enter them — they just approach the horizon and then “boil off” in the distant future due to Hawking radiation.
There is this obsession in modern physics of clinging to overly simplified models and then treating the edges of their capabilities as real things to discover instead of modelling failures to get past with better models.
You can’t climb the mountain range represented by the fold crease in your map.
>Real black holes form over time, and worldlines don’t enter them
That's not correct. It only appears that way in some forms of diagrams because of their choice of coordinate system or the way they display lines of constant time/space. But all black holes (eternal Penrose and normal ones) have worldlines entering them normally. The claim that these worldlines would suddenly stop at the horizon for no particular reason is pretty wild (but not completely new). I don't think the majority of physicists would doubt that an observer could fall inside a black hole (particularly large black holes with low tidal forces), because from their point of view there would be absolutely nothing unusual happening at the horizon.
From the point of view of a distant observer, objects falling into a black hole become infinitely redshifted as they approach the horizon. Nobody outside the black hole observes anything completely falling in.
Penrose diagrams aren't drawn from the perspective of an in-universe observer. They're drawn from the perspective of a god, looking "down" at the universe from outside it.
Most such diagrams -- irrespective of coordinate system -- are drawn with these world lines continued into the black hole, typically ignoring the change-over-time of the destination itself.
If you were to "ride the world line" into the black hole in the physical universe, you will slow down so that the hole in front of you appears to speed up, evaporating faster and faster as you approach.
You never observe falling into a "black" hole, it'll become brighter and brighter and explode in your face as you approach.
A black hole is a supernova explosion smeared out in time.
That's not true. You can check out chapter 32 in Misner's Gravitation for a detailed discussion of a free falling observer's worldline into a black hole.
In that section "infinite" is used a bunch of times, which is a non-physical simplification. All of the sections using Penrose diagrams talk explicitly about "the point at infinity", but Hawking showed that black holes don't last for infinite time. You have to keep reading past the overly simplified models.
E.g.: right after that section:
"But from an external vantage point the star requires infinite time to reach the horizon".
In other words, observers don't ever see a black hole form completely, which is an observation that infalling observers can't contradict.
Either you believe that there are two different universes for non-infalling and infalling observers, or you have to accept a single physical reality that black holes never completely form, and then evaporate in a finite time.
I really like Penrose's new theory that the singularity is a torus within a bubble of normal space inside the event horizon. The point-like singularity predicted by GR seems obviously implausible, it makes more sense for it to be smeared throughout some volume of space. If I understand correctly, he proposes that the singularity is a compact object rotating about the center so fast that it's more or less a solid torus.
Intuitively, it also makes sense that space within a black hole could be more or less normal. Or at least have a consistent curvature that approximates normal space.
Conversely, I find the notion of space-time equivalence as illustrated by Penrose diagrams to be quite unconvincing. All we really have is high order approximations of how GR might behave in extreme conditions. I believe that if it ever becomes testable, we'll see some kind of limit to how far space can translate to time. Or that time is more complex than we think.
No one really knows for sure, but it's a lot of fun to speculate about!
> The point-like singularity predicted by GR seems obviously implausible, it makes more sense for it to be smeared throughout some volume of space. If I understand correctly, he proposes that the singularity is a compact object rotating about the center so fast that it's more or less a solid torus.
I think the idea there is that there is no singularity. He's taken the prediction for a rotating blackhole (all mass collapses into a 0-volume ring) and supposed that there is a yet to be discovered force that resists infinite compaction leaving a ring of greater than 0 volume.
Space-time equivalence is pretty key to the functioning of the Global Positioning System, and it comes from special relativity. It shows up well before black hole gravitational conditions.
In the article they describe a 2D wrapper that can represent 3D objects in the 3D bulk space between it with mathematical equivalence. Can we not extrapolate this to mean that it is possible our 3D universe encapsulates a 4D bulk space?
I don't know the answer, but a couple of things to keep in mind with such speculations:
* Properties of lower dimensions don't always extrapolate to higher dimensions. An example that comes to mind is the result in probability theory that a 2D random walk will always return to the home position an infinite number of times, whereas a 3D random walk has a 2/3 chance of never returning.
* Physics is interested in what is observable and testable. In your grant application, what are you saying are the testable aspects of this 4D bulk space which would validate your theory?
On the other hand, we can already perfectly represent 4D objects in 3D space. Just write down a bunch of 4-vectors. If this seems like it's trivializing what you're saying, then it means you need to provide a clearer definition of the objects which you have in mind, and what it means to represent them.
So overall I think you'd want to be careful with how you're defining the objects you're interested in, and what is the mathematical form of the claim you want to make about those objects, and how to test whether it has any physical relevance.
For example, the Banach-Tarski paradox can't happen in two dimensions, it needs at least three. Terence Tao's 2nd book on analysis explains it, tho I can't yet understand the explanation. (It (the weird measure expanding rearrangement) can happen for countable # of sets in 2, even 1 dimension, but not for finite # of sets).
The goal is to make a prediction about the boundaries or interiors of black holes that can be tested. It’s based on the common sense belief that information is conserved even if it reaches the center of a black hole.
The black hole information paradox presents an apparent theoretical contradiction between black hole radiation, which Hawking predicted carries no information about its past, and quantum theory, which conserves information at all times. The common sense belief would be that information is conserved, perhaps because there is something measurable inside a black hole and not a singularity.
Maybe I should have said that the ideas in the article are an approach to resolving the paradox, rather than invoking common sense.
> The common sense belief would be that information is conserved
I'll say it once more, since you didn't seem to understand my point the first time: "information is conserved" is not a "commonsense belief". (Unless you mean "commonsense within the quantum mechanics community".)
Commonsense is "I've seen a shredder. I know information can be destroyed."
Ok, well, would you expect a shredded book to be indistinguishable from a shredded bowl of jello, or a shredded pot of coffee, or a shredded parcel of ionized plasma?
But a shredder doesn't actually destroy information.
Back to the original discussion, though: Time and again I'm fascinated by the fact that it's the quantum mechanics community that believes in unitarity / information being preserved so strongly. After all, the collapse of the wave function violates unitarity, too.
> But a shredder doesn't actually destroy information.
See, people keep talking about the "common sense" idea. Well, in the common sense view, yes, a shredder actually does destroy information, because you can't read the page any longer. The "common sense" people don't have any notion of information in the quantum sense, or unitarity, or waveform collapse, or any of that stuff.
"Common sense" does not mean "common sense among postgrads in quantum mechanics". Those are two completely different things.
The "common sense" level is probably something like, black holes exist, and maybe some notion that something called "quantum mechanics" exists. Most people don't have any kind of "common sense" about the details.
Oh I didn't mean to say a shredder not destroying information was common sense. Maybe I just didn't like this particular example too much because I think even for non-physicists it's still imaginable someone could – in theory – piece together the shredded paper again.
I think this should be retitled to specify it's a story about the key players or the history of this subject, not so much subject matter itself.
Often I click something related to Quanta and it seems to occupy a very strange target audience. It starts with a title or link that seems lures a reader with in-depth knowledge of some subjects. I think most readers see that title and get excited about all kinds of things like a simplified introduction to subjects they weren't able to grasp in the past or someones unique perspective that might help them better understand the topic.
Instead the article is actually the story and history of some people involved. That's an interesting article as well, but I think it's a different title.
Agree with your point. The actual title, "If the Universe Is a Hologram, This Long-Forgotten Math Could Decode It," has another issue: physicists may have "forgotten" operator algebras, but it's certainly been a very active part of mathematics.
Wikipedia calls them by a different name! As for history, Tomita is said to be the most important guy* behind the elucidation of Type III but gets no credit…
Feels like this is a classic case of blind men and elephant
>These were further developed later by Takesaki, and the theory is called the Tomita–Takesaki theory. It has great influence in statistical mechanics too. That was the beginning part, but in Tomita’s papers, he didn’t write proofs.
I: Mathematicians usually like proofs. Is Tomita a mathematician?
A: [Minoru] Tomita is a pure mathematician. There are a lot of algebraists in Japan, including [Masamichi] Takesaki, but Tomita is a completely different kind of person, very “singular”.
An operator can be something like the differentation operator (d/dx), the integral operator, gradient, curl, or other structure. Operators are similar to functions but work more generally, like mapping functions to other functions.
IIUC, Operator algebra is the study of the properties of operators just like real, complex, matrix, and linear algebra are the study of the properties of those objects/constructs. As such, once you have defined an operator for e.g. Schroedinger's wave equation you can manipulate and explore that using the rules and principles of operator algebra. Thus, making them easier to work with.
von Neumann algebras are operator algebras. They are algebras whose elements are operators on some Hilbert space. They also satisfy some other conditions. What I just said is not meant as a definition.
>Physicists have taken this to mean that the contorted space-time fabric of a black hole may be made of atomlike parts, just like a gas.
as far as i see the "singularity" at the center of black hole is just a mathematical artifact of the smoothness of the GR. And while that smoothness is a valid approximation at macro scales, by all the accounts the world isn't that smooth at the micro scales, and similarly to white dwarfs and neutron stars it seems naturally for a black hole core to be some next step of degenerate matter, something like quark-gluon soup.
It's more tricky than that, because inside a black hole things turn really weird. Of course things might get unpredictable close to the singularity if you really assume it is just a ton of matter squished into a tiny point as seen from the outside, but the overall spacetime geometry inside black holes would be untouched by that. Especially if you consider large, supermassive black holes that even have comfortable tidal forces at the horizon. If you now look at black hole geometries in GR, you'll find that the singularity is not a point in space, it is actually a moment in time. Once you cross the event horizon, that moment becomes part of your future, which means that there is absolutely nothing you could do to escape it. So a more accurate description of a singularity would not say "super dense point in space" - it would literally describe it as "the end of time." As in the opposite of the big bang.
Personally, I’m not convinced that there is an “inside”. From the outside there doesn’t appear to be one — you never see anything cross the horizon, including the substance that originally formed the black hole! Conversely, observers see the black hole evaporating from just above the surface, with all infalling matter recovered in this way. World lines approach the horizon, hug it closely, then leave. They don’t fall “in”.
In my mind scientists filled in a blank with their imaginations, but the blank they are filling in may not even exist as a location. It’s like complex (imaginary) numbers, you can talk meaningfully about solutions to equations: but ordinary +, -, *, / arithmetic “can’t get you there”. A black hole could be like a pinhole stretched out into a larger hole in a stretchy sheet of fabric. From the point of view of an ant walking on the surface the hole is a boundary that can be approached, but not entered, and around it the fabric is highly distorted.
There have been some good papers published recently on related topics. For example, Penrose diagrams as typically drawn use a simplification that black holes extend forever forward and backward in time. This allows infinitesimals to add up over infinity, which is a non-physical sleight of hand. Real black holes form over time, and worldlines don’t enter them — they just approach the horizon and then “boil off” in the distant future due to Hawking radiation.
There is this obsession in modern physics of clinging to overly simplified models and then treating the edges of their capabilities as real things to discover instead of modelling failures to get past with better models.
You can’t climb the mountain range represented by the fold crease in your map.
>Real black holes form over time, and worldlines don’t enter them
That's not correct. It only appears that way in some forms of diagrams because of their choice of coordinate system or the way they display lines of constant time/space. But all black holes (eternal Penrose and normal ones) have worldlines entering them normally. The claim that these worldlines would suddenly stop at the horizon for no particular reason is pretty wild (but not completely new). I don't think the majority of physicists would doubt that an observer could fall inside a black hole (particularly large black holes with low tidal forces), because from their point of view there would be absolutely nothing unusual happening at the horizon.
From the point of view of a distant observer, objects falling into a black hole become infinitely redshifted as they approach the horizon. Nobody outside the black hole observes anything completely falling in.
Penrose diagrams aren't drawn from the perspective of an in-universe observer. They're drawn from the perspective of a god, looking "down" at the universe from outside it.
Most such diagrams -- irrespective of coordinate system -- are drawn with these world lines continued into the black hole, typically ignoring the change-over-time of the destination itself.
If you were to "ride the world line" into the black hole in the physical universe, you will slow down so that the hole in front of you appears to speed up, evaporating faster and faster as you approach.
You never observe falling into a "black" hole, it'll become brighter and brighter and explode in your face as you approach.
A black hole is a supernova explosion smeared out in time.
That's not true. You can check out chapter 32 in Misner's Gravitation for a detailed discussion of a free falling observer's worldline into a black hole.
In that section "infinite" is used a bunch of times, which is a non-physical simplification. All of the sections using Penrose diagrams talk explicitly about "the point at infinity", but Hawking showed that black holes don't last for infinite time. You have to keep reading past the overly simplified models.
E.g.: right after that section:
"But from an external vantage point the star requires infinite time to reach the horizon".
In other words, observers don't ever see a black hole form completely, which is an observation that infalling observers can't contradict.
Either you believe that there are two different universes for non-infalling and infalling observers, or you have to accept a single physical reality that black holes never completely form, and then evaporate in a finite time.
I really like Penrose's new theory that the singularity is a torus within a bubble of normal space inside the event horizon. The point-like singularity predicted by GR seems obviously implausible, it makes more sense for it to be smeared throughout some volume of space. If I understand correctly, he proposes that the singularity is a compact object rotating about the center so fast that it's more or less a solid torus.
Intuitively, it also makes sense that space within a black hole could be more or less normal. Or at least have a consistent curvature that approximates normal space.
Conversely, I find the notion of space-time equivalence as illustrated by Penrose diagrams to be quite unconvincing. All we really have is high order approximations of how GR might behave in extreme conditions. I believe that if it ever becomes testable, we'll see some kind of limit to how far space can translate to time. Or that time is more complex than we think.
No one really knows for sure, but it's a lot of fun to speculate about!
> The point-like singularity predicted by GR seems obviously implausible, it makes more sense for it to be smeared throughout some volume of space. If I understand correctly, he proposes that the singularity is a compact object rotating about the center so fast that it's more or less a solid torus.
I think the idea there is that there is no singularity. He's taken the prediction for a rotating blackhole (all mass collapses into a 0-volume ring) and supposed that there is a yet to be discovered force that resists infinite compaction leaving a ring of greater than 0 volume.
Space-time equivalence is pretty key to the functioning of the Global Positioning System, and it comes from special relativity. It shows up well before black hole gravitational conditions.
In the article they describe a 2D wrapper that can represent 3D objects in the 3D bulk space between it with mathematical equivalence. Can we not extrapolate this to mean that it is possible our 3D universe encapsulates a 4D bulk space?
I don't know the answer, but a couple of things to keep in mind with such speculations:
* Properties of lower dimensions don't always extrapolate to higher dimensions. An example that comes to mind is the result in probability theory that a 2D random walk will always return to the home position an infinite number of times, whereas a 3D random walk has a 2/3 chance of never returning.
* Physics is interested in what is observable and testable. In your grant application, what are you saying are the testable aspects of this 4D bulk space which would validate your theory?
On the other hand, we can already perfectly represent 4D objects in 3D space. Just write down a bunch of 4-vectors. If this seems like it's trivializing what you're saying, then it means you need to provide a clearer definition of the objects which you have in mind, and what it means to represent them.
So overall I think you'd want to be careful with how you're defining the objects you're interested in, and what is the mathematical form of the claim you want to make about those objects, and how to test whether it has any physical relevance.
For example, the Banach-Tarski paradox can't happen in two dimensions, it needs at least three. Terence Tao's 2nd book on analysis explains it, tho I can't yet understand the explanation. (It (the weird measure expanding rearrangement) can happen for countable # of sets in 2, even 1 dimension, but not for finite # of sets).
https://terrytao.files.wordpress.com/2010/02/epsilon.pdf
That's how it would generally work, assuming you don't need tiny curled up dimensions to stuff unwanted particles into.
Assuming you mean 3D boundary, 4D bulk.
Is any of this experimentally falsifiable? Or rather, where downstream of this does something experimentally falsifiable pop out?
The goal is to make a prediction about the boundaries or interiors of black holes that can be tested. It’s based on the common sense belief that information is conserved even if it reaches the center of a black hole.
If I understand correctly, the idea that information is preserved is based on QM using unitary matrices. That's hardly a "common sense" belief.
The belief that that applies even at the center of a black hole is an additional non-common-sense step.
The black hole information paradox presents an apparent theoretical contradiction between black hole radiation, which Hawking predicted carries no information about its past, and quantum theory, which conserves information at all times. The common sense belief would be that information is conserved, perhaps because there is something measurable inside a black hole and not a singularity.
Maybe I should have said that the ideas in the article are an approach to resolving the paradox, rather than invoking common sense.
> The common sense belief would be that information is conserved
I'll say it once more, since you didn't seem to understand my point the first time: "information is conserved" is not a "commonsense belief". (Unless you mean "commonsense within the quantum mechanics community".)
Commonsense is "I've seen a shredder. I know information can be destroyed."
Ok, well, would you expect a shredded book to be indistinguishable from a shredded bowl of jello, or a shredded pot of coffee, or a shredded parcel of ionized plasma?
But a shredder doesn't actually destroy information.
Back to the original discussion, though: Time and again I'm fascinated by the fact that it's the quantum mechanics community that believes in unitarity / information being preserved so strongly. After all, the collapse of the wave function violates unitarity, too.
> But a shredder doesn't actually destroy information.
See, people keep talking about the "common sense" idea. Well, in the common sense view, yes, a shredder actually does destroy information, because you can't read the page any longer. The "common sense" people don't have any notion of information in the quantum sense, or unitarity, or waveform collapse, or any of that stuff.
"Common sense" does not mean "common sense among postgrads in quantum mechanics". Those are two completely different things.
The "common sense" level is probably something like, black holes exist, and maybe some notion that something called "quantum mechanics" exists. Most people don't have any kind of "common sense" about the details.
Oh I didn't mean to say a shredder not destroying information was common sense. Maybe I just didn't like this particular example too much because I think even for non-physicists it's still imaginable someone could – in theory – piece together the shredded paper again.
Anyway, I agree with your overall point.
I think this should be retitled to specify it's a story about the key players or the history of this subject, not so much subject matter itself.
Often I click something related to Quanta and it seems to occupy a very strange target audience. It starts with a title or link that seems lures a reader with in-depth knowledge of some subjects. I think most readers see that title and get excited about all kinds of things like a simplified introduction to subjects they weren't able to grasp in the past or someones unique perspective that might help them better understand the topic.
Instead the article is actually the story and history of some people involved. That's an interesting article as well, but I think it's a different title.
Agree with your point. The actual title, "If the Universe Is a Hologram, This Long-Forgotten Math Could Decode It," has another issue: physicists may have "forgotten" operator algebras, but it's certainly been a very active part of mathematics.
Can someone tell me why quantmag’s coverage of these algebras is so different from
https://en.wikipedia.org/wiki/Von_Neumann_algebra#Factors ?
Wikipedia calls them by a different name! As for history, Tomita is said to be the most important guy* behind the elucidation of Type III but gets no credit…
Feels like this is a classic case of blind men and elephant
* https://en.wikipedia.org/wiki/Tomita–Takesaki_theory
>These were further developed later by Takesaki, and the theory is called the Tomita–Takesaki theory. It has great influence in statistical mechanics too. That was the beginning part, but in Tomita’s papers, he didn’t write proofs. I: Mathematicians usually like proofs. Is Tomita a mathematician? A: [Minoru] Tomita is a pure mathematician. There are a lot of algebraists in Japan, including [Masamichi] Takesaki, but Tomita is a completely different kind of person, very “singular”.
http://www.asiapacific-mathnews.com/04/0402/0012_0018.pdf
Operator algebra refers to https://en.wikipedia.org/wiki/Operator_(mathematics).
An operator can be something like the differentation operator (d/dx), the integral operator, gradient, curl, or other structure. Operators are similar to functions but work more generally, like mapping functions to other functions.
See https://en.wikipedia.org/wiki/Quantum_field_theory#Canonical... for some of the applications/definitions in quantum mechanics.
IIUC, Operator algebra is the study of the properties of operators just like real, complex, matrix, and linear algebra are the study of the properties of those objects/constructs. As such, once you have defined an operator for e.g. Schroedinger's wave equation you can manipulate and explore that using the rules and principles of operator algebra. Thus, making them easier to work with.
von Neumann algebras are operator algebras. They are algebras whose elements are operators on some Hilbert space. They also satisfy some other conditions. What I just said is not meant as a definition.