Operator algebras and the substructure of space and time
quantamagazine.org
quantamagazine.org
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.
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!
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.
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.
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.
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.
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.
Assuming you mean 3D boundary, 4D bulk.
* 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.
The belief that that applies even at the center of a black hole is an additional non-common-sense step.
Maybe I should have said that the ideas in the article are an approach to resolving the paradox, rather than invoking common sense.
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."
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.
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.
Anyway, I agree with your overall point.
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.
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”.
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.