If you're not a theoretical physicist yourself, you might think that physics is physics -- we ask questions about the Universe, do experiments/make observations, and get the answers -- and math is just a tool that we use to help us get there.
But that really sells the power of mathematics short. For a physical theory to be valid, there are a whole host of mathematical properties that theory needs to possess, including being free of logical inconsistencies, making predictions about observables, and that those predictions agree with observations. Yet when we look at our theory of gravitation at the smallest scales and with the strongest gravitational fields, our theory itself fails.
Interesting - but is there a reason the equations used in the snippets are from statistical theory rather than physics?
@dean #1: Stock photo, much?
I realize that Michael, but are there none of physics? (I like the stat stuff, since that's my area: just thought it was an odd choice).
Lack of preview bites me again: the reason I think it is an odd choice: the topic is the success of pure mathematics as a tool for investigating physical theories, yet the math in the photo doesn't relate to that. No criticism of the content, it just jumped out at me.
I have so many questions on this that I almost don't know where to begin, but I'll start with 2 related questions:
1 - Why is the requirement to retain information so absolute that any theory in which information is irretrievably lost is almost immediately discredited?
2 - Keeping in mind the requirement for preservation of information, why do physicists believe singularities exist?
#3 added to Denier's list.
What's the difference between regular singularity and circular/curvature singularity?
@Denier #5: "Retain information" is really a consequence, not a requirement in itself. The underlying issue is what is called "unitary evolution" (that's the keyword you should use), which may also be expressed as "conservation of probability."
If I have a six-sided die, and I roll it, there's 1 in 6 chance of any particular number coming up. But there's exactly a 1 in 1 chance (unity) of *some* number coming up.
In quantum mechanics, or more generically quantum field theory, the wavefunction for an initial state is normalized to unity, so the probability of the initial state is 1. As the wavefunction evolves, it must remain normalized, so that the total probability of _any_ of the different possible final states remains unity. In other words, _something_ must happen, even if we can't predict in advance which something it might be.
If the evolution isn't unitary, then you could evolve a system to a state which has a 95% probability of one of it's final states happening, but a 5% probability of .... what, exactly? "Disappearing"? But the vacuum is a valid final state, so it should have been included in the final wavefunction. Then what?
Singularities are mathematical features, not physics. A simple kind of singularity (called a coordinate singularity) occurs at the origin of a polar coordinate system. Every point in the plane may be uniquely identified by two numbers, R and phi, the distance from the origin and the angle away from the reference direction ("x axis"). But at the origin itself, where R=0, the angle is undefined! Any value of phi at R=0 corresponds to the same point, which violates the uniqueness of the coordinate system. That's one kind of singularity.
The math of general relativity permits the appearance of singularities (not just coordinate singularities, but more complex forms), so classically, we presume they can exist physically. The math of quantum field theory does not (I believe, but I am NOT a theorist!) permit singularities in the same way, so this is one of the places where GR and QFT need to be reconciled.
There are many physicists who have developed legitimate theories which preclude (or forbid) the physical existence of singularities. String theory is one such, as is loop quantum gravity.
Well you can't know the position of something without some other error in measurement according to quantum theory.
But at a singularity, you can know the position precisely (it's precisely at the singularity, a precise single point) yet also NOT know it precisely (all points exist at the same point, therefore an error in one point makes no difference to the location).
This is logically inconsistent.
Partly a problem theoretically, partly a pragmatic issue.
Theoretically, every model should be logical otherwise it cannot be extended. The model therefore gets changed. To take Michael's example, the polar coordinate singularity at R=0 does not exist in cartesian coordinate models of location.
Pragmatically, though the universe doesn't HAVE to be logical, if it weren't we'd be buggered in understanding it. Giving in to it means we stop trying to understand things. Every time before a lack of understanding has been the result of not thinking clearly rather than some mystical or magical event.
The math of quantum field theory does not (I believe, but I am NOT a theorist!) permit singularities in the same way, so this is one of the places where GR and QFT need to be reconciled.
IIRC, QM does allow for nodes in wavefunctions where the calculated probability is exactly zero. You see those, for example, in electron shells with a principle quantum number >1 (aka L, M, and N shells). To my way of thinking, such nodes would be fairly analogous to coordinate singularities. However I have no idea if this observation is relevant to black hole physics, and if it is, how it might be relevant.
eric, the wavefunction is not the object. A black hole singularity would be the object.
@Michael Kelsey #7
If my understanding of what you wrote and of what Sabine Hossenfelder wrote is correct, there may be no singularity in the center of any black hole. It is not just me, but actual smart people think the center of a black hole may contain a tiny, but not infinitely small, super-dense, but not infinitely dense ball, of stuff.
The singularity may in fact be nothing more than a mathematical artifact from taking a theory of approximation too far, and is not real in our universe.
That is huge. Am I understanding that correctly?
@Denier #11: Yes, you're understanding correctly. Whether that is true or not is an extremely active area of research, which is not yet resolved.
@eric #9: Nodes in a wave are not singularities at all. I would recommend the Wikipedia article https://en.wikipedia.org/wiki/Singularity_(mathematics) to understand the math.
@Michael Kelsey #12
In a previous post I can't seem to find, I believe Ethan wrote that cosmological expansion redshift did not violate the conservation of energy because the process was reversible. If the universe were to start shrinking, the energy would go right be into the traveling photon and so the energy isn't really "lost". That we don't currently know of any way to shrink the universe seems irrelevant to the reversibility argument.
Does the same logic hold true for information in a black hole? If there is no singularity, and all matter that falls in is just crunched down into being part of a super-dense tiny ball, is the event horizon irrelevant? If reversing the gravity of a black hole in the same way as hypothetically reversing the expansion of the universe, thereby causing everything that ever fell into a black hole to come exploding out enough to say information isn't being lost?
Regardless of Hawking Radiation, does the dense-ball black hole model itself solve the information destruction problem?
"Mathematics is in the first line a discipline of thought. Cleaned of the vagueness of human language, mathematics is a tool to deduce consequences from assumptions"
What about a Godel sentence ?
The next question I have is completely different.
How does the detector in a 2-slit experiment work?
In trying to find an answer to this question I found this YouTube video that makes a claim between 2:40 and 3:30 that can't be right.
I know QM can be counter-intuitive, but detector has to be collapsing the wave form in the monitored slit. That is the only way that experiment makes sense, so how does the detector work?
"How does the detector in a 2-slit experiment work?"
By being excited at meeting the thing that came through the slit.
The detector isn't "collapsing the wave" since there isn't any wave, it's a way to work out where the particle can go. It's somewhere, but if you try to find exactly where, you've changed where it is going.
The two slit "problem" is entirely one of mixing models. If you're trying to measure the wave properties, you'll see wavelike properties. If you're trying to measure particle like properties, you need a different experiment, and you get a particle like properties.
Did you see this one recently?
Click on the poscast to hear her explanation.
Also from your video,
in that max Plank quote there at the 8:10 mark "Science cannot solve the ultimate mystery of nature. And that is because, in the last analysis, we ourselves are part of nature and therefore part of the mystery that we are trying to solve."
Does that also follow along with Kurt Godel's Incompleteness theorem?
So from what I am reading here, we can never find "The Missing Piece"
Based on Plank, and Godel input is this true?
The puzzle picture is a perfect analogy to raise a question - we don't know what the missing pieces are, but we don't know either what the missing pieces aren't -- so could it not be the case that blocks the road to completing the puzzle of a TOE, that we are already familiar with ingredients of it that we are convinced don't belong inside the puzzle, so that we can't imagine them there? A logical way for such a situation to arise, would be if those missing pieces were familiar to a scientific specialty that's not understood to be a provider of physics (easy since none are, except math).
"but we don’t know either what the missing pieces aren’t "
For any meaningful meaning of "don't know".
We know it isn't mass hallucination caused by the Flying Spaghetti Monster having gas.
And so on.
"Does that also follow along with Kurt Godel’s Incompleteness theorem?"
Godel's two incompleteness theorems apply to mathematical systems, not the physical universe.
"Godel’s two incompleteness theorems apply to mathematical systems, not the physical universe."
So don't put much weight in this guy Max Tegmark's book/work on "Our Mathematical Universe" then ( He Is A crackpot like one guy asked) ?
In the Q&A Section:
Q: Isn't the mathematical universe hypothesis ruled out by Gödel's incompleteness theorem?
A: No, not as far as we know. Given any sufficiently powerful formal system, Gödel showed that we cannot use it to prove its own consistency, but his doesn't mean that it is inconsistent or that we have a problem. Indeed, our cosmos doesn't show any signs of being inconsistent or ill-defined, despite showing hints that it may be a mathematical structure. Moreover, what were we hoping for? If a mathematical system could be used to prove its own consistency, we'd remain unconvinced that it actually was consistent, since an inconsistent system can prove anything. We'd only be somewhat convinced if a simpler system that we have better reason to trust the consistency of could prove the consistency of a more powerful system - unsurprisingly, that's impossible, as Gödel also proved. Of the many mathematicians with whom I'm friends, I've never heard anyone suggest that the mathematical structures that dominate modern physics (pseudo-Riemannian manifolds, Calabi-Yau manifolds, Hilbert spaces, etc.) are actually inconsistent or ill-defined.
@Ragtag Media #18
Quantum entanglement is an entirely different bag of cats about which I have a key question to ask, but am waiting for Ethan to post something relevant before I blurt it out.
With regards to Kurt Godel’s Incompleteness theorem, I treat it like Han Solo treats of odds of successfully navigating an asteroid field.
I like to play with ideas until I find the edges. In this post I've already found 1 edge and possibly another.
Edge #1 - Physicists don't all agree that there is a a singularity in the center of a black hole.
Possible Edge #2 - It is very possible that PhD Michael Kelsey and PhD Ethan Siegel disagree completely on reversibility and conservation of stuff in our universe. I would love to get those 2 in the same backyard with some adult beverages and a GoPro camera when that subject was brought up.
The mathematical universe hypothesis is a mathematical system: it asserts that the universe is a mathematical system, so it is possible that Godel's theorems could apply, but it should be noted Tegmark makes an assumption to get there: he says that the only types of mathematical systems (the name mathematical structure is often used) that can represent reality are those that are "Godel Complete" ("Godel decidable" also used).
He has to do a lot of hand-waving to make that argument (see his CUH, Computable Universe Hypothesis work) and as he himself has noted, there isn't a lot of evidence in its favor, and it doesn't account for a good amount of the known mathematical structure of the observable universe. Schmidhuber has an interesting probability-based objection to this idea.
But the point I was trying to make is: you can't discuss Godel's two theorems as applying to the universe as a physical thing: IF you propose that it is a manifestation of a mathematical system, and if that system has the appropriate structure and complexity, then the theorems could apply.
Jürgen Schmidhuber THAT's It.. .
Haha Thanks Dean, I just saw this guy last night on through the wormhole but didn't catch his name, he had a computation for a program to run the universe on one page. Well, now I at least know you know what I was getting at
I had about 50 tabs open looking for this guy.. thanks