The h Bar - 2015-02-26 (page 5 of 6)

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7:00 PM

@GlenTheUdderboat to see something different? Ask Matthew McConaughey...

@StanShunpike He's being a bit passive-agressive to me because I have commented on several of his answers that they're not very good.

Glen The Udderboat

When we can't literally see, isn't it scientific to assume that the same rules apply as where we can see (unless a mathematical contradiction occurs)?

good question. if we trust induction from the past to the future, and from the here to the elsewhere, why don't we trust induction from the macroscopic to the microscopic i.e. to high-energies that we haven't tested?

is "anti-realism" in science just a particular kind of skepticism about induction?

@GlenTheUdderboat This is not the basis of QM. Observing effects things.

@innisfree Because that has turned out to be a bad idea, see the discovery of quantum physics?

Glen The Udderboat

7:03 PM

@0celo7 Fair point. I think.

@acurious certainly induction can go wrong!

@innisfree Yes, exactly. Isn't that what you are arguing for...? I think I misunderstood you.

Induction?

Electromagnetic induction

Electromagnetic induction is the production of an electromotive force across a conductor when it is exposed to a varying magnetic field. It is described mathematically by Faraday's law of induction, named after Michael Faraday who is generally credited with the discovery of induction in 1831. == History == Electromagnetic induction was discovered independently by Michael Faraday in 1831 and Joseph Henry in 1832. Faraday was the first to publish the results of his experiments. In Faraday's first experimental demonstration (August 29, 1831), he wrapped two wires around opposite sides of an iron...

:D

@ACuriousMind Is $(g\circ f)_*=g_*\circ f_*$ just the chain rule in disguise?

@0celo7 Yes, I believe it is

Alternatively, it's the statement that $()_*$ is functorial, but that's not a very helpful thing to say, is it?

7:10 PM

No, I realise that induction is fallible, but in some instances it's trusted much more readily than in others. For example, will the Standard Model still be correct when we turn on the LHC again? Most certainly. Will the Standard Model be correct if we perform particle collisions at 100 TeV or 10000 TeV? Agnostic.

Not helpful, but I do know what that means.

@innisfree We don't expect the laws of physics to be a function of cosmic time or whatever.

But of the energy level, absolutely.

They are both inductions - one forward in time, one to higher energies - neither is logically sound. But one we trust, the second we don't.

Glen The Udderboat

But it seems fair to assume that something that is (by definition) unobservable (the inside of a black hole) is like the observable. If not, it is merely mathematical fanciness. (And also not quite useful in any conceivable way.)

@innisfree We have evidence of stuff at higher energies, though. Take quark deconfinement.

@innisfree Isn't the reason just that it has turned out that the former almost always works, while the latter might work, but if it doesn't, it fails spectacularly?

7:12 PM

but @acurious, don't you see? that is itself an inductive argument! from the past to the future.

@innisfree Ah, yes

I see

Inductive, but with a good reason, perhaps?

Well. You have to start somewhere, believing in temporal induction seems a good thing to do

Because if you don't believe in temporal induction, you can't make any predictions, and you can't do science. What does knowledge even mean when you are sceptical of temporal indiction?

@ACuriousMind "believing" Is science a religion now?

Jimdalf the Grey

@0celo7 Some non-scientists would tell you it is. And some non-scientists believe it like you would a religion

7:14 PM

@0celo7 This is...one level below science, I'd say, we've just opened the epistemological engine hood

@ACuriousMind Yay epistimology!

@acuriousmind, you're quite wrong (this may be the only time I'm able to say that). Popper's model of science (which is sort of what we're all doing) is built upon only deductive reasoning. One of his motivations was to solve this so-called "problem of induction".

@innisfree The only people who get scientific evidence from long ago are the cosmologists, who use models which are constant in time to evaluate them.

So no one can say for sure.

This is where the battle lines are drawn: do you go with Popper, abandon inductive reasoning, and build science on falsificationism (which is entirely deductive)? or do you try to rehabilitate inductive reasoning? You can use Bayes' theorem, for example.

@innisfree I think only a mix can work.

7:18 PM

@ocelo what can no one say for sure?

That the laws of physics and fundamental constants are invariant.

Mainstream physics has induced this, as you put it.

@0celo7 Consider situations in which there are competing theories (e.g., f(R) metric gravity vs dark matter). There is no consensus or proof of one vs the other, so we can choose (if we want) to believe one over the other. QED: science does indeed allow for belief

@KyleKanos I was commenting more on the word belief than the action thereof.

Off to shovel snow for the final time this year.

@innisfree Hm. Well. You're right, the Popperian conception of knowledge is really deductive.

Now I am very confused because I thought I had understood how we use Popperian knowledge as a basis for our actions, but what the hell do you do with knowledge, then?

Jimdalf the Grey

@KyleKanos Ahh, I love how my field is the prime example of being able to decide for yourself what the correct science is

7:21 PM

You can't base any of your actions on the assumptions that anything will happen again as it once did, right?

@JimdalftheGrey There's also ACuriousMind's favorite: Bohmian vs Copenhagen

Because that'd be induction

@KyleKanos Shut up! (You don't necessarily need to calculate something ;) )

But I want to work with real things :D

I think Popper realized it didn't work. He tried to salvage something by claiming theories could be "corroborated" by experiment, and that we should have greater faith in theories with higher degrees of corroboration. But logically that's all over the place, it's just sneaking induction in through the back door.

Popper put forward a set of rules for a game called "science", but just like e.g. chess, the rules don't tell you why anyone would bother spending their life playing. Popper couldn't fully justify why his rules would lead us to knowledge quicker than any other set of rules.

The solution is to embrace induction via Bayes' theorem, and build a philosophy of science around it.

I can live with that, Bayes has always seemed quite reasonable to me

7:28 PM

but is has its critics too. Some people insist you just cannot assign numerical degrees of belief (probabilities) to arbitrary propositions; others claim prior dependence will always be an issue.

The question is - is there anything better? And, does it, after all, matter why we follow the rules of the game called science? Perhaps just success proves us right?

(for a suitable notion of proof, of course ;) )

How fringe is Bohmian? On a scale of 1 to 10 where 1 is magnetic monopoles and 10 is Lamarckian evolution.

I've been scrawling some math the last day or so and my hand hurts from all the writing :(

@Jiminion Lamarckian evolution is not fringe, it's wrong.

Jimdalf the Grey

@ACuriousMind So 11?

7:35 PM

(at least, that's what I learned in biology class)

@JimdalftheGrey Yes, on a scala where Lamarckian evolution is 10, Lamarckian evolution is 11

I'm also not sure of the scale

I'm not sure how fringe magnetic monopoles are

one louder, ok, then how about 10 is intelligent design

Is that even a scientific theory?

Yes, to be a Larmarckian nowadays is simply wrong; but when proposed it was a plausible theory. Bohmian mechanics, on the other hand, had problems and was treated with scorn from day 1.

Or is it philosophic?

7:36 PM

@KyleKanos I actually wrote an answer about them today. They're not really fringe, it might just turn out that they don't exist - but they're perfectly possible within our current frameworks.

Why is Lamarkism wrong again?

because of all the mice tails that were cut off.

@ACuriousMind You have too many math things for me to contemplate right now. I'll read it later

So bohmian is plausible but unproven and possibly able to be proven wrong someday? Could it someday be proven right?

@ACuriousMind $\mathbb{R}^3-\{0\}$ is homotopic to the sphere? :: cries because topology is so confusing ::

7:39 PM

@Jiminion Theories aren't proven, remember? ;)

Also, I have never understood what it means to say that an interpretation of QM is true, because they seem to me, by definition, to be indistinguishable experimentally

@Jiminion According to Sofia, we'll have some experimental proof in a few months.

Also, cosmic ray pressure terms suck

@ACuriousMind but they can be disproven? OK, then somday could Bohmian displace Copenhagen as the more accepted theory?

Screws up the jump conditions

@Jiminion We'd have a lot of textbooks to replace if it does ;)

Bohmian mechanics isn't a mere interpretation of QM; it's a hidden variables theory

How could there NOT be hidden variables (IMO)?

7:41 PM

@innisfree Have you seen an actually distinct prediction for an actually in principle possible experiment? No one seems to be able to give me a clear answer to that.

@ACuriousMind How is a 3-space homotopic to a surface??

@0celo7 Well, you can start with $\mathbb{R}$, and observe that it, when you remove a point, is just two points, homotopically, right?

Si

The two points are actually the 0-sphere

OH

Noice

7:43 PM

Now, you take $\mathbb{R}^2$, and remove a point. That's the circle, or the 1-sphere

I get it now.

@ACuriousMind Is contraction to a point a homotopy?

Jimdalf the Grey

@Jiminion why should there be?

No, but I don't think it can be formulated for finite Hilbert spaces, so it's no good as replacement for all of QM.

@JimdalftheGrey Because God is not a gambler.

Jimdalf the Grey

@0celo7 He has to have fun some time

7:44 PM

but i see better why you call it an interpretation of QM

@0celo7 Somethings missing there

(just figured out you can highlight and determine the precise reference.....)

@0celo7 how's that measure theory going?

@StanShunpike Abandoned.

No need for it really.

@0celo7 Yes, all retracts are homotopic to the identity

7:46 PM

Zeidler uses it, but I don't have time to read all of Zeidler.

@JimdalftheGrey I heard he drinks quite a bit too....

@Jiminion Yeah, he peed for 40 days and nights once.

@0celo7 how did you reach that conclusion so quickly? No use at all?

Jimdalf the Grey

@Jiminion Apparently he drinks so much that even his son's blood is just wine

Just curious

7:47 PM

@StanShunpike Collective consensus from h bar.

@innisfree What? I thought the main problem was doing QFT/proper relativistic QM with it? They can't even do pure spin spaces?

I didn't find a single person who told me I should learn it.

I actually won $104 last night.

Interesting. Good to know!

(It's weird. Ever since I said I don't care one whit about which quantum interpretation is right, I find myself here constantly talking about them :D )

7:51 PM

I am so excited about the Kepler mission. This is the second most important thing our species has ever done, right behind inventing the concept of delivery pizza.

@ACuriousMind You know Copenhagen is right.

No, I don't know what it means for an interpretation to be right. I keep saying that, and no one believes me ;(

@ACuriousMind no, i don't think so. i can't remember, but i think you need a continuous, differentiable wave-function. it's only because QM works so well that people want to interpret it.

@ACuriousMind You know, you just don't know it yet.

If a theory can't be proven can't it at least be substantially more suitable than others? (maybe that is subjective).

Jimdalf the Grey

@0celo7 If that's the way our perspective is doomed to be, then at least we can say we thoroughly exhausted our narrow field of view

7:54 PM

@ACuriousMind Schoredinger's ACuriousMind.

@Jiminion if you like reading, the stanford encyclopedia is superb plato.stanford.edu/entries/confirmation

@0celo7 ? I like Heisenberg better, I never liked wavefunctions to begin with :P

And I don't think I'm in any kind of superposition here

@ACuriousMind You're in a superposition of knowing and not knowing :D

Oooooh, @DanielSank , pleeeeease do your duty and go through the close queue. We're all out of votes here

It looks good, but it needs more postfixins.

Jimdalf the Grey

7:58 PM

Well, I'm not out of votes but there's nothing left in the queues for me

Do physicists use RPN?

I'd have 8 things in the queue, but no votes left

@0celo7 What's that?

Reverse Polish Notation?

Yes, my family $\subset$ engineers all swear by it.

Reverse Polish notation

Reverse Polish notation (RPN) is a mathematical notation in which every operator follows all of its operands, in contrast to Polish notation, which puts the operator in the prefix position. It is also known as postfix notation and is parenthesis-free as long as operator arities are fixed. The description "Polish" refers to the nationality of logician Jan Łukasiewicz, who invented (prefix) Polish notation in the 1920s. The reverse Polish scheme was proposed in 1954 by Burks, Warren, and Wright and was independently reinvented by F. L. Bauer and E. W. Dijkstra in the early 1960s to reduce computer...

Basically instead of writing 3+4, you write 3 4 +

8:02 PM

That sounds confusing

That confusing sounds

I get confused enough always whether $f\circ g$ is f after g or g after f, I don't need another confusing notation :P

Forth is RPN and Forth => Postscript.

It's a big thing in programming, particularly with stack-oriented languages (e.g. Forth)

So writing 3 + 4 is semi reversed?

@infinitesimal It's normal, for crying out loud!

8:03 PM

I personally find it dumb

Jimdalf the Grey

Okay people: sanity check time

@ACuriousMind :(

Forth is incredibly powerful and incredibly hard to read.

@JimdalftheGrey Yep, you're slightly insane.

Jimdalf the Grey

Am I crazy or is finding the schwarzschild radius through newtonian gravity completely the wrong way to do it?

8:05 PM

Semi insane ;-)

@JimdalftheGrey It's the wrong way to do it, but doesn't it lead to the same answer?

It works, but is wrong.

@JimdalftheGrey Couldn't it be "both" ?

Jimdalf the Grey

@KyleKanos Yes, but I'd call that a coincidence

@KyleKanos If it's wrong, then why would it lead to the same answer?

Jimdalf the Grey

8:06 PM

I ask because of this answer:

1

A: What is the reasoning behind the idea that light cannot escape from a black hole?

The speed of light is invariant, so it is not possible to accelerate it above $c$. From Newton's law we know that the escape velocity $v$ from any given mass is $v_{esc}=\sqrt{\frac{2 G M}{r}}$ so if the mass $M$ is high enough and the distance $r$ to the mass small enough $v>c$. Since nothing ca...

@Jiminion Because try it for other metrics & it doesn't work?

It's wrong because in Newtonian mechanics light must be treated massively.

@JimdalftheGrey I agree.

Jimdalf the Grey

How has it gotten more votes and agreement than my answer which is basically "GR happened and gave us this"

I'm being fought

seriously

The escape velocity formula requires that which is escaping to be massive!

Jimdalf the Grey

8:07 PM

wot?

Massive, sorry.

@KyleKanos but what are the odds that it coincidentally gives the correct result?

Jimdalf the Grey

phew

@Jiminion hindsight says 100%

$\frac{1}{2}mv^2$ doesn't make sense for a photon.

@Jiminion No idea. Probably small

8:09 PM

lol, the question we undeleted is hot now

Rising from the ashes :D

@KyleKanos I did get called out... somebody must be touchy about the movie, it's pretty easy to see how a different number changes things. At least I wasn't downvoted for it!

@tpg2114 I personally found your answer to be better than the others.

Thanks, but people have a hard time believing that approximation would ever work. The same was true on the platform diver question

Nooo, I don't want a badge for saying "Momentum is always conserved". :(

8:12 PM

But I also haven't tried to go to great lengths to explain how/why it works

@ACuriousMind Hah

And it looks kind of like magic

Jimdalf the Grey

@tpg2114 did I hear someone mention magnets?

I think that is the same root word.

8:36 PM

@ACuriousMind Beautiful exercise: Prove that the Poincare lemma holds globally on a contractible manifold by making use the homotopy operator.

:: weeps at the beautiful union of geometry and topology ::

Ooh, you have use the exponentiated Lie derivative. Fancy.

@ACuriousMind Hmm. What is the pullback from a point? Surely it is zero, right?

@0celo7 What is a "pullback from a point"? I know many things you can pullback, a point is not one of them

@ACuriousMind Define the operator $\Phi_t$ such that it contracts all the points on the manifold: $$\lim_{t\rightarrow\infty}\Phi_t(x)=x_0$$ Let $\alpha\in\Omega^p(M)$. Then what is $\Phi^*_\infty \alpha$?

i.e. $\Phi_\infty$ is the trivial map $M\rightarrow x_0$

That's going in the wrong direction, the pullback would take a form on $x_0$, not on $M$.

@ACuriousMind So $\Phi^*_\infty\alpha$ does not make sense?

That why it's the pullback - it takes something living on the image and pulls it back along the map

Nope, $\alpha$ lives on $M$, you can only push it forward.

8:45 PM

I thought we pullback forms?

Yes, you're right, you can't pushforward a form, but given a map $M\to N$, you can only push things on $M$ forward and pull things on $N$ back

So you can't do jack with your $\alpha$ and the map given

(sry for typos, if you saw that)

Hmm. The definition of the homotopy operator is given as $$\hat h=-\int_0^\infty dt\,\Phi^*_t i_\xi$$ where $\Phi_t$ is the flow of $\xi$. I need to check that $d\circ \hat h+\hat h\circ d=\operatorname{id}_{\Omega^p(M)}$

So you're saying that applying that operator to $\alpha$ does not make sense?

I have no idea what that thing is, my understanding of homotopy is completely intuitive - continuously deform a thing.

But yes, I am saying that, given a map $M\to N$, it does not make sense to apply the pullback to something that lives on $M$

I get what you're saying. I have to double check the notation of the book.

Hmm. Does $\Phi^*_0=\operatorname{id}_M^*=\operatorname{id}_{\Omega^p(M)}$ make sense?

I doesn't to me.

@0celo7 It does

8:53 PM

How?

$id_M$ is a map $M\to M$, so you take forms living on the image ($M$), and pull them back to forms on the source ($M$). It's kinda evident that pulling back along the identity will do nothing to the forms, so $id_M^* = id_\Omega$.

I'm going through the derivation of the operator right now.

Jimdalf the Grey

come on guys, I know you're out of reviews in the queue, but you can still VTC. This question is so obviously a dupe. We can get it closed. Why is my vote the only one so far?

@ACuriousMind Ayy, you've done a number on my poor soul. If $\phi:M\rightarrow N$, then what is up with this integral: $$\int_M \phi^*\omega=\int_N\omega$$ Is $\omega$ in the exterior algebra of $M$ or $N$?

If $\phi$ is a diffeomorphism, then can't we take the pullback of $\omega$ even if it is in the exterior algebra of $M$?

Answers to questions: Yes. Yes. Think, and look at the integral. The answer what $\omega$ is is right there

9:06 PM

@ACuriousMind The problem is that the author says it is in $M$.

That's wrong

No matter what he says, that's wrong, you can't integrate a form on $M$ as $\int_N\omega$ unless $N$ is a submanifold of $M$ whose dimension matches the degree of $\omega$.

Either you're reading the text wrong, or the author has managed to confuse himself about how pullbacks and pushforwards and the whole shebang work.

@JimdalftheGrey Nope, I can't VTC.

I've run out of votes, not reviews.

I actually always run out of votes, not reviews, because I'm apparently one of the few people who actually cast the first votes on questions

@ACuriousMind Are flows diffeomorphisms?

@0celo7 Only for nice vector fields

Define nice?

Nice = complete, and: A vector field is complete if its flow curves exist for all time.

See, a physicist would never ask what's nice there. Physicists just nod and are happy with nice things.

9:16 PM

@ACuriousMind Do diffeomorphisms preserve the dimension of the manifold?

@0celo7 What kind of question is that?

What is a diffeomorphism for you?

@ACuriousMind Invertible, differentiable map

I get the feeling you're just looking at a bunch of formulae and memorizing them instead of seeing any concepts

Okay, and what does it, on the local charts, for the map mean to be invertible?

@ACuriousMind Bijective?

Exactly

Does that answer your question?

9:19 PM

@ACuriousMind I'm seeing contradictory information within the same book. Forgive me for trying to figure out what I know is correct.

@0celo7 I'm just trying to see where the problem lies - you know that a diffeo is bijective on the local charts. Can it therefore happen that one of the charts is $\mathbb{R}^2$ and the target chart is $\mathbb{R}^3$, for example?

(Yeah, strictly speaking, bijectiveness is not enough here, but smoothness + bijectivenes does the trick)

@ACuriousMind I guess not. Hence why my question was confirming what I thought I knew.

Any opinions on this question? physics.stackexchange.com/questions/162744/… I answered it but

I don't understand why the question has 6 upvotes and was posted by a 12.4k user!

@NeuroFuzzy I actually don't understand the question

Integral laws are simply ugly

And so I don't understand the reasoning about retardation and such in the question.

The fact that there is a time derivative under the surface integral on the RHS tells me that I simply shouldn't try to intuitively reason with this thing in any way

But people like intuition, and that's where the upvotes come from, I think, it happens all the time

user54412

9:35 PM

@JimdalftheGrey Newtonian approaches to black holes are completely wrong. I've harped on this before, but to no avail.

@ACuriousMind What if I define a $p$-form at that point? Can I then pull it back onto the full manifold using the flow?

Jimdalf the Grey

@ChrisWhite at least I'm not crazy...... about this

@0celo7 How many different $p$-forms can you define living on a point?

@ACuriousMind 0-forms. Sigh.

I have no clue what this guy is talking about.

How is a diffeomorphism mapping a whole map into a point?

Is homotopy a diffeomorphism?

@0celo7 It can't, a diffeo is bijective

What are you reading?

No, being a homotopy (equivalence) is far weaker than being a diffeo

9:38 PM

@ACuriousMind Fecko "Differential Geometry and Lie Groups for Physicists"

It was shady at first but this is pretty bad.

@0celo7 Well, if he writes what you say, which I don't doubt, then either he or I are horribly wrong.

@ACuriousMind I checked Lee, which should be right, correct?

I think Lee agrees with you.

And what you say makes sense.

Yay!

And I agree with you.

And yes, Lee is pretty good, as I hear

9:45 PM

@0celo7 this ncatlab.org/nlab/show/Poincare+lemma has a really nice proof of the Poincare lemma the way you're trying to do it (I think)

But I'm trying to figure out what the hell is happening here.

@bolbteppa I've seen a proof, I just have a major issue with this one.

If you are integrating $\int_N \omega$ then you are integrating a function on a manifold $N$, nobody has mentioned exterior algebras

@ACuriousMind I guess I missed something fun?

@ACuriousMind Wtf. Look at the last equation in that article. Is that a form being pulled back onto the first space??

@ACuriousMind It's stuff like this that kills self-studyers.

@DanielSank It's just that most active users seem to be out of review actions/close votes, and the queue is quite full

9:51 PM

Hey. I am working on an electrostatic potential problem (first year physics). In the scenario, there is a solid sphere of radius $R$ inside of a thick conducting spherical shell (with a charge $+q_{shell}$. The potential at the center of the sphere is the same as the potential at infinity. The questions asks for the ratio between the charge of the inner sphere to the charge of the outer shell when the sphere is conducting and when the sphere is nonconducting (with uniform charge distribution).

What is the significance of the potential at the center?

@0celo7 Yeah, they pull the form back onto the "homotopy cylinder" and then integrate the interval out

@DanielSank is there a way to make a note to a user from his profile site? I have a question the Savanna, but it's not a from the types of questions asked in this site, it's something else.

@ACuriousMind Is the homotopy cylinder $[0,1]\times X$?

@0celo7 Yes

My topology lecturer always called it that, I find the picture quite nice

But that could also be because the spaces I think of most are just circles, spheres, and cylinders, so everything that has an interval in it is a cylinder :P

I'm interested in the flyby anomalies - en.wikipedia.org/wiki/Flyby_anomaly

Does anyone know if there is any flyby data from the Juno spacecraft? It made a flyby, but I can't find any data.

9:56 PM

Alright, skipping this chapter. I think in his attempt to make this accessible he killed himself mathematically. He mentions that this is a simplified version of the proof given in Flanders, perhaps I should look there.

I don't know why he did this.

@0celo7 here math.stackexchange.com/a/697674/82615 is an explicit example of what a pullback is, you see the intuition for a pullback is just to talk about a change of variables

@bolbteppa I know what the pullback is. I got confused by a bad book is all.

Well, I don't so I'll check it out lol

That book is actually really good, it explained conformal Killing fields to me better than any other source for example

@bolbteppa Can you explain what the heck is going on in chap 9 then??

How are we pulling back a form from a point, when it is not even defined on the point?

9:59 PM

I think you guys are taking manifolds too seriously, tangent vectors as equivalence classes of curves is terrfying until you ask what classical baby concept it's just formalizing, same for pullbacks and pushforwards and forms

A form is defined at a point

The pullback is supposed to be zero, but I have no clue how to show that.

@bolbteppa But homotopic contraction to a point gives a 0-dimensional manifold. No way that is generated by a diffeomorphism (flow of a vector).

A differential form is a function on a manifold $\omega : M \rightarrow \mathcal{L}(T(M);\mathbb{R})$ (something like that)

I know what a form is. But you can't define $p$-forms on a 0-dimensional manifold for $p >0$.

It maps points on a manifold to forms, $\omega(p)$ that take in $k$ vectors

user54412

Everyone's been complaining about running out of close votes. I just reviewed 16 questions: 8 I voted to leave open, 3 I skipped, and I only closed 5 (2 of which I thought were being closed for the wrong reason).

10:02 PM

@bolbteppa Do you have access to the book right now?

What page of Fecko is that on?

193

I guess I don't understand in general how the flow, a diffeomorphism, creates homotopy to a point.

Okay one minute, yeah diff forms is a subject for which no good book actually exists, but they are all good for certain things, this is the best thing I've ever seen on forms lol em.groups.et.byu.net/pdfs/publications/formsj.pdf

@ChrisWhite Well, the others complained about running out of reviews, not votes, so it seems somehow a lot of questions are unnecessarily flagged

I'll go check if I'm the culprit..

@ChrisWhite I ran out of both.

user54412

10:06 PM

Can one run out of reviews without votes?

Yes

You have 20 reviews and 24 close votes

You can spend all 20 reviews on Leave Open

user54412

Ah, but you also can't review if you've cast all those close votes outside the queue, right?

Right

Correct

user54412

Well, I wouldn't be surprised if I differ from the majority in my reviews.

10:07 PM

Anyway, something's filling up the queue far faster than usual

@ChrisWhite We cant tell yet, you've been the first or second reviewer for quite a lot of them

@bolbteppa that PDF is like the best intuitions of MTW but written in a way that makes sense.

MTW is the only thing I've seen really make an effort to use visual representations of forms besides Wikipedia

Yeah I like MTW but it's so scattered and compared to Landau's GR I don't think the trade off is good enough, but for EM I think forms really illuminate things

Jimdalf the Grey

@ACuriousMind That seems to indicate there's far more than we can handle in a day

@0celo7 the homotopy map he defined on page 193 is $H(t,x) = e^{-t}x$ and then the end of the proof has the pullback of $H(\infty,x) = 0$ right?

@bolbteppa No the one in exercise 9.2.3

The integral

10:15 PM

@JimdalftheGrey Right, but why? What happened today?

Jimdalf the Grey

@ACuriousMind Not just today, this whole week

You said "I guess I don't understand in general how the flow, a diffeomorphism, creates homotopy to a point", but the book says "the flow itself (the shrinking homotopy) is $\Phi_t : x^i \rightarrow e^{-t}x^i$" which means that $\Phi(t,x) = e^{-t}x^i$ is the flow/homotopy shrinking any point on $\mathbb{R}^n$ to the origin as $t \rightarrow \infty$, this is what they are exploiting in the proof, albeit through the pullback of this

user54412

@ACuriousMind Mercury is approaching greatest elongation?

@bolbteppa Alpha is on $M$, so how are they pulling back from that point?

@ChrisWhite Damn, you're right, and I forgot to sacrifice a goat.

10:19 PM

@KyleKanos is there a way to make a note to a user from his profile site? I have a question the Savanna, but it's not a from the types of questions asked in this site, it's something else.

@Sofia No, we are explicitly not a social network. The only way to talk to users is through comments or in chat, if they are there.

@0celo7 The flow $\Phi : M \rightarrow M$ is defined on $M$ as stated beside the 9.2.3 box, so the pullback makes sense right?

@bolbteppa But $M\ne$ a point, right?

@bolbteppa suppose I have a manifold and at each point a tangent space. Let's now suppose I form the tangent bundle of all the tangent spaces. Are the points in tangent bundle the same thing as the points on the original manifold?

Doesn't the contraction to a point mean we get a 0-dimensional manifold?

Oh the double tangent bundle?

There's a wiki article for that.

10:27 PM

Always think of the circle, people.

Say what? I'm talking about the regular tangent bundle.

The tangent bundle of the circle is the cylinder.

(infinitely long cylinder, but cylinder)

@StanShunpike Do you mean this? en.wikipedia.org/wiki/Double_tangent_bundle

@0celo7 no i meant the regular one but thats cool.

@ACuriousMind If I contract a manifold to a point, is it a zero-dimensional manifold?

10:30 PM

@0celo7 The point is zero-dimensional, yes

@0celo7 No contraction to a point just means you suck every point in your space to a certain point, it's just the normal geometric image you know, so contracting all of 3-d space to the origin is just mapping all your points to the origin, it's not changing dimensions or touching that question it's just mapping points to points, so contracting a circle of radius 2 to a circle of radius 1 is just a map. Why do things all of a sudden change when you map everything to a circle of radius 0?

Well now you are disagreeing with @ACuriousMind.

You can't contract a circle to a point.

It is possible that you are misapplying the word "contraction"

@ACuriousMind Contraction is homotopically shrinking the manifold to a point.

@ACuriousMind I regret it so much. Savanna has wide knowledge, I wanted to ask a couple of things, quite difficult and highly professional.

10:33 PM

Except the people filtering in late are the partiers, so you end up with drunken makeouts in the living room and the next roommate to return home has to sleep in the hall lounge orbital.

The shrinking homotopy is defined as a map on page 193, it tells you that you are just shrinking the space but only in the special case of $t = \infty$ do you shrink everything to a point, that is the point of 9.2.1

Since bolbteppa seems to have your book in front of him, he's probably better suited than I to decipher what the book is telling you

@0celo7 contraction is not homotopically shrinking the manifold to a point, contraction is literally just "contracting" geometric objects, i.e. stretching or shrinking them, it's more general than just turning everything into a point, for example you have to be able to talk about contraction on manifolds with holes on them where you can't map curves to a point

@0celo7 Pauli didn't speak of additional room-mates, his concern was only the pairs. So, you are "not even wrong".

@bolbteppa He defines contractible on p.192 with regards to the point.

@Sofia The lone electron can't be in the other orbitals.

10:37 PM

@0celo7 I think this version of the principle is not necessarily as strict as the real one. :P

@ACuriousMind ;P

Jimdalf the Grey

@DavidZ Bam! He was dead before he hit the ground. Fastest mod in the west, eh?

@0celo7 Yeah but he defines it using the limit and look he's taking a limit of the image of any point $\Phi_t : M \rightarrow M | x \mapsto \Phi_t(x)$ i.e. he's taking a limit, w.r.t. of the image value of this map, and saying that the map is contractible to a point if the limit $\lim_{t \rightarrow \infty} \Phi_t(x) = x_0$ holds for every $x$ in $M$

@JimdalftheGrey @DavidZ That was pretty impressive.

I refreshed and saw if for the first time, refreshed again and it was gone.

@JimdalftheGrey That user is asking increasingly weird questions. Like, the questions get exponentially weirder, and they linked me a site about ectoplasm to show that momentum isn't conserved

Jimdalf the Grey

10:40 PM

Still.... Lightning reflexes with the close button

He must have been crouched and ready to pounce for a while

poor little antelope

@ACuriousMind I think that guy just wanted to know that conservation of momentum arises naturally when the Lagrangian modelling your system allows for translational invariance, i.e. he was basically asking if Noether's theorem can fail even when it's hypotheses are satisfied

@bolbteppa And then they gave me a link about ectoplasm

@bolbteppa Is this a different definition of the one usually used in topology?

@StanShunpike the tangent bundle is just the union of all the tangent spaces right? It needs to be isomorphic to your manifold in the sense that a map from the manifold to the tangent bundle maps a point to the tangent space at that point

@ACuriousMind Well? Did you answer his ectoplasm question?

10:47 PM

@ACuriousMind re the integral form question: Good that it's not just me.

@0celo7 it's more just a special case of the $H : [0,1] \times X \rightarrow X | (t,x) \mapsto H(t,x)$ homotopy map in en.wikipedia.org/wiki/Homotopy but instead of $H(0,x) = f(x)$, $H(1,x) = g(x)$. The intuition is to picture $f$ as the top arc of the unit circle, $H(0,x) = f(x) = \sqrt{1-x^2}$, $g$ as the bottom arc $H(1,x) = - \sqrt{1-x^2}$ and $H(t,x)$ as any curve inside the circle. You picture $H$ as just continuously deforming the top arc into the bottom arc, it's very natural.

They just defined it using $H : [0,\infty] \times X \rightarrow X$ such that $H(0,x) = e^{-0}x = x = f(x)$ and $H(\infty,x) = e^{- \infty}x = 0 = g(x)$ as a way to say we deform any curve into a single point (interpreting it as a function mapping all points to zero, so this may have caused the confusion).

The Poincare proof on the page I first linked you to uses the more standard $[0,1]$ notation but same idea

Change "but instead of" to "with", edited my post lol

So we're still just mapping $M\rightarrow M$?

And thus the pullback is acceptable?

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