The h Bar

ACuriousMind

4:00 PM

Alright, chat session time!

David Z

@ACuriousMind so it is! Who's here?

Secret

there's heaps of people in the session

Balarka Sen

If there's any good mathematics, I am in. Otherwise, I've left this place 10 minutes ago.

David Z

@Secret the idea is that all those people can stick up their virtual hands and say hi

yuggib

@BalarkaSen there can always be good mathematics

Qmechanic

4:04 PM

Btw, I plan to unlock the question mentioned in this meta post on Dec 3rd. Then it is to old to migrate, but reviewers can of course still vote to close, if they want.

Balarka Sen

adding "good" before "mathematics" is redundant.

however, there can be bad physics.

Terry Bollinger

@BalarkaSen Massive elaboration of math without insight or objectives is good?

Balarka Sen

How can one do math without insight or objective?

yuggib

let's talk about the optimization of gagliardo-kohn-niremberg constants by means of fast diffusion equations

Terry Bollinger

@BalarkaSen Easily, I'm from the software world, I just write a program.

David Z

4:07 PM

@TerryBollinger I approve

Balarka Sen

Fair enough.

Terry Bollinger

@BalarkaSen That said, elaboration can be deeply insightful at times, so I'm not knocking it in general. I'm having great fun elaborating permutation matrices right now just to get familiar with their properties.

David Z

Well, anyway. While I have this conversation with @theMan going on, is anyone here new to chat or new to chat sessions?

I see a lot of unfamiliar avatars lurking in the background

Balarka Sen

Since there's a computer scientist around, let's talk about homotopy type theory.

Oh wait, but that'd remind me of $(\infty, n)$-topos.

0celo7

Bad mathematics = set theory, topology, analysis, algebra

Balarka Sen

4:09 PM

Oye.

0celo7

Category theory

I'm sure there are more. Those are just the common offenders.

Terry Bollinger

@BalarkaSen Hah, I wish! Deep is my respect for the range of math, but you'll need to go to someone else for that.

Balarka Sen

General Relativity would not exist without topology and analysis, mr. math-hater.

@TerryBollinger Are you aware of type theory? If so, I'm willing to hear.

John Rennie

@BalarkaSen Topology? I'm not sure I see how that was relevant to Einstein's work.

Balarka Sen

You have to define what a topological manifold is before defining what a Riemannian manifold is :)

yuggib

4:12 PM

@BalarkaSen Let's say that the physical theory would still be there, but we would not be able to explain or develop it...

David Z

→ 36 messages moved to Discussion on question about skaters with theMan and David Z

0celo7

A manifold is something that looks like R^n

Balarka Sen

Yeah, I mean, it would have taken more time had Riemann not already done it.

yuggib

@BalarkaSen and with that reasoning, then set theory and logic is fundamental for GR as well

Balarka Sen

@0celo7 *locally.

ACuriousMind

4:13 PM

@BalarkaSen You can say what continuity on the reals is without ever touching actual topology, so you can define that without ever really doing topology.

Balarka Sen

Well, you need to define topological spaces to define what a topological manifold is.

0celo7

@BalarkaSen implied.

Or you just don't define things?

It's like you've never read an intro GR book

John Rennie

I doubt Einstein would have concerned himself much with topology

He just took it as a given.

Slereah

I think the first guy to care about topology in GR was Herman Weyl

0celo7

Does the GDP of GR depend on topology

Slereah

4:16 PM

GR has no GDP

ACuriousMind

@BalarkaSen You can circumvent that pretty well, just look to any physics treatment of manifolds. Just always talk about charts and coordinates, you don't really need the idea of a topology on the whole manifold. I'm not saying that is a good way to do it, but it works well enough.

John Rennie

@Slereah GPS generates lots of GDP ...

ACuriousMind

Essentially, you just talk about submanifolds of $\mathbb{R}^n$.

Slereah

According to a SE answer, GPSs don't actually use GR :p

Terry Bollinger

@BalarkaSen because I've been pretty deep into neural-inspired cognition research in recent years, combined with a penchant for defining information as forming the dynamic boundary between quantum and classical physics, my perspective on 'types' and 'sets' is, um, very different from most folks.

Slereah

4:17 PM

The effect exists but is not used by the gPS

John Rennie

@Slereah it depends on what you mean by use I suppose

Secret

Whenevr I just learnt a new mathematical object (usually from abstract algebra), the first thing I do is try to draw part of a cayley table, or wrote some simple expressions to see what I end up with in order to learn the rules of that mathematical structure

It's like exploring a new place, but with pencil (and computers)

ACuriousMind

By the embedding theorems, that's not a narrower notion of manifold than the abstract viewpoint, and then you don't need to know what a topological space is, all the topology comes from knowing the topology on the reals.

David Z

Well, GPS is not reliant on GR. There's that myth going around that without accounting for GR corrections, the system would drift by 11m per day or some such thing, and that I've heard is not correct

Slereah

See? No GDP at all

John Rennie

4:18 PM

Well the clocks would drift if you didn't correct for time dilation

Slereah

Einstein is not rolling in GR money

All the money he got was from his fridge patent!

ACuriousMind

@Slereah No, he's rolling in his grave.

John Rennie

Einstein died a very wealthy man!!

Huy

still the same discussion?

Slereah

heheh

Terry Bollinger

4:19 PM

@Slereah he was not good fiscally, he tried to market a liquid-metal refrigerator once along with another famous physicist, I forget who...

Wo, you added the fridge, cool!

Slereah

Einstein's fridge is quite famous

Even if awful

Terry Bollinger

@JohnRennie from fame? Poor Tesla, there's a sad case...

ACuriousMind

I've never heard of Einstein's fridge.

Terry Bollinger

@ACuriousMind very noisy due to cavitation within the liquid metal coolant.

Slereah

Einstein had a patent on a fridge model

yeah it was quite noisy

John Rennie

4:21 PM

@TerryBollinger Bsically yes. For example he used his fame to bully Princeton into paying him a full salary in retirement.

Einstein refrigerator

The Einstein–Szilard or Einstein refrigerator is an absorption refrigerator which has no moving parts, operates at constant pressure, and requires only a heat source to operate. It was jointly invented in 1926 by Albert Einstein and his former student Leó Szilárd and patented in the US on November 11, 1930 (U.S. Patent 1,781,541). This is an alternative design from the original invention of 1922 by the Swedish inventors Baltzar von Platen and Carl Munters. == History == From 1926 until 1933 Einstein and Szilárd collaborated on ways to improve home refrigeration technology. The two were motivated...

Balarka Sen

Sorry folks, I have to run. Carry on with the physics!

Qmechanic

0

Q: When will the Winter Bash 2015 begin?

When will the Winter Bash 2015 begin? Any info on the non-secret hats?

2

Terry Bollinger

The cavitation is interesting because liquid metal bubble surface tension should increase the intensity of sonoluminescence-style heating at the final collapse. Hard to study when embedded in liquid metal, though.

Slereah

"Gas refrigerators have largely been phased out, because of the danger of carbon monoxide poisoning. In 1998, the Consumer Product Safety Commission (CPSC) warned about 22 carbon monoxide-related deaths in the US and 60 in Canada, even though a product recall had been started as early as 1990."

Einstein : A murderer???

Terry Bollinger

Carbon monoxide? Hmm, are we talking the same device?

Slereah

4:24 PM

Yeah they might not be talking about his fridge specifically

I dunno

John Rennie

@Qmechanic Ooh, hats!

Slereah

you know who was afraid of fridges?

Terry Bollinger

@JohnRennie Eh?

Slereah

Kurt Gödel.

Terry Bollinger

@Slereah now there's an amazing dude...

Slereah

4:24 PM

Among his various neurosis he was afraid of dying of monoxide poisoning from his fridge

John Rennie

@TerryBollinger Hats! The Christmas hats!

ACuriousMind

winterbash2015.stackexchange.com

Hm

Slereah

(he was also afraid someone would try to poison his food so he would only ate what his wife cooked)

Terry Bollinger

@JohnRennie ah, sorry, forgot 1 Dec!

Slereah

(he died of starvation, on a related topic)

Terry Bollinger

4:25 PM

Tesla was weird too, he and Goedel would have hit it off nicely.

ACuriousMind

The little people form 11134 on my screen.

Is that a countdown in minutes?

Terry Bollinger

Tesla liked to do triplets of everything...

David Z

@ACuriousMind wow I'm not quite sure what's going on there

@ACuriousMind I wouldn't think so, Winterbash should be starting really soon, and 11134 minutes is almost 10 weeks

wait no

10 days

Maybe it is, then...

Qmechanic

@ACuriousMind : It's counting down the minutes.

John Rennie

More on the Einstein fridge:

books.google.co.uk/…

Terry Bollinger

4:28 PM

Countdown? You must have a different screen.

ACuriousMind

hm, I have trouble deciding if those are 1s or 7s

@TerryBollinger You have to wait for the stick figures to arrange themselves into numbers

Terry Bollinger

@JohnRennie I was serious about liquid-metal cavitation as having some really interesting non-linear dynamics potential at the very end of the collapse. Alas, again, hard to study.

John Rennie

If we have nothing better to discuss I'd like to ask for views on a (highly downvoted) answer of mine ...

Terry Bollinger

@ACuriousMind ah... I have no idea what you are referring to? I'm not seeing anything different from the usual screen.

David Z

@JohnRennie works for me

John Rennie

4:30 PM

1

A: Quantum Joke (not a real joke, not a riddle)

The state $-|\text{Down}\rangle$ is the same as $|\text{Down}\rangle$ just multiplied by a phase factor (of $\pi$). So they are the same physical state and it doesn't matter which sign you use. I wonder if you are getting mixed up with the LCAO method for approximating molecular orbitals, for ex...

Terry Bollinger

@DavidZ me too

ACuriousMind

@TerryBollinger Do you have JavaScript disabled, perchance?

Terry Bollinger

@ACuriousMind probably!

John Rennie

This was a rather rushed, and possibly rather incomplete answer, but is it really three downvotes worth of bad?

If it is then feel free to say so - don't be shy :-)

David Z

@JohnRennie I wouldn't think so, but the question seems crappy to me

"crappy" being a technical term for close-worthy

Terry Bollinger

4:32 PM

"this would not apply to your coffee machine" does sound like downvote bait... :)

ACuriousMind

@JohnRennie Although I dislike the question much more, yes, that answer is bad (but it didn't downvote it because I didn't look at the answers after I VTC'd the question). You are correct that $-\lvert \psi \rangle$ and $\lvert \psi \rangle$ are the same state - but $\lvert \phi \rangle + \lvert \psi \rangle$ is a different state from $\lvert \phi \rangle - \lvert \psi \rangle$. (It's the difference between an overall and a relative phase).

Your answer sounds, when you say "it doesn't matter", as if you believed those two states are the same.

John Rennie

@ACuriousMind Ah OK, thanks.

David Z

Oh, I see, that makes sense. Yeah I missed that.

John Rennie

Hmm, now do i delete it when I'm only a few votes away from the [quantum-mechanics] gold badge?

I suppose I could wait till I've got the badge, then delete it :-)

David Z

Way to game the system

John Rennie

4:37 PM

Oh what the hell - deleted now. I'll just have to work a bit harder for the badge ...

David Z

A few votes shouldn't be hard to come by

ACuriousMind

Delete it! And don't answer any QM questions in the next few months so I can overtake you!

Terry Bollinger

@JohnRennie I just upvoted it because it demonstrates a common misconception in a way that is useful for other readers looking at that same issue... :) Oh shoot, you deleted it?

John Rennie

@TerryBollinger Well, it was wrong

Terry Bollinger

@JohnRennie ah... true, as nicely explained by A. Curious...

David Z

4:39 PM

While we're on it, what about the question? I'm tempted to vote to close but it'd be nice to get that fourth non-mod close vote, if anyone thinks it's warranted

John Rennie

Quantum mechanics seems far harder than GR

In the sense that you need far more precision when discussing things quantum mechanical

@DavidZ I'd feel a bit of a spoilsport voting to close it

Terry Bollinger

(Slowly I realize you answered your own question... it was only the question I upvoted, for reasons described.)

ACuriousMind

@TerryBollinger huh? You may be talking about another question, that question wasn't John's...

David Z

@JohnRennie Ah, well, up to you of course. I tend not to worry about such things myself.

Hazard of being a mod I suppose

ACuriousMind

Always remember, we hate fun.

Terry Bollinger

4:44 PM

@ACuriousMind correct, hmm, can I blame having JavaScript off for major brain toots?

ACuriousMind

Sure, if you're a robot.

David Z

I wouldn't have thought you'd be using chat without Javascript!

Or at least if you were, you'd know you have a script blocker

ACuriousMind

Maybe you weren't looking at the correct site for the countdown, either, @TerryBollinger? It's: winterbash2015.stackexchange.com

Terry Bollinger

@ACuriousMind AssessProbability[Assert->Robot(Bollinger)] ==> {[<1%].ExternalPerspective, [<0.0001%].InternalPerspective]. Nope, alas, I don't think I can use that excuse effectively... :)

@ACuriousMind that worked, thanks! Wow, that's... kind of interesting and kind of weird...

@JohnRennie QM is atemporal, GM is temporal to the max, QM is free of ordinary "info" (it has its own kind), GM is pure and unadulterated info to the max. Fascinating contrasts.

ACuriousMind

@TerryBollinger What do you mean by "QM is atemporal"?

The Hamiltonian and its exponential would like a word with you...

David Z

4:54 PM

OK, well we're a few minutes from the end but I should get going - it's 1 AM here

See everyone in two weeks!

Terry Bollinger

@ACuriousMind Every example of QM -- and I do mean every -- can be defined as a lack of specific resolution in the form of classical information, an ongoing state of the "integral of all possible histories". It's not just a characteristic, it's a way of defining QM.

John Rennie

@DavidZ You're in the far east?

ACuriousMind

@TerryBollinger The lack of a unique classical history is very different from being "atemporal", I'd say. Quantum mechanics has a well-defined notion of stepping forward in time, and it is essentially the same as in classical (Hamiltonian) mechanics, what is different is what kind of state space we operate on.

Terry Bollinger

@ACuriousMind The Hamiltonian is the temporal way of analyzing QM, a very classical perspective, and applying it to the max has interesting consequences...

ACuriousMind

Whereas you can turn this on its head again by Koopman-von Neumann mechanics and say the state spaces are essentially the same, but it are the operator relations which are different.

Terry Bollinger

4:57 PM

@ACuriousMind yes, but look at what you just did: You added classical observers to the QM world. All QM in our universe is bound and boxed in.

ACuriousMind

Because of these many different formulation, I think a blanket statement like "QM is atemporal" is not warranted.

Terry Bollinger

@ACuriousMind Interesting! But I'll have to look that up

ACuriousMind

@TerryBollinger Where did I add observers?

vzn

@ACuriousMind amusing, obviously half tongue-in-cheek by you & the writer (SE co CEO atwood), had not seen that & have seen quite a few old classic SE blog posts. can you give an example of that policy in play here on Physics?

Terry Bollinger

@ACuriousMind I won't disagree, in the sense that "atemporal" is a poor word for describing the lack of specific resolution of multiple time paths. That's the real issue: The nature of how the event evolved over time remains unresolved until observation.

ACuriousMind

5:01 PM

@TerryBollinger In my view, even thinking about "multiple time path" is clinging to a classical description of the world and trying to phrase QM in classical terms. QM doesn't know of your classical paths (and the path integral is kind of a red herring, most paths it integrates over you'll never see as solution to a classical non-stochastic motion because they're not differentiable), it only knows the "quantum path" - the path a state traces out in the quantum space of states under time evolution.

Terry Bollinger

@ACuriousMind any attempt to "describe" a quantum event unavoidably implies detection and thus the creation of specific historical information. Very few of our models are careful about that, making it hard to pin down information creation at times.

ACuriousMind

@TerryBollinger I'm not describing "events", I'm talking about a theory of mechanics - how the world works, how it evolves in time. Until we specifically talk about measurement and "collapse", QM does not know observers any more than classical mechanics does.

Terry Bollinger

@ACuriousMind wow, I would not disagree with that at all! My point is only that when we as classical creatures start poking such systems, they become subject to our concept of "classical" histories. Those in turn are most easily comprehensible to us as integrals of multiple histories -- a fiction, yes.

@ACuriousMind again, I think I agree fully, and perhaps am just saying it a bit differently?

vzn

re QM as "atemporal" have recently been trying to figure out a question that cuts into that. re periodic detection of photons... does anyone know experiments that look at time oriented statistics? ...

Terry Bollinger

@vzn There is a fellow who has had some interesting papers lately on "maxing out" wave functions, argh, German name, not well known...

ACuriousMind

5:06 PM

@TerryBollinger Hm. Maybe I'm a bit too used to people disagreeing. You're right, you didn't say anything against that (I was anticipating views you apparently don't hold, I apologize).

Terry Bollinger

@ACuriousMind don't apologize, I liked your characterizations better than my own! You are stating the key points better on the QM side.

Must be going now! It's been fun.

ACuriousMind

5:18 PM

@vzn Maybe our ruling that even solving problems more intricate than the usual homework problem is off-topic as homework-like, e.g. exemplified in this discussion about Kapitza problems

Especially JohnRennie's statement there:

> But then lots of people are interested in crossword puzzles, and my views expressed here shouldn't put off anyone who wants to ask or answer such problems here. I'm just not convinced it would make the site a more interesting place to be.

John Duffield

5:29 PM

@0celo7 : yes, I can prove that there are no CTCs in physical spacetime. Because spacetime itself isn't physical. We live in a world of space and motion, and we model this using the abstract mathematical "space" called spacetime. This depicts space at all times, so there is no motion in it. World lines only exist in this spacetime, they do not exist in space, and it's space that's physics. And CTCs are just special-case worldlines.

0celo7

@JohnDuffield Can I have an Einstein quote for that?

John Duffield

Probably. But better still, step outside later and point up to the starry sky. Then say to your companion: "Hey, there's a worldline! Just next to that light cone".

0celo7

I see worldlines all the time...

@BalarkaSen why

the morphisms are shitty notation

John Duffield

@I'mmostlyjustanidiot : You might be interested in this answer on the astronomy stack exchange.

@DavidZ : you should read A World Without Time: The Forgotten Legacy of Godel and Einstein.

vzn

@ACuriousMind thx for that, the idea behind the atwood policy post seems to be that "occasional scattered exceptions of stuff that is 'fun' is permissible..." (otherwise life would be near robotic...?) personally havent seen that around these parts at all. :| ... one of the old SE FAQ pgs used to say "bring your sense of humor" but maybe thats been deleted by now... :| ofc chat is quite facetious, but the main site(s) SE wide seem to have a low-tolerance :|

ACuriousMind

5:48 PM

@vzn I think (non-malicious) humor is not "banned" here, but...physics doesn't lend itself to "humorous" things, imo. Sure, you can make puns, and perhaps make some witty jokes, but I cannot think of an example that would be both funny and teaches something about physics.

And that's what posts on the site should do - they should teach something, you should know more after reading them than before. One can make jokes while doing that, but they will be largely unconnected from the meat of one's answer, and trying to be funny generally ends in one not being funny at all

vzn

@ACuriousMind agreed on one level although one might say the same thing about coding... ofc the policy post by atwood was about a coding site. find SE culture nuances interesting to study. the chat room is chock-full of humorous physics.

not even sure atwood is the one still working at SE... :|

ACuriousMind

@vzn I think he resigned some while ago to devote time to other projects? Note it says "emeritus" at the author tag of the "We hate fun" post

vzn

ah yes it was him who left awhile back. to be with his family or something.

0celo7

6:17 PM

@ACuriousMind Zee's book is pretty funny.

"after Lagrange invented the Lagrangian, Hamilton invented the Hamiltonian"

Or the solution for an exercise

"I'm not going to teach you how to do integrals"

Slereah

Apparently all quantum fields of finite extent in phase space obey some sort of quantum inequality

How disappointing

Although quantum inequalities only put technical limits on weird spacetimes

They do not forbid them outright

that would explain why that domain wall solution violated it with such glee

"This is equivalent to state that local measurements cannot allow any distinction between the vacuum state and, say, a N-particle state. "

:O

0celo7

6:37 PM

I still don't see why counting particles is an issue...

Slereah

"Then the Malament theorem asserts that, if the conditions 1-4 hold, $E_\Delta = 0$ for all ∆. In other words, it is impossible to detect the particle in any spatial region. No localization is possible!"

QFT gets a bit weird once you get under the hood

" In other terms, if a particle is localized, no dynamics is possible (except the trivial one)."

:D

"Moreover, by adding the further condition of absence of an absolute velocity, it can be shown that the theorem entails $E_\Delta = 0$ for all ∆ (see alvorson and Clifton, 2002, Lemma 2 of Appendix A), like in Malament theorem. "

0celo7

Is that just the Heisenberg thingie

Slereah

No it's a theorem for localized theories of QFT

Basically the reasons for people to use AQFT

0celo7

Why even bother with QFT

GR is so much nicer

Slereah

QFT increases the GDP

Huy

6:45 PM

stop with the GDP already holy shit

0celo7

^ the fuk

Slereah

Blasphemer!

0celo7

I don't know what that guy's problem is, LOL

How does this kind of QFT increase GDP?

How does any QFT increase the GDP

Slereah

QFT is used for solid state physics

Electronics and whatnot

all the good stuff

0celo7

Really

Explicitly or in theory? I don't need to understand quantum biology to plant food.

Slereah

6:57 PM

We don't really live in an agrarian society anymore

And explicitely nowadays, totally

Graphene and such use plenty of QFT

0celo7

7:21 PM

do they

source?

Slereah

arxiv.org/pdf/1511.08829.pdf

Bass

7:49 PM

hi guys, to prove that the Riemann curvature tensor is indeed a tensor, why is it enough to show that $R(X,Y)Z=X^\lambda Y^\mu Z^\nu R(e_\lambda,e_\mu)e_\nu$?

0celo7

@Bass I think because that shows that it's a multilinear map.

Bass

@0celo7 well, maybe I'm lacking some understanding, but the Christoffel symbols can be interpreted as some sort of multilinear map too? But they don't transform as a tensor..?

and, isn't the property above more than multilinearity, it is $C^\infty(M)$-linearity?

0celo7

Ok, that shows you that $R(X,Y)Z$ depends uniquely on the values of $X,Y,Z$ and the functions $R_{ijkl}$ at some point p, right?

Bass

$X,Y,Z$ being vector fields

0celo7

ok do Carmo defines a tensor by that property

so let's see if he says why the LC connection components are not tensors

@Bass well for tensors on smooth manifolds we take the vectors to be smooth

everything is smooth

Bass

8:00 PM

@0celo7 yep

0celo7

Jost defines everything in terms of vector bundles, not too helpful here

Bass

@0celo7 $R(X,Y)Z$ depends on the components of $X,Y,Z$, but also on $R(e_\lambda,e_\mu)e_\nu$. Doesn't this property make $R$ a tensor? If it depended on the components of $X,Y,Z$ only, it might depend on the coordinate system..?

0celo7

@Bass I'm...not sure

Getting confused by do Carmo now

@Bass what book are you using?

Bass

Zee's gravity book and Nakahara

ACuriousMind

@Bass Since the l.h.s is manifestly coordinate-invariant and you know the transformation behaviour of the $X,Y,Z$ components, this shows the transformation behaviour of the components $R(e_\lambda,e_\mu)e_\nu$ is precisely that of the components of a tensor.

0celo7

8:06 PM

Zee won't explain it like that

@ACuriousMind oh, derp

@ACuriousMind is $(X,Y,Z)\mapsto \langle \nabla_XY,Z\rangle$ a tensor

it's not linear in $Y$

ACuriousMind

@0celo7 it's not?

Both the covariant derivative and the...whatever inner product that is are linear, no?

0celo7

@ACuriousMind $\nabla_X(fY)=f\nabla_XY+\mathrm{d}f(X)Y$

ACuriousMind

Oh, you mean $C^\infty$-linear. I thought $\mathbb{R}$-linear!

Bass

@ACuriousMind oookay... the LHS is coordinate invariant, so it does not transform at all under coordinate transformations. $X,Y,Z$ transform contravariantly, so they each "receive" a coordinate transform matrix $T^\mu_\nu=\frac{\partial x^\mu}{\partial x^\nu}$ to transform. So on the RHS we have three times the $T$ matrix, which means that $R(e_\lambda,e_\mu)e_\nu$ must transform with three $T^{-1}$ matrices to cancel the $(T)^3$.

right so far?

This means that $R(e_\lambda,e_\mu)e_\nu$ transforms as a $(0,3)$-tensor, but the Riemann tensor is $(1,3)$. What am I missing?

ACuriousMind

@0celo7 You're correct it is not a tensor field because it is not $C^\infty$-linear.

@Bass Well...what is your definition of the components of the Riemann tensor?

@Bass Yes

Bass

8:19 PM

@ACuriousMind ${R^\kappa}_{\lambda\mu\nu}=\langle dx^\kappa,R(e_\mu,e_\nu)e_\lambda\rangle$?

okay, so the fourth transformation should come out there..?

ACuriousMind

@Bass Yep. So, now you know the transformation behaviour of the r.h.s, and you can deduce that of the l.h.s.

In slightly related news, does someone know a nice proof of the second structure equation for the curvature of a principal (or, better, Ehresmann) connection?

In other words, that $\omega([X-\omega(X),Y-\omega(Y)]) = \mathrm{d}\omega(X,Y) + [\omega\wedge\omega](X,Y)$ (modula prefactors) for a connection form $\omega$.

Cartan's magic formula looks as if it should give the result, but I can't get it quite to work

Bass

@ACuriousMind thx, as always!

ACuriousMind

@Bass No problemo :)

Also, nvm my above question, Cartan's formula works.

0celo7

8:56 PM

@ACuriousMind the Lie derivative one?

Huy

u wat m8

ACuriousMind

@0celo7 Yes

Hm

Now I end up with $\mathrm{d}\omega(X,Y) = \omega([X,Y])$.

Bass

9:11 PM

is it correct that $\nabla_{fX}Y=f\,\nabla_X$ implies $\nabla_{fX}\nabla_{gY}(Z)=f\,\nabla_X\left\{g\nabla_Y(Z)\right\}=(fX[g]+fg\nabl‌a_X)\left\{\nabla_Y(Z)\right\}$?

and, wtf am I doing wrong in that tex?

0celo7

@Bass most things

Bass

@0celo7 the $\nabla$, why does it appear as $\nabl\,a$

0celo7

user was removed, I got some rep back

ACuriousMind

@Bass ChatJaX is a bit weird sometimes, I don't see why it does that in this case

0celo7

@ACuriousMind what

pls show me how this happens

Bass

9:27 PM

must be some invisible character.. when I export it as TeX and insert it in a PSE question, it appears too.. but you cannot see anything inside the \nabla

ACuriousMind

@0celo7 A bit of patience, pls

Bass

another attempt, a bit simpler $\nabla_{X}\nabla_{gY}(Z)=\nabla_X(g\nabla_Y(Z))=X[g]\nabla_Y(Z)+g\nabla_X\nabla‌_Y(Z)$. looks correct, doesn't it?

Sir Cumference - Pies

Oh awesome, got my points back

ACuriousMind

9:45 PM

@0celo7 Here you go

0celo7

9:57 PM

@ACuriousMind Ah!

@ACuriousMind Why are you reading about basic things

ACuriousMind

@0celo7 I am not exactly reading about it.

0celo7

@ACuriousMind what are you doing then

ACuriousMind

10:58 PM

1

The concept of "light" described by modern physics seems, to me at least, an incomprehensible bundle of properties: a "universal constant," the "maximum speed," a "wave-particle" phenomenon, the invisible source of "visibility," the equivalent of the square root of "energy divided by matter," a l...

philosophy-of-science metaphysics reference-request

I have absolutely no idea what this question is asking for

An explanation of the physics without an explanation of the physics?

0celo7

@ACuriousMind Stupid question: why do we sometimes worry about horizontal/vertical vectors when talking about connections and other times...not?

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