The h Bar

0celo7

17:07

@ACuriousMind Let $v_1,\dotsc,v_k$ be lin. indep. elements of of a vec. space $V$ and $w_1,\dotsc,w_k$ be elements of $V$ s.t. $\sum_{i=1}^k w_i\wedge v_i=0$. Since $\{v_i\}$ is a minimal spanning set, it's a basis. Thus there are scalars $h_{ij}$ s.t. $w_i=\sum_j h_{ij}v_j$. Then $\sum_{ij}h_{ij}v_i\wedge v_j=\sum_{ij}h_{[ij]}v_i\wedge v_j=0$. Now how does this imply that $h_{[ij]}=0$, instead of just the whole sum vanishing?

Is it because $v_i\wedge v_j$ span $\Lambda^2V$, so they're linearly independent?

@ACuriousMind My physics class unanimously agreed that real people don't write in cursive.

FenderLesPaul

17:31

@0celo7 sup

0celo7

@FenderLesPaul Preparing my mind to learn about Cartan geometries

FenderLesPaul

Ooh fun

0celo7

@FenderLesPaul Actually it should be.

FenderLesPaul

@0celo7 dude I'm in a pickle

0celo7

It's a more abstract way of classifying geometries

FenderLesPaul

17:34

So I really wanna go to UCSB

0celo7

The sad fact is that most of the motivation comes from algebraic geometry

ACuriousMind

@FenderLesPaul I don't see a pickle yet.

FenderLesPaul

sorry

mom called

but I got this email from Wald

which, among other things, said "I was very pleased to learn that you have been admitted to our Ph.D. program...So, I very much hope that you will accept our offer."

and now if I pick UCSB I'm going to feel so shitty

:(

ACuriousMind

That's what we Germans call "whining at high level", but I understand you

0celo7

@ACuriousMind Please translate

ACuriousMind

17:42

Heulen auf hohem Niveau

FenderLesPaul

@ACuriousMind I don't mean it like that

0celo7

Don't know that one either.

FenderLesPaul

:(

ACuriousMind

@FenderLesPaul You mean that you're going to feel shitty because you're disappointing Wald although you're picking an excellent option, right?

Bernard Meurer

@FenderLesPaul But if you don't pick UCSB @DanielSank will be sad too

@FenderLesPaul Heartbreaker

2

Slereah

17:47

I am pretty bad at asking clear questions

0celo7

@ACuriousMind So a G submodule is a subspace of a G module that is closed under the action of the representation of G?

Slereah

Also I think I got a crazyman answer

Time dilation is the counter part of Length contraction. If you hold these both simultaneously within your calculations you find that your curve is closed. I have my own paper; researchgate.net/publication/… -Chapters 2.3 and 2.4. But it might be too complicated as the whole physic is explained without a mass, so even the need to discuss such a topics is limited. Basically the "timelike curve" REMAINS closed through Radioactivity-decay which causes long-wavelength"gravity"-photons. -But nevermind. — JokelaTurbine 46 mins ago

"I made the paper myself" and Researchgate are pretty big red flags

Also the title

ACuriousMind

@Slereah That user routinely pushes non-mainstream views. Just ignore it.

Slereah

I'll give him a lil flag

Another answer on the QFT time thing and it's the OTHER time assymetry I wasn't asking about :p

Fortunately I don't think there's any more things related to time assymetry

0celo7

@Slereah abuse

Slereah

17:53

The third answer should be the charm

@0celo7 I spit on Einstein and the evidence

0celo7

@Slereah :o

Don't let Mr. Duffield hear that

Maybe we should downvote Einstein some more

Slereah

shakes fist at old man Einstein

NeuroFuzzy

18:19

Need duplicate close votes plx physics.stackexchange.com/questions/240093/…

Satwik Pasani

Can someone explain the joke here?

smbc-comics.com/index.php?id=3919

NeuroFuzzy

@SatwikPasani it's just trying to be dumb. If P=0 then N * P=0 so P=N * P.

There's not much of a joke, heh

Satwik Pasani

I didn't get why the QED reference there

The dumb part, i figured. :P

NeuroFuzzy

Q.E.D.

Q.E.D. is an initialism of the Latin phrase quod erat demonstrandum, meaning "which is what had to be proven". The phrase is traditionally placed in its abbreviated form at the end of a mathematical proof or philosophical argument when what was specified in the enunciation—and in the setting-out—has been exactly restated as the conclusion of the demonstration. The abbreviation thus signals the completion of the proof. == Etymology and early use == The phrase quod erat demonstrandum is a translation into Latin from the Greek ὅπερ ἔδει δεῖξαι (hoper edei deixai; abbreviated as ΟΕΔ). Translating from...

Satwik Pasani

Oh. Thanks. I misinterpreted quantum electrodynamics.

*it to be

NeuroFuzzy

18:24

NP! :p

Satwik Pasani

But some of this guys comics are pretty great.

0celo7

19:58

timesfreepress.com/news/politics/state/story/2016/feb/24/…

Bwahahahaha

Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ

20:17

They considered making the Bible the official state book?

0celo7

20:34

@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ apparently

Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ

Sounds more like something Utah would consider.

Qmechanic

58

Q: A list of inconveniences between quantum mechanics and (general) relativity?

It is well known that quantum mechanics and (general) relativity do not fit well. I am wondering whether it is possible to make a list of contradictions or problems between them? E.g. relativity theory uses a space-time continuum, while quantum theory uses discrete states. I am not merely looki...

This question has already 3 close-votes. Should it be locked as well or just closed?

0celo7

@Qmechanic Was it bumped today?

Qmechanic

It says: active: 2 years ago.

0celo7

And you just happened to stumble across it?

ACuriousMind

20:38

@Qmechanic I actually don't think it should be closed. Sure, it asks for a "list", but it might as well ask the equivalent question: "Why are general relativity and quantum mechanics incompatible?"

@0celo7 It's rather clear what happened: Someone linked it on one of the LIGO or quantum gravity questions, someone else cast a close vote on it, Qmechanic saw it in the close vote review queue.

Qmechanic

@ACuriousMind : I have often linked to it. It is quite useful FAQ. Let's hear what other's think.

0celo7

@ACuriousMind How is that clear?

ACuriousMind

@0celo7 How else would it have gotten 3 close votes?

bolbteppa

@Qmechanic great answer to my Nambu-Goto/Polyakov question, but I could have done my calculation with Lagrange multipliers also, just hoping to find a quicker way to it.

If you hand-wave and say that the amount of $h_{ab}$ per unit volume $\sqrt{-h}$, w/ $h_{ab}=\partial_a X^{\mu} \partial_b X_{\mu}$, $\frac{h_{ab}}{\sqrt{-h}}$ is equal to some other magical $\gamma_{ab}$ per unit $\sqrt{-\gamma}$ such that $\gamma_{ab} \gamma^{ab} = 2$, then the calculation is one line, and physically it kind of makes sense. Is there anything to this?

ACuriousMind

How does that "physically make kind of sense"? What is an "amount of $h_{ab}$"?

The whole point is that that thing doesn't actually contain physical d.o.f.

bolbteppa

20:51

It makes some sense if you think of $\gamma$ as some other crazy metric which, although different, behaves, in a unit volume, exactly as $h_{ab}$ does, and since this stuff is plugged into path integrals and summed over volumes that's all that matters in the end?

Qmechanic

@bolbteppa : Also it does not explain/illuminate why $h_{ab}$ can be taken as independent variables, and not just a trivial renaming of the quantity $\partial_a X^{\mu} \partial_b X_{\mu}$.

Slereah

I put a bounty on a question and bam

Suddenly everyone upvotes it!

ACuriousMind

@Slereah Completely normal phenomenon

Slereah

But is it a fair one, tho!

ACuriousMind

Not really if you bountied your own question, I'd say. If you bounty someone else's question, I'd say it is fair - after all, you mostly put bounties on questions which you think are good, so the asker should get a bit of rep from it too.

0celo7

21:01

@ACuriousMind People smarter than me, apparently.

bolbteppa

Maybe it amounts to switching from viewing, say, a sphere as embedded in $\mathbb{R}^2$ via it's metric to viewing the same sphere being viewed as a hole missing from $\mathbb{R}^2$ via a new metric on $\mathbb{R^2}-{S^1}$, on average the amount of the line element in a certain direction per unit area remains the same at any point :\

That kind of idea

ACuriousMind

Uh...how does "viewing a sphere as a hole" give it a different metric?

How do you even talk about the "line element" of a hole?

(The defining characteristic of a hole kinda is that it is not there)

bolbteppa

The equation $\frac{\gamma_{ab}}{\sqrt{-\gamma}} = \frac{h_{ab}}{\sqrt{-h}}$ is 1.2.7 of Polchinski and is derived from the Polyakov action so it's just a question of understanding the meaning of it, maybe you can skip Polyakov and physically motivate this equation to get what you want

1.2.17*

FenderLesPaul

@ACuriousMind Yeah exactly

emotions n stuff

:(

Slereah

btw is the Polyakov action used for strings or is it for n-branes in general?

From my book on domain walls I suspect it's generalized to n dimensions?

0celo7

21:10

@Slereah you can generalize it

bolbteppa

@Slereah page 26 problem 4 has it math.berkeley.edu/~kwray/papers/string_theory.pdf (also page 21)

Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ

So you have made your final decision? @FenderLesPaul

Slereah

b/c apparently they use Polyakov action as an approximation of domain walls

Which are 2+1D

0celo7

what's a domain wall

Slereah

The transition between two vacuum states that have a...

The vacuum structure is like

Discontinuous

It's a homotopy thing

FenderLesPaul

21:13

@Sᴋᴜʟʟᴘᴇᴛʀᴏʟ I still have to visit all the schools

but I'm most likely going to pick UCSB

and work for Joseph Polchinski or Gary Horowitz

ACuriousMind

@bolbteppa The meaning is just that on-shell, the worldsheet metric is actually proportional to the metric the sheet has as a submanifold

0celo7

@Slereah what?

FenderLesPaul

or Don Marolf

ACuriousMind

But you have to note that it is an equation of motion

Slereah

The vacuum's toplogy has a 0-th homotopy that isn't trivial

ACuriousMind

21:14

Derived already from thinking the metric on the world sheet is actually made of independent d.o.f.

Slereah

$Z_n$ for instance

0celo7

@Slereah what are you talking about

bolbteppa

The meaning of $\frac{\gamma_{ab}}{\sqrt{-\gamma}} = \frac{h_{ab}}{\sqrt{-h}}$ is just that on-shell, the worldsheet metric is actually proportional to the metric the sheet has as a submanifold?

Slereah

Well the basic domain wall is a $Z_2$ theory, like $\varphi^4$

0celo7

@bolbteppa yes, up to a Weyl transformation

Slereah

21:16

$\phi^4$ is invariant under $\varphi \rightarrow -\varphi$

0celo7

@Slereah what does any of that mean

FenderLesPaul

@0celo7 life

it means life

Slereah

Then you have two different values for the vacuum

$\pm \varphi_0$

ACuriousMind

I guess if you think that the worldsheet has some sort of "independent existence" prior to embedding it into the target space of the fields living on it, you could say that this is naturally forced upon you upon embedding and identifying the $X$ with the actual coordinates.

But that's again starting from the Polyakov action and getting NG, not the way you seem to want

Slereah

If you have a space where $\phi(\infty) = \phi_0$ and $\phi(-\infty) = -\phi_0$

bolbteppa

21:17

So $h_{ab}$ is the intrinsic metric of the world-sheet right? Then $\gamma_{ab}$ is just the same (though extrinsic) metric viewing the world-sheet as embedded into some other space?

Slereah

Then you will get a domain wall in the middle

0celo7

please decide on $\phi$ or $\varphi$ and stick to it

ACuriousMind

It think NG -> P can only happen through Dirac-Bergmann because NG just has less d.o.f. How could the off-shell P action be encoded in less d.o.f.?

Slereah

I prefer $\varphi$ but $\phi$ is shorter to type

FenderLesPaul

@ACuriousMind if you have kids

will you tell them string theory stories for bedtime?

0celo7

21:18

@ACuriousMind what is Dirac-Bergmann?

ACuriousMind

@0celo7 The canonical recipe for turning a Lagrangian with singular Legendre trafo into a constrained Hamiltonian system

0celo7

@ACuriousMind as usual, cf. QoGS?

ACuriousMind

@FenderLesPaul Only after I've run out of sheaf stories ;)

Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ

Who needs kids when you have the Internet

:P

ACuriousMind

@bolbteppa No, the $\gamma$ is the worldsheet metric, the $h$ is the induced one

FenderLesPaul

21:20

@ACuriousMind "The Little Sheaf Who Could"

bolbteppa

Okay thanks, I think I can see why they only need to agree on shell now too

ACuriousMind

The statement of Polchinski's 1.2.17 is that the equation of motion for the "intrinsic" $\gamma$ is to be conformally equivalent to the induced $h$

@FenderLesPaul lol

I'll so use that some day.

Probably when I'm a crazy hobo ranting about math in the streets, but still...

0celo7

Sigh...

I don't even know what that's referencing.

Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ

Or like me ranting about self help on the Internet :P

bolbteppa

So let me see, if I think of terms like $h_{ab}$ as 'distortion factors' of the line element in cetain directions, then $\frac{\gamma_{ab}}{\sqrt{-\gamma}} = \frac{h_{ab}}{\sqrt{-h}}$ is saying that (on shell) the distortion per unit volume of the intrinsic metric is equal to the distortion per unit volume of the intrinsic metric, i.e. the area's are equal?

Sᴋᴜʟʟ ᴘᴇᴛʀᴏʟ

21:25

I think I can, I think I can,...

ACuriousMind

21:37

@bolbteppa No, the areas are not equal, that would be $\sqrt{-g} = \sqrt{-h}$.

bolbteppa

Okay forget about the i.e., how about the rest of it?

0celo7

@ACuriousMind ::puts on suit::

Oh yes.

ACuriousMind

@bolbteppa I'm not sure what "distortion per unit volume" means. (If you say that it is $h_{ab}/\sqrt{-h}$, then of course it is true, but I don't really get the information content of the statement)

bolbteppa

23:21

@ACuriousMind it means that, taking $ab$ as area with $a$ and $b$ as lengths:
$\frac{a_{extrinsic}}{a_{extrinsic}b_{extrinsic}} = \frac{a_{intrinsic}}{a_{intrinsic}b_{intrinsic}} \rightarrow b_{intrinsic} = b_{extrinsic}$
on shell

ACuriousMind

A component of a metric would be length squared (i.e. $a^2,b^2$ instead of just $a,b$ in the numerator), and the determinant contains crossterms. I really think you're trying too hard. The metrics are related by a Weyl transformation. If you really want an explanation in words of what that means, you should say that they measure angles to be the same.

0celo7

Didn't I say earlier that they're conformally related?

ACuriousMind

You did, and I did, too.

We should really say "related by a Weyl transformation", though

0celo7

i thought I said exactly that

Or they're the same up to a Weyl trafo

ACuriousMind

Scrolling up, yes, you did.

0celo7

23:33

@ACuriousMind But I have a terrible memory and don't remember any string theory

ACuriousMind

Doesn't mean you can't be correct once in a while ;P

0celo7

Broken clock.

@ACuriousMind about to see an opera and all I can think about are normal vectors.

I've caught a bad case of nerd sniping.

ACuriousMind

23:49

Given the choice between an opera and differential geometry, I'd choose differential geometry, too :P

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