The h Bar

obe

00:02

well... sorry I was forced to say that. please forgive.

ACuriousMind

@obe You can mimic 0celo7's awkward apologies for saying silly stuff all you like, it's not the same :'(

3

obe

I know. :(

obe

00:20

@ACuriousMind I don't remember him apologizing though, in fact it's difficult to imagine him saying sorry.

DanielSank

02:33

What data indicates that the universe is expanding via the metric as opposed to things just moving away from each other?

Secret

03:11

NbCeID 15

iopscience.iop.org/article/10.1088/0264-9381/32/17/175003/meta

Secret

NbCeID 16
http://www.nature.com/ncomms/2015/150804/ncomms8917/full/ncomms8917.html

Danu

06:46

@DanielSank What do you mean?

You do understand that, if it was not metric expansion, there would be a preferred "center of the universe", right?

Think about a fixed box with balls moving around in it.

Or, maybe more generally, you wouldn't have everything moving away uniformly (homogeneous & isotropic universe).

DanielSank

07:44

@Danu I never understood that. If I pop a balloon full of particles, all of the particles wind up moving away from all the others. What is the "kinematic" difference between moving particles and what astrofolks mean when they say "expansion"?

Danu

@DanielSank You cannot make everything move away from everything in the same way without expanding the volume, no.

Just think about particles in a box (no metric expansion)

Fix a single point in the box.

Now, you require that everything moves away from it---this is backed by observations of the universe.

Then pick a different point in the box. Your particles will not all be moving away from it.

Also it's not astrofolk but cosmofolk

Astronomy does not concern itself with GR (much)

(neither do astrophysicists)

DanielSank

08:37

@Danu gf calls herself astrophysicist, uses GR.

Ghost

hello!

a good friend of mine is also an 'astrofolk' and uses GR as well

Danu

@DanielSank I'm just basing that on the fact that they don't teach courses on GR in most astro degrees---nothing personal :P

and of course "our own" Kyle Kanos :D

Ghost

interesting... several astro based degrees here do teach GR

hello?

lol... just saw an article about 3D-printed pancakes.... love it!

Slereah

09:01

hello

Danu

@Ghost Hmm, maybe my sample size was simply too small ^^

Ghost

@Danu different places, different processes

Slereah

"Zee combined the idea of the BD theory with the Higgs-Mechanism of Symmetry Breakdown for mass generation, which led to a scalar-tensor theory with Higgs field as scalar field, in which the scalar field is massive (short-ranged). An example of this theory was proposed by H. Dehnen and H. Frommert 1991, parting from the nature of Higgs field interacting gravitational- and Yukawa (long-ranged)-like with the particles that get mass through it"

neat

Slereah

09:19

"In language that rarely appeared in the unpolluted pages of Annalen der Physik, Abraham (1912c, 1056) accused Einstein’s theory of relativity of having “exerted an hypnotic influence especially on the youngest mathematical physicists which threatened to hamper the healthy development of theoretical physics.”"

My god

He denied Einstein and the Evidence!

Ahahah

This paper talks about Einstein's theory of gravitation using a variable speed of light

(this was 3 years before GR)

Good old JD, always at the forefront of physics

Physics Meta

11:00

0

Q: Limitations of human learning

Are there any proposed solutions to the following problem: People can only learn a limited amount of knowledge in their lifetime. Physics and math theories are becoming more complex, and it becomes harder to understand them as time passes. On the other hand, a person's learning capacity is limit...

discussion

Slereah

He confused Physics Meta with metaphysics I think!

3

Danu

@Slereah This is pretty hilarious

yuggib

11:29

@Slereah I have been always confused by the "english" usage of the word Meta. In ancient greek, is a preposition that means "after"

Danu

@yuggib You can only have a meta discussion after a discussion ;D

yuggib

and metaphysics was called like that because the books by Aristotle were anthologized after the one about physics

@Danu :-P

but the meaning it has now is of self-reference

I find it quite misleading

I knew about the greek usage first, so when I first encountered words like metamathematics

I thought it was some mystic concept beyond maths :-D

(maybe it actually is, at least in many people's opinion)

John Duffield

11:51

@DanielSank : IMHO a better analogy is squeezing a stress-ball down in your fist, and then letting go. Or the raisin cake. IMHO it's important to appreciate that space isn't nothing, and that there is no evidential reason for assuming that the universe has no centre or edge.

Danu

@JohnDuffield There is very strong evidence against the universe having a center.

It's called the CMB

But its discovery came after Einstein's time, so this shouldn't come as a surprise to me.

Slereah

The Einstein universe also did not have a center

Nor does the FRWL metric

Danu

@Slereah But it's known not to be a good model

Slereah

well yes but if we must talk of EINSTEIN AND THE EVIDENCE

Danu

@Slereah The status of FLRW wasn't so obvious early on, I assume.

The huge confirmation came with the CMB

Slereah

11:56

I think pretty much all early cosmological models had no center

Because a center is a weird thing to have in GR

Danu

It goes against the spirit of GR

John Duffield

@Danu : it isn't evidence for the universe not having a centre. It's evidence for the raisin-cake expansion of space, wherein being in space can be likened to being in a fog. The expanding universe isn't like Daniel Sank's moving particles. But a raisin cake does have a centre.

Danu

@JohnDuffield That's hilarious (and wrong)---it's evidence for the homogeneity and isotropy of space---which implies that there is no center.

Slereah

the homogeneity part, specifically

yuggib

"Dios es una esfera inteligible, cuyo centro está en todas partes y su circunferencia en ninguna"

Danu

12:01

right

God in the center

John Duffield

@Danu: there's no evidence that space is isotropic. This is an assumption. For all you know some guy somewhere is looking up at the clear night sky, and half of it is black.

Danu

@JohnDuffield That's, in fact, irrelevant because homogeneity already implies no center.

John Duffield

@Danu : it doesn't. Note that a gravitational field is inhomogeneous space. There is no overall gravitational field in the universe, it didn't collapse when it was dense. The fact that we're here is evidence that space is homogeneous on the large scale. But that isn't evidence that space has no centre.

Danu

@JohnDuffield It's pretty obvious that a presupposed "center" of the universe would imply that not all points are the same.

I hope you're able to understand it

John Duffield

@Danu : I understand it. It's like being in a fog. Everything around you looks the same, a white-out. But you cannot deduce from this that the fog has no centre.

Slereah

12:10

ahah what

John Duffield

I would go so far as to say that the expanding universe is hard scientific evidence that space is not infinite. (How anybody can square an infinite universe with the Big Bang beats me). And if it isn't infinite and if it's flat as WMAP suggests, with no magical mysterious intrinsic curvature, that means space is somehow bounded. And that means it has a centre. The thing I don't understand is why Einstein didn't predict it.

Danu

@JohnDuffield You're falling for the beginner's fallacy of believing the Big Bang was a place, not a time.

If it's flat it would be very unnatural to assume a center (in the view of every professional physicist), regardless of what you may want to believe.

Secret

Is it possible that every points in the universe look the same, except there might be a centre (and its neighbourhood) that is not, but that region of inhomogeneity just happen to lie outside our hubble horizon?

Or is the CMB already have this information encoded in it since it corresponds to the period in the history of the universe where it just became transparent (the last scattering surface), hence any evidence of a inhomogenious pathc of spacetime must already be "projected" onto it?

and the homogenity of the CMB thus rules out the presence of a centre?

Slereah

What does a center even mean in the context of spacetime, though

Unless it has boundaries it is a rather odd idea

even with boundaries it's not that trivial to define

John Duffield

@Danu : no I'm not falling for a beginner's fallacy. And what's very unnatural, is for professional physicists to make assumptions that are unsupported by scientific evidence. Perhaps you're referring to cosmologists? Some of whom have only a sketchy understanding of general relativity?

@Secret : it doesn't rule out the presence of a centre. The CMB is akin to fog. The presence of this fog, wherein you see white everywhere you go, doesn't mean there's no centre to the fog you're in.

Danu

12:21

@JohnDuffield Right... You know what's sketchy? Claiming to understand the universe without (1) any serious physics education (2) any apparent mathematical knowledge (3) any published research (4) any regards for the professionals who do fulfill all other requirements for believability (5) any willingness to learn from others.

John Duffield

@Slereah : it means what you think. See the NASA picture here. They've dropped one space dimension and they're showing the time dimension going from left to right. Each vertical grid line is a billion years. Imagine you're looking into this "stemless wine flute" from the top. Where the sides are is the edge of the universe.

Secret

Danu

@Secret Hey, look, a drawing that I do understand! Maybe John will get it too ;-)

Secret

@Slereah The left case does not work because one need to mention what all that nonspacetime "stuff" in between is

The right case will imply the presence of a small region of inhimogeneous spacetime, which we have not observed

But...

@Danu Does the CMB contains the global details of our entire universe, or just up to the hubble horizon/observable universe?

Because if that's the latter, then it is possibel that the inhomogenous region may just happen to fall outside our observable universe and thus we then cannot simply say the homogenity of the CMB rule out a centre

So does the CMB contains global details of the entire universe at the time of the last scattering event?

Danu

@JohnDuffield, I can even refute your claim in your terms. Imagine walking in a fog. If there is a center, one can simulate this by a point that generates fog and blows it outwards in all directions. If you're standing at this point, everything moves away from you. Now, move one step. If you look back at where you came from, the fog is now moving towards you. This flatly contradicts homogeneity.

@Secret Of course we can, by definition, not say anything about things outside the observable universe from an experimental perspective.

You can assume all you like about what's outside of what we can observe, but that'd be the biggest cop-out possible.

Secret

12:31

@Danu So the CMB only maps the observable universe at the time of the last scattering event, and thus the homogeneity of the CMB only told us that "within the observable universe", there is no centre?

and the observable universe is the thing that is experimentally relevant to us, while anything outside is pretty much unreachable and not measurable?

John Duffield

@Danu : I give you a reference to Einstein describing a gravitational field as inhomogeneous space. I point out that WMAP indicates that space is flat. And that the FLRW metric assumes the homogeneity and isotropy of space. If all you can offer in return is featherspitting hubris because you have no adequate response, well, it doesn't look too prefessional, now does it?

Danu

@Secret Of course; one can philosophically even take the stance that things outside the observable universe don't really exist, although this is not a very popular point of view among physicists.

Secret

Hmm, ok

John Duffield

@Danu : this "refutation" is nothing of the sort. I'm not saying the fog is blowing outwards. Particles are not blasting out from the centre through space. Space itself is expanding. It's like this gin-clear ghostly elastic solid. With an innate pressure. It expands.

Danu

@JohnDuffield Obviously FLRW assumes homogeneity and isotropy. The point is that we measured things in the Universe and found that it is, indeed, well-described by FLRW. It's not that hard: This is how all of science works, in case you hadn't ever noticed.

I don't think references to Einstein are relevant because Einstein did not understand how to properly describe our universe by means of GR.

John Duffield

12:36

@Danu : and suppose I snapped my magic fingers, such that you were no longer in Denmark, but instead were standing on a planet 46 billion years away. Looking up at the clear night sky. Wondering why half the sky is black?

Danu

@JohnDuffield I've put enough effort into talking to you about this; I'll stop responding.

You're just demonstrating once more that you're unwilling to listen to anything else than your preconceived ideas.

@JohnDuffield I'm not in Denmark. No snapping needed.

Also, years are not a measure of distance.

Bye for now.

John Duffield

@Danu : you're the one who is unwilling to listen, you're the one running away from the references that you can't counter. And by the way, a light year is a measure of distance.

If anybody is interested, see this answer which relates to the above.

Danu

@JohnDuffield light year, yes.

Secret

@Danu @Slereah
That CMB question I asked above shows how my mind is more of a mathematician than a physicist

When I chat with my physics professors and peers, often they have to state explicitly that "blahblahblah is experimentally relevant" in order for me to be able to understand why the very general mathematical case is ruled out

That is, at least for the past few years until the present, my way of learning physics is like this:

1. Observed a phenomenon, obtain some data
2. Try to find some patterns in the data

Secret

One example of the above is back in my 2nd year, I do the particle in a box problem as if it is a ODE problem for 3 times, and that rule out the trivial cases and other physically irerelevant cases by substituting the assumptions or constraints one by one into my general maths equation

It is only after this I can do this problem like a physicists did, by skipping over the physically irrelevant cases and still be confident that I am not cheating or overlooking or missing out something

So basically, as of now, to me physics is like maths but with a special type of constraints that are based on observations and cannot be derived from axioms, which is needed to make the equations work in the physical context

correction: "back at physical intuition" should be "bad at physical intuition"

Secret

12:55

In short, my mathematical oriented mindset means physics to me is a sub(multi)set of maths that there exists a type of constraints that are derived from an object called "experiments" and this is needed to ensure the solution the problem remains closed in physics under the usual operations

and this is why I generally struggle in physics, because I am bad at figuring out those experimentally derived constraints

thus end up writing pages and pages of algebra when doing problems

ACuriousMind

Um, @Secret, this isn't your diary, it's a chat. We talk to each other here, and posting giant walls of texts isn't really conducive to that.

Secret

I am just trying to tell you guys about my learning methods, which might help on refining the answer when I ask questions later in physics in the chat

ACuriousMind

@JamalS, haven't seen you here in a while, how have you been?

Secret

I am not sure how to shorten them more though to get the point across. For one thing I am not a native speaker thus my English is not perfect. For another, the timing out of the messages means I usually cannot fix my previous message and end up need to clarify it on th next block of message

ACuriousMind

@Secret That's supposing the people you ask these questions read and memorize what you write there even if they may not even have been here when you posted it.

Slereah

13:15

It's a gh-gh-ghost D:

Ghost

boo!

ACuriousMind

It's not even a ghost-of-ghosts, not that scary

Ghost

not scary at all

ACuriousMind

Ah, a little bit scary since even normal ghosts mean quantization will be a bit tricky

Secret

negative kinetic term?

Ghost

13:19

that's what she said....

ACuriousMind

@Secret Mh, yes, that's what their terms in the action look like

One introduces that term precisely because it leads to the correct result at the end, though

0celo7

Two tokamak cores go round the outside, round the outside, round the outside

Guess who’s back, back again, Ossi’s back, tell a friend

I said this looks like a jobber me, So everybody, just follow me

Cause we need a little, controversy, Cause it feels to empty without me

Ghost

welcome back @0celo7

ACuriousMind

What's a "jobber me"?

Secret

I wish I am already at advanced QFT understanding so I can calculate myself how this negative KE term of the Lagrangian affect the possible histories in the path integrals (so as to get some kind of rough geometric picture on how they act differently ftom ordinary fields)

Well, I guess I have to be patient

Anyway, currently reading some basic quantum stuff and two questions are in production in the pipeline...

0celo7

13:23

@ACuriousMind Huh

That was supposed to be "job for me"

Secret

JamalS

@ACuriousMind Hi - yes, I've had a break from physics for a few months, just occasionally watching lectures. I'm getting round to being more active on the site again, just need a mathematical physics question that I can answer to get me hooked again :)

Slereah

@ACuriousMind Then again even EM has spooky ghosts

But they are non interacting :p

Ghost

I ain't normal anything

Secret

@ACuriousMind
https://en.wikipedia.org/wiki/Matrix_mechanics
How to derive the fundamental commutation relation? cause I am trying to understand why the RHS is as shown. Since there's no notion of orbits in matrix mechanics, then there is nothing that suggest something is rotating or in a circle thus no justification on the $2\pi$ factor.

While the wikipedia article have used a prove by contradiction to show that why position and momentum eigenstates don't commute (by showing that if they do then you get $[x,p]=0$ but we need $[x,p]=i\hbar$), it does not address why it has to be that parti…

ACuriousMind

13:31

@JamalS Might take a while, these have not become more frequent :/

yuggib

@ACuriousMind They had become more unfrequent I would say... T_T

So not much to do when I am bored/slacking

ACuriousMind

@Secret Well, one might say that if it was $8\hbar$ (you can't have the square because of units) then $8\hbar$ would be what we would call $\hbar$

Secret

@ACuriousMind so what really physically matters is just whether it is nonzero, and not what value it takes (even if negative?)

yuggib

@ACuriousMind Yes but beware of the $2\pi$, otherwise no Fourier transform, and therefore quite a mess...

ACuriousMind

@Secret Well, the actual value of $\hbar$ somewhat sets the scale at which quantum effects become relevant. And it has to be positive - the commutator of $x$ and $p$ being a (positive) multiple of the identity has qualitatively different features from it being a negative multiple.

JamalS

13:35

@ACuriousMind I guess I'll have to post one then :)

ACuriousMind

@JamalS Why, that's the spirit! :D

Emilio Pisanty

@Secret The canonical commutation relations are really just the de Broglie hypothesis in disguise. (More formally, the Stone-von Neumann theorem means that you need to identify p with a derivative, and this gives you de Broglie when applied to waves.) If you change the value of ħ in the CCR it's equivalent to changing it in de Broglie's relation.

ACuriousMind

@yuggib Hmmm? What do you mean?

yuggib

@ACuriousMind That it is quite important that position and momentum operators are "Fourier transforms one of the other" (if you pardon me the inaccuracy)

ACuriousMind

@yuggib Yes, of course - but they remain Fourier transforms when one rescales that relation, don't they? The $xp/\hbar$ in the exponent of the transform should just become $xp/(8\hbar)$, right?

I mean, we usually scale $\hbar = 1$, anyway

yuggib

13:40

@ACuriousMind Yes yes...of course ;)

Slereah

you know

ghost fields in EM do not interact

But

They do contribute to the stress energy tensor, IIRC

ACuriousMind

@Slereah How do you define the stress-energy tensor?

"the variation of the action w.r.t. the metric" is too vague in this context

Because you need to say which action

The action we started from? The action with ghosts? The action with ghosts and Lagrange multipliers for the constraints? The gauge-fixed action?

Slereah

@ACuriousMind : The values of the state applied to the stress energy operator

(semiclassically)

ACuriousMind

@Slereah Um...how do you define that operator?

Slereah

Ugh

That is the worst question

ACuriousMind

13:52

lol

Slereah

The stress energy tensor in semiclassical gravity is defined as a limiting point splitting process on the stress energy bitensor quanticized

Through this procedure you can also define it as a differential operator on the propagator

And you gotta use

The WALD AXIOMS

0celo7

wtf

why does physics suck so much

yuggib

because it's not math :-D

just kidding...math sucks even more

0celo7

that leaves... S TE M

engineering wins

plus it's the only useful one

Slereah

it raises the GDP

0celo7

13:59

yeah :D

hmm, General Relativity for Mathematicians is available for MyCopy...

Slereah

arxiv.org/pdf/1202.5107.pdf

@ACuriousMind

yuggib

VM

0celo7

>not posting the abstract

@yuggib did you miss me

or do you like a quiet chat D:

yuggib

@0celo7 I missed you ;-)

0celo7

suspicious winking

yuggib

14:04

:D

no seriously, it was too silent

ACuriousMind

@Slereah Um...maybe I was not entirely clear on what I meant - I wasn't talking about the technical details, but what action you use to begin with.

0celo7

@yuggib hmm

ACuriousMind

They do a lot of technical stuff in there, but they don't consider the cases of gauge theories where you have quite a lot of different actions to choose from

0celo7

there's only one good action

yuggib

@0celo7 and you should read this:

press.princeton.edu/titles/5159.html

0celo7

14:06

the heck is that

reading_list ++

seriously

what is that

yuggib

a rigorous result in GR

0celo7

what are the prerequisites for reading that

yuggib

klainerman and christodolou are two biiiig mathematicians

(and after that work, it seems that now they don't speak to each other anymore :D )

@0celo7 well, that I do not know exactly; probably some knowledge of analysis

it should concern PDEs of GR

0celo7

ok

I will put it on my list

but will read after I've taken the pde course

yuggib

@0celo7 probably is better; however they have a very geometrical approach to PDEs

(that may be of help to you)

0celo7

14:11

I'm still deciding which advanced courses to take

I'm not a pure math major so I have to choose wisely

no time for everything

Slereah

@ACuriousMind EM + gauge fixing + spooky ghost

idi.harvard.edu/uploads/investigators/…

from this paper IIRC

"Since we will see in Section 5 below that the expectations4 of the ghost and gauge-breaking contributions to the stress-energy tensor precisely cancel, "

Oh wait I guess not

Makes sense

ACuriousMind

I would've been quite surprised if the ghosts made an actual contribution

Slereah

Booooo

Maybe we should make a question for halloween

about ghosts and phantoms

what is the ghost term for a phantom field

0celo7

phantom field?

Slereah

Hm, what's the spookiest gauge group

0celo7

14:16

seriously?

@Slereah Monster group :D

Slereah

@0celo7 : Field with a negative energy kinetic term

Monster group isn't a gauge group!

0celo7

why

Slereah

It's not a manifold

ACuriousMind

@0celo7 It's finite.

Not a Lie group

0celo7

did not know that!

but can't you have finite groups as gauge groups in condensed matter?

Slereah

14:18

Eeeeh

ACuriousMind

Yeah, they call them that

Slereah

kinda

0celo7

$E_8^{\times\infty}$

ACuriousMind

I'm not sure to what extent that's actually a "gauge theory" in the non-condensed matter sense

Slereah

I don't think those groups are really gauge groups, though

Do they have a connection or whatever

plus no associated gauge field

0celo7

14:19

dunno

just another part of my sphere of ignorance

Slereah

Hm

What is the spookiest manifold

0celo7

@ACuriousMind so all Galilean spaces are isomorphic

Slereah

Maybe the Loch Ness Manifold

0celo7

hint pls

Slereah

Loch Ness monster surface

In mathematics, the Loch Ness monster is a surface with infinite genus but only one end. It appears named this way already in a 1981-year article by Sullivan & Phillips (1981). The surface can be constructed by starting with a plane (which can be thought of as the surface of Loch Ness) and adding an infinite number of handles (which can be thought of as loops of the Loch Ness monster). == See also == Jacob's ladder surface Cantor tree surface == References == Sullivan, Dennis; Phillips, Anthony (1981), "Geometry of leaves", Topology 20: 209–218, doi:10.1016/0040-9383(81)90039-2, ISSN 0040-9383...

0celo7

14:20

@Slereah you must be fun at trivia

ACuriousMind

@0celo7 What do you mean by a "Galilean space"?

Slereah

GHOSTS OF PHANTOMS ON THE LOCH NESS MONSTER

Let's make a paper

0celo7

@ACuriousMind it might be best for you to download Arnold

gonna have a lot more questions from it :)

Slereah

👻

^host operator

0celo7

link.springer.com/book/10.1007%2F978-1-4757-2063-1

@ACuriousMind page 5

second problem on the following page

Secret

14:24

Big Rip

The Big Rip is a cosmological hypothesis first published in 2003, about the ultimate fate of the universe, in which the matter of the universe, from stars and galaxies to atoms and subatomic particles, is progressively torn apart by the expansion of the universe at a certain time in the future. According to the hypothesis, the scale factor of the universe and with it all distances in the universe will become infinite at a finite time in the future. The possibility of sudden singularities and crunch or rip singularities at late times occur only for hypothetical matter with implausible physic...

The phantom rip us apart

ACuriousMind

@0celo7 Do you know why all $\mathbb{R}^n$ with an inner product are isomorphic to the standard Euclidean $\mathbb{R}^n$?

Because this is very similar

0celo7

I do not :(

ACuriousMind

Well, then think about it ;)

0celo7

Conceptually, I'm not sure what there is to show

ACuriousMind

Well, you have to construct the isomorphism

0celo7

14:32

well the map from $\mathbb{R}^n\to\mathbb{R}^n$ is trivial

I think

not sure about the inner product part...

I need a hint of what to construct

I know what an isomorphism is

ACuriousMind

@0celo7 Remember that the isomorphism must respect the structure/inner product, so sending $(1,0,\dots)$ to $(1,0,\dots)$ doesn't work.

0celo7

but I don't know what it should look like

@ACuriousMind what?

ACuriousMind

@0celo7 We are given an $n$-dimensional vector space $V$ with inner product $(\dot{},\dot{})_V$ and want to have a map $\phi : V \to \mathbb{E}^n$ such that $(v,w)_V = (\phi(v),\phi(w))_\mathbb{E}$ (and it is bijective and linear).

Isomorphisms of inner product spaces are isometries, in other words

0celo7

I still don't know what you want :/

show that $\phi$ is bijective and linear?

ACuriousMind

@0celo7 You have to define $\phi$ first!

0celo7

14:39

well you just defined it!

ACuriousMind

Where?

0celo7

$(v,w)_V=(\phi(v),\phi(w))_E$

ACuriousMind

that's not a definition of a map, or can you tell me from that what $\phi(v)$ for some $v\in V$ is?

0celo7

I'd need to know what $(.,.)_V$ is

I think

ACuriousMind

@0celo7 that's just the inner product on $V$.

0celo7

14:40

yes, of course

that was phrased poorly

ACuriousMind

And no, we don't need to know anything more about it except that its an inner product

0celo7

so $g_{ij}v^i w^j=\delta_{ij}\phi^i{}_k\phi^j{}_lv^kw^l$

ACuriousMind

Geez, you've been tainted by physics :P

0celo7

no

abstract indices :^)

so $g_{ij}=\delta_{kl}\phi^k{}_i\phi^l{}_j$

ACuriousMind

Okay, here's what one does:

0celo7

14:44

well that's not gonna work

WAIT

is $\phi$ the bein

ACuriousMind

Since we have an inner product on $V$, we can construct some orthonormal basis $e_i$.

0celo7

stop

ACuriousMind

k

0celo7

isn't $\phi$ just the vielbein??

ACuriousMind

WTF is a vielbein?

0celo7

14:45

@Slereah

ACuriousMind

$\phi$ is a map between two vector spaces

0celo7

@Slereah @Slereah

@ACuriousMind GR thing

ACuriousMind

@0celo7 That much I could've told you :P

Slereah

a vielbein is a tetrad :p

but germanier

0celo7

one writes $g_{\mu\nu}=\eta_{ab}e^a{}_\mu e^b{}_\nu$

Slereah

14:47

well it's the generalization of a tetrad

0celo7

so phi is linear because it's a tensor thingie

Slereah

since tetrad is specifically 4D

0celo7

and invertible because $g$ is nondegenerate

ACuriousMind

@0celo7 dude, you still haven't defined $\phi$.

0celo7

INSPECTION

just solve the above matrix equation

Slereah

14:49

a section of the orthonormal frame bundle~

ACuriousMind

@0celo7 Can you guarantee existence of the solution?

0celo7

@ACuriousMind if $g$ is nondegenerate...maybe?

ACuriousMind

There is a way in which you can just define $\phi$ and it is obviously bijective, linear and isometric.

0celo7

ok

give me some time to work it out

Secret

bolbteppa

14:55

$\phi$ is given by the first problem

Secret

by cyclic do they mean the cyclic rule of partial derivatives is obeyed?

ACuriousMind

12

Q: Why are they called "cyclic" coordinates?

In Lagrangian formalism, when $\frac{\partial L}{\partial q} = 0$, the coordinate $q$ is called cyclic and a corresponding conserved quantity exists. But why is it called cyclic?

soft-question lagrangian-formalism definition

Slereah

Cyclic coordinate is when the Lagrangian is independant from it

bolbteppa

@0celo7 Arnold is basically a re-write of Landau's mechanics, stripped of lots of intuition and replaced with formalism, with some cool added stuff

@Secret the word cyclic (as explained in Arnold's mechanics coincidentally!) comes from thinking about a lagrangian in, say, polar coordinates (when solving the 2-body problem, e.g. planetary motion and scattering) with no $\theta$ variable so that the $\theta$ variable can cycle around without affecting anything

Secret

nice

bolbteppa

15:00

@Secret if you are wondering about adding a potential to the relativistic Lagrangian, I recommend the discussion in Lanczos & Zee's Gravity book

Secret

I might read about this later after getting the basics from wikipedia
As of this page so far, I found the relativistic lagrangian more straightforward than the classical mech one, but my opinion might change once you get to the bits where christoffel symbols start to appear

Secret

15:20

Slereah

@Secret : You don't need to take a screenshot every time

You can just post the quote

Secret

ok
---
for the case of massless particles, is the affine parameter $\sigma$ will only be a parameter in the calculation and will not have physical consequence, similar to how since we cannot define a 4-velocity for a massless particle, thus we use an affine parameter that is not proper time to define something similar?

Slereah

yes.

user54412

15:43

@Danu Are you so sure...?

user54412

@Danu :(

0celo7

@ACuriousMind is it possible to give a hint but not solve it?

obe

0celo7 is back :D

0celo7

someone in the lab is reading Sakurai

ACuriousMind

1 hour ago, by ACuriousMind

Since we have an inner product on $V$, we can construct some orthonormal basis $e_i$.

user54412

15:46

@DanielSank Subtle. If not for special relativity, I'm not sure off the top of my head we can distinguish those cases. But metric expansion isn't constrained by the speed of light.

0celo7

so you want me to find $\phi(e_i)$

$(e_i,e_j)_V=\delta_{ij}$

user54412

@Danu With the right definition of "observable" I'm pretty sure things outside the observable universe don't have any place in scientific ontology, by definition ;)

ACuriousMind

@0celo7 Yes, you need to define that in the "correct" way

Transcript for