what are you two talking about

Steenrod

I think I have a rough idea of the proof

A lot of little proofs in between to do, but a rough idea

I haven't seen you talk about obstructions yet

According to @BalarkaSen sempai it is not necessary

possibly, anyway

why not

Because apparently if you have a subbundle section, then the parent bundle also has a section

what page is it on?

In class right now

No Steenrod on hand

well how do you know that you have a subbundle section?

Well that's another proof

...which requires obstruction theory

Prove that the Grassmanian bundle section is equivalent to a nowhere vanishing vector section

And then prove that a nowhere vanishing vector section requires for compact manifolds $\chi = 0$

btw steenrod will only prove this for triangulated manifolds

so you need to learn that proof

I'm guessing that the hairy ball theorem probably has a lot more books talking about it

the hairy ball theorem is trivial

more than "one"

just use morse theory

$\cdot \cdot -- \cdot --$

just me or is google not working?

it's you

but all other sites are working

google in chrome isn't

Now I can put a sticker on the book that says "Contains the Steenrod proof!"

@Slereah See page 133 of Hirsch.

Suppose one day everyone lost all of their emotions and ability to feel

how would this effect people?

@Slereah I look forward to you trying to understand triangulations.

will all motivation be lost?

I assume you draw little triangles on the manifold

That's not a proof

are emotions the only real valuable currency of life?

is all human endeavour simply about emotion or a means to achieving a particular emotion?

No it's US dollars

but don't people seek US dollars to feel emotions like happiness, self esteem etc.

and avoid negative emotions like pain etc.

@Slereah Is "continuous by pieces" a French thing

@Kenshin Of course one can be interested in science, and follow the scientific method, and maybe draw conclusions based on that all by himself just for the sake of it (and eventually tell anyone). But I would not say that it is common practice to call such a person a scientist. Maybe a science amateur, or enthusiast. I think that almost everybody uses the word scientist to define a person whose main occupation in life is doing science.

@0celouvskyopoulo7 Continu par morceau?

It means piecewise continuous

yes

I was wondering if it's the proper french term

it is

@yuggib I disagree and so does wikipedia. Many are referred to as scientists for simply follwing the scientific method

Stop your wanking over terminology and do actual science

@Kenshin urge to achieve a particular emotion is also an emotion :P

@Slereah soooooooo

@yuggib even actuarial is considered a science these days "actuarial science"

the metric is not $C^\infty$

it has to be discontinuous for a star

because $G=T$, and $T$ is discontinuous

@BalarkaSen I didn't say urge, I said endeavour

so $G$, being $\sim\partial^2 g$, is discontinuous

so $g$ can only be $C^1$

Well sure, if you do a cube with harsh edges

@Slereah I'm talking about a fluid ball

although with corners, I'm not sure if you can do a smooth cube

Oh

@BalarkaSen but now I think furterh I see your point

reading Yvonne right now

why also do harsh boundaries there?

@Kenshin Endeavor to achieve an emotion is a resultant of the urge or feel of necessity to achieve it.

You can just do a decay

@BalarkaSen if you think about physics, a particle always responds to it's local viscinity (e.g. the field in the local viscinity). Similarly humans respond to emotions locally

@Slereah Like you... ;-P

@Slereah I think that would be even worse

so even if we have big aims, following a means to an end is still following a local emotino to do so

you'd have to worry about how fast it decays

How about "very fast"

Like exponential decay

@BalarkaSen yeah cos if doing action A will lead to result B, and B gives me an emotion, I still wont' do action A unless there is an emotion now compelling me to do so.

@Kenshin I think so. But I do think human beings are capable of shutting down emotional urges via cerebral means.

Logistic function sort of thing

@BalarkaSen can they really?

@Slereah Really fast decay isn't the issue

it's a function you have to worry about though

or does our cerebrum actually cause us to feel good about thinking?

I still need to read some papers to decide what BCs I want

there's a lot that I don't understand :(

@BalarkaSen my theory is when you try to think long term, you get an emotional reward for doing so

I don't think everything is about feeling good. But the boundaries of the realms we are speaking of are so thin that it almost feels useless to talk about it.

but surely you agree that everything is ultimately about feeling good?

I don't :)

or do you think a life without emotions is still worth living?

:/

so what are we aiming for? @BalarkaSen

how?

I am just saying, it is easy to state and justify that human emotions are fundamentally about feeling good. But thinking about it, I feel like that's not it.

lol

Note the "I", though. It's a personal opinion, almost a belief.

@BalarkaSen I have thought about it too, and sure it would be nice if there were more than emotions (I hate the idea of peopel being satisfied by a pill)

@Kenshin I don't think human beings fundamentally aim for something. Sure, some believe in some certain philosophies and make a fundamental aim.

but I just can't seem to find anything else that makes esense

@BalarkaSen but every action that isn't for a purpose is deeemed "irrational"?

and most people consider "irrational" action as negative

so doesn't this imply most people do see a purpose?

and that purpose to me seems to be achiving emotional currency

it could be somethign else but nothing else seems to fit

@Kenshin You should read some mind melting stuff and step out of the abstract philosophical realm for a while IMO. Over the course of years I have found literature very enlightening.

like what

@Slereah jesus christ, what does stationary mean again?

@0celouvskyopoulo7 pens, paper, rulers

in the context of general relativity

it's a relative concept

@Kenshin And you mean stationery.

yeah i though tyou mistyped it

;P

well I told you to read notes from the underground. if you want short, but solid mindmelts, read any short story from Borges's "Labyrinths".

I can recommend some if you get that book

@KyleKanos It was evil.

He won't get pinged

@BalarkaSen do you think it odd that most people don't care about these questions

what happened?

like what the purpose is etc.

(i can also recommend good movies if you watch them)

@Kenshin I guess I do.

@Slereah Yvonne has the incorrect proof that trying to boost out of a black hole decreases survival time

pretty sad

Really low energy

I don' think there's any book that explains that correctly, actually.

It's so subtle

also do you have to do a discontinuous stress energy tensor for the sphere

You can just do $C^0$

That way the metric is $C^2$

Or even just do it as a bump function

no worries about the falloff

And another of my bounties that'll just expire :/

Hm guys, I still don't understand one thing about behind normalising our exponential decay function. How do we know that the mean of the density function will give us the expected lifetime? Like, we first have a function that gives us the amount of particles on a given time, and then by normalising it, we suddenly get a probability density function. What is the meaning of $P(x)dx$ here then? The probability that the particle decays at that time?

I'm afraid I'm not very good at SUSY

@Slereah What sort of SUSY? I've learned a fair deal about it by now

The type you just linked :p

I do have a SUSY book

Oh, that

But I have so many books

It's hard to do it all

anyways I think that "SUSY Q by CCR" is a captivating name for a paper, not only for a song

@Slereah Damn GIFs

@ShaVuklia I'm lacking some context here? What exponential decay function and what density function?

by Creedence Clearwater Revival?

@Slereah they sang it at least

@yuggib That...actually almost makes sense :D

@ACuriousMind :-D indeed

@ACuriousMind Well you could consider consider discrete elements in a set, or for instance the amplitude of oscillations. They are given by $N(t)=N_0e^{-\lambda t}$. The exponential decay function is a scalar multiple of the density function of the exponential distribution, which we get by normalising our exponential decay function.

@Slereah Good ol' Twilight Zone

So $P(t)=\lambda e^{-\lambda t}$

It has aged fairly well, for the most part

@Slereah My advisor really likes Weinberg for some reason

@ACuriousMind Like, apparently we're talking about the distribution of the lifetimes, but I don't see why that would hold.

Do you mean as a person or one of his book

@Slereah does Stephani have the Tolman metric?

also which book

I do not know

@Slereah He says his GR book is one of his favorite physics books

I can look once I'm home

I only know Tolman for the Tolman paradox

yvonne is not making sense right now

"the general dust solution admitting a $G_3$ on a $V_2$"

wot

@Slereah what does Stephani mean

It means it has 3 Killing vector fields

An isometry group of 3 generators

I forget what $V$ is

Stephani is big on Killing vectors

The first 500 pages or so is just a laundry list of spacetimes by descending order of Killing vectors

@ShaVuklia Given a number of particles $n(t)$ as a function of time and assuming each particle is equally likely to decay, the probability $p(t_1,t_0)$ that any given particle decays between $t_0$ and $t_1$ is $\frac{n(t_0)-n(t_1)}{n(t_0)}$. Choosing $t_0 = 0$, we find that $p(t_1,0) = 1 - \exp(-\lambda t)$. The probability density associated with this probability is simply its derivative (since integrating the density yields the probability), which is $P(t) = \lambda\exp(-\lambda t)$.

I'd have to read this whole book to understand the notation, damn.

Don't worry, the notation is only the first... 9 chapters or so

although you can probably skip the chapter on the Penrose Newman formalism

@ACuriousMind oh wow, this already helps a lot. I will have to go through it slowly.

No one should ever have to look at it

damn yvonne

is not not opening to others too?

Also the algebra classification of spacetimes is mostly not that useful for the chapters on symmetric spacetimes

I'm 100 pages in and only 1/8 of the way through

You can probably skip the Petrov classification

@Fawad Works fine for me

It is opening now. Thanks for attention

@ACuriousMind ohhhhmgoodness, I think I understand it now. Honestly, I thank you sooooooo much. I was almost crying because of this, because I've been trying to understand this for hours :P I even wrote in my book "... I'm sorry I don't understand it", as an apology note to my (future) self who might reread the text again :P

aww how sweet

:P

@ShaVuklia You're welcome :)

@ShaVuklia now edit that sorry and replace with script link.

hahahahah, I will!!

@0celouvskyopoulo7 what u want is the $G_3$ spacetimes

That is where all the sweet spherically symmetric static spacetimes are

Or $G_2$ if you want non-static ones

$G_3$ also has static cylindrical symmetry and plane symmetry

What is the meaning of the matrix mechanics in QFT, anyway

If I represent states as vectors in $\mathfrak l^2$, what is a component of $\mathfrak l^2$

Normalized number states, maybe?

@Qmechanic, you know q mechanics

do you know

matrix mechanics is just treating the diff ops of wave mechanics as operators on Hilbert space

then representing them as matrices

it's completely wrong

What is even the isomorphism between $\mathfrak l^2$ and $L^2(\Bbb R)$

@Slereah Pick a Hilbert basis for both, map one to the other.

Oh I see

I guess the convergence of the series is for the state to be normalized

what?

What is the representation of the rigged Hilbert space, though?

What's $\vert x \rangle$ in $\mathfrak l^2$

So Yvonne mentions the RN metric in one paragraph, then dismisses it as physically uninteresting

Low energy!

@Slereah what is $\ket{x}$?

To be fair, RN is 100% unphysical

@0celouvskyopoulo7 Eigenvalue of $\hat x$

RN is static, topologically weird, it's not globally hyperbolic, it has timelike singularities and closed timelike curves and it's unstable

Total trash

Also real black holes are unlikely to have any significant charge

@Slereah As explained here, the notion of rigged Hilbert space is relative to an algebra of observables, and the two "rigging" spaces are essentially the subspace of smooth (maybe even analytic) vectors for the representation of the algebra and its dual.

@Slereah it's not clear to me why

@ACuriousMind what's a smooth vector?

Isn't there only one algebra of observables for non-relativistic QM?

Up to unitary etc

@0celouvskyopoulo7 $H^\infty = \{ h\in H\mid A\to H, g\mapsto \pi(g)h\text{ is smooth}\}$

what?

Where the notion of differentiability is that of Frechet differentiability, $A$ is a Banach algebra, $\pi$ its representation, and $H$ the Hilbert space and $H^\infty$ is the set of smooth vectors we wish to define.

What brings a person the greatest happiness?

oh

@Kenshin Dopamine and serotonine

yes

@Slereah but that is a response to something

How does one do that exercise in the PSE post?

$\mathcal S$ is dense in $\mathcal H$

@Slereah e.g. to power, or fame

or music or whatever

what actually is the best?

As a response to injecting serotonine and dopamine in your brain

lol

aw man

Heroin can do the trick if you do not have access to those

so you'd rather have serotonine and dopamine injection than be the richest most powerful, most loved man in the world?

Well, physiologically speaking, I would feel exactly the same

Quite possibly better

isn't that mad

won't your cognitive faculties dull the effects of the dopamine? since ur brain knows it's not real

Well do you want happiness or a sharp mind

Make up your mind

no i want happiness

but

won't the cognitive functions instinctively supress the happiness

You may get used to it, yes, but the same is true of actual happiness

because it is evolved to seek happiness from the real world

You get used to a hapy life and will get less happy from the same situation

although the same is true of unhappiness

People get used to misery

:(

Why are you so negative?

yes, but the cognitive brain can influence the emotional brain

not that much

can't reason your way to happiness

e.g. if you win money, the cognitive brain recognises that money is good, and thus tells the emotional brain to react

@Slereah the cognitive brain is too smart that's why you can't

you won't get happy no matter how much you try to reason it if you're continuously stabbed in the balls

@0celouvskyopoulo7 If you assume the representation is topologically irreducible (which one does), then it follows from the set of smooth vectors being non-empty and invariant under the group/algebra action.

@Slereah but very abstract things can make us happy, thanks to our cogntive influence on the emotional brain

so similarly, if the cognitive brain isn't that impressed with dopamine injections (due to it not enhancing survival) it may tell the emotional brain to shut down

@ACuriousMind must the action be transitive?

The rational mind doesn't regulate that as far as I'm aware

@0celouvskyopoulo7 no

Then your comment does not make sense to me.

Oh, topologically irreducible.

@0celouvskyopoulo7 "Topologically irreducible" means there is no non-trivial closed invariant subspace. If $H^\infty$ is non-empty and invariant, it follows its closure is the full space.

But he didn't assume that in what he said.

@ACuriousMind Yes.

But that's not the situatation in the PSE post.

He says $\mathcal A$ is an algebra of ops with continuous spectrum, whatever that means.

@0celouvskyopoulo7 You'll note the post did not make any assumptions on the representation of the algebra, which one must in generally do, which leads me to conclude the post is not meant to list every detail :P

I take it you learned about this in that representation theory class?

Yup

You need the smooth/analytic vectors and their density to use nice formulae like $\pi(\mathrm{e}^{A}) h = \mathrm{e}^{\mathrm{d}\pi(A)} h$ for $\pi$ the representation of a Lie group, which you need to reduce the question of the unitary representations of the Lie group to a question about the algebra

^@Slereah question for you

Non-static black hole solutions are a migraine

I mean, I know that all black holes must eventually settle into a Kerr black hole

But in between

Who the fuck knows

@Slereah Yes, but are they a giraffe?

I don't know of any theorem that would prevent the event horizon from being shaped like a giraffe

A giraffe is still topologically a sphere, right? as I recall from the discussion with JohnRennie, you cannot have nontrivial event horizon topololgies in 4 spacetime dimensions

is that for static spacetimes, though

that I don't remember, I think we have not delved into that aspect

I don't think there's even an analytic non-static black hole solution that isn't axisymmetric

The most dynamic black hole I know is the Valaya metric

Wrong spelling

I forget the name

It's just Scharzschild with $m(t)$

It's a black hole that radiates (or takes in) radiation

Guys, the text says that slow decay makes sense when the damping is very strong. But I would think that it makes sense that we have fast decay when we have strong damping?

oh wait, I get it

arxiv.org/pdf/gr-qc/9809034.pdf

Maybe relevant?

it doesn't even have the chance to go back quickly, because of all the friction

@Slereah Vaidya metric

@Kaumudi.H Every now and then I'll settle down and reread them.

C and H?

calvin and hobbes?

If I leave it long enough I forget enough of it to make the reread still enjoyable.

@Avantgarde Yes.

I got it! ha

It is such a wonderful comic strip. It never fails to leave me in a good mood :-)

Oh yeah? I should look at it sometime then

:D

@Kaumudi.H less than ten years ago but more than five years ago. I wasn't aware of it when it was running as a weekly strip, only when I saw the collection.

oops, yes.

The Dilbert cartoons are also good, though round about 2008 they tailed off a bit. To be fair Scott Adams has been drawing them since 1989 so I think he just ran out of ideas.

Some of the early Dilbert cartoons have me literally laughing out loud, even though i must have read them a dozen times.

Have you seen the Foxtrot cartoons? They also make me laugh, though again they've tailed off a bit after the first decade.

Foxtrot

spurious states?

@Kaumudi.H All the cartoons are the same family, so you feel you get to know them and that makes them funnier. Just reading one or two doesn't really give you the feel for them. Let me know when you've finished C&H and I can give you some Foxtrot collections.

Is the homogeneity and isotropy of the universe independent from all refrence of frames?

@001 No. For example in the rest frame of the Solar System the CMB has a dipole anisotropy because the Solar System is moving relative to the average mass distribution.

The universe is homogeneous and isotropic for comoving observers.

Ok ty

Is the litterature the art of soiling?

@yuggib In British English Soiling means to make dirty and that isn't really the same as littering. The work soiling tends to imply the dirt is of an offensive nature e.g. bodily substances not just dust.

@Kaumudi.H I doubt it would fit.

@JohnRennie so what would you choose as a synonym of littering?

@Kaumudi.H Pretty much everything that was inside my body and exited via one of my bodily orifices isn't a material with which you would want a close aquaintance.

And home

@Kaumudi.H Only a cover specifically made for your model of laptop will fit well. Anything else might sort of fit, but it would be a bad enough fit to make it annoying.

@Slereah the two bibles

@Kaumudi.H The trouble is that yours is a business laptop so doesn't have much available in the way of consumer frills.

well there's no C&H official merchandise

Bill Waterson was always opposed to it

He does the retirement

@Slereah wonder if Yvonne will explain cosmology nicely

did you ever get an answer to your homogeneity question?

I did not

Cosmology is a lie

what physics isn't a lie?

lololo

"there is no evidence"

even she agrees with you

@JohnRennie, hello

@yashas, hello

@Slereah she basically says exactly what you did

it seems to be an unresolved issue that people just accept as fact

it's not clear to me how one is supposed to verify such things, anyway

there are no experiments one can perform, are there?

well as I said

Just some notion of averaging would be nice?

@0celouvskyopoulo7 there is work going on at the moment to calculate the geometry taking into account the non-uniform distribution of matter. Part of the motivation is to see if this offers an explanation for the accelerated expansion.

Like if you consider the averaging of the metric and the averaging of the SET on some volume

"an homothetic metric"

it should be roughly like FRW

why an

is it some French thing?

Yeah french people don't really pronounce h

@JohnRennie that seems horribly complicated

@0celouvskyopoulo7 nevertheless it is being attempted. I could try and find references if you're interested, though I can remember where I read about it. Scientific American or New scientist probably.

Hello @JohnRennie

Hi Albert.

"we will return to the question of local homogeneity, regardless of isotropy, in a later section"

@JohnRennie maybe Yvonne will have a little of that here

@JohnRennie, could you please tell me what is the reason behind gravitational time dilation?

@JohnRennie do you not have the papers?

@0celouvskyopoulo7 no. It is something I've read articles about, but not something I'm sufficiently interested in to read the original papers.

I don't think it's likely to be very exciting. Yes the inhomogeneity will create differences from the simple FLRW metric, but I suspect those differences will be small deviations rather than qualitatively different behaviour.

Agreed.

@AlbertEinstein that simply isn't possible to explain in any meaningful way at your level - sorry.

savage