The h Bar

AccidentalFourierTransform

11:00 PM

hes our Ouija board

anonymous

Ask Bernardo directly na XD

It seems Bernardo is the owl here

vignette2.wikia.nocookie.net/harrypotter/images/a/af/…

@Kaumudi.H i see ! great!

ACuriousMind

@Danu not yet

I see the quality of the star board has deteriorated since I left :P

6

heather

@ACuriousMind, you know general relativity, would you mind explaining what indices are?

AccidentalFourierTransform

@ACuriousMind now its better?

heather

@ACuriousMind, you know, it's strange, the starboard just improved all of a sudden, i wonder what makes me think that...

anonymous

11:05 PM

@ACuriousMind Probably you are holding your judgement scale upside down XD

AccidentalFourierTransform

@heather youre a coder, you know what indices are

heather

like the 0th thing in a list, the 1st, the 2nd, the 3rd...

but that doesn't make sense in the context of the sentence i heard.

AccidentalFourierTransform

yes, see, C++ and GR are essentially the same thing

ACuriousMind

@heather I can explain them, but I need to first determine what you already know: Do you know what a vector is (abstractly), what its components are and how to change the basi of a vector space?

AccidentalFourierTransform

@heather what sentence did you hear?

heather

11:07 PM

@ACuriousMind, depending on how abstractly you mean, yes, i think i know what a vector is/what its components are, how to change the basis of a vector space, i think so.

ACuriousMind

@anonymous I was about to add that this deterioration is quite a feat given that it prominently discussed testicles when I left :P

4

heather

@AccidentalFourierTransform, when referencing the Einstein field equations, the guy said "It looks like it is only one equation, but in fact it is not, you'll notice the indices, $\mu$ and $\nu$ that appear several times in the equations."

AccidentalFourierTransform

@ACuriousMind we are like hollywood: our standards for whom to give stars is pretty low

anonymous

@heather Do you know tensors ? Indices are related to tensor mathematics in GR.

Raising and lowering indices

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type. Raising and lowering indices are a form of index manipulation in tensor expressions. == Tensor type == Given a tensor field on a manifold M, in the presence of a nonsingular form on M (such as a Riemannian metric or Minkowski metric), one can raise or lower indices to change a type (a, b) tensor to a (a + 1, b − 1) tensor (raise index) or to a (a − 1, b + 1) tensor (lower index), where the notation (a, b) has been used to denote the tensor order a + b with a upper indices and...

AccidentalFourierTransform

@heather well yes: if there are indices in both sides of an equation, then it means that the equation is true for all the possible values these indices can take

heather

11:11 PM

@anonymous, no, i guess i'll google those

ACuriousMind

@heather Okay. So when I say "Let $v$ be a vector and $e_i$ for $i = 1,\dots,n$ a basis of the vector space, then $v = \sum_i v_i e_i$", do you know what I mean?

My internet connection seems a bit flaky currently, sorry if I don't respond immediately

heather

@AccidentalFourierTransform, but...but...i've been always been taught that the subscripts don't do anything. $\mu$ and $\nu$ are subscripts.

@ACuriousMind not a problem, thanks for helping.

@ACuriousMind erm, i think that means that $e_i$ is your basis, and $i$ equals something between 1 and $n$, and then the vector is equal to the sum of the components of the vector (where the last part is a big i think)

AccidentalFourierTransform

@heather well, if you saw a piece of pseudocode with a[i]=b[i]+c[i]

heather

well the ith thing in the first list equals the ith thing in the second list plus the ith thing in the third list @AccidentalFourierTransform

AccidentalFourierTransform

it is exactly the same in GR

I can write e.g., $a_i=b_i+c_i$

though the standard notation is $i\to\mu$, but its the same

heather

11:14 PM

but...$R$ (for example) is a list?!

AccidentalFourierTransform

$a_\mu=b_\mu+c_\mu$ means that $a_0=b_0+c_0$, and $a_1=b_1+c_1$, etc

heather

i thought it was the, um...*(::looks at wikipedia::)* Ricci curvature tensor

AccidentalFourierTransform

and if you have something with several indices, its just a multidimensional array

@heather the proper name is "tensor", but yes: it is a two dimensional array

a list of numbers

heather

oh...seriously?

AccidentalFourierTransform

well, almost

heather

11:15 PM

@AccidentalFourierTransform like a matrix

AccidentalFourierTransform

yes

ACuriousMind

@heather Well, quite good. More precisely I meant that I have $n$ basis vectors which I call $e_1,e_2,\dots,e_n$ and then I find the components $v_i$ (which are numbers) of the vector $v$ in that basis. By definition the components are the numbers such that $v_1 e_1 + v_2 e_2 + \dots + v_n e_n = v$.

AccidentalFourierTransform

@heather it is actually more useful to think in abstract, but that is more complex

tensors not exactly matrices

heather

hmm...are they kind of generalized, like bra-ket? abstracted?

AccidentalFourierTransform

yes yes yes

heather

11:17 PM

@ACuriousMind okay, i think i follow

@AccidentalFourierTransform tensors vs. vectors...

i am unsure of the difference.

ACuriousMind

@heather Okay, so the first step to "index notation" is now that the components $v_i$ encode everything I know about the vector. The abstract object $v$ is of no use if I want to do some computation, I need its components. So the physicists from now on simply writes $v_i$, with the tacit understanding that, unless something is said to the contrary, everything that carries such a subscript (or index) is meant to be the $i$-th component of a vector.

For the addition of two vectors, $a+b=c$, we would now write $a_i + b_i = c_i$ (if you're wondering why we'd go to the trouble of adding the $i$s to the notation, that's fine at this stage)

heather

so, i am somewhat unsure here: when you write $a_i$, is $a$ the $i$th component of the vector, or is $a_i$ the $i$th component of $a$?

(if that makes any sense)

AccidentalFourierTransform

$a$ is the abstract vector, $a_i$ is its $i$th component

ACuriousMind

$a_i$ is the $i$-th component of the vector - $a_1$ is the first, $a_3$ is the third, and so on.

heather

okay, that makes sense.

ACuriousMind

11:22 PM

If I write $i$ in an equation, it means that the equation is supposed to hold for every possible choice of $i$

(likewise for other indices, I need not take $i$ as the variable)

heather

oh...do you have to specify $i\in\mathbb{R}$, for instance? is there a standard assumption?

AccidentalFourierTransform

$i\in\mathbb N$

heather

@AccidentalFourierTransform en.wikipedia.org/wiki/Tensor - i read the first paragraph five times and i'm still not sure what it's talking about =)

ACuriousMind

@heather Well, you usually know the dimension of the vector space, so $i$ is an integer between 1 (or sometimes 0) and the dimension.

heather

@ACuriousMind okay

AccidentalFourierTransform

11:23 PM

i=1,2,3,...,n

or i=0,1,2,...,n-1

heather

different question - can $i$ ever equal a decimal, or $\pi$, or something funky like that?

AccidentalFourierTransform

nope

heather

okay, that makes sense.

AccidentalFourierTransform

btw the first paragraph of the wikipedia article is not really useful if you dont already know what a tensor is

heather

@AccidentalFourierTransform okay, that makes me feel a lot better =)

AccidentalFourierTransform

11:26 PM

ok ACM go on... :-P

heather

okay, so it seems like my big problem is that i don't understand what a tensor is - indices don't seem too bad. (::waits for a bombshell about indices to drop::)

ACuriousMind

@heather Very well! Now, what happens to the components $v_i$ of a vector when I change the basis $e_1,\dots,e_n$ to another basis $f_1,\dots,f_n$? (Note that here the subscripts are not indices, but simply enumerate the basis vectors; also, do you know that every base change can be encoded in a matrix?)

heather

@ACuriousMind yeah, i read about that (base change -> matrix). so you multiply the vector by the matrix?

^correction: i watched a video that explained that.

the other thing: how do you know whether subscripts are indices or just enumerating?

AccidentalFourierTransform

what's the difference?

heather

@AccidentalFourierTransform, like what ACM said - the subscripts there are enumerating the basis vectors, not acting as indices.

AccidentalFourierTransform

11:34 PM

@heather ok. $e_i$ is the only object where the $i$ is not an index

everywhere else, it is

actually, let ACM explain that

heather

@AccidentalFourierTransform okay

AccidentalFourierTransform

cmon ACM say sumfin

heather

he did say he had a slow connection @AccidentalFourierTransform

AccidentalFourierTransform

connection! let's move on to geodesics

(its a joke that you probably wont get)

heather

according to wikipedia, "in the presence of an affine connection, a geodesic is defined to be" - i'm guessing it has to do with that affine connection thing?

AccidentalFourierTransform

11:39 PM

yep. Hilarious, right?

^.^

heather

very hilarious =P

i'll probably laugh harder once i learn about geodesics

=)

AccidentalFourierTransform

(Im trying to come up with some pun, wait for it)

no, sorry, I got nothing

heather

there must be $\sum$ pun

AccidentalFourierTransform

anyway, its almost 1am here

I should leave

bye peeps

heather

bye

brb

ACuriousMind

11:50 PM

@heather You know it because the author is careful enough to tell you; and later on from context.

My connection just died for a few minutes

Maybe I should ask my flatmate to stop downloading whatever he's downloading :P

Sir Cumference

@ACuriousMind Tengo una pregunta

Bernardo Meurer

@ACuriousMind halp

ACuriousMind

@heather Yes, exactly. The columns of the matrix are the components of the new basis vectors in terms of the old basis.

Bernardo Meurer

if I do one more K-map I will asplode

Sir Cumference

@ACuriousMind OK, this is gonna sound dumb, but I oughta ask

ACuriousMind

11:54 PM

@SirCumference I don't speak any living languages apart from German and English, sorry :P

Sir Cumference

@ACuriousMind "I have a question" -spanish

Does dark energy exist in a Big Crunch universe?

ACuriousMind

...perhaps?

Sir Cumference

OK, then more importantly, can phantom energy exist in a Big Crunch universe?

I'm going to assume no

ACuriousMind

I mean, it might conceivably not, but you could always add a non-zero but neglegible amount of anything to a universe without changing anything, no?

Bernardo Meurer

@SirCumference Didn't you ask him this exact question the other day?

Sir Cumference

11:56 PM

@BernardoMeurer Wait, what?

Either I'm losing my mind or I never did

ACuriousMind

@BernardoMeurer No, but he's been asking stuff about dark/phantom energy for a while

Bernardo Meurer

And I said he didn't like any physics greater than ~5nm

Sir Cumference

@ACuriousMind The class moved on from cosmology

Bernardo Meurer

To which he felt he should've been insulted

Sir Cumference

I'm still confused

It's gonna be on the final

Bernardo Meurer

11:57 PM

Somewhere inbetween I cried for Christ White to come back

2

and, on that note,

CHRIS WHITE COME BACK

Sir Cumference

@BernardoMeurer "Christ" White, please come back!

Bernardo Meurer

Either I dreamt or I've seen you asking this before lol

Sir Cumference

;)

Bernardo Meurer

lol

Fortunate typo

Sir Cumference

@ACuriousMind Well doesn't phantom energy's pressure increase as its volume increases?

And increasing the volume causes the pressure to increase

Shouldn't it cause a Big Rip eventually?

ACuriousMind

11:59 PM

@SirCumference Sure, but I have no trouble thinking that if there's enough conventional stuff there, you never get to the point where its pressure dominates, and the thing just crunches with our tiny amount of phantom energy never getting to rule the universe

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