The h Bar

0celo7

21:00

Why does the chat think a reasonably active conversation is spam?

Sir Cumference

It's stupid

heather

@SirCumference, yeah, that makes sense.

bolbteppa

'Ed Witten was a journalism or history major at Brandeis University. After he graduated, he became interested in theoretical physics.

He was already at Princeton, and he started talking with one of the physics professors about his interests. The professor told him that if he was really interested, he should read Jackson, which is the standard E/M textbook for physics majors and grad students. However, this book is notoriously difficult to read. The professor obviously told him this to get rid of him.

heather

Ugh, now this Schwarzschild problem is going to be bothering me.

Sir Cumference

@heather Yep, and if we look back, we'd always find the Big Bang as the lowest point of entropy the Universe has ever been in. That removes the motivation for believing in an oscillating universe, which is supposed to answer what happened before the Big Bang.

heather

21:02

Yes, but then you get into the whole Boltzmann brain issue and why the universe started at a low entropy...

Sir Cumference

Anways, the Big Crunch hypothesis implies gravity will eventually bring all objects together and they will form one gigantic black hole, right?

@heather Haha, love the concept of a Boltzmann brain.

heather

@SirCumference, I know, right? And, yes, I think so. Double checking real quick...

Sir Cumference

Then we'd have a problem. You guys said Kerr black holes have an upper mass limit, or otherwise we'd have naked singularities

heather

Yeah (well 0celo7 said, but yeah)

FWIW, I'm no expert on this at all.

Sir Cumference

So does that throw away the idea that all of the Universe's mass will coalese into one black hole?

0celo7

21:05

No, lower mass limit.

Upper mass limit wouldn't make sense

Sir Cumference

We talking about Schwarzschild or Kerr black holes?

0celo7

kerr.

Sir Cumference

I'm talking about Kerr

Oh wait, you did only mention a lower mass limit

heather

@0celo7, you said Kerr black holes have an upper mass limit because otherwise there would be a naked singularity.

Sir Cumference

Nvm, I'm going crazy

heather

21:06

@SirCumference, okay, then I'm going crazy too, because I thought he'd said that.

0celo7

where?

Sir Cumference

So we're back to square one. Is there a mass limit for Kerr (or Kerr–Newman too, why not) black holes?

0celo7

14 mins ago, by 0celo7

So there's a lower mass limit

Sir Cumference

You mentioned there were none for Schwarzschild black holes

Ugh my head

heather

35 mins ago, by Sir Cumference

@0celo7 So does there appear to be a mass limit in the Kerr metric?

34 mins ago, by 0celo7

If you don't want a naked singularity, yes

0celo7

21:08

Mass limit =/=> upper mass limit.

Indeed, there's a lower mass limit.

Sir Cumference

@0celo7 Ooooohhhhh...misunderstood you.

heather

Yeah, but earlier SirCumference posted

40 mins ago, by Sir Cumference

Is there a maximum mass limit for black holes in any GR solution?

which set the stage for the other two comments

0celo7

That's a completely different question.

heather

making everyone involved think you were talking about an upper mass limit.

Sir Cumference

Okay, for any solution, is there any upper mass limit?

0celo7

21:09

"any GR solution" is very vague

Sir Cumference

@0celo7 Well, any you know of

0celo7

@SirCumference Yes, I'm sure there are stability issues with very large black holes. Or maybe they get more stable.

Too bad Chris White got banned

Sir Cumference

:(

He'd know this stuff best

heather

After Sir asked that, you then said "not in static ones but maybe in some dynamic ones"

Sir Cumference

@0celo7 "More stable"?

heather

21:10

And on from there.

Who is Chris White

Sir Cumference

@heather I don't think it matters too much, he probably misunderstood

0celo7

@SirCumference I'm being as vague as I want to be

@heather Before your time young one

heather

@SirCumference, yeah, probably. Well, sorry.

0celo7

Before your time...

Sir Cumference

@heather A physicist here who used to have > 30,000 rep

Very knowledgeable on GR

heather

21:11

Whoa

cool

Sir Cumference

Left after some event, not too sure on the details

Now his account is gone from stack exchange

0celo7

it involved the Daemon

Sir Cumference

Huh?

0celo7

the GR Daemon

heather

That's unfortunate. It seems, well, strange.

Sir Cumference

21:12

you mean "demon"?

heather

Sounds like he was a good contributor.

0celo7

@SirCumference Nay

Sir Cumference

Yep

Wait what

Oh

Wait what on earth?

0celo7

ok @heather do you want to know what a manifold is

Sir Cumference

@heather You're in middle school?

How do you have 3.7k rep here?

0celo7

21:13

yeah and she knows more GR than you :P

Sir Cumference

Wh-wha...

heather

@0celo7, yeah, that'd be cool!

0celo7

@heather do you know what $\Bbb R$ is?

Sir Cumference

@0celo7 Jesus, who is this person?

heather

@SirCumference, two things: yes, I'm in middle school. And you definitely know more about GR than me =P

0celo7

21:14

@SirCumference Heather, apparently.

Sir Cumference

@0celo7 You should probably ask if she knows what a set is, or a real number

heather

@0celo7, the set of all real numbers.

0celo7

can'y you read?

Sir Cumference

@heather Wait, do you know calc?

heather

@SirCumference, I believe a set is a group of numbers, like A = {1, 2, 3} and a number can be expressed as an element of a set.

And no, I do not know calc. But I'm working on it; I'm currently learning trigonometry.

Sir Cumference

21:15

Jesus christ...

0celo7

@heather what?

heather

@0celo7, what?

0celo7

wait

Sir Cumference

@0celo7 Made sense to me

0celo7

my itunes is broken

heather

21:16

Did I get that wrong?

Sir Cumference

Though the wording is a bit off

0celo7

@SirCumference well it's wrong

heather

Oh, okay.

Sorry.

0celo7

a set is a basic object, you can't really define it

although there are some sets that are not sets

heather

Here's what wikipedia says: "a collection of objects" - is that right?

0celo7

21:17

but that's not important

@heather NO

Sir Cumference

@heather Ehhhhhhhhhhh

0celo7

there are collections of objects that are not sets

Sir Cumference

That's very basic

heather

It also says "Although any type of object can be collected into a set, set theory is applied most often to objects that are relevant to mathematics."

Obliv

@0celo7 is a basket of strawberries considered a set?

0celo7

21:18

yes

heather

Okay, now I'm confused. What is a set then?

Obliv

@0celo7 is a basket of sets considered a set?

0celo7

what you wrote is good enough for 75% of math

@Obliv no

heather

Okay...but the 25%...=P

0celo7

is abstract nonsense

which is uninteresting

Sir Cumference

21:19

@0celo7 Quit teasing her

heather

Ah, gotcha.

=)

0celo7

@SirCumference what?

It becomes important when talking about small/locally small categories but that's completely irrelevant right now

Sir Cumference

@0celo7 Saying "You're definition is wrong!", then saying "the reason isn't important"

heather

@SirCumference, generally, I operate under the assumption that a lot of what people say is sarcasm. =P

Sir Cumference

I meant "teasing" by mentioning something interesting, and then shrugging it off

heather

21:20

And really, it makes sense: my definition is a loose definition is wrong in specific cases that aren't quite important in the current situation.

0celo7

If you define sets naively then you get Russel's paradox.

Sir Cumference

@0celo7 Oh yeah, that fun

heather

@0celo7, oh, yeah, I've read about that!

0celo7

But we're not going to get Russel's paradox here

Because we're being reasonable

heather

Yup

0celo7

21:21

@heather ok do you know what an ordered pair is

heather

(x, y) on the cartesian plane

or (x, y, z)

or (x, y, z, other dimensions)

0celo7

that's not a pair :)

heather

So just the first one?

or none of them?

0celo7

yes, but we want something more general

heather

okay, how about:

0celo7

21:22

First, facts: Let $A$ be a set. Then $A\subset A$ and $\emptyset\subset A$.

heather

right

the second one, that's the void set, right? Just {}

0celo7

Ok, so we're doing to define the product of sets

heather

Okay

0celo7

@heather empty set

heather

yeah, that thing

0celo7

21:23

$A\times B=\{\text{all ordered pairs $(a,b)$ with $a\in A$ and $b\in B$}\}$.

Exercise: compute $\{a,b\}\times \{c,d\}$.

heather

$\{(a,c),(b,d)\}$?

0celo7

That's half of it

heather

The other half...oh, duh!

$\{(a,c),(b,d),(c,a),(d,b)\}$?

0celo7

Wrong other half

heather

Ergh

Oh wait, that makes sense, because $a$ wasn't in $A$ in the second half

oh, question, how do you make the "is an element of" symbol in MathJax?

Obliv

21:26

\in

heather

@Obliv, thanks

@0celo7, I'm unsure what the other half is.

Would it be something like (0, b)?

And stuff? But 0 isn't in A or B...

0celo7

@heather the answer is $\{(a,c),(a,d),(b,c),(b,d)\}$

Obliv

@heather #1 rule now that you're learning elementary set theory is to use quantifier notation in normal conversation.

0celo7

Wait

heather

Oh...I was just going in order. Hmm.

0celo7

21:27

Typo

heather

What is quantifier notation?

Obliv

ex: @0celo7 so I $\in$ class the other day and my professor said the funniest thing..

0celo7

@Obliv But not overuse it.

@Obliv don't do this

heather

Oh, gotcha

Obliv

do what? 8^]

0celo7

21:28

^that either

Obliv

hour left till class and I'm still doing this lab report. sorry I'm a little bored :(

heather

Well, $\in$ class today...=P okay, I'll stop now.

back to multiplication of sets.

0celo7

@heather so are you confused?

heather

I don't think so, I think that makes sense.

0celo7

Ok, next thing on the docket

Adding sets

This is fairly straightforward

heather

21:29

Okay

0celo7

$A\cup B=\{a\in A\text{ or }b\in B\}$

make sense?

heather

One moment, I think I've got it

0celo7

Compute $A\cup B$ from earlier.

heather

Okay

$\{a,b,c,d\}$

0celo7

notation: $\exists$ - exists, $\forall$ - for all, $\land$ - and, $\vee$ - or

heather

21:32

okay

0celo7

@heather yes!

heather

victory!

Obliv

\forall , \exists , \land , \vee

0celo7

Ok, next one is intersection, which I don't this has an arithmetic analogue

$A\cap B=\{a\in A\land b\in B\}$.

heather

okay $\cap$, right?

okay, yeah

0celo7

21:33

Ok, suppose $b=c$, $a\ne d$.

Now what is $A\cap C$?

heather

Oh, {b,c}

It's all the members of A that are also members of B, right?

Like in a Venn diagram, the middle part.

0celo7

Well $b=c$

So that set isn't "reduced" enough

heather

so {b} I guess, or {c}

0celo7

Either one

heather

okay, that makes a whole lot of sense

0celo7

21:35

Ok, $\times$ is called the Cartesian product

So what is $\Bbb R\times\Bbb R$?

heather

right...anything like the cross product, or no?

0celo7

nothing at all like the cross product

heather

k, just double-checking

$\R^2$?

0celo7

that's what $\Bbb R^2$ means, but what is that?

heather

I'm not sure...

0celo7

21:37

what is $\Bbb R\times\Bbb R$

we just defined $\times$

heather

We did? Is it $\R^2$ or something else?

0celo7

what do you mean by $\Bbb R^2$?

Obliv

it's \mathbb{R}

heather

I have no idea =)

Obliv

and scroll up when you defined A \times B

heather

21:37

@Obliv, thanks

0celo7

@Obliv No, \Bbb R

Obliv

$\Bbb{R}$

well shit, I've been typing \mathbb{R} this whole time when I could do it that way.

heather

$\Bbb R$

oh, okay.

@0celo7, well, anyway, would that be the set of all real coordinate pairs, then? Or is that going to far?

0celo7

that's what I want

heather

Okay

0celo7

21:39

So we can define $A\times B\times C\times \cdots$ inductively

heather

Inductively?

vague notion of it have something to do with proofs

0celo7

$A\times B\times C$ is $(A\times B)\times C$

Obliv

@0celo7 proof?

0celo7

you keep building up the chain from the basic definition

@Obliv Clear

heather

Okay, I think I get enough of it to understand.

0celo7

21:40

you have to check that there's no ambiguity but meh

@heather Ok so now $\Bbb R^n=\times_{k=1}^n\Bbb R$.

heather

whoa, whoa, what?

0celo7

$\Bbb R\times\Bbb R\times\cdots\times\Bbb R$ $n$ times

heather

oh, okay. gotcha now.

0celo7

So now what a manifold is...

heather

$\times$ is like a summation then for multiplication in this case

0celo7

21:42

yes

Do you know what a countable set is?

heather

hmm

nope

0celo7

welp, we have to define functions

do you want the formal definition?

heather

I guess...?

Obliv

@heather en.wikipedia.org/wiki/Cantor%27s_diagonal_argument

that's the gist of it without needing function definitions

0celo7

huh?

heather

21:44

Well, that makes some kind of sense...I've read about that before but never realized that was what it was called

0celo7

@heather ok, basically this

you know what $\Bbb N$ is, right?

heather

the set of natural numbers

0celo7

right

heather

would it include zero, or no?

0celo7

Never unless you're in the 3rd world like Italy or wherever

heather

21:45

gotcha

carry on.

0celo7

(some people put 0 in)

Obliv

lol..

heather

okay

0celo7

So a set $A$ is countable if for each $x\in A$ we can "assign" a unique $n_a\in\Bbb N$

It's a notion of "size"

any finite set is countable

$\Bbb N$ is of course countable

surprising things: $\Bbb Q$ is countable. $\Bbb R$ is not countable

heather

Wait...um...so it's if we can say, for each thing in A, there is also a thing in N...like drawing a line between the first two in each set, and then the second two, and so on.

0celo7

21:48

yeah

heather

Oh, okay

0celo7

and you have to be able to do it for every member of $A$

heather

And that's where the whole "there are bigger and smaller infinities" thing comes from?

0celo7

Countable things are nice!

heather

They sound nice

0celo7

21:48

Because we can list them

starting with 1,2,3 etc.

heather

But infinitely countable things...?

0celo7

You can have infinite countable sets

Like $\Bbb N$

You can certainly list the natural numbers

@heather yes

heather

Oh, okay.

0celo7

So a manifold

An n-dim manifold is a countable number of $\Bbb R^n$s glued together

heather

Whoa, what do you mean by gluing together the set of all real numbers?

And real quick, intuitively, what is a manifold? Is it like a blob that is n dimensions? Or what exactly?

Obliv

21:54

@0celo7 I forgot what it was exactly but something along the lines of $|\Bbb{N}| + |\Bbb{N}| = |\Bbb{N}|$ is true, right?

or like $|\Bbb{R}| - |\Bbb{N}| = |\mathbb{R}|$?

0celo7

I'm not a PhD set theorist

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