The h Bar

So the definition of M is a topology where, for every point in the topology, there exists an open set containing p, and a function f that maps members of that open set to R^4 (in our case)?

0celo7

i.e. not to $\mathbb{R}^3$, etc.

@barrycarter yes and $f$ is bijective and bicontinuous

barrycarter

OK, continue.

0celo7

So let $U\subset M$ containing $p$ be the open set in question and let $U'$ be its image in $\mathbb{R}^4$ under $f$.

barrycarter

Oh, I see where you're going with this. Because the image is an open set in R^4, we can assign dt, dx, dy, dz by f^-1 ?

0celo7

Yes.

Exactly.

Wait

What are dt, etc.?

Vectors?

barrycarter

6:16 PM

Numbers at the moment.

0celo7

We can assign coordinates this way

barrycarter

There's an open ball in R^4 that is a subset of the image of p's neighborhood in M, right?

0celo7

The function $f$ is called a chart.

@barrycarter Yes, I think so.

barrycarter

OK.

0celo7

One can replace "open set" by "open ball" IIRC.

barrycarter

6:16 PM

OK, wait.

Yes, in R^n the open sets must contain open balls.

0celo7

Yes.

By definition.

barrycarter

They don't have to be open balls themselves, like open sets on the real line don't have to be intervals.

0celo7

Yes.

@barrycarter It's a nontrivial proof.

barrycarter

OK, but for a given p, there are multiple f possible, right?

0celo7

But it can be done.

@barrycarter Exactly!

barrycarter

6:18 PM

Ocelot, I don't think it's nontrivial, but OK.

0celo7

That's why I objected to @JohnRennie saying "unique"

barrycarter

So, which f do we choose as our 'chart'?

Oh geez.

0celo7

@barrycarter which part?

barrycarter

Trivial to show than an open set in R^n must contain a ball in R^n

0celo7

Proving that one can define manifolds with open balls instead of open sets is nontrivial.

@barrycarter That's literally what "open" means.

barrycarter

6:19 PM

Ummm, can't agree with you there, but OK.

0celo7

How do you define the metric topology otherwise?

John Rennie

@barrycarter you can choose any f. Different choices of f give different coordinate systems.

0celo7

A set is open wrt. the metric topology if it contains an epsilon-ball around every point.

barrycarter

By the raw definition of topology: any collection of sets that includes R, the empty set, the arbitrary union of sets in T, and the finite intersection of sets in T.

0celo7

@barrycarter Dude, you have to define the topology of the reals first

The metric topology on the reals is not automatic

@barrycarter True or false: by definition, an open set in $\mathbb{R}^n$ with the standard topology is one that contains an epsilon-ball around every point which is completely contained in the open set.

barrycarter

6:22 PM

(sorry, breif break, back now, reading)

Ocelot, I would argue false. I think you can prove that though.

Ocelot, any T2 space on the reals should have that property.

0celo7

What the fuck do you think the standard topology on $\mathbb{R}^n$ is if not that

@ACuriousMind I need you

@barrycarter Not every T2 space has a metric o.O

barrycarter

Oh, standard topology, ok that might work.

Ocelot, T2 is Hausdorff, right?

0celo7

Yes

barrycarter

You should be able to create a metric then.

OK, wait, you're right. The open sets on R^n are the cross product of intervals on R.

0celo7

@barrycarter No.

barrycarter

6:25 PM

Er, those are the basis open sets.

It's the same thing. You can prove the epsilon condition from there.

0celo7

Uhhhhhhhhhhh, the usual basis we take are the balls with rational radii at rational coordinates

barrycarter

Really?

0celo7

Yes

barrycarter

That's a more minimal basis.

0celo7

It's a countable basis

barrycarter

6:26 PM

True, but why does that help us?

I've always heard the standard topology on R is (a,b) for any reals a and b.

ie, the basis, I mean

John Rennie

This has gone beyond my ability to usefully contribute. I'll see you later ...

barrycarter

Thanks @JohnRennie

(for helping not for leaving)

Ocelot, we've also strayed somewhat from the topic.

0celo7

You should refresh your topology

barrycarter

If you say so...

ACuriousMind

Actually, the boxes (products of open intervals) are also a basis of the topology on $\mathbb{R}^n$, so you two are disagreeing over nothing.

barrycarter

6:30 PM

We know this, the key word is 'standard'

We understand our definitions are equivalent.

ACuriousMind

They are the natural basis if you define the topology as the product topology, while the balls are the natural basis if you define the topology via the Euclidean metric.

barrycarter

For any finite number of intervals, there must be a minimum epsilon whose ball is contained in R^n

ACuriousMind

"Usual" or "standard" entirely depends on which of the two you are doing

barrycarter

Ocelot, at the moment, we're getting off topic :)

I'm so glad I asked @JohnRennie instead :)

Ocelot, are you saying @JohnRennie's argument is incomplete, but correct?

0celo7

Maybe

barrycarter

6:36 PM

OK, I guess my point is: we're talking about physics here. If it gives the right answers, is rigor important?

0celo7

Yes.

barrycarter

Why?

0celo7

You of all people should understand that

barrycarter

Not really. In science, there are no degenerate functions.

We don't need to worry about corner cases.

I'm not inventing a new type of relativity.

All functions are C-infinity in reality.

0celo7

No

They need not even be C^0

barrycarter

6:43 PM

Explain?

0celo7

@barrycarter cf. @ACuriousMind

barrycarter

Which of the things he said?

0celo7

He'll tell you about discontinuous wave functions

barrycarter

OK. But bashing ahead with relativity, where do you disagree with @JohnRennie

ACuriousMind

@0celo7 I won't, since that is completely irrelevant to your current discussion.

0celo7

6:48 PM

I'm not sure what the discussion is

barrycarter

Ocelot, you complained that @JohnRennie's explanation was inaccurate, or at least hinted at it.

Are you now saying you believe the explanation was correct (in the sense of giving the right answers), but not rigorous?

0celo7

Did I ever say anything different?

barrycarter

Before @JohnRennie had finished explaining, you seemed to be complaining about his accuracy, not juts his methodology.

I'll settle for accurate but not ri... ooh, I summoned him by mistake.

I must have said it 3 times.

You don't need mathematical rigor for science.

0celo7

Yes you do, otherwise how can yo be sure your mathematical manipulations are correct?

SR generally escapes the difficulties of GR because the manifold is a vector space with a nice metric.

barrycarter

Ocelot, because, in reality, you never hit the corner cases you do in pure math. And besides, you already agree that @JohnRennie's method was valid.

I'm anxious to apply it to accelerating objects and see what happens.

0celo7

6:57 PM

Rigor is more than "corner cases" -.-

barrycarter

It can be, yes.

But this is reality... there are no infinite series, there are no epsilons, there are no square roots of negative numbers.

Reality is a very limited subset of mathematics.

0celo7

@barrycarter I don't know what this means

barrycarter

The mathematics we use to solve science problems is much simpler than the entire structure of pure mathematics.

0celo7

Depends on the science.

barrycarter

Maybe, but I think relativity doesn't require the rigor you want to give it.

0celo7

6:59 PM

Maybe not special

But general does, definitely

John Rennie

If you think it's worth it I'll rejoin the conversation ...

barrycarter

@JohnRennie Sure, but I think Ocelot is about to concede the point that what you told me is correct. I do have some additional q's for you, but those will be after I play around with what you've already told me.

Ocelot, science is applied math.

0celo7

I never said that @JohnRennie was incorrect.

Incorrect about certain things, yes.

barrycarter

I'm not going to search through the chat logs, but, early on, when you were "interrupting", I think you did.

0celo7

@barrycarter So?

John Rennie

7:04 PM

@0celo7 Most of the key theorems in GR cannot be proved without the rigorous approach that 0celo7 is describing

barrycarter

Ocelot, but is he fundamentally correct about the invariance of ds^2

0celo7

Why does applied math deserve less rigor?

barrycarter

You're kidding, right?

0celo7

@barrycarter No.

John Rennie

That means no Hawking-Penrose singularity theorems

0celo7

7:05 PM

@barrycarter Yes.

barrycarter

Final answer, please.

0celo7

I'm replying to different messages.

barrycarter

So, you're saying @JohnRennie is incorrect, but you agree that Applied Math requires less rigor?

0celo7

John is correct about the physics, but the mathematics can be refined.

barrycarter

No Scharwzchild radius, either? ;)

0celo7

7:06 PM

I don't see any reason why applied math needs any less rigor.

barrycarter

Wow, I can't really answer that.

If it works, it works.

John Rennie

@barrycarter the Schwarzschild radius is easy to derive. Even I can do that. Want to see how?

barrycarter

Will you use the formula for the final size of a star that goes supernova?

0celo7

@barrycarter How do you know that it works?

@JohnRennie >easy

barrycarter

This is science. You test.

If a methodology works in science, you use it.

0celo7

7:08 PM

@barrycarter How do you know that it works?

barrycarter

You don't have to prove why it works.

You test it.

Crap, I'm almost arguing against myself in another thread.

John Rennie

Well, OK, yes, you need to provide that no null trajectory from the event horizon can real one of the conformal infinities (I can't remember which)

barrycarter

If you find a formula that gives you the correct results, the formula is correct by the definition of science.

Obliv

Until you observe something different, then the math can be reformulated

barrycarter

Science isn't rigorous.

@Obliv Correct!

Obliv

7:09 PM

but if it were rigorous in the first place it wouldn't have had to be reformulated

maybe

barrycarter

@Obliv But the concept of rigor makes no sense in science.

0celo7

@barrycarter Can you test the singularity theorems?

John Rennie

I don't think you're being fair.

Obliv

I like the quote on @acuriousmind 's profile. something along the lines of "rigor cleans the dirt that lets intuition shine through" or something

John Rennie

The point of a theory is that it allows us to extrapolate from experimental measurements

0celo7

7:10 PM

@barrycarter What makes sense is the rigor of the mathematical model.

@JohnRennie Yes, this.

barrycarter

OK, that's where you do need to consider special cases....

OK, hold on.

Science = find a hypothesis, if it works, declare it good. Yes?

John Rennie

A rigorous approach guarantees there aren't inconsistencies that will mess up our attempts to extrapolate

barrycarter

Yes, but how can you have rigor when your formula itself comes from observations?

John Rennie

@barrycarter That's precisely wrong

barrycarter

Explain please?

John Rennie

7:12 PM

We invent a mathematical model then see if it fits the data. If it does we book our flight to Stockholm. If it doesn't we try again with a different model.

barrycarter

Agreed.

John Rennie

Rigor is the way of guaranteeing our model is internally consistent

Obliv

hey someone in the math chat says he'll give anyone $100 if you can solve the borel conjecture for any specific dimension greater than 3. Sounds like easy money for my boy @0celo7 . we can split profits 60-40 since I told you about it.

barrycarter

OK, that's a little different.

John Rennie

Experiment is the way we find out if it describes the real world

Both are necessary

barrycarter

7:14 PM

Yes, your mathematical model has to be consistent, but that just means you check for where it might break down.

But those are extreme cases.

John Rennie

There are people on this site who claim there is nothing inside th event horizon of a black hole

and any attempt to extend physics inside the horizon is meaningless.

barrycarter

You can extend the mathematical model, but you no longer know if it applies.

John Rennie

But because we have a self consistent and experimentally tested theory that works outside a horizon, we feel confident extending it to regions where we can never do experiments

barrycarter

And I disagree with you there.

You can't extrapolate mathematical models.

You build a model to describe a little piece of reality.

It doesn't necessarily describe reality beyond the confines for which it was built.

John Rennie

Extrapolation has been the driver for progress in science since neanderthals started banging rocks together

barrycarter

7:17 PM

Extrapolation with testing, though, not extrapolation without testing.

0celo7

GR might be the most extrapolated theory of all time.

John Rennie

Yes, extrapolate then test.

barrycarter

Yes, you're free to guess what will happen by extrapolating your model, but it isn't science until it's observed and confirmed.

0celo7

We went from the fucking orbit of mercury to black hole singularities.

John Rennie

But the testing may have to wait decades or centuries.

barrycarter

7:18 PM

And in the mean time, the theory remains unproven.

John Rennie

Gravitational waves were an extrapolation until a month ago

barrycarter

Actually, they were a logical necessity, of sorts.

John Rennie

No, GR could have been wrong

barrycarter

True. They were necessary if we accept information can't travel faster than light.

John Rennie

They were a logical necessity only because GR demands they exist

I can't believe I'm agreeing with 0celo7 ...

barrycarter

7:19 PM

OK, let's back up a step. If I have a statistical model of something, would you agree that it's not necessarily accurate to interpolate or extrapolate from such a model?

John Rennie

That's not the point I'm making and I'm not going to comment on it.

My point is that extrapolation tells us what experiments are going to be interesting to do next.

The experiments aren't guaranteed to match the extrapolation ...

and of course the big steps have been when they didn't - relativity and quantum theory

But unless we extrapolate our existing models we're reduced to doing experiments at random in the hope of stumbling across something interesting

barrycarter

OK, I agree with you there, but I think I've lost the main point.

Where does mathematical rigor enter the picture?

John Rennie

It guarantees our extrapolations are a sensible direction to pursue

barrycarter

It almost seems like you're saying mathematical rigor is a bad thing because we'd have to reject any formulas that could give meaningless results?

John Rennie

as opposed to random stumblings around

barrycarter

7:23 PM

Not really. If your formula describes reality, it works. Why does it have to be "rigorous"?

John Rennie

Because we aren't content with describing reality

barrycarter

Are you saying: if we can't extrapolate from a formula, then we can't use this formula for the cases where it does work?

ACuriousMind

@barrycarter Because just finding formulae after formula that describes a tiny piece of reality would be stamp collecting, not science.

John Rennie

We always want to go further

barrycarter

@ACuriousMind I disagree. I believe that's how we got to where we are now.

ACuriousMind

7:24 PM

We don't want a collection of disconnected formulae that describe reality, we want a coherent theoretical framework of how the world works.

barrycarter

@ACuriousMind Which comes from stamp collecting formulas and then gluing them together.

ACuriousMind

@barrycarter Sometimes, yes.

barrycarter

So, if I have a non-rigorous mathematical model that happens to describe reality for a limited set of values, you're saying that model is invalid?

John Rennie

@barrycarter This is getting into navel gazing territory. Unless you want me to teach you GR I'm going to bed. I start work in eight hours.

Obliv

it'd be better if it described reality for a larger set of values.

doesn't mean it's invalid

barrycarter

7:26 PM

@JohnRennie It's Friday night, dude.

@Obliv Yes, but you can't reject the theory for lack of rigor.

@JohnRennie No, my next GR questions will be no earlier than tomorrow.

John Rennie

@barrycarter Like you academics ever do any real work anyway :-)

barrycarter

I wholeheartedly agree with that statement :)

I meant for you.. you work Saturdays?

ACuriousMind

@barrycarter Just having a formula that describes some data is not what we call a "theory".

John Rennie

I work seven days a week - welcome to the real world :-)

But to be fair I work for only the first few hours of each day.

barrycarter

@ACuriousMind I would argue otherwise. If the model fits the data and is predictive, it's a theory!

@JohnRennie Sleep tight, try not think about people in the rest of Europe who only work 5/7.

I see science as people coming up with math formulas/models that work first, and then try to expand them later.

You don't say "this model won't work when [unrealistic situation], so we must discard it"

ACuriousMind

7:31 PM

I feel we're talking past each other, because no one denies that's part of science.

barrycarter

OK, to me, a non-rigorous model is one that gives undefined results for certain parameters. Do we agree on that?

ACuriousMind

No

barrycarter

OK, your definition please?

ACuriousMind

A non-rigorous model is one that has not been rigorously derived from first principles.

Where what the "first principles" are varies from subfield to subfield.

barrycarter

OK. I'll agree with that.

In other words, in deriving the model, we made certain assumptions that aren't necessarily true?

All series converge, all functions are continuous, etc?

ACuriousMind

7:35 PM

Something like that, yes

barrycarter

But the resulting model does work for the currently observed cases?

ACuriousMind

Well, if it doesn't, it's not only non-rigorous, it's wrong.

barrycarter

Yes, I'm leading up to a possible point, maybe.

But we're talking about a non-rigorously derived model that is applicable to the observed cases, right?

ACuriousMind

I don't know

I wasn't following the entire conversation :P

barrycarter

Yeah, me neither :P

Ultimately, I'll say this: in applied math, if it works for some cases, it works with maybe a little star saying "based on a bunch of assumptions that hold true for our purposes".

ACuriousMind

7:39 PM

@barrycarter Let's just stop then lest we start disagreeing about unspecified things again ;P

barrycarter

LOL :)

Bass

@ACuriousMind Re this. Is that just "we have determined experimentally that the electron's electric charge is three times that of the down quark etc, and the most elementary way this fits into Yang-Mills theory is that the down quark's charge is $q=1$ (which determines the coupling constant $e$), while the electron must be $q=3$..." or is there some deeper meaning to this?

In other words, are there other (physical, theoretical, whatever) consequences of the different fermion representations than just their electric charge?

Obliv

tfw you type up a long ass paragraph asking about something and understand the answer when you finish typing.

I wonder what that's called. Probably just idiocy. If it helps it helps I guess.

barrycarter

That happens to me a lot. Writing down a problem often solves it.

About half the questions I plan to ask Stack are solved in the writing it down stage.

Obliv

that makes me feel better :')

barrycarter

7:45 PM

Because it forces you to be specific.

Obliv

yes. that definitely got me.

barrycarter

You can't hand wave and say "well, obviously..." (well, you can, but people will call you on it).

Then you realize "hey, that's not obvious.... it's not even true"

Obliv

all of this set notation stuff really makes the words in a sentence valuable. I have to pay very close attention to detail when learning this.

Bass

@Obliv Yep, happens to me a lot :)

ACuriousMind

@Bass I...don't want you to take my word for it, but I would say no.

Bass

7:49 PM

@ACuriousMind So if there was some particle whose electric charge was e.g. a 13th of the electron's charge, would theorists just adapt $e$ to the greatest common divisor of all these charges and then adapt all the particles' $q$ value to "restore" their charges?

DanielSank

@ACuriousMind I dunno, but I think it does.

Bass

And, if we find a particle whose electric charge is an irrational number times the electron's charge, the whole theory would collapse?

ACuriousMind

@Bass I think so

@Bass :O

Fortunately, you can never tell if a physical quantity is rational or irrational since the rationals are dense in the reals, so that's not a concern.

Bass

@ACuriousMind Yep :D

ACuriousMind

@Bass My thoughts exactly ;)

@DanielSank I have realized that I don't actually know what being entangled means for a mixed state. :/

Obliv

8:03 PM

what does this mean: $a,b \in \mathbb{Z} - \{0\}$?

the minus {0} part

Kyle Kanos

this guy deleted the one question I answered recently that would have given me the silver discussion badge :(

Anyone want to help me undelete it for completely selfish reasons?

ACuriousMind

@KyleKanos Only if you had gotten 4 upvotes for that mess ;)

Also, hi!

Kyle Kanos

Hi

I'm home sick today :(

Obliv

I summon thee @barrycarter

Kyle Kanos

My sister in law brought some ailment with her during her visit 10 days ago.

Passed through 3 of the 4 kids and myself :(

ACuriousMind

8:06 PM

Well, get well soon!

0celo7

@KyleKanos that sucks

ACuriousMind

That sounds strange. How does one wish better health to a sick person in English?

Kyle Kanos

@ACuriousMind "Get well soon" is correct

@0celo7 Yes indeed. Fortunately it's just a general soreness and headache, not vomiting and the like

0celo7

@ACuriousMind How does one do it in German?

ACuriousMind

@0celo7 "Gute Besserung!"

0celo7

8:11 PM

@ACuriousMind Vielen dank

Kyle Kanos

Also: Brandon hit 1k reviews in Late Answers yesterday, making him the 2nd to get all the Steward badges

barrycarter

8:44 PM

@Obliv I am summoned.

I tried to move a comment thread to chat but it only moved the first few messages?

David Z

9:14 PM

@barrycarter I think that tool only pays attention to a conversation between two specific people, and only affects comments they've posted, but not other people's comments... was that it?

barrycarter

9:26 PM

@DavidZ physics.stackexchange.com/questions/248072/… and just the two of us. Me and @curiousone

DanielSank

9:56 PM

@ACuriousMind I don't think I do either.

0celo7

10:17 PM

@KyleKanos how are you trying to heal yourself

Kyle Kanos

By resting. Laying down & watching twitch stuff