The h Bar

12:00 AM

I guess for $F_{\phi\theta}$ the expression might be actually correct, since then you don't need to worry about how the curl of vectors looks in different coordinates

0celo7

@lılostafa You asked ACM about electromagnetism and get exterior algebra and Hodge duality.

pZombie

If you were allowed to show someone a single experiment. Nothing else. Out of which he were to conclude GR, him having figured out SR. Which experiment would that be? Assume that he has never seen gravity or knows anything about it. He only worked in flat space until then...

0celo7

And you people think I overanswer

ACuriousMind

But it's 2 am here and I'm kinda exhausted, I'll think about what's going on there are bit more tomorrow

pZombie

there has to be a michelson morley experiment equivalent for GR

or several i guess

Slereah

12:06 AM

there's no single experiment that showcases all of GR

light deflection is a pretty good one

lılostafa

ResearchGate isn't allowing me to access a page on it! It wants to make sure I'm a researcher!

Slereah

Well

ARE YOU?

dmckee

@lılostafa Your main mistake is trying to access a page on ResearchGate.

Or are my biases showing?

0celo7

@dmckee more like your saltiness

Slereah

Research Gate has its uses

Like if some author published a paper on some journal with no electronics archive

but has it on his research gate page

dmckee

12:14 AM

Coming, as I do, from a field that has long had both a pre-print archive and a person database on-line the only thing stranger than the idea that someone would try to start a for-profit research aggregator is that anyone would use it.

I'm forced to assume that it mainly targeted at fields that are impoverished in proper internet era resourcing.

Slereah

Well you know

Sometimes they do

bolbteppa

@pZombie Landau goes into the distinction between non-inertial and gravitational fields by noting they are equivalent locally but not globally as you go to infinity, then derives all of GR by setting up and minimizing an action based on this in volume 2

Slereah

not as great as vixra, of course

lılostafa

@Slereah If anything, I feel like [an idiot](hat_it_is_like_to_be_a_Mathematical_Idiot/links/58b5abcf45851591c5d18458/‌What-it-is-like-to-be-a-Mathematical-Idiot.pdf) now.

^ proof (the link)

dmckee

And of course, the perfectly predictable has occurred: every know-nothing, wannabe out there has a resaerchgate page and they "publish" things on it.

0celo7

12:17 AM

no-nothing?

Isn't that a double negative?

dmckee

@0celo7 Psssthththt!

0celo7

oh my god, someone call an ambulance

david is having a stroke

dmckee

@0celo7 Oddly though my original typing has it "no-knowthing".

And I 'fix'ed it.

0celo7

@Slereah does MTW talk about causality?

Slereah

I don't remember MTW talking about causality much

It's more into very practical concerns

0celo7

12:20 AM

GR isn't practical

like at all

Semiclassical

tell that to GPS

0celo7

@Semiclassical urban legend

Semiclassical

though that's the only example i know if i'm honest

dmckee

@Semiclassical All that GR does in GPS is require more code. It doesn't improve it at all.

0celo7

@dmckee According to the late Chris White, GPS completely ignores GR.

Semiclassical

12:21 AM

hmm

is it only SR that GPS needs?

0celo7

You just synchronize the satellites and they all have the same error so it doesn't matter at all.

bolbteppa

Some interesting looking videos/notes which go into the whole Cauchy causality stuff in GR uwaterloo.ca/physics-of-information-lab/teaching/…

Semiclassical

:/

pZombie

@bolbteppa locally "everything" is equivalent, is SR or so it appears to me. Be it inertial, non-inertial, free falling in a field, or on a planet

bolbteppa

Needs GR

Semiclassical

12:22 AM

@0celo7 That's...disappointing

0celo7

@Semiclassical Chris White had an authoritative explanation on the site, but since he was compelled to delete his account it's hard to find.

@Semiclassical see for instance

19

A: Why does GPS depend on relativity?

There's the article from Ohio State University http://www.astronomy.ohio-state.edu/~pogge/Ast162/Unit5/gps.html which explains quite well why the clocks on a GPS satellite are faster by about 38 microseconds every day. The article then claims that not compensation for these 38 microseconds per da...

Semiclassical

yeah, i noticed that one

Secret

Last night dream is a strange one which sounds almost like something Markus Zuckerberg would said when he first had the idea to create Facebook:

0celo7

Markus?

Secret

Sorry, Mark Zuckerberg, misremembered his name

"Consider the following: A person has an offer for you, in exchange for a sum. Now consider another scenario: you entered a shop to purchase a product.

Imagine a vast network where a person has a digital identity. Now recruit multiple people to promote one aspect of this identity. Supported by the huge data, one can easily monetise this resource

In the first case, authorisation is optional, but in the second case, authorisation is necessary. The second case showed there isn't a physical medium but only a number indicating value and thus how price is a construct. However, individual data i…

(And the above is just the introduction narration in whatever movie I was watching within the dream)

Semiclassical

12:29 AM

ugh, mathematica

do this fourier transform you dumb thing

Secret

It gets even more weird when another set of narrations with a scene that is said to be some interior of some USA national security give the following:

"After the survey and these interviews were done, we will have you entire chemical makeup. Via (forgot) checks, we would have your entire proteome makeup. This allows more effective medicine. For example, should a flu is caught, we can based on your biological makeup to search for the most effective drug."

Slereah

We don't care about your dreams.

5

pZombie

are we borg now?

Secret

And consider that what I did in the past 3 days is nothing but foundations of mathematics, it is not clear how on earth did my dreams get these weird ideas that has nothing to do with type or set theory

bolbteppa

Hmm, maybe gps doesn't need GR because of the mechanics of how it's set up, but a lot of sources say it does and this claim it doesn't is on a lot of crank websites, so hmm...

0celo7

12:35 AM

@bolbteppa Yeah idk. We need to talk to someone from Garmin

bolbteppa

'gps relativity myth' is like a thing to these people

pZombie

@bolbteppa doesn't sound right. A lot of mass is involved, hence clocks tick slower because of it. Maybe the percentage GR affects compared to SR is small, but if you want the most accurate results, GPS definitely needs GR

Slereah

The effects of GR are measurable on a GPS but not necessary

bolbteppa

@pZombie numbers on the mass are here en.wikipedia.org/wiki/…

Semiclassical

I suppose it comes to people wanting to have a nice go-to example of GR being useful/relevant

bolbteppa

12:39 AM

"The effect of gravitational frequency shift on the GPS due to general relativity is that a clock closer to a massive object will be slower than a clock farther away. Applied to the GPS, the receivers are much closer to Earth than the satellites, causing the GPS clocks to be faster by a factor of 5×10^(−10), or about 45.9 μs/day. This gravitational frequency shift is noticeable.

When combining the time dilation and gravitational frequency shift, the discrepancy is about 38 microseconds per day, a difference of 4.465 parts in 1010.[15] Without correction, errors in the initial pseudorange o…

0celo7

Suppose I have $\Omega\subset\Bbb R^3$ bounded s.t. $\partial\Omega\approx S^2$. Is $\Omega\approx D^3$?

i.e. $\Omega$ is the "region bounded by" something diffeomorphic to $S^2$.

@BalarkaSen

Slereah

@0celo7 It could be $\mathbb R^3 \setminus D^3$

0celo7

I said bounded

Slereah

Hm

In that case I'd say yes

Can't think of a counterexample

Semiclassical

this Physics Today article goes into some detail about the subject, though I can't vouch for its authority: uam.es/personal_pdi/ciencias/jcuevas/Teaching/…

0celo7

12:43 AM

Is it at least simply connected?

bolbteppa

$D^3 = \{(x,y,z) \in \mathbb{R}^3 \ | \ x^2 + y^2 + z^2 \leq 1 \}$, $S^2 = \{(x,y,z) \in \mathbb{R}^3 \ | \ x^2 + y^2 + z^2 =1 \}$

Change coordinates and done, idk what the problem is

Guy who wrote that article is used as a football back and forth in these types of arguments heh dealingwithcreationisminastronomy.blogspot.ie/2012/08/…

If GR wasn't hard enough on it's own this would be an easy one

pZombie

how do mirrors work exactly? They actually bounce the photons back or do they absorb and emit the photons?

bolbteppa

@pZombie absorb and emit youtube.com/watch?v=iE6I52Th9DE

pZombie

If you want a bright room without too many bulbs, mirrors instead of tapestry should be better

so there could exist some other "stuff" which instead of absorbing photons and emitting them at the front side at a slightly less frequency, would rather absorb and emit them at the back side at a slightly higher frequency, hence in a sense get pulled by photons rather than pushed

0celo7

1:16 AM

amazon.com/Conformal-Structure-Space-Times-Geometry-Analysis/dp/…

@Slereah

@Semiclassical argh, this has to be trivial

if $\Omega\subset\Bbb R^3$ is bounded by something $\approx S^2$, then $\Omega$ has to be a ball

Slereah

Is it any good?

0celo7

@Slereah It has an article relevant to my research

Slereah

It's not on genlib

0celo7

skype

Captain Bohemian

how are causal relations and conformal structure related?

Slereah

1:28 AM

Conformal transformations preserve the causal structure

Iffff the spacetimes are distinguishable, I think?

0celo7

conformal transformations leave light cones invariant

pZombie

If you draw two arbitrary lines crossing each other within an x-t spacetime diagram, and you draw those lines within another x-t diagram, representing another inertial frame of reference, the two lines will cross at the same event in both frames. What's more, if you mark two points on any of the two lines and move from one point along the line to the other, the ordering of the events you will pass will be the same in both frames.

Captain Bohemian

but does the conformal invariance define the causal structure completely?

Slereah

Captain Bohemian

does the conformal invariance determine the causal structure completely?

Slereah

1:33 AM

Does this imply that there's non-distinguishing spacetimes where two conformal spacetimes don't have the same causal structure?

@CaptainBohemian No

A causal structure is an equivalence class of every conformally related spacetime

pZombie

So i ask myself, if we really need geometry to map that which we call spacetime, or if a mapping would be possible without any geometry, based on the above i wrote.

such a mapping would resemble absolute "spacetime" except it would not be called spacetime anymore, because there would be neither space nor time intervals, but rather an ordering of events along arbitrary lines

Argon

If I have a beam of atoms and send them through 3 Stern-Gerlach devices that respectively accept only $S_z=\hbar/2$, $S_x=\hbar/2$ and $S_z=\hbar/2$, is the beam intensity at the end proportional to $|\langle x+|z+\rangle \langle z+|x+\rangle|^2$?

(Beam normalized to unity after the first SG)

Or am I approaching this wrong

pZombie

Draw an x-t diagram and draw any amount of lines inside, crossing each other at various angles, each crossing representing some event. Filling up almost the whole diagram. Draw some light beams on tops which are also just lines. Then you draw another x-t diagram representing a different inertial frame. All those lines will cross at the same points of course in both frames. Every of those lines if you were to follow one, would have the same ordering of events when followed

This appears to be enough for creating a map without involving any space or time intervals at all which create the appearance of observers in different frames looking at a different spacetime object, when it is actually the same.

DanielSank

2:07 AM

@Slereah In what sense?

I'm not sure what you're asking.

Slereah

2:20 AM

@DanielSank Can you model a quantum computer simply with path integrals rather than wavefunctions

Secret

Does QFT makes sense in quantum computing?

urbandictionary.com/define.php?term=wdc

peterh

3:16 AM

@Blue The oppression can be tricked, gamed and fighted, but all of them are very boring. You have to find a compromise, where you get what you want, even against the downwind.

Slereah

3:49 AM

arxiv.org/pdf/quant-ph/0607151v1.pdf

John Rennie

I have a great deal of sympathy for your point of view. Your original comment about India shouldn't have been flagged, and indeed the flags on it were rejected by the mods/10K users. But your follow up comment went too far and was genuinely offensive.

For the record the message wasn't flagged after I moved it to trash. I saw the flag on it, rejected the flag then moved the message to trash in an attempt to stop the flag being confirmed. However I was just too late and you ended up with a suspension.

0celo7

For the record, it wasn't me.

John Rennie

To be fair the flag on the India is barely Independent comment was reviewed and rejected, so to that extent the flagging system is working. The flag on the follow up comment got confirmed, but then the follow up comment did go too far.

The problem isn't really the flags, because a flag on its own doesn't do anything. It's when the mods/10K users who review the flag don't take the trouble to read the flagged post in context.

0celo7

Balarka is just a bad boi

John Rennie

It's you high spirited young fellows disrupting the peace and quiet of the chat room :-)

0celo7

4:01 AM

I'm a very peaceful person

I can't help that 95% of the flags against me are malicious

Secret

hmm... I see, Danielsank or Emilo will definitely be a better person for that question

John Rennie

Remember that a flag doesn't do anything until it's confirmed, and the external users who review the flag don't know you from Adam. So if you got suspended you can't blame it all on the flagger.

0celo7

@Slereah are you up late or up early?

@JohnRennie adam?

are flags given without a username attached?

John Rennie

The one who got thrown out of Eden for pre-marital sex

0celo7

Is that the British version of the tale?

John Rennie

4:05 AM

@0celo7 I can't see who flagged. I think mods can see the flagger while the flag is active, i.e. before it's reviewed, but once the flag is gone it's hard to find out who did the flagging.

0celo7

No, I mean do they see who was flagged?

John Rennie

@0celo7 British version? I suppose British in the sense that it was Brits who produced the King James bible.

0celo7

I don't remember Adam leaving Eden.

John Rennie

@0celo7 reviewers see which message was flagged. A blue marker appears next to the flagged post.

0celo7

@JohnRennie do they see a username of the flagee?

John Rennie

4:07 AM

@0celo7 I'd have to check, but I thought the pair of them got thrown out and an angel with a flaming sword stood guard to stop them getting back in.

@0celo7 yes, because they see the post and obviously it has the username next to the post.

0celo7

Oh, maybe. But not for premarital sex?

@JohnRennie Ok so I can blame the flagger and his buddies (or maybe sock accounts).

John Rennie

@0celo7 Ah well, that depends on what you think eating from the tree of knowledge is a metaphor for. Hint, it doesn't mean learning about differential geometry :-)

@0celo7 only mods and 10K users can review flags

0celo7

::looks up from differential geometry::

hmm

@JohnRennie your point?

John Rennie

@0celo7 the point is the reviewing is done by well established users not some drive by sock puppet account.

0celo7

@JohnRennie A sock puppet can be well-established

John Rennie

4:12 AM

Yes, if someone hated you that badly they could create a sock puppet and then work to build up that account to a 10K rep. But they'd have to really hate you!! :-)

0celo7

@JohnRennie yeah people are insane

John Rennie

Sweet mother of God I've got a lot of servers down this morning. Bloody Windows updates :-( Oh well, I'll have to go offline for a bit to deal with all the debris. Back in a while.

John Rennie

5:05 AM

@Blue that doesn't prove anything. I have a PhD from Cambridge and I'm a layabout computer hacker :-)

0celo7

@JohnRennie link.springer.com/book/10.1007/978-1-4612-2432-7 would I cite this just as "Topics in Geometry"?

John Rennie

No idea. I'm well out of touch with current conventions in academic writing. Sorry :-(

Balarka Sen

5:49 AM

@JohnRennie That is true, the current system of reviewing flags is moot.

John Rennie

Everyone agrees the current flagging system isn't very good. It's just that chat isn't a high priority for the SE so I doubt it will be fixed any time soon.

Balarka Sen

Yeah I can understand that. I just think it's trolled way too often

John Rennie

Though actually in this case the flagging system worked OK i.e. when someone flagged your comment about India that flag was rejected.

Balarka Sen

And I don't really have a polite message for the trolls :P

@JohnRennie Are you still fighting with your servers?

John Rennie

Everything is back online now :-)

dmckee

5:56 AM

Offered without comment:

-1

A: Twin Paradox Without Acceleration

The idea of "proper time", is a fudge-factor appended to the Special Theory in an attempt to get it to seem to work. In Einstein's original presentation, the phenomenon of time-dilatation is derived purely as a function of (relative) speed, and owes nothing to a geometrical account, such as that ...

Balarka Sen

Ah great

John Rennie

@BalarkaSen Microsoft release Windows updates on the second Tuesday of the month, and our servers are configured to install the updates the following Saturday night i.e. last night. With 600 or so servers updating and restarted there are always a few caualties.

Balarka Sen

Quite naturally. What are these servers that you babysit for, again?

John Rennie

I used to work for a company that does IT support for companies too small to have their own IT department.

I wrote the monitoring system that checks all the servers, and when I decided to retire no-one wanted to take over running the monitoring.

So now I do the monitoring first thing in the morning as a contractor. It takes only an hour or two a day of my time and pays for my laptop addiction :-)

Balarka Sen

Ahh I see. So suddenly the kids you babysitted for are now yours to take care of.

Fun

@0celo7 Nope nope nope nope nope Alexander's horned sphere

John Rennie

6:09 AM

Actually I quite enjoy it :-)

Balarka Sen

Well, you are a computer nerd, so I am not surprised :)

@bolbteppa He's not asking whether the unit ball bounds a sphere. He's asking whether every topological sphere in R^3 is bounded by a topological ball, which is false

There's a totally F-ed up object called the Alexander's gored ball which bounds a topological sphere and is not even simply connected (cc @0celo7)

If your embedded sphere is locally flat, then it does bound a ball however. Generalization of Jordan-Schoenflies

proved by our homeboy Mazur

Secret

6:27 AM

[Random]
(from one of the chat rooms: Secret: 35 secs ago, by Secret 7 secs ago, by Secret 8 secs ago, by Secret 1 ...)

Anonymous

@JohnRennie Nah, it proves that a PhD from Cambridge makes one a weirdo =P Well, someone could argue that only weirdoes do PhD anyway. :)

Anonymous

(But truly speaking I think that professor is now aging and has decided to stay closer to home with his family)

John Rennie

:-)

If the guy is willing to help you learn GR and QFT then that's a great opportunity. But be careful you don't start liking it to much or it may derail your plans to become a tech millionaire :-)

Anonymous

@JohnRennie Haha. I will think of ways to utilize GR and QFT to become a tech millionaire! BTW I will meet him on Monday. Let's see how it goes (I need to convince him to teach me one or two hours a week)

John Rennie

6:45 AM

You should be able to teach yourself GR - there are loads of good books. The great thing about about having an expert to hand is that he can help you when you run into a something you can't get past.

One or two hours a week isn't enough to teach you, but it would be enough to helpget you past any roadblocks.

Balarka Sen

@Blue should learn full fledged Lorentzian geometry

Secret

I am weird, very weird, nonsensically weird, therefore I am a PhD

Anonymous

@JohnRennie Yes, exactly. I am totally willing to work through it myself. But I tend to get stuck a lot of times and am clueless how to proceed further when studying alone. Another problem is that during self-study one is not as focused as while studying for an exam. So, meeting someone to discuss problems helps to maintain regularity.

Anonymous

@BalarkaSen I know a bit of Lorentz transforms :P

Anonymous

But Lorentzian geometry, no

Balarka Sen

6:57 AM

Those are isometries in the standard Lorentzian metric in R^4, or something, I think

Did you ever read the physics paper I linked here a while back

Abhikumbale

Hey guys I'm a physics undergrad any general approach towards physics you guys can recommend?

I'm currently taking a classical mechanics course and going through parts of goldstein

Anonymous

@BalarkaSen A part of it. I will read it well when I will do differential geometry next year. For the time being I'm focusing on Multivariable Calc

Balarka Sen

OK

It's mostly an pop-exposition of the math, but that's fine

Anonymous

@Abhikumbale I got bored with goldstein :P

Anonymous

But I've heard it is the golden book of mechanics

Abhikumbale

7:03 AM

@Blue I'm just starting out and I've covered 2 chapters. Yes it is a highly recommended book.

Balarka Sen

Do you know the inverse/implicit function theorem yet

That's probably the most important theorem in mult.calc.

Anonymous

@BalarkaSen No, but it's there in syllabus. I'll get to that part soon

Anonymous

@Abhikumbale Qmechanic is the Classical Mechanics gawd here. You can ask him any Goldstein question :P

Physics Meta

0

Q: Meson Mass solving

I am totally helpless with the following problem. Any kind of help is appreciated. In a classical model, a scalar meson consists of a quark and an antiquark bound by a potential:V(r)=ar+(b/r)here(a=100MeV−V(r)=ar+(b/r)here(a=100MeV−fm^{-1}$ and b = 200MeV-fm.) If the masses of quarks and antiquar...

discussion

Balarka Sen

9:03 AM

@0celo7 lmao scare pewdiepie season 2 is getting replaced by scare jake paul

Slereah

10:22 AM

ittyon

Noun: ittyon (plural ittyons)

(physics, rare) tardyon...

what

Ced

Is this correct ? Force necessary to give a velocity of 10km/h to an object of 10kg.

10km/h = 10000 m/ h = 10000/3600 s = 2.777777 m / s

F = 27777 N

Last time I did physics was in school a decade ago and I'm not sure I'm doing this right

It's assuming 0 friction

and the force is applied during 1 second

pZombie

Ced - What's a newtonmeter?

@Ced a = m/F and s = 1/2 a * t^2

sorry a= F/m

Ced

10:47 AM

A newton is the force applied to 1g to give it an accel of 1m/s². I figured if I give an accel of 2.77m/s² during 1 sec it'd have a v of 2.77 isn't it ?

pZombie

not newton. Newtonmeter

pZombie

11:03 AM

@Ced sry i misread the initial question. Yes, that is correct

@ced v= a * t, a = F/m => F = v*m / t

Secret

in Mathematics, 2 mins ago, by Secret

I see, so we can always prove something is independent, or is there exists predicates that we cannot show it has any of the following truth values: Undecidable, True, False, Partially true, Unprovable, Independent, a constructive proof that gives the object?

in Mathematics, 1 min ago, by Secret

Put it simply, is there exist predicates that has a truth value, but we don't have any way to determine what it is, and we don't even have a way to determine that we don't have any way to determine what the truth value is, (similar to nearly all things in real life)?

Real life is different from anything that appear on paper and theory:

There exists something called "you don't know and you are not allowed to know"

The only thing in real life I hate is when things refused to explain itself when it is given the chance, and real life has heaps of these, unlike mathematics

ACuriousMind

@lılostafa I looked at it again and it appears both of Tong's formulae are correct. We have $F_{\theta\phi} = F_{ij}\frac{\partial B^i}{\partial \theta}\frac{\partial B^j}{\partial \phi}$. Starting from the expression for $F_{ij}$ he gives, I managed to arrive at $\sin(\theta)$ with that, after a lot of sign errors :/

Ced

1:00 PM

@pZombie thanks

bolbteppa

1:25 PM

@BalarkaSen This en.wikipedia.org/wiki/Alexander_horned_sphere#Impact_on_theory says "The horned sphere, together with its inside, is a topological 3-ball, the Alexander horned ball, and so is simply connected;" i.e. it "is" a topological ball, "and its boundary is therefore a sphere" mathworld.wolfram.com/AlexandersHornedSphere.html how can this be false, wouldn't mathematics break?

This thing is on: chat.stackexchange.com/rooms/66708/physics-workshops

0celo7

@BalarkaSen What is this locally flat bullshit?

I have a smoothly embedded sphere

lılostafa

1:40 PM

@ACuriousMind If $B_i$s are cartesian (unit) vectors, isn't $\frac{\partial B^i}{\partial \theta}$ or $\frac{\partial B^i}{\partial \phi}$ zero?!

ACuriousMind

1:56 PM

@lılostafa No, e.g. $B^1 = B\sin\theta\cos\phi$

Balarka Sen

2:30 PM

@bolbteppa This is what happens when you make up conclusions when learning mathematics from wikipedia and wolfram mathworld.

The interior is indeed a topological ball. The exterior is not.

That is precisely what the "Alexander gored ball" is.

@0celo7 Then it's true.

0celo7

@BalarkaSen He knows his shit. He can derive the Riemann tensor from memory (whatever that means)

bolbteppa

Sure, why do you care about the exterior

0celo7

And also in 5 different ways

Balarka Sen

@bolbteppa Because the exterior bounds the sphere, and is not a topological ball :P

0celo7

@BalarkaSen Is the proof easy?

Balarka Sen

2:32 PM

@0celo7 I don't think so. Look in Hatcher's 3-manifold notes; there's probably a Morse theory proof.

0celo7

Compact Morse theory is easy

ZeroTheHero

@Abhikumbale Much better and more modern alternatives are Marion&Thornton and Hand&Finch. There is also a nice digest by Calkin that is really well done

CooperCape

-.-. --- -- .--. .- -.-. - theory

bolbteppa

@BalarkaSen ahhh, exterior...

Balarka Sen

@bolbteppa Indeed. It is actually a standard issue in 3-manifold theory.

0celo7

2:36 PM

@BalarkaSen I probably don't want to know what happens in $\Bbb R^4$, do I

Balarka Sen

Quite shocking, so your disbelief was justified.

@0celo7 Um, what do you want to know about R^4?

0celo7

If you have an embedded $S^3$, what can the interior look like

Balarka Sen

It can be pretty bad lmao

You can get exotic R^4's

Not to say things which are not topological balls

0celo7

@BalarkaSen Whenever I say some word that can be intepreted in different cateogries, always assume I mean smooth unless I'm doing GMT.

Balarka Sen

Well, exotic objects are bad objects in the smooth category...

You can get things bounding S^3 which are homeomorphic to 4-balls but not diffeomorphic to them

0celo7

2:40 PM

@BalarkaSen I'm reading a paper by someone in my group. He just needs the interior to be simply connected.

Is that always true?

bolbteppa

@Abhikumbale some slides following Goldstein users.physics.harvard.edu/~morii/phys151/lectures and some videos youtube.com/playlist?list=PLaNkJORnlhZmDOd4KxfvlkJh3zJP8sEMP

Solenodon Paradoxus

Hi dudes. Quick question about the Lorentz group. I remember an answer on PSE describing in details the quadratic invariants of SO(n,1). How many there are and what are their expressions in terms of the Lie algebra generators. E.g. for SO(3,1) there's $K^2 - L^2$ and $K \cdot L$. I think this answer was by @ACuriousMind but can't seem to find it..

So either pointing me to that answer or simply giving the answer straight away would be greatly appreciated. Thankns

Balarka Sen

@0celo7 van Kampen should tell you that if it's a smoothly embedded S^3

0celo7

@BalarkaSen How so?

Balarka Sen

just do it - Shia Labeouf

Oh wait it doesn't

boo

@0celo7 I guess you have no other way than to prove Alexander's theorem :(

Namely, complement of a smoothly embedded S^(n-1) in S^n disconnects it into two D^n's

0celo7

2:55 PM

@BalarkaSen I imagine h-cobordism saves the day in high dimensions?

Balarka Sen

@0celo7 No, it's a Morse theory proof, it should not be hard.

Look in Hatcher's notes, like I said

0celo7

I did

got bored

need to do more GR

Balarka Sen

I kinda felt like it would be easy to prove the connected components of S^n - S(n-1) are simply connected though

@0celo7 lol sucker

0celo7

@BalarkaSen why a sucker

Balarka Sen

@0celo7 Meh... I was right about the van Kampen thing. Say $X$ and $Y$ are thickening of the two components of $S^n - S^{n-1}$. $X$ and $Y$ intersect in a tubular neighborhood of $S^{n-1}$, which deformation retracts to $S^{n-1}$, which is simply connected.

Then $0 = \pi_1(S^n) = \pi_1(X) * \pi_1(Y)$

That means $\pi_1(X) = \pi_1(Y) = 0$.

Note that $S^{n-1}$ being smoothly embedded in $S^n$ is crucial for having a tame neighborhood

0celo7

3:23 PM

@BalarkaSen Hmm

@BalarkaSen I will think about this later and send this to my colleague. Much easier than what he was proposing

Balarka Sen

Ok.

ACuriousMind

@SolenodonParadoxus Do you mean this answer (not by me)?

Constantine Black

3:47 PM

Hi to all. Does anyone know how to handle a trace over dirac spinors and isospin(two flavours so we only have a proton-neutron). For example I have a trace of the form $Tr[\bar{u_1 } \bar{u_1}' u_1 ' u_1] $ over both spinor indices and spacetime. For the spinor part, technically I will average of the spin polarizations due to Casimir's trick.

Should I do something similar with the isospin(I don't think I have seen this anywhere) or is there a standard result to be found for the isospin trace? Any advice will be helpful-thanks.

Solenodon Paradoxus

@ACuriousMind yes, thanks! I dunno why I thought it was by you.

Constantine Black

Note-The primed spinors just mean the momentum of the particles after scattering.

0celo7

@BalarkaSen Ok so what about in $\Bbb R^5$? Suppose one has an embedded compact simply connected 4-fold $M$. Is the region bounded by $M$ simply connected?

In one less dimension one knows that $M\approx S^3$, but here?

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