The h Bar

ACuriousMind

00:00

Because the cover consisting of all balls with radius smaller than its radius doesn't have a finite subcover?

0celo7

@ACuriousMind Huh?

How are the standard balls in $\mathbb{R}^n$ compact then?

ACuriousMind

Oh

::facepalm::

It's still not compact, though

But the cover which doesn't have a finite subcover is the cover of "annuli" centered at $r/2$ with thicknesses $\epsilon < r/2$.

DanielSank

@MGA Noooooooooooooooooooooo

Use Munkres!

0celo7

@ACuriousMind Damn.

DanielSank

00:15

@ChrisWhite There are two better ones: Analysis on Manifolds by Munkres, and Vector Calculus, Linear Algebra, and Differential Forms: A Unified Approach by Hubbard and Hubbard.

0celo7

@DanielSank Or Lee's Smooth Manifolds.

user54412

I actually don't own any books that prove Stokes.

user54412

wonder where I learned it

0celo7

Frickin lie.

I know you own Arnold.

user54412

@ACuriousMind Which webcomic was about not hiring anyone who got their degree post-wikipedia?

user54412

00:22

@0celo7 Oh yeah. Couldn't spot it among all the yellow on my math shelf.

0celo7

@ChrisWhite How many books do you own?

ACuriousMind

@ChrisWhite I...don't know what you mean :(

user54412

@ACuriousMind I swear it exists. It's also impossible to google for.

ACuriousMind

You probably don't even know which one it was, i.e. xkcd, SMBC, PhDComics, or something else?

user54412

@0celo7 Maybe 15 math, 30 physics, 15 more astrophysics, 15 philosophy, and 10 random science

ACuriousMind

00:26

Random science is best science.

user54412

that comes at the bottom (yes, I actually sort my books by purity)

0celo7

@ChrisWhite Pic?

What are the 15 math books on

user54412

let's see

user54412

differential equations x3, statistics x2, algebra, graph theory, real analysis, complex analysis, differential geometry, algebraic topology, differential topology, more algebra, even more algebra, mathematical physics, dynamical systems, number theory

0celo7

Random math books?

Since when do you do algebra

user54412

00:33

basically

0celo7

or graph theory

or algebraic topology

user54412

I was a math major you know

0celo7

You took a graph theory class?

Also just one diff geo book?

user54412

yeah -- not a terribly advanced one

user54412

and I learned my diff geo from GR books

user54412

00:35

also, I did more algebra than anything else in math

0celo7

@ChrisWhite that explains a lot

why, algebra is pretty much the worst math

it literally tells you nothing important

user54412

had the best instructor

ACuriousMind

@0celo7 Math is not about telling you "something important".

0celo7

here we go

ACuriousMind

Also, once you know them, you start seeing groups, rings and modules everywhere else :P

user54412

00:38

^ so true

0celo7

so

no need to have 5 algebra books

@ChrisWhite do you have a book that proves Gauss-Bonnet

Qmechanic

1

Q: Does it make sense for physics to predict the unphysical?

I hear people say things like "inside a black hole the laws of physics are not valid" or "there can be parallel universes with different physical laws" or "before the big bang there was nothing". Can physics really make such predictions? Is not there a logical issue here, namely, is it p...

soft-question

Is this primarily opinion-based? Or too broad?

ACuriousMind

@Qmechanic Was debating that myself, decided I don't feel strongly one way or the other

user54412

@0celo7 apparently Milnor.

user54412

surprising it has anything at all, being only 57 pages long

0celo7

00:43

the diff topology one?

or morse theory

how much did you pay for that pamphlet

user54412

differential topology

ACuriousMind

@ChrisWhite I have the impression Milnor has an extremely dense style

user54412

I hope it was cheap

user54412

@ACuriousMind after reading him for a while, you do begin to wonder what everyone else is saying with all those words

0celo7

what did you need that book for

what does it even accomplish

ACuriousMind

00:46

@ChrisWhite Nothing relevant, apparently :P

dmckee

@Qmechanic Frankly "People say things like" isn't a good basis for a question. Who says these things? In what context? For what audience?

I can't think of a reputable scientist who would go with #1 or #3 in a serious context, and #2 is only sensible with certain understandings of what "parallel universe" means (and it is not the one the average layman imagines).

Qmechanic

@dmckee : Ok, I close it.

0celo7

01:11

@ACuriousMind Is the growling really necessary?

@dmckee I'll listen to one Pink Floyd song of your choosing.

ACuriousMind

@0celo7 Well, for that style, it is. You'll have to go more towards power or symphonic metal to lose the growling.

Qmechanic

01:29

@0celo7 : There are so many good PF songs to chose from! How about dogs?

0celo7

@Qmechanic People claim this but I have yet to see the proof.

@Qmechanic 17 minutes!

(This is literally the first time I've ever listened to this band.)

ACuriousMind

Ah, the short attention span of the youth :P

0celo7

Meh.

What's so special about this?

ACuriousMind

@0celo7 How you managed to not hear at least Wish You Were Here is beyond me

0celo7

@ACuriousMind We're the same generation, buddy.

@ACuriousMind Maybe? Sounds a tiny bit familiar. Not something I'd remember.

Not impressed. Looks like @Qmechanic and @ACuriousMind blew it, @dmckee .

ACuriousMind

01:42

Good thing I didn't put those pitchforks from yesterday away yet.

0celo7

Meh. I'm not scared. You Germans don't know how to fight.

NeuroFuzzy

@0celo7 youtube.com/watch?v=_FrOQC-zEog my recc

0celo7

You got one shot. I listened to two just because I owe @ACuriousMind one. No more.

NeuroFuzzy

@0celo7 Hey, $10^7$ views on youtube isn't too shabby.

0celo7

That one crappy Korean pop thing has more. Numbers are irrelevant.

dmckee

01:57

@NeuroFuzzy Comfortably Numb and The Gravy Train are indictments of the record industry, but not the strongest social commentary. My usual recommendation would be The Wall (that is the whole album), but if @0celo7 wants only one song it is a little harder to choose.

0celo7

02:29

Anyone know what this error means?

ACuriousMind

@0celo7 It means you can't write on the file LinSys_Final_Project_curr.log :P

dmckee

@0celo7 It's trying to write the log file to the same directory as the source file and has encountered as OS error of some kind (locked media, insufficient premission, device full, etc...)

ACuriousMind

As for why...do you have it opened in a text editor?

0celo7

@ACuriousMind It's opened in my TeX editor.

I can edit the source just fine.

I can't typeset it.

ACuriousMind

@0celo7 Not the source, the log file

0celo7

02:34

@ACuriousMind Uh, this isn't mine. All I have is the .tex.

Also, what are the advantages of  over $$ for in-line goodness?

dmckee

@0celo7 That's because tex wants to write the log file and has failed to do so. That's what it is complaining about.

ACuriousMind

@0celo7 Hmm...then it seems you don't have permission to create files in that directory.

0celo7

@dmckee Ok, how do I fix that?

ACuriousMind

Where is the .tex file located?

0celo7

@ACuriousMind OH!

dmckee

02:36

@0celo7 You figure out what is stopping it, then you fix it...

0celo7

I took this from my Windows partition, and I can't write on it using Mac.

@dmckee Typical prof response.

ACuriousMind

@0celo7 None, AFAIK $...$ is fully equivalent to $...$ .

Which is not the case for \[...\] and $$...$$, as you probably know when you ask that question.

0celo7

@ACuriousMind What's the difference.

ACuriousMind

@0celo7 Look here

0celo7

@ACuriousMind We've had this discussion...

ACuriousMind

02:43

Yes. It was easier to just post the link again than discuss your lacking memory :P

0celo7

@ACuriousMind So should I always use \[ \]

dmckee

@0celo7 Or \include{amsmath} and use the align environments which do the right thing with one or more lines. Then you don't have to think as much.

ACuriousMind

@0celo7 I actually like \begin{equation}...\end{equation} better because then it's easier to change whether or not the equation is tagged.

And, as dmckee says, use amsmath.

0celo7

Can't be bothered right now.

Reading something for a friend, then gonna play some ME.

user54412

03:02

@tpg2114 Here's a question for you: How would you quantitatively define turbulence?

user54412

Turbulence is like pornography: hard to describe, but I know it when I see it.

user54412

Should it be characterized by a cascade of something to different scales? Vorticity (however that generalizes beyond fluids)? Chaoticness?

user54412

The best I could come up with was a high convection to diffusion ratio of energy/momentum, where somehow magically we can measure convection, diffusion, and advection separately.

0celo7

03:17

@ACuriousMind How the heck do I know what my current objectives in ME are?

Slereah

@ChrisWhite Oh so like solitons

ACuriousMind

@0celo7 By, uh, looking in your quest log?

Usually called the "Journal"

0celo7

@ACuriousMind Can't find it.

ACuriousMind

@0celo7 Press Escape.

0celo7

@ACuriousMind Is that a button?

ACuriousMind

03:25

@0celo7 Are you joking?

0celo7

Oh, it says Journal!

Did not see that.

Thanks.

0celo7

03:37

@Slereah Or a crackpot.

1

Q: How does GR determine the topology of spacetime?

The crux of GR is the action $$ S=\int _\mathcal M d^n x \sqrt{|g|}\,R $$ Varying this and setting $\delta S=0$ gives you the Einstein field equations. However, that only determines the metric, not the manifold, and a metric doesn't uniquely determine a manifold. For instance, $$ ds^2 = du^2 ...

general-relativity variational-principle action topology

@ACuriousMind A kindred spirit?

ACuriousMind

@0celo7 Yep, and the question is a dupe

tpg2114

04:12

@ChrisWhite That's a very tricky question... and you're right, it's just like porn (in more ways than one -- just like sex, I think we spend far more time/resources simulating it than experiencing it).

But given that -- and I think this is how Tennekes and Lumley describe it but my copy is in the lab -- there are a few defining characteristics:

Seemingly random and chaotic behavior

A wide range of scales

Scales which are isotropic

The first one is totally qualitative and we can't really use it

A wide range of scales we can, in some sense, use. The scales which add energy are far from the scales which dissipate energy. Naturally "far" is subjective so that's not very satisfying

Slereah

can't you use like liapounov exponents or something

tpg2114

But with Kolmogorov's hypothesis that the energy cascade follows a -5/3 law and the assumption that small scales are isotropic, you can argue that "far" enough is when a -5/3 slope appears.

So something is turbulent when there is an inertial range to the energy cascade separating the energy-containing scales from the energy-dissipating scales

Which is quantitative only with some sort of hypothesis on what the slope should be. Kolmogorov was pretty genius when he added the "at sufficiently high Reynolds number" condition to his hypothesis because that can literally never be disproved. "Oh, it doesn't follow -5/3? Your Re was too low sucker!"

@ChrisWhite If your interest is in extending the definition to something other than fluids (so we're not looking at momentum and energy per se), I would say that turbulence is: (seemingly) chaotic; may be described by random variables; has a "large" separation between scales which produce something and scales which dissipate something; has a cascade of something from large to small scales.

Where that something depends on what you are using to define things (energy, or fluctuations in a variable, or vorticity or curvature or... whatever) and "large" is also dependent on what you are looking at. Perhaps another thing to add is limited interaction between scales

So it is turbulent when the biggest and smallest scales are separated by enough in some measure (frequency, wave number... maybe something else since I don't know if either of those is well defined for relativistic situations since my question about it was never answered) that there is basically only a one-way feeding of whatever your something is from large to small

No back-scatter, "energy" or "vorticity" or whatever only goes from big to small

Slereah

05:34

"And famously, 1≠0."

I want to answer "source"

0celo7

06:17

@Slereah Proof?

Slereah

@0celo7 : I just answered that post, actually

grid.ece.ntua.gr/sites/metamathsite/mpeuni/ax1ne0.html

^with the proof

Physics Meta

07:10

0

Q: A canonical relativistic rocket question?

John Baez's old web site has disappeared, presumably since he's no longer associated with the university whose server they were on. In particular his page on the relativistic rocket equations is gone, and that was a useful article for gathering all the equations together. I was thinking of askin...

discussion

0celo7

07:37

@Slereah PhD level set theory is insane

yuggib

08:20

@0celo7 the proof of $1\neq 0$ is not strictly set theory; and that proof above is just useless stuff

the proof of $1\neq 0$ follows trivially by definition of $\neq$ (in set theories), $1$ and $0$ (and basic logic inferences)

that website that @Slereah likes so much shows the way that no mathematician would use to do proofs :-P

Slereah

Obviously you've never read the Principia Mathematica :p

yuggib

mathematicians prefer words to obscure symbols

it is an actual practice of mathematics to avoid quantifiers and other stuff in the text of theorems

Slereah

Words are for the layman

Damn plebs

yuggib

many times, words are much clearer than a string of symbols, and mathematicians know that very well

take for example "$\Bigl((W\subset V)\land W \text{subspace} \land (\exists d\in \mathbb{N})\text{dim}W=d\Bigr)\Rightarrow$" as opposed to "For any finite dimensional subspace $W$ of $V$ then"

Slereah

08:55

Well there's a balance between the two

Secret

11:45

(Reading the yellow sentence) O it's John Rennie again, so another guy who ask homework questions I guess
(After reading beyond the yellow part), wait a second...

Explanation of the above: John Rennie has a template whenever he remind users that they posted homewrk questions on PSE, thus it is not suprising for me to get confused when I found 50% of his template there in that comment

David Z

It's a pretty generic phrase

That being said, there are templates

300

Q: AutoReviewComments - Pro-forma comments for SE

No more re-typing the same comments over and over! This script adds a little 'auto' link next to all comments boxes. When you click the link, you see a popup with 6 configurable auto-comments (canned responses), which you can easily click to insert. This script was inspired by answers to thi...

script comments

yuggib

15:34

what a silent chat today...

almost desolate

where are the chatty muricans?

0celo7

I feel hungover.

@yuggib What exactly is the Lebesgue integral?

ACuriousMind

1

Q: General Relativity finally Quantized?

I am not an expert in the field so i would really appreciate the opinion of any experts. There has been a new paper on arXiv recently, "Metric Redefinition and UV Divergences in Quantum Einstein Gravity" by Sergey N. Solodukhin, where he proposes a new method on tackling the divergence problem o...

general-relativity renormalization quantum-gravity

on-topic?

0celo7

Like how would one of you smart people integrate $f:x\mapsto x^2$ on $[0,1]$ in the Lebesgue fashion?

ACuriousMind

@0celo7 By using that it's Riemann integrable and computing the Riemann integral :P

0celo7

@ACuriousMind How do you know that it is Riemann integrable?

ACuriousMind

15:42

The point of the Lebesgue integral is not so much computing the actual integral - it's that it is far better behaved w.r.t. swapping it with limits and such than the Riemann integral

@0celo7 Write down the Riemann sums, see that they converge.

But that polynomials are integrable is easy because you can just write down the antiderivative.

So no need to actually compute anything, just use the fundamental theorem of calculus

0celo7

Can you?

How do we know FTC even works?

One has to assume that the Riemann sum converges.

ACuriousMind

@0celo7 Stop trolling :P

0celo7

@ACuriousMind Really tired and achy for some reason. Why do you think that's trolling?

How do we know the FTC works, say, for polynomials?

ACuriousMind

Ah, you're worried that a function might possess an antiderivative but not be integrable?

0celo7

Worried? No. Curious? Maybe.

0

Q: Why do many people say that virtual particles do not conserve energy?

I've seen this claim made all over the Internet. It's on Wikipedia. It's in John Baez's FAQ on virtual particles, it's in many popular books. I've even seen it mentioned offhand in academic papers. So I assume there must be some truth to it. And yet, whenever I have looked at textbooks descri...

quantum-field-theory virtual-particles

One for the Bajoran.

ACuriousMind

15:48

@0celo7 Well, either you need to write down the Riemann sums explicitly and check convergence, or you need to know that every continuous function is integrable.

0celo7

@ACuriousMind Yeah. I mean in physics we see crazy stuff done with divergent series. It would not surprise me that one could somehow rig an an infinite sum to give $F(b)-F(a)<\infty$.

ACuriousMind

@0celo7 Indeed, there are strange functions which are derivatives but which are not integrable.

0celo7

@ACuriousMind I know that. Don't know the proof.

And now that I think about it...I should probably just wait.

Taking analysis next semester.

ACuriousMind

@0celo7 Well, the proof is essentially writing down the Riemann sum and using continuity to show it converges, I think

0celo7

Well, time to study for my mechanics final and do some spinor GR.

^ My studying is really messed up.

ACuriousMind

15:52

Funnily enough, the thing with the fundamental theorem works if you go to the Henstock-Kurzweil integral, which coincides with the Riemann integral on Riemann integrable functions.

0celo7

So I've been thinking about something. The common canonical example for a spin 2 field is Einstein gravity. We then say that spin 2 fields are only attractive. But one can show easily enough that gravity can be repulsive if the ADM mass is negative. However, the positive energy theorem states that the ADM mass is positive, and therefore gravity is never repulsive, iff the dominant energy condition holds.

Does QG assume the dominant energy condition?

Does ST? After all, we take the symmetric bosonic mode of the string and reduce it to the spin 2 rep in 4 dimensions. Does this imply the dominant energy condition?

@Slereah What are some things that violate the dominant energy condition?

ACuriousMind

@0celo7 Side remark: "Common canonical example" is too weak - I seem to recall that any spin-2 field essentially must couple to the energy-momentum tensor, making it indistinguishable from gravity.

0celo7

@ACuriousMind Yes, that seems right.

That does not change my question.

yuggib

https://math.berkeley.edu/~arveson/Dvi/105/note1.pdf
Just to see that there are theorems relating Riemann and Lebesgue integrals

0celo7

@ACuriousMind Does my thought even make sense?

yuggib

15:58

Anyways, despite Riemann integration being intuitive, measure theory is much more fun ;-)

ACuriousMind

@0celo7 Yes, it does make sense. I have no qualified opinion, but I'd guess that it resolves like all other "paradoxa" between classical and quantum physics - the classical derivation is simply not valid physics (i.e. while you would see repulsive gravity for negative ADM masses classically, you just don't quantumly).

yuggib

and on a side note, Bourbaki dedicates 5 books to integration :-D

ACuriousMind

(Another possibility could indeed be that the quantum theory somehow ensures those energy conditions)

0celo7

@yuggib I head bad things of Bourbaki from one of my math professors.

"To few pictures." "Too little intuition."

Something along those lines.

@ACuriousMind Should I throw this out as a PSE post?

ACuriousMind

@0celo7 Sure

(Unless we already have it, that is, I didn't check)

yuggib

16:04

@0celo7 pictures and intuition?? heresy :-D

0celo7

@ACuriousMind How does one check?

@yuggib Shocking: this was not from a geometer.

yuggib

bourbaki is not so much appreciated by logicians and people doing analysis I think

0celo7

Are PDEs "analysis"?

yuggib

yes

ACuriousMind

@0celo7 I have heard rumors about using a newly developed feature called a "search".

yuggib

16:05

PDEs are not at all present in bourbaki series

(afaik)

0celo7

@ACuriousMind What would one search for?

1

Q: Any examples of negative ADM energy solutions with WEC but not DEC satisfied?

Any examples of negative ADM energy solutions with weak energy condition (WEC) but not dominant energy condition (DEC) satisfied? Witten's proof of the positive energy theorem requires the dominant energy condition. Is there any counterexample of a metric satisfying the weak energy condition, bu...

general-relativity energy

Secret

http://iopscience.iop.org/article/10.1088/0143-0807/36/2/025005/meta
Wait, so negative mass, once being treated with quantum mechanics, will not display that curious "mass chasing mass" behaviour mentioned in this article?

yuggib

however, for algebra, topological vector spaces (or topology in general) and also measure theory (to some extent) it is a very thorough reference

if you need a classical theorem in its more general form (in those fields), you will probably find it on Bourbaki...

like e.g. the most general form of Ascoli theorem that I ever read

0celo7

Nope, doesn't look like it.

@ACuriousMind I shall ask the question. But first I want to see if the DEC is really "iff" for the PET.

Also I need Mr. @Slereah to let me know what the DEC implies.

ACuriousMind

@Secret I said "I have no qualified opinion, but I'd guess "

Also, I'm not convinced "negative mass" is a sensible thing to even have. Most formally, mass is the value of a quadratic Casimir of the Poincare group, and that simply can't be negative.

0celo7

16:12

What is the Poincare group on curved spacetime?

ACuriousMind

@0celo7 I'm talking standard QM/QFT here.

0celo7

@ACuriousMind Sensible? Have you seen the crap @Slereah has been up to lately? Did you see Wald's definition of asymptotic flatness? People interested in GR are not sensible.

ACuriousMind

@0celo7 QFT on curved spacetimes is, well, a mess. When we talk about the spin-2 field being always attractive, that's a theorem of QFT on flat space (i.e. with Poincare invariance).

@0celo7 I've seen it, and I'm largely not impressed :P

0celo7

I wonder what's wrong with "stuff becomes flat at large $r$".

ACuriousMind

@0celo7 Ew, that's a coordinate-dependent definition :P

0celo7

16:17

@ACuriousMind Ok have your covariant derivatives on the conformal boundary and the topology of the null future and other crap :P

@ACuriousMind Does algebraic geometry do anything in GR?

What the heck does algebraic geometry do at all?

ACuriousMind

@0celo7 I don't know

@0celo7 Moduli spaces

0celo7

@ACuriousMind Sad.

@ACuriousMind I've used those once.

Still not sure what that was about tbh.

ACuriousMind

They modulate stuff :)

0celo7

That explains everything.

Is the category of categories a thing?

ACuriousMind

@0celo7 Yes. It's the archetypical 2-category

0celo7

16:23

That's enough category theory for today.

yuggib

@ACuriousMind yes but you need to fix a Grothendieck universe that contains all the interesting categories and the category of categories as sets...if else it is something closer to metaphysics than science :-D

I mean, else I have the feeling you cannot do anything on the 2-category

0celo7

$A\longrightarrow B$

yuggib

for example, you cannot quantify over categories belonging to the 2-category

0celo7

I'm now a category theorist

ACuriousMind

@yuggib Why not? Grothendieck universes are sets. Add the axiom that for every $x$ there is a universe that has it as an element. Fix a universe $U$ and take the category of categories whose collections of objects and morphisms are sets that are an elements of $U$. Now there is a universe $V$ such that $U\in V$. The categories with objects in $U$ are elements of $P(U)\times P(U)$ (I think?), which is an element of $V$, thus is a set and hence can be quantified over

0celo7

16:40

Fun fact: a particle in the orbit r=e^θ has zero energy.

Slereah

@0celo7 : I think the whole "spin 2 must always be attractive" thing is a vertex thing

It applies only to one particle momentum eigenstates

Which I think always have nice energy conditions

Also the DEC is pretty easily violated IIRC

Like even classical scalar fields can violate them

yuggib

@ACuriousMind Categories $(O,m)$ with $O\subset U$ and $m\subset U$ are in $U\times U$. Now the collection $\{(O,m), O\subset U \land m\subset U\}\subset 2^{U\times U}$. Therefore you need e.g. the universe, to be assured that the $2$-category of categories is a set contained in it, $2^{2^{U\times U}}$ (I would say)

ACuriousMind

Yeah, so pick that. Every set is contained in a universe :)

yuggib

the cardinality of it is not going to be nice... @_@

I agree that this is the way to do it correctly...but a friend of mine (that works with categories) has told me that you have to be careful in doing manipulations

because you have to be sure that you do not "go out of the universe"

because if else you need to take a bigger universe that contains both at the beginning

of course I do not know any detail :P

@0celo7 what is your opinion as a category theorist?

Slereah

Oh wait

Classical scalar field violates the SEC, not DEC

Wondering if there's something that violates the DEC but not the WEC

yuggib

16:55

no one violates the SEC

ACuriousMind

@yuggib I think that's the reason why some people eschew using set theory altogether and use type theory instead

(of course I do not know any detail :/)

yuggib

@ACuriousMind I don't know type theory, but a quick web skim suggests me it is more appealing for computer scientists than mathematicians...

Slereah

@ACuriousMind ADM mass isn't quite the same as regular mass

Qmechanic

0

Q: Homotopy and homology groups in physics

What is the connection between homotopy and homology groups and physics? When does one want or need to find invariants of manifolds in physics?

topology

Too broad list-type question?

Slereah

a bit yes

0celo7

17:08

ADM mass is the energy of spacetime...or at least one definition of it

"The ADM mass of a particle" is nonsense

So yes, it's not really a mass.

@yuggib I think more category theoreists should focus on the important things, like GDP.

@yuggib How do you know about the SEC? I guarantee @ACuriousMind knows nada about it.

yuggib

@0celo7 I like football

0celo7

@yuggib what

@ACuriousMind is this normal for a Euro

are there really Europoors who like good things

yuggib

@0celo7 I actually played football for more than ten years in my home country

at the extremely low level that is available there of course ;-P

0celo7

they have football in Yugoslavia? wow

yuggib

I was a pretty good TE (relatively to my country's level)

Slereah

17:20

Telekinetic Eel

0celo7

12

A: Global Properties of Spacetime Manifolds

Well, the simplest case is that some topologies of spacetime may only allow a particular class of metrics. But unfortunately, it usually requires the knowledge of the metric at every point to be quite certain. Here's a few thing we can probably assume about the spacetime manifold : All the us...

WHOA

Slereah

who is that handsome gentleman who did such a great answer

0celo7

The "globally hyperbolic $\Leftrightarrow$ no naked singularities" is just a conjecture, no?

Slereah

No, it's part of the definition of globally hyperbolic

0celo7

Where

Show me the page in HE

Slereah

17:23

IIRC it's the $J^-(p) \cap J^+(q)$ being compact part

This is not true if you allow naked singularities

0celo7

source?

Slereah

arxiv.org/pdf/gr-qc/0609119v3.pdf

p. 43

0celo7

seems to be in Wald somewhere

Slereah

probably

0celo7

oh

it's a conjecture that there are no naked singularities

Slereah

17:27

Yes

0celo7

and therefore things are globally hyperbolic

I thought there were two conjectures there

whoops

Slereah

It's kinda "Things would really be nice if there were no naked singularities"

Unlike CTCs I don't think things go quite as bad with naked singularities

Although you still lose the Cauchy problem and quantum unitarity

ACuriousMind

@Qmechanic Yes, but I'm out of close votes for today

Slereah

@ACuriousMind serial hater

0celo7

@ACuriousMind I voted in your stead.

ACuriousMind

17:32

@0celo7 I don't care about sports in general :P

0celo7

@ACuriousMind y

0

Q: Is the speed of gravity still discovered ?if it is what is the speed?

Gravity is the most confusing force which humans do not know alot about I had a confusion in mind what is the speed of gravity and if it is just the similar to the speed of light and its consequence of the curvature of spacetime caused by the uneven distribution of mass/energy; and resulting in...

quantum-mechanics gravity

ACuriousMind

@yuggib homotopy type theory is quite en vogue among category theorists, I think

Secret

Visualising the klein bagel (one of the immersion of klein bottle) without the intersection

Priliminary guess suggest it is formed by two sudanse moebius stripes

Sudanese Möbius band

►

Now to check if the maths agrees with my guess...

ACuriousMind

@0celo7 lol@ your comment and VTC as dupe, please ;)

0celo7

@ACuriousMind done

@ACuriousMind ok so the other day you said that $(X,Y,Z)\mapsto \langle \nabla_XY,Z\rangle$ is not a tensor

Secret

17:38

If my guess is correct ,the maths should show that the parametrisation of that figure 8 shaped hole is the same as a circle after applying some transformation that unbends it

0celo7

but $\nabla_\mu$ totally is, right?

Slereah

No, that's a map :V

ACuriousMind

@0celo7 ...not exactly. Given a tensor $T$, $\nabla T$ is also a tensor. $\nabla$ on its own is not a tensor because it doesn't act on either vectors or covectors to produce a number.

Slereah

Also what would $\langle \nabla_X Y, Z \rangle$ even be

$\nabla_X Y$ is a scalar

0celo7

@Slereah wtf

it's $X^iZ_j\nabla_i Y^j$

Slereah

17:41

Oh right

nvm

I am p. tired

0celo7

@ACuriousMind ok, so what does that statement actually mean then

ACuriousMind

@0celo7 What statement?

0celo7

3 mins ago, by 0celo7

@ACuriousMind ok so the other day you said that $(X,Y,Z)\mapsto \langle \nabla_XY,Z\rangle$ is not a tensor

ACuriousMind

@0celo7 It's not a tensor (because, although it maps vectors to a number, it is not $C^\infty$-linear, as you correctly remarked). What else could it mean?

0celo7

but what does that mean...in...coordinate notation

I don't see the significance

I think the mathy notation is obscuring something that I should be noticing

ACuriousMind

17:44

@0celo7 If you would examine its transformation under diffeomorphisms, you'd find it doesn't transform like a tensor

(because of the product rule applied to the transformation of $Y^\mu$)

0celo7

18:01

@ACuriousMind Yes, I understand all of that. But $X^iZ_j\nabla_i Y^j$ is obviously a scalar...

So I'm not sure "what" is not a tensor here

@ACuriousMind and if you say "the above mapping"

I'll smack you

@ACuriousMind i.e. what is the coordinate representation of the $\mapsto$ above?

ACuriousMind

I'm trying to find way to state the issue here such that it is unambiguous

You are right that your expression is a scalar. It is an expression of the tensor $\nabla Y: TM\times TM\to \mathbb{R}, (X,Z)\mapsto \langle \nabla_X Y,Z\rangle$.

This is a proper tensor - because it is $C^\infty$ linear in both $X$ and $Z$. It's component expression is $\nabla_i Y^j$.

0celo7

I know.

@ACuriousMind >it's

ACuriousMind

18:17

Now, the map $(X,Y,Z)\mapsto\langle \nabla_X Y,Z\rangle$ acts on three objects. Its component expression must thus have three different indices, therefore your expression doesn't relate to this map at all.

0celo7

I know all of this.

ACuriousMind

So what is the issue?

0celo7

So what is the component representation?

Or is there none?

ACuriousMind

Ah

0celo7

I'm asking what the three indexed object is, sorry that was unclear.

ACuriousMind

18:18

Well, since it is not a tensor, you can't expect there to be one, I think

0celo7

Ok, but other things like connection forms are not tensors, but they have coordinate expressions.

ACuriousMind

Yes, that's why I said "expect" and not that there is none

I would have to examine this a bit more closely, but I have to go

0celo7

@ACuriousMind Cheerio.

0celo7

18:35

Carson's behind Rubio now. Rubio had better watch out for a shanking.

0celo7

18:47

> For example, the "coordinate" speed of light at the event horizon is zero. So if the black hole is spinning at half the speed of light, how fast is it spinning? What's half of zero?

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