it is not true in the least

By definition the metric is required to have the specific signature

Which is indeed locally Minkowski

@Slereah can I ask you to set the commenter straight? or for a simple authoritative reference that'll take it beyond I say vs he says?

Well any GR book starts with the declaration of spacetime's signature, really

@EmilioPisanty that's an "i'm an idiot" comment

You literally can't do physics without SR let alone GR

Well, you can set up the POLA or the axioms of QM but it's all formalism until you bring in the symmetry groups

what about causal sets

Lol @Ryan's comment

@BalarkaSen it's correct.

btw projecteuclid.org/download/pdf_1/euclid.jdg/1214431480 is another neat little paper

you can use this to show that if a metric on a space form $S^n/\Gamma$ has positive scalar curvature, the metric can be deformed to have constant sectional curvature

$\mathbb{C}^n$ is path connected right

I mean it's just $\mathbb{R}^{2n}$

It should be

unless you have discovered a secret number

@SirCumference yes

disconnected from it

@Slereah Welp intuitively yep

Wanted to make sure there wasn't something weird I'd be missing tho

@SirCumference math.stackexchange.com/questions/715720/…

@RyanUnger Huh nice

@BalarkaSen you need this fact in Marques' psc 3-manifolds paper

I'm surprised I didn't know that fact before

@Sir You can write down a path from $(x_0, \cdots, x_n)$ to $(y_0, \cdots, y_n)$. Just do $(tx_0 + (1 - t)y_0, \cdots, tx_n + (1 - t)y_n)$ :P

on that note, any convex subset of a vector space is path connected

isn't that like

Sort of a definition of a convex subset?

in fact, any star-shaped

pretty much

path connected by lines

the pathiest of path

pathetic path

That was a Dilbert joke

Am I supposed to know what Dilbert is

@BalarkaSen Weyl showed that conformal classes of psc metrics are contractible

the proof is...literally what was said above lmao

Sigh why does topology have so many terms

@BalarkaSen Only if you are aware of comics

that they exist

this is turning into a memory game

(that is, they're convex)

maketh sense

@Slereah I only read big brains comic

like Tintin

this paper is actually really complicated

Tintin is nothing but pulpy trash!

Also unrelated question, can a norm be defined on any vector space (regardless of field)

@BalarkaSen do you know what paper I'm talking about

@SirCumference depends

@SirCumference Certainly for finite dimensional ones

Dunno about in general

Oh wait

Regardless of field

Then no I think?

Vector space of hyperreals is not metrizable

@Ryan no

I think no non-archimedian field is metrizable

@BalarkaSen Let $M$ be an orientable closed 3-manifold and $\mathcal R_+(M):=\{g\in \mathrm{Met}(M):\mathrm{scal}(g)>0\}$. Then $\mathcal R_+(M)/\mathrm{Diff}(M)$ is path-connected.

Huh pretty cool

it's complicated because you have to use Ricci flow with surgery

you prove it by backwards induction on surgery times

Oh wait

the fact above (the short JDG paper) is essentially the "base case" for the induction

Is backward induction if it works for $n = \infty$

And then you prove $n - 1$

@Slereah what?

It a joke

it's easy to see that there's only finitely many surgeries if you start with a metric of positive scalar curvature

@Slereah voldemort responded to my comment

@ACuriousMind what's the protocol here? He who shall not be named responded to a comment of mine on the main site. Am I allowed to discuss it here with Slereah?

@RyanUnger Yeah... I barely understand GR/SR; but the statement he quoted is pretty clearly agreeing with what you said.

What's the purpose of directional derivatives?

to differentiate in a certain direction

Omg yes!

got it

lol

more seriously, you want to know how a function is changing along a curve

if $\gamma$ is your curve, then you have $(f\circ\gamma)'=D_vf|_\gamma,$ where $v=\gamma'$ is the tangent vector of $\gamma$

How to make the inverted delta thing, how to put the vector thingy over a letter, how to write the curly d?

1. \nabla 2. don't do that 3. \partial

$\nabla_w f(x, y) = a\frac{\partial f}{\partial x} + b\frac{\partial f}{\partial y}$

That's what I know.

Anyone who lives in San Francisco?

If your textbook uses $\vec{\nabla}$ abandon it

Lol it doesn't

I use Khan Academy

Khan Academy's multivariable calc articles are actually extremely good

A lot better than the videos

Have you watched the vids and read the articles?

The gradient kinda unifies the partial derivative wrt x and partial derivative wrt y into a vector-like structure no?

@NovaliumCompany The articles basically got me through my multivariable class

They should be used to supplement a real textbook though

As far as a I know, the gradient shows the steepness (fastest ascent/descent) of the f(x, y) on a 2D plane?

They are excellent at offering intuition, but you'll only get the full details in a textbook

I'm not gonna become a mathematician nor a physicist, so intuition is enough :D

I just hate not understanding complicated things.

Sal Khan must be quite smart uh?

Like, he knows a lot about everything

@NovaliumCompany Again, learn single variable calc strongly first :P

@SirCumference Strongly is relative

calculus is not complicated

agreed

I don't know why I am learning calculus. I wanna be an entrepreneur lol

I guess to stimulate my mind with some new info that is not so easy to grasp

I don't see why an entrepreneur wouldn't benefit from calculus

@JMac Very true indeed

^ tru dat

there you go

I took a few microeconomics courses that had legit calculus in em

1 + 1 = 3 Prove me wrong.

3 -2 = 1 = 0

done

one apple is different than no apple

1 != 0

Omg wait. There is mutlivariable integrals :cc

Finding the area under a 3D function?

you mean volume?

yah xDD

me stupid

@Gyromagnetic I'm pretty sure Kyle Kanos does finance and calculus as his job

Now I realize programming is easier than maths

why integrate when you can add trapezoids :)?

And in the end of the day I ask myself, why would anyone need to find the directional derivative of a 2 input, 1 output function? Are there practical examples?

whats the force on ball sitting on a surface f(x,y) where the f is the height?

gravity exists, ofc

none

if the ball isn't moving

no acceleration, no force

welllllllllllllllll.........

thats not really true

a swinging pendulum has 0 velocity twice per period

so no movement doesnt imply no acceleration

wut

I am confusion

@Gyromagnetic Depends on what you mean by "no movement". If you only mean "velocity at an instant is 0"; then sure. If you mean "the ball is not in motion", that's a different argument

anyways... if youre looking into partial derivatives and wanna learn something cool that I used a bunch while studying economics, look into Lagrange multipliers en.wikipedia.org/wiki/Lagrange_multiplier :p

its good fun

@Gyromagnetic How old r u?

and not hard to grasp

28, im an oldie :(

@Gyromagnetic Might look at calculus in economics.

@Gyromagnetic Better pack it up, retirement is coming.

watcha do for living?

physics

research @ an institute

damn

watcha plan doing living?

keep doing that forever, im very happy

watcha research?

a lot of different projects, but all involve nuclear magnetic resonance

damn

nice

I'm glad ur happy

thx mate ;)

can u hook me up with one NMR to put it up in my home?

kinda like a pre-christmas present

actually, its not that hard to make a cw nmr spectrometer

but it would probably get expensive kinda quick, sooo you gotta begin enterpreneuring ;)

Can u tell me what NMR does, wikipedia is too high level for me :D

@Gyromagnetic I'm 17, entrepreneuring has to wait 2 years.

NMR is basically moving nuclear spins around with microwaves and then measuring the magnetic fields they make

theres a lot of information you can get from a system from that

the cliche example is brain imaging through MRI

Oh, cool.

MRI measures the flow of blood in the brain right?

its one of the many things you can measure ;)

cooool

you can measure oxygen consumption (which people like to relate with activity) which is called fMRI for functional

also different tissues give you different relaxation times

Exactly. I'm a big neuroscientist fan, read a lot of books

I want to get invoved in future brain tech.

so you can map out the different tissues

@Gyromagnetic 20s is not old... D:

@SirCumference +8 tho :(, now i want to stop ageing for a bit haha

Gotto go, cya guys

@NovaliumCompany cool paper i read recently: They took patients in a coma and told them: "think of playing tennis for YES and of you in your house for NO" and a few of the coma patients could answer a bunch of questions through the differently lit areas in their brain (seen using fMRI)

cya ;)

@Gyromagnetic awesome!

Hilarious

@Slereah why do you hate Belgium?

=P

@Slereah seriously though. That thread requires the attention of a specialist. Ideally one who can engage in comment threads without tone problems =).

@EmilioPisanty I'm a specialist but cannot engage without tone problems.

The presence of John Duffield disturbs me.

You mean the self-proclaimed specialist

He who shall not be named

The voice of Einstein in the 21st century

Imagine engaging people on the internet about GR and not understanding the equivalence principle

That old Einstein book was super confusing when I tried to read it before knowing basic calculus in fairness

amazon.com/General-Relativity-Without-Calculus-Undergraduate/dp/…

'2011th Edition'

@bolbteppa have you read this book

No

@RyanUnger I know

like I said the comment thread is asinine

it needs to be nuked

I imagine the zero metric thing in quantum gravity is due to the zero metric being a limit of Lorentzian metrics

say $\varepsilon\eta_{\mu\nu}$

so it's a "pole" for curvature functionals

I imagine that claim that was made is not true in the slightest

The zero metric is not even a metric

which claim

@bolbteppa of course not

The claim about perturbations about the zero metric are common in the quantum gravity community

but I can see how the metric degenerating can be of importance when considering path integrals over 'the space of metrics'

hmm so for the bosonic string polyakov path integral I don't think you integrate over degenerate metrics (since they're not even metrics)

yeah of course you don't integrate over them, they're in the boundary of the region you're integrating over

like integrating $1/z$ over $\Bbb C^*$ or whatever

Turns out first year linear algebra steers people away from egregious mistakes :p

How can you even write the Ricci scalar $R = g^{\mu \nu} R_{\mu \nu}$ to derive the EFE if $g_{\mu \nu}$ is degenerate, can't even derive the EFE to plug the zero metric into it, the thesis has to start going into pseudo-inverses to try to circumvent this issue

I asked the guy what the Ricci curvature of the zero metric is

pretty hard to write down the Christoffel symbols too

@RyanUnger On the bright side, if JD agrees with a post about relativity, that's a good indication that it's probably wrong. ;) Pity that the converse isn't always true...

The sad thing is, his quote agrees with me :((((((((

That's one of his standard tricks: quoting something that actually says almost the exact opposite of what he claims it means.

I have much experience debating the duffield

old age has not made him wiser

@RyanUnger i thought you said you admired him

He's our local expert in bozonic gravity theory.

@SirCumference I was probably trolling you when I said that.

"Yes, they laughed at Einstein, but they also laughed at Bozo the Clown".

Where is that last one from again

I better stop before ACM slaps me.

@bolbteppa Sagan, IIRC.

shortly after his ban

I believe I was banned for a year shortly after that

Be Nice

@RyanUnger I argued to make them let you back a good few times fyi

I know, thanks

Ppl sent me links to comments sometimes

haha

JD actually sent me an email

@RyanUnger welp perhaps it was in poor taste

but in seriousness, he actually did get a 3000 year ban after you were gone

got "fixed" shortly after though

wtf

@RyanUnger "Subject: Sup homie, what's poppin?" I assume

no, titled "the h bar"

close enough

Hmm

Does anyone know his blog?

it's on his profile iirc

yeah just found it on his profile

amazing

oh wow there's comments too

this is amazing

'independent investigators like us'

@RyanUnger What is the history of this JD and yourself?

@AaronStevens he’s been posting annoyingly wrong answers on this site for years

But calling him a crackpot isn’t allowed

@RyanUnger Ah ok got it. There was a similar user a couple of months ago like that. Got into comment wars too. Currently on a one year ban. I can't wait for it to end to flag more comments by them haha