I dunno, what do you want to write about?

Sobolev spaces.

Hm

Not sure it would fit well thematically?

Though it's pretty broad to be fair

there's a lot of building up GR stuff

Really I think in the end I could cut out the weird causality part and make a GR book out of it if I ever write everything I want to

I need to prove that on manifolds

oh no...it depends on the dimension

I give up.

In the end I just want a book with GR proofs in it

That doesn't just refer to 30 different books

that's a tall task

It is

Especially with my poor understanding of topology

But part of the benefit of the book is that it forces me to learn

y tho

What is even K

and A

some constants.

I think I will call random manifolds $M$ and fancy spacetimes $\mathcal M$

I should write down my notation somewhere else, until I can get the glossary package to work

There's a lot piling up and I'd like to keep everything consistent

I can't seem to find the convention table in MTW

Which is weird, because I have it

Right there

But I can't find it in the book

do you own the book?

Do I look like a millionaire

I'll buy the reprint when it's out

I see what the Riemann sign convention is but I don't know what the Einstein sign convention is

Also, on the topic of aesthetic, which is best, ${R^a}_{bcd}$ or ${R_{abc}}^d$

Hey, Latex has a \thanks tag

Let's see what it does

\thanks{My friend Jesus} does not appear in the document

Very sad

Apparently the correct usage is \author{ABC\thanks{XYZ} \and DEF\thanks{UVW} \and GHI\footnotemark[1]}

@Slereah first

I'd follow Straumann for notation and conventions.

But with abstract indices.

maybe

why only maybe?

Straumann's notation makes sense from the coordinate free perspective

and we all know GR should be coordinate free as much as possible

Well ideally i'd like to have the same conventions as most books

or at least the big ones

HE and MTW both use the + convention for all three conventions, and Penrose uses - for all three

make up your mind, GR people

Let's see if every possible combination is in it

+++, ---, -++, -+-, --+, +--, ++-

All combinations but +-+

what are you talking about?

Welp, I have no reason to believe this soblev inequality

the signature of the metric, the sign of the Ricci tensor and the sign of the Einstein tensor

Let's check two books with opposing Einstein tensor convention to see what that's about

HE is +, Synge is -

Perfect

Ah yes, it's $G = T$ versus $G = -T$

I guess it's opposing signs in the definition of $T$

I think +++ is best

What does Straumann use?

Let's check

$(-+++)$ for the metric, $G = T$ for the EFE

and same definition for the Ricci scalar

and same for Wald

Probably best to use that one

@Slereah Choquet bruhat has 200 pages of appendices

this book is too damn long

you could make a decent size book out of that

add an index, an intro and a table of content and that's a full fledged math book

Ugh, why is this so hard?

If $(\phi_n)\subset \mathscr D(M)$ converge in Sobolev $H^{1,p}$ norm a constant function, that function is zero.

This seems to be impossible to prove

Wait

A constant function is $C^\infty$

most certainly

...

I need traces

I need a Sobolev trace theory on manifolds

It would be nice to never have to worry about Sobolev spaces, I must admit

they are horrid objects

Last night dream, h bar relevant thing: For some reason, the Kerr Cube becomes a circular cornered cubic frame known as a Krasinov cube (which is also known as a Krube in short form)

@Slereah so what did you take from Krasinov's email?

How do I put a circle over a quantity?

Aubin has: Let $\bar W_n$ be a compact Riemannian manifold with boundary of class $C^r$. If $f\in C^{k-1}(\bar W)\cap \mathring{H}^p_k(\bar W)$, then $f$ and its derivatives up to order $k-1$ vanish on the boundary $\partial W$.

So if my $f$ has $\nabla f=0$ a.e., then $f=c$ a.e. Hence $f$ has a rep in $C^{k-1}(\bar W)$, namely $c$. But if $c=0$ on $\partial W$, then $c=0$ everywhere since $c$ is constant.

Hence $f=0$ a.e.

Wonderful!

@ACuriousMind After 4 months of PDE and functional analysis I finally proved what I told you about in December :D

@0celouvskyopoulo7 just the confirmation that the operator was a relation?

Also he gave me another paper where things are more developped

@0celouvskyopoulo7 Yay!

So the good news is that Sobolev spaces on manifolds make sense.

he literally said I was wasting my time reading that other paper

a bit harsh

it's not that bad a paper

If I were him I'd be happy someone was reading my ramblings

I do like Krasnikov

He is enthusiastic about weird spacetimes

If there's a paper that says "Wait, there's still a loophole for weird spacetimes to be possible!", odds are 80% chance it's Krasnikov

@ACuriousMind Schoen and Yau prove a Poincare inequality of the form $C\int |f|^p\le \int |\nabla f|^p$, but $C$ can be zero as they define it

Quite a terrible estimate tbh

it is zero for a ball, for instance...

Well...I guess at least it's not wrong ;)

I'm fairly sure the integral of a norm will be indeed superior or equal to $0$

@ACuriousMind Yeah...they manage to never have to divide by it. I'll have to read their book to see what's going on.

I'm ashamed to say but I still put little hats on my quantum operators

That's a good way to get your book discarded.

What am I supposed to do, just use $\phi$?

I have like 5 other things named $\phi$

I should do a little appendix on Hilbert spaces

Don't wanna sprawl too much on the topic

@Slereah $\phi_1, \phi_2, \phi_3 \ldots$

That has 11 different $C_i$ constants

I really, really don't like this circuit layout program I'm using.

It's really hard to do stuff.

Maybe I actually should send an email to Smith, the guy with the stings and the q-loops

If he's still alive

Although at a glance, I don't think he has a public email

his last papers (the crazy papers) were in the 90's

Also he was born in 1930

he was a boy during WWII

Investigating complex systems using nano particle aggregates controlled by laser