The induction of rotations on objects near rotating masses

@BalarkaSen when [redacted] so fat that when she spins spacetime warps

@Slereah I’m not sure why, but time travel nonsense bothers me less than quantum mysticism nonsense

"huh"

Maybe because time travel isn’t as easy to use to sell pseudoscience products

Say that to my new brand of closed timelike curve remedies

i still don't get how the hecker can the metric change due to anything that's happening on the spacetime 4-manifold

I can’t afford the rocket fuel required to use them

@BalarkaSen Add a mass

Bam

You changed spacetime

There’s a very nice one-sentence description of GR which Einstein gave to s reporter once

Which I can’t find yet :(

@BalarkaSen what?

The metric changes from time slice to time slice

the metric on the 4-fold is static in some sense

because time is a "direction"

what does that mean. is the 4-fold M x R?

@BalarkaSen For many applications it is

and R is a time coordinate

ic

you can show (well, argue, it's a conjecture) that if there are no naked singularities, then the spacetime splits as RxM

Is it a conjecture?

(also you need causality)

Strong cosmic censorship is a conjecture

The conjecture would be "there are no naked singularities"

I think there's a conjecture no naked singularities ==> globally hyperbolic

+ causality and time orientability and some other assumptions

what is a naked singularity

@BalarkaSen Go out in the street naked and you'll understand

naturalists do that everyday

that does not break the spacetime

they are weirdos

normies have to do it for things to break

AdS isn't GH and it has none of this

Blah I can’t fimd it

but that's related to the closedness of the lightcone

I did fimd this line by Wheeler, though: “Spacetime tells matter how to move; matter tells spacetime how to curve.“

There's also this notion of spacetimes having holes, despite being globally hyperbolic and singularity free, but I still don't quite grasp that notion

it's fairly rarely explained

@Slereah is that your boy smith's work?

Nah

It's some Geroch and company thing

It's related to the domain of dependance

Krasnikov wrote a bunch of stuff about it, too

although it's a tricky thing to define because if you don't do it properly then even Minkowski space has holes

Ah I remember that

Minkowski space should ideally have no holes at all

@BalarkaSen basically the dynamical parts of GR are the induced metric and second fundamental form on Cauchy surfaces

Everything else is determined by a constraint or algebraic equations

And matter fields are in there too if you're crazy

I tried playing the new South Park game but it was so buggy that I had to ask for a refund

I'll just wait a while for everything to get patched

I'm watching the LP rn

This is edgy and inappropriate over 9000

well that's South Park

It is to be expected

That's fine with me

What is less fine is being stuck on the first screen due to a control bug

@BalarkaSen my little pony?

First two seasons were ok

lets play

What's that?

watching playthroughs

It is a video where a fat man films a video game he plays and comments on it

^

What game I mean

Also incredibly sexist @Slereah

@0celo7 the new south park, the one slereah mentioned

Well I don't know any fat women doing let's play

the only significantly fat man who does let's play that i know of is Daz

Time for crepes baby

Look a bit... Crep to me... ha. ha. ha. ha.

@BalarkaSen I should take complex analysis next semester

It's taught by an operator algebraist, so no god damn sheaves :)

then u can work on Kahler spacetimes

Those seem to be popular for no reason

Stringy theory probs

@CooperCape wrong dimension

Meh

whatevs

I remember the word Kahler in my book

Kahler was a huge nazi

somewhere...

I refuse to learn his stuff

Wasn't that kind of book.

Teichmuller was a dedicated Nazi, but I love that stuff

I don't have to love the person

Nazi's huh...

You like IUT??

That's the Mochizuki crap

I want to learn it

Is it hard?

Good luck

No, that's inter-universal Teichmuller theory

@BalarkaSen thank you

Teichmuller theory has to do with foliations on Riemann surfaces? That’s my off the top of my head recollection

IUT, by contrast, is ... something

@Semiclassical It's about constructing configuration spaces of Riemann surface upto biholomorphism type

And studying those

Hmm. I wonder why I was thinking filiations then

What's a Hodge theater?

I think I need to learn stack theory

I do think singular foliations on Riemann surfaces have things to do with Teichmuller theory

What the hell is that

@0celo7 is going down the Mochizuki rabbithole

I don't understand. How is studying prime numbers so hard?

I don’t know a simple summary of IUT

Why do I need an etale sheaf Hodge cohomological stack to do prime numbers

I’m not even sure what that could even mean

This is why Analysis is the best math. You don't need to do some fucking ridiculous abstraction/generalization to get the job done

Most of the etale stacky shit is generalizing the analysis from C or R to other fields

Want to prove the Poincaré conjecture? Just flow the manifold and cut out the bad bits. No bad bits? Sphere.

Trivial!

etale maps = local homeomorphisms for arbitrary algebraically closed fields

But want to prove there's twin primes involve thousands of pages of categories and stacks and whatnot

No, Zhang's approach to twin primes conjecture is combinatorial number theory. I suspect it's just lots of combinatorics and analysis

Not all of number theory is the horrid algebraic business

I think one of the reasons Zhang became an instant hero is the idea of his proof is relatively simple

@BalarkaSen is perelman your hero?

no

I thought you had a thing for crazy commies

i don't believe in heroes, that's a bourgeois mentality

i believe in komrads

Oh, you got me

Good.

@BalarkaSen so Comrade Perelman is your comrade?

So what the hell are the Bianchi spaces that aren't flat space, a 3-sphere or 3-hyperbolic space

Check stephani?

Well he enumerates a lot of properties and all

But they're all like Lie group properties

i don't really have a good feel of the topology of the manifold itself

I don't even know what the 2D Bianchi space that isn't $R^2$ is

It's apparently "The solvable Lie algebra of 2×2 upper triangular matrices of trace 0."

But what the hell does that mean, as a manifold

"Types I, VII0, V, VIIh and IX have received particular attention as they contain the isotropic FLRW flat, flat, open, open and closed universes respectively. "

@ACuriousMind would know.

Why are both type I and type VII0 flat

@ACuriousMind we summon thee

Hm

Apparently the Taub-NUT metric is a type of Bianchi spacetime

I think I got summoned by accident

Is the volume of the universe finite?

the answer is most certainly maybe

concerning

How do you define the 'ground state' energy content of the universe that I've heard about? I figured that with finite volume it would make sense but I don't understand at all how the energy content of the universe could be infinite. Am I misunderstanding ground state? Most likely I am

energy density?

Yeah. I assume then that it's not as simple as E = pV, if V is infinite

You just subtract off infinity from your Hamiltonian B)

I can appreciate V tending to infinity but I can't understand it BEING infinity

Because you can't separate off a section of real numbers out of infinity can you? Since it's not a number. I.e. afaik you couldn't work backwards from V is infinite, and E is infinite and arrive at a finite energy density

Idk I really don't understand infinity.

the energy of the ground state of field theory is somewhat arbitrary

We just require that it be finite

Outside of GR its exact value isn't very interesting

While I'm asking poorly formed questions about things I barely understand: where do the fluctuations come from

Quantum mechanics

Measurement is always probabilistic

so it fluctuates

Right, with only the expectation value being constant with a time independent hamiltonian right?

but then

How would the expectation value ever be determined to be zero if the measurements are constrained to nonnegative numbers

How would you go about relating that distribution of energy values [which to the extent of my limited knowledge are $\in \mathcal R^+ $U$ {0}$] to an expectation value of <0>? (i dont know how to write nonnegative R efficiently rip)

whoops typo

ah nevermind sorry I'm back to my usual shit. Thanks anyway Slereah

you can't measure absolute energy

at least not without considering gravitation

You can only measure energy differences

Guys, I don't know if any of you know Taylor's derivation of the Euler-Lagrange equations by heart, but I basically don't see where they use the fact that they evaluate at $\alpha=0$

here is the exact context:

Like, I get the idea that we're using just any $\eta(x)$ and all the technicalities that follow, but it seems to me we should evaluate at $\alpha=0$ at some point, for this to hold for the specific case we're interested in?

otherwise, it almost seems like the integral is always stationary, no matter the value of $\alpha$

hm,

I am 80% unfamiliar with Euler-Lagrange equations, but the fact that your variation functional $S$ is minimum at $\alpha = 0$ is the thing needed to set $dS/d\alpha|_{\alpha = 0} = 0$, or something, right?

maybe it's the fact that we consider $f$ as a function of $y$ and $y'$, in the sense that we evaluate it at $y$ and $y'$ when we write down the Euler-Lagrange equation

yea, that's right

like, you want the derivative to be zero, whenever there is no variation, because then you have found the 'right path'

@Slereah any Lie algebra is a vector space, so that's...$\mathbb{R}^2$ :P

@ACuriousMind The manifold is the resulting Lie group :p

I mean, yes, it's Fermat's theorem

you butt

Not Nice

right we haven't had this in math yet, but luckily the idea is straightforward

but I think I'm happy enough with my answer, unless anyone would object. the evaluation sort of explicitly happens at 6.13

is the sphere not a lie group by the way?

2-sphere is not

It cannot be given any Lie group structure

@Slereah The resulting simply connected Lie group is still isomorphic to $\mathbb{R}^2$ (but with a non-commutative group structure)

@Slereah The only spheres that admit a Lie group structure are the 1-, 3- and 7-sphere

Hm

The 15-sphere admits a nonassociative "group" structure

One way to see that is to know that Lie groups are parallelizable, i.e. must have trivial tangent bundle.

What's the difference between two spacetimes with the same topology but a different Lie group?

Oh wait

Different metric, right

Since it's linked to the exponential map

Yes, the Haar measure on that $\mathbb{R}^2$ is probably not the usual flat metric

What's the metric of a Lie group from the Lie algebra?

I don't think you get a canonical biinvariant metric on a Lie group

Oh, you do

@Slereah Biinvariant metrics on the Lie group are in bijection to Ad-invariant inner products on the Lie algebra.

There is an averaging construction isn't there

it hurts my eyes reading that sentence

The canonical choice of Ad-invariant inner product is the Killing form, which up to a factor agrees with the ordinary matrix trace for matrix algebras

What is ad-invariance

Adjoint?

Adventure?

Pick an inner product on the Lie algebra and extend it to the Lie group by left-multiplication. That's a left-invariant metric. Now define the bi-invariant guy to be the average of the pullback of the left-invariant metrics to the Lie algebra by right multiplication wrt a left-invariant volume form

@Slereah adjoint representation I guess. For Lie algebra elements $u,v$ $\mathrm{ad}_u(v) = [u.v]$.

I never understood Ad

It's just a fancy name for $[u,-]$ so we can write things like $\mathrm{tr}(\mathrm{ad}_u\circ\mathrm{ad}_v)$ (that's the Killing form).

Nothing mysterious about it, just the natural action of the Lie group/algebra on the algebra

Ah...

Ah right so I need to pick the inner product on the Lie algebra that I am starting with in my construction to be ad-invariant I guess

That should establish the bijection you mentioned

So $g(a,b)$ would be $\text{Tr}(\text{ad}_u(a) \circ \text{ad}_v(b))$?

No, literally g(u, v) = tr(ad_u $\circ$ ad_v) I think

The thing inside is a matrix/linear transformation $\mathfrak{g} \to \mathfrak{g}$

Take it's trace

Hm

^^yes, that

I guess for Euclidian space, $\text{ad}_u = \text{diag}(1)$

Yep

Wait, how does the composition work here

Is it just matrix multiplication?

matrix multiplication :P

Ok I got sniped

What would be the ad of the two basis vectors?

I'm not sure how to get that $g(\partial_x, \partial_y) = 0$

I mean, $[\partial_i, \partial_j] = \delta_{ij}$ in $\Bbb R^n$

The rest is computation

@Slereah Why are there partials?

Tangent vectors?

He just means the coordinate vectors, I think

No need to think like that about Lie algebras

But I'm not sure what $[\partial_x, -] \circ [\partial_y, -]$ is

Just observe that the algebra of an abelian $\mathbb{R}^2$ is itself abelian, i.e. the bracket is always zero, i.e. $\mathrm{ad}_v = 0$ for all algebra element

Oh, sorry, your $\mathrm{ad}_u = \diag(1)$ up there was wrong, it's equal to 0

Ah

But then where do you get $g(\partial_x, \partial_x) = 1$

No I am dumb

@Slereah You don't

pretty odd for euclidian space

The Killing form is zero, there is no metric induced

The bracket is always zero in R^n. That's just because partials commute always ever.

I need sleep

Those are all zero matrices

but then in what way is that formula a metric

It is a metric when the algebra has no non-trivial Abelian ideals

Not a clue if the Bianchi spaces do :p

Is the metric generated for $S^3$ the canonical sphere metric?

That algebra of triangular matrices you mentioned at least doesn't

@Slereah No clue, but I suspect not

I should find a book on Bianchi spacetimes because I have no clue what's going on

a few books kinda talkabout them but don't go into much details

@BalarkaSen I think I found the paper I had in mind regarding Teichmuller theory and foliations:

Apparently it's based on the Lie bracket of the Killing vectors

Since all the Bianchi universes are homogeneous

It's a bit hard to trace the history of the Bianchi universe because most papers just refer back to Bianchi

But Bianchi wasn't doing GR at all

He was just classifying those groups

There's some 60's papers on the topic but it's hard to say if they're the first

@BalarkaSen projecteuclid.org/euclid.acta/1485890020

I think Levi-Civitta's 1917 papers talk about it?

But it's hard to say

Last night dream is about three things: Quaternion power series, quarternion integral (transforms) and uncountable well ordering (plus some pine trees in a stormy narrow roadside island.) In particular, the main theme of the dream is how Semiclassical said he love quarternions and me asking him about whether a power series of quarternions makes sense except for some google search (within the dream) I found about decomposing complicated engineering components and rotational

bearings into different components

Reality check: I don't remember whether the reality counterpart of Semiclassical likes quarternions

...

"A Bianchi spacetime is a spacetime $(M, g)$ with $M = \mathbb R \times G$, $g = -dt^2 + h_t$ where $G$ is a 3-dimensional Lie-group and $h_t$ is a left-invariant Riemannian metric on $G$"

I think they’re neat but I don’t have an abiding opinion towards them

Does that mean that $g(u,v) = g(au, v)$, with $a \in G$?

Though I do like Pauli matrices, and those are pretty much just matrix reps of quatertnions

Well not the Pauli matrices

But quaternions are the Dirac matrices

Same exact algebra

Guys, maybe a stupid question - but is $U(r_1,r_2)$ the same as $U(r_1)+U(r_2)$?

Eh, multiply the Pauli matrices by i

@ShaVuklia It is not

I wonder if anyone wrote the Dirac equation in quaternions

Let's see

But why isn't it then @Slereah? Because we do simply consider $T_1+T_2$. So if there like a basic/intuitive answer?

Because a single term can contain both that can't be split

About that stormy pine tree scene, it is implied in the dream by using a random distribution to assign a zig zag pairing, I somehow end up getting an explicit well ordering of the reals. The next thing I remembered during my sleep, is then at that moment in dream when that happens near the end, the outside of my bedroom is raining, and my first thought that came to mind after waking up is that I must have broke reality (given how suddenly strong the rain is outside)

Slereah: I think it should make it more compact. I also vaguely recall seeing something like that once in chemistry when they talked about the Doglas Hess DFT methods.

For instance, if there's gravitational interactions

$$U(r_1, r_2) \propto \frac{1}{|r_1 - r_2|}$$

oh alright, and the kinetic energy is never intertwined?

It is not

ohhh, okay that example really helps

@Slereah judging from the Wiki page on Pauli matrices, the linkage between Pauli matrices and quaternions isn’t quite what I thought

@Semiclassical Ah let me have a look

Oh yeah I have seen this correspondence

It's in Farb-Margalit IIRC

I can’t say I ever managed to understand it, mind

So when do we get the inter-universal Teichmuller quantum gravity anyway

But if you can grok it, cool

The quantum gravity approach to IUT: figure out how to travel to other universes and ask them if they understand what the hell Michizuki means

And interestingly, google searches on "quaternion integrals" found they are mainly used in an engineering context: math.stackexchange.com/questions/912353/quaternion-integration

You know it's coming

@Secret The word you are looking for is "gimbal".

@dmckee Actually, it's a lot more complicated. The stuff I saw in the dream is some kind of huge macinery with many rotating parts that is drawn in exploded diagram like this:

and in the dream, each term in a quaternion power seires describes one component, as if it is a rotational generalisation to the frequency domain

(not sure whether reality has the same idea...)

So imagine a component that has 2 fold roational symmetry like a peanut shape, then it corresponds to some lower order term while those which has many lobes or more asymmetry corresponds to higher order terms

@Slereah think the Dirac eq in quaternions is here visualphysics.org/preprints/qmn10096241 wouldn't trust this page though, written by youtube.com/watch?v=vir71hYyTlY

I mean, I think the equation would basically just be a substitution

$\gamma^1 = i$, etc

Or something equivalent

I always avoid quaternions as much as possible

There is a quaternion version of GR

it was pretty much $e^\mu = e^0 + i e^1 + j e^2 + k e^3$

or something stupid like that

'William Clifford invented his algebras in 1876 as an attempt to generalize the quaternions to higher dimensions' hmm

Yeah Clifford algebras are basically that

Like $\mathbb C$ is just the Clifford algebra of $\mathbb R$

and so on

Quaternions are the Clifford algebra of $\mathbb R^4$

The algebra of real space is the Clifford algebra of $\mathbb R^{1,3}$

And the Dirac algebra is its complexification

Incredible way to get Clifford algebras in general from representing rotations as reflections ma.utexas.edu/users/dafr/M392C-2015/Notes/lecture11.pdf

Want to tie it into quaternions someday but meh

The manifold people in here will like the "quaternionic structure" at the end of that pdf

Oh man youtube.com/watch?v=QdCwstJSxqw

These things are just so funny

I think I remember that lady

She wrote an article on her selling time to cranks

Which sounds like a good business model

Although I'm not sure I could do it without using swears

@Slereah You'd also need a PhD so the cranks think you can help them :P

It actually is a great idea if you think about it

@ACuriousMind the cranks know very well that you don't need a degree to be a genius

@Slereah Yeah, but I think her selling point was that she was part of the mainstream academia these people want to convince. They're not so much looking for a physics expert than for validation.

This is all lqg quaternion-gut hilarious stuff to look at every couple of months

LQG quaternions sounds like the worst thing

@Semiclassical they're isomorphic as multiplicative groups, no?

Holy crap @ACuriousMind @Slereah what is with that algebra hell up there

I'm looking into Bianchi spacetimes

It involves a bit of the old Algebra

In trying to do physics with quaternions, is it not like trying to do physics with vectors, where instead of a vector you use the matrix $\begin{bmatrix} z & x + iy \\ x - iy & - z \end{bmatrix}$ as your 'vector'?

@0celo7 that's what i thought. but Wikipedia has the following statement: "Quaternions form a division algebra—every non-zero element has an inverse—whereas Pauli matrices do not."

@Semiclassical Ah, right.

They're viewing quaternions as an algebra, I thought you were talking about $\{1,i,j,k\}$ with the products and stuff

that's merely a group

right.

group-isomorphic, etc.

Quaternions are kind of shit to deal with because to replicate a bunch of the matrix stuff you have to use the weird quaternion automorphisms

Had my fill of these things for a year