And the global solutions are done with whatever property that it implies

Like missing points or boundaries or cyclical coordinates

So, before you go and solve the case of Schwarzschild, you assume that you are working on $S^2\times \mathbb{R}^2$?

Well ideally, yes

But that's a flawed approach! You are assuming that which you want to derive

Although historically I think it was more because, if you solved it on $R^4$, it was not geodesically complete

One thing that bothers me really is what real black holes look like, topologically

But what tells you that's not the solution? You just say "I don't like it being incomplete, let's extend it"?

There's a piece missing here, I feel

Well yes

I thought most of these old definitions of mass were flawed and ignored in modern day treatments, now JD is saying they make sense, and some books seem to use them, while others ignore them?

There's no reason why a local solution should have a particular topology

You can put De Sitter space in $R^4$ if you want

You'll just have a singularity

BTW, the same piece is missing in almost all treatments of gauge theories, where one usually says one is working one a fixed principal bundle over spacetime, and then later on the instanton number - a topological invariant of the bundle - is suddenly allowed to change dynamically in the theory.

I mean you can take Minkowski space technically and do all the identifications and point removal you want

It will still be Minkowski space

@Slereah Exactly, so...does GR not determine the actual spacetime manifold at all?

No

Other arguments are necessary to determine it

I feel that is unsatisfying. The manifold itself should be determined dynamically, not by "other arguments".

Well the manifold can't be determined dynamically

Since it already span all spacetime

It doesn't change with time, it is time

You can change the topology of the spatial hyperslice dynamically

Although most theorems forbid it

Hm, yes, but the variations of the stuff in the action don't happen in time. Stuff doesn't "move" towards least action, it just takes least action. So if one included enough global quantities of the manifold into the action and varied them, one could, in principle, think that the global structure could arise as the solution of an action principle

No metatime or whatever required for that

Well as said, you can do some topological stuff for that

But it's not a very good idea

Summing over manifolds is a bit tricky mathematically

And when it's possible, it is pretty badly divergent

unless you really restrict the class of manifolds you allow

Well, the string perturbation series sums over all worldsheets effortlessly ;)

Worldsheets are 2D

2D topologies are fine

4D is another pair of trousers

And "It's pretty badly divergent" has never stopped anyone from using a path integral :D

IIRC it's double exponential divergent :p

Also, I don't want to go quantum. I'd be fine - for the moment - if the classical theory cared about the manifold as such

Can't think of any

Usually it's more

You try to solve it in $R^4$ or something else simple

Then if it seems weird you might switch manifold

Or you just decide the manifold from the beginning

Yeah, okay, you've convinced me that that's the usual modus operandi

Still, I don't like it one bit now that I've thought about it

Well write a paper :p

Also I think that some topologies don't even care

A metric will be basically the same if you do any number of identifications

as long as it is along some killing vector thing

@Slereah Y'see, the issue is the few leads on considering such global structures formally as dynamical at all come from crazy people talking about $\infty$-jet bundles and cohesive topoi and whatnot :/ I was hoping the local crazy GR person would have some more down-to-earth approaches ready ;)

Well I did my best :p

And I thank you for indulging me :P

"The causal structure of the Godel universe is of particular interest: there exist closed timelike curves through each spacetime point [24, 19]. In addition, the model contains no spacelike hypersufaces without boundary."

Whaaaaat

Plot twist: The "closed timelike curves through each point" are one gigantic fractal space-filling curve :P

There are actually spacetimes without CTCs but with curves like that

Not even sure if that'd surprise me :D

@Slereah See, knew it!

Totally trapped curves

They are curves that are stuck in some compact region of spacetime

Most of them are CTCs but you can contrive a spacetime where that isn't the case

Are they "only" dense or actually space-filling?

Hell if I know

I doubt physicists care about the nuance :p

Sigh...probably right

If you want to check, here's one

$ds^2 = (\cosh(t) - 1)^2 (dt^2 - dy^2) + dtdy - dz^2$

The manifold is $R \times S \times S$ with the identification $(t,y,z) = (t,y,z+1)$ and $(t,y,z) = (y,y+1,z+a)$

And $a$ is irrational

Is that a torus with irrational modular parameter (in the spatial part)? That thing is also a frequent example in classical contexts for producing dense instead of closed curves

Kinda yes

@JohnDuffield okay so now I finally think I see what you're doing, you are just using relativistic mass to define a notion of 'inertial mass' for a photon and giving it obsolete old names and mixing up concepts, you can use $E = mc^2$ to define a notion of mass for a photon so long as you write it as $E = mc^2 = \gamma m_0 c^2 = \frac{0}{0} c^2$ and then using $E = h v$, now I'd love to throw away hours finding out what you mean when you say a photon can accelerate, but...

Of course you say $E = mc^2$ doesn't apply, instead you prefer to use some Newtonian mechanics concept of inertial mass to justify analyzing a photon, though nobody knows how you defined inertial mass as $m = hv/c^2$ :\

@bolbteppa please stop

haha yes, I threw away a whole day of string theory, qft and cft on this nonsense :(

just don't try arguing with Duffield

@Slereah This kind of thing is what got you suspended last time. Stop.

Well maybe I'm silly but I have faith and hope that he'll eventually pick up a serious book and study it out of interest

@ArtOfCode It is no laughing matter.

Duffield is rather disruptive and argumentative.

@Slereah I'm well aware, but being mean to the guy is not the way to solve it.

It is commendable to at least try and go back to read Einstein's papers

Indeed it's not.

My method is to get people to stop indulging him.

@Slereah are you not interested to find out @JohnDuffield explanation as to why he claimed physicists usually define energy in a circular high-school-level-textbook fashion, then had no idea of the link between Energy and Noether's theorem?

No.

@ArtOfCode Well, what is the way to solve it?

That conversation took place many times.

Duffield does not actually know physics and will not actually listen to any argument

It's a rather pointless endeavour to try

Nor is it particularly intriguing

@ACuriousMind He does something, flag 'im. Or come find me or one of the Physics mods, since we know the case.

Because while I am rather supportive of the "Be nice." code, I am feeling stuck in groundhog day, watching the same play unfold over and over again.

@bolbteppa I'm definitely not. Last time I argued with him I spent two or three hours repeating "I don't disagree with that statement. What exactly do you disagree with?" before I got it cornered that he disagreed with an assumption I explicitly spelled out.

@ArtOfCode The thing is, a lot of his stuff isn't flag-able, I feel.

It's just not good discourse anywhere.

@ArtOfCode Well...the issue here is not that he's being offensive or something (at least most of the time)

Yeah there isn't really an explicit rule against cranks

You can't flag a user's personality, can you?

@HDE226868 Mod flags are always an option, though. Mod flag and explain what's new.

Nothing is new, though

It's the same pattern for months

@ArtOfCode The mod would have no justification to act, though.

He's just being annoying.

Yet not actually trolling.

Or if he's trolling, he is quite dedicated

Well, quite honestly I've never seen a more bland blatant case of someone being 100% wrong yet repeating an incorrect statement, not only in their book, but in a video and on this thread, a statement pivotal to the whole shtick (the energy one), and actually admitting they didn't know the right answer (link to Noether), I meant to just press on with getting some recognition of this before getting sucked into the whole $E = mc^2$ stuff haha

Once rules are bent enough, they can be considered broken. And I know I and some others consider this particular brand of annoyance rather a bend of the Be Nice policy.

@Slereah What I mean is, next time a big argument starts up, come find one of the aforementioned mods. We are trying to... educate.

The policy that a suspension gets longer each time applies on chat, too.

In some particular cases I've seen - not all - I disagree. But I certainly agree that certain responses that have been used are . . . inappropriate.

I tend to ban cranks pretty fast because it really never ends well :p

@HDE226868 Those build up, and those are the grounds we've used for past suspensions. And will use again if they're needed.

Alright, I guess I'll be a bit more inclined to click the flag button in the future.

@ArtOfCode Yeah, I've used that rationale in the past (in this particular case, too).

@ACuriousMind Aye. Don't flag as offensive unless it actually is, but feel free to mod flag and ask an involved mod to come take a look.

Or maybe just stop trying to argue, really :p

You can put someone on ignore

Click on the icon, "Hide posts"

or just plain "ignore this user"

Hide posts is temporary.

I have a mean thing to say but I cannot say it, so try to imagine it

imagines mean things

::imagines teasing @0celo7 about solving a quadratic equation::

Hehe

imagines banning everyone in this room without reason - that's mean, right?

Well yes, but that is not a thing you say

Hm, more like megalomaniac, I'd say

Hmmm. True. I'm stuck on imagining mean words then.

"meanie" comes to mind

Four instances of that word in the transcript

Three by me

you big meanie

Ohh, we had that query about the meanest downvoters.

I was 8th

Now I am 3rd (among those with at least 100 up and 100 downvotes).

:|

Oh

I am the meanest by absolute downvotes.

@ACuriousMind : You might be mean but are you truly evil? :)

@Qmechanic I don't know, but I just got a job offer from a certain S. Atan.

Dr. Acula

@ACuriousMind quant stuff?

@NeuroFuzzy I'm...not sure, but it says here I should bring my best asbestos suit.

↑ hi all this Q had a physics angle, ie statistical physics/ mechanics, evaluation of phase transitions via machine learning, wondering if anyone here has heard of anything like it...

I'm baaaaaaack

@ACuriousMind you're the meanest by sheer meanness...

@NeuroFuzzy Don't remind us of KK ;_;

@ACuriousMind wtf is a "jet bundle" anyhow

@ACuriousMind Wow, you got a job out of the blue!? That happens to physicists?

@ACuriousMind you punk

@ACuriousMind what's the difference

@ACuriousMind what? how is this not a philosophical question

@0celo7 dude, I don't literally mean "who".

You have a pattern of missing the point today :P

@ACuriousMind no shit

but still, why give a rat's ass if we can't calculate the action

you have a pattern of giving rat asses

@ACuriousMind what does that mean

I think you construct the manifold.

Given the initial data, $M$ is unique, that's the whole point of the solution theorem.

Now, I haven't read this section thoroughly because I'm a PDE ignoramus.

@ACuriousMind Does that theorem help you some?

@0celo7 That formulation has a chance of doing what I want. However, note the restrictions which have no real origin except to get a "nice" resulting spacetime such a global hyperbolicity and the stuff about extrinsic curvature

Additionally, this formulation is completely unlike the usual physical striving for a principle of least action. (This is just an aesthetic argument)

@ACuriousMind "what? how is this not a philosophical question"

aesthetic $\cong$ philosophical

@0celo7 Yes, I don't deny that that part is not a physical argument

It is not my main point.

@ACuriousMind Stuff about extrinsic curvature?

You mean the constraint?

@0celo7 Yes, I mean the conditions (ii) and (iii)

Well, (ii) has a very physical origin.

Although I guess (iii) is really just that they are initial conditions

You can't have stuff coming in from behind a Cauchy horizon and messing things up.

@ACuriousMind Yes.

@0celo7 There's a very good reason for not wanting non-hyperbolic manifolds. That is not generally seen as an argument to exclude them from the theory entirely

@ACuriousMind You don't exclude them!

There's no unique solution if you have them!

So the problem "solve the EFE" can't be solved if you have them.

So you should ask "how do you know if you have them"

@0celo7 Yes, exactly. But no one excludes Norton's dome from being a classical mechanical system because it fails to have unique solutions of Newton's equations, either.

Furthermore, multiple possible classical solutions of the equation of motion are basically what generates instanton sectors. Restricting to unique solutions of the equations of motion is not even always possible.

@ACuriousMind I'm afraid I don't know what dome that is. (I googled.)

@ACuriousMind Isn't that quantum bullshit?

I'm trying to find the theorem in HE, maybe it has some more info.

@0celo7 I am actually not being facetious

@ACuriousMind I didn't say you were....but aren't instantons quantum things?

Don't shoot me if that's wrong.

@0celo7 It's just a special shape such that the e.o.m. for a ball sitting right at its top does not have a unique solution - both starting to roll and staying at rest are valid solutions

@ACuriousMind How does that work?

I read the (short) Wiki article.

@0celo7 Yeah, kind of because you usually don't have non-Abelian gauge theories clasiscally, but in principle they occur also in classical field theory

@ACuriousMind Well, no one is saying that one should exclude unhyperbolic spacetimes.

@0celo7 The ODE fails the conditions of Picard-Lindelöf, so it doesn't have a unique solution

Ok, let me explain...

Let's take a nice unhyperbolic spacetime

Yes, I just coined that term.

Maximally extended...Kerr.

Do you know what the Penrose diagram for that looks like?

Don't know and I cannot read Penrose diagrams, anyway

Never learned them, never cared

"never cared"

Why do you hurt me

Light goes at 45 degrees.

This is not globally hyperbolic.

hmm, that's not what I'm looking for.

Gimme a moment to find the right picture.

BTW, I have the same kind of issue with classical gauge field theories, so any answer to my conundrum that relies heavily on weird Lorentzian properties is unlikely to be what I'm looking for

I just posed it as a GR question because there are more GR than gauge theory people here

Aha!

I need the electrovac one

for dynamical systems did you say arnold?

you mean the physics arnold?

Er, that's the same thing lol

@ACuriousMind OK

Now I shall explain

Imagine we are in the part labeled "universe"

@ACuriousMind Are you with me?

@obe yes

@0celo7 I...am looking at wavy lines labeled with GR terminology. Does that count as "with you"?

Sure

Now imagine we want to use GR to determine what happens in "white hole"

Make sense?

Continue

Energy and momentum can come from the parallel antiuniverse

This is not a part of the initial data in "universe"

So GR cannot predict what happens in "white hole"

Okay, yes. This generates some other questions about global topology which I will postpone until I've fleshed them out.

Continue

I'm trying to figure out what Hawking is going on about :P

Then I'll continue

I need to figure out what a Cauchy horizon is again

Also, I am beginning to think a lot of weird pop-sci ideas come from just looking at Penrose diagrams without knowing what's going on :D

You should have asked me this 4 months ago

@ACuriousMind Oh, this has no basis in reality

@0celo7 I suspected that, but someone just seeing the diagram doesn't know that

Extended Penrose diagrams are BS, there's too many stability issues.

This is just the "textbook example" that HE uses

Although it's not a textbook, so I have to prove when they say "there is a horizon"

Ok, continue?

Uh, what do you want me to continue with?

Unfortunately, I don't know where exactly in the proof of that above theorem the cauchy surface thing comes in.

I don't know enough about PDEs. Sorry.

I'm not sure I adequately communicated what my problem is either. Also, I'll have to think about initial value formulations in this context even if I don't like them

@ACuriousMind If you think about the above diagram "time reversed", you can see that you lose information into the "antiverse" and "parallel antiverse" and so not all of the info makes it back to "universe"

So if we try to evolve it forward, we know that we can't have possible gotten all of the info

That's what we mean by "nonunique"

@0celo7 I'm not even sure where time is in that diagram

@ACuriousMind up

The blue curve is a timelike worline

This is like the third Penrose diagram I've looked at in my life, it doesn't convey a lot of information to me

your curved quantum fields class wasn't full of them?

Nope. They only appeared in the stupid information paradox section

stupid?

It sounded to me like people being surprised that semiclassical approximations and weird virtual particle arguments are not consistent with what we expect of a full quantum theory

Hmm, maybe you should read Wald's book

It's algebraic QFT

I can't say whether the presenters were bad or it really is like what I heard, but either way, the talk about that was mostly wasted time to me.

and he throws out the notion of particle in the second chapter

I would love to read it, but the Hilbert space stuff goes over my head

Our discussions will become a lot better after my second year :)

Should we move on?

^ sound reasonable?

@0celo7 "don't cross" where? In configuration space or in phase space?

@ACuriousMind Configuration.

Books on the GR Cauchy problem suck.

@ACuriousMind You sure you wanna bother with that, lol

@0celo7 Hm, then I neither see a way to prove that nor do I think that's actually true.

@ACuriousMind What's wrong with my proof!!?

The extremals are geodesics, this is true. And in a sufficiently small nbd, they don't cross. Another fact.

So the extremals don't cross, in a sufficiently small nbd.

@0celo7 You did not give me the "sufficiently small" information at all.

@ACuriousMind $t<t_0$!

Says in the proof!

And what does that mean? What is $t,t_0$?

A simple counter example is $q=-\ddot q$

@ACuriousMind $t$ is the curve parameter, $t_0$ is some value for it

Okay, then I at least see how it can be true

So my proof is alright?

But I do not see a straightforward way to see it, I guess your proof is not that much overkill

Perhaps not overkill at all if Arnold doesn't prove it

@ACuriousMind Yeah.

He doesn't talk about geodesics much, it's just something that occurred to me earlier.

@HDE226868 gave me a hint

I've also never seen that statement before. Far more common is the statement that Hamiltonian phase space trajectories don't cross (for Hamiltonians for which the Hamiltonian e.o.m. fulfill the conditions of Picard-Lindelöf)

Hmm, lemme check the book

Maybe I've made a discovery!

Nope, he says configuration.

Does he do anything with that statement?

@ACuriousMind There, I taught you something new today :)

unless you're not convinced D:

@ACuriousMind I think he uses it to prove the uniqueness of the solution of the Hamilton-Jacobi equation.

But I haven't read that part thoroughly.

@0celo7 About . . . ? I understand ~10% of the conversation right now.

You lost me at "Picard-Lindelöf".

Well, before that, too.

I thought you knew GR