The h Bar

Wow, that is tedious. Should just let you not write it lol. I've had professors like that. If you do well enough, you are excused from like end of term finals/projects.

0celo7

The teachers are forced to exempt you from the final -- they just make the final the paper + presentation, and the research proposal and actual research is a fourth quarter test grade.

Icosahedron

12:44 AM

@0celo7 What level is it?

0celo7

@Icosahedron Advanced.

Icosahedron

Is it introductory at least?

0celo7

No.

It doesn't cover any of the basics.

Icosahedron

Then... you linked that because of what?

0celo7

Put it on the list after Wald.

Wald is a good prep.

Icosahedron

12:46 AM

Will do, he also has lectures online for computational relativity.

0celo7

Now that's boring.

Icosahedron

By that I mean making calculations (coordinate system gr).

0celo7

Yes, it's a really complicated PDE analysis.

That's not something you learn on a whim.

Icosahedron

:/

0celo7

I mean, I'm sure you can watch some videos, but keep in mind that numerical relativity is what Chris White does for a living.

It's a huge field.

Icosahedron

12:52 AM

You didn't tell me what to read after Zee.

0celo7

That's because you're 5% of the way through it.

Icosahedron

Then why are you telling me what to read after Wald?

0celo7

I think you should finish Shankar and Zee at the same time.

@Icosahedron I have my reasons.

Icosahedron

I will, by october or sooner.

0celo7

If you really want to read multiple books at once, you can do it with those two. Neither is terribly hard.

Icosahedron

12:54 AM

Assuming consistency until then.

0celo7

They both have easy and hard chapters.

After Zee... What do you want to do?

You could go to QFT, or do more GR.

Icosahedron

Both.

0celo7

Or both I guess.

Maybe skim Sakurai.

Icosahedron

What did you read after Zee again? Wald?

0celo7

Pick up the parts that Shankar glosses over.

I read Wald after Straumann.

I haven't completely finished either.

Icosahedron

12:56 AM

And after Zee?

0celo7

I read Shankar after Zee.

Icosahedron

GR I mean.

0celo7

Then I read Zee's QFT, which was a mistake.

Weinberg.

Icosahedron

Ok, I'll do HE after Zee.

0celo7

Good one.

Icosahedron

12:57 AM

;)

0celo7

I'm having trouble with it. If you can take it, good for you, but it's a very hard book.

Assuming you go through every proof like I am.

Once you've taken topology and learnt a bit of differential geometry, you should be able to read Wald. (Wald is not the best place to learn geometry, the treatment is rather incomplete.)

This makes sense to me after Zee.

bolbteppa

1:12 AM

Any of you guys know how to derive $e^{iS/\bar{h}}$?

From classical physics

0celo7

@bolbteppa Can't derive a postulate.

bolbteppa

Maybe forget $\bar{h}$

So derive $e^{iS}$

0celo7

@bolbteppa You know there is an \hbar, right?

bolbteppa

But $\hbar$ is a unit conversion factor

0celo7

I guess the thought of the day is nothing.

@bolbteppa $\hbar$ sets the scale for quantum effects, in a sense.

The $\hbar$ in the exponential is critical for the idea of the path integral.

bolbteppa

1:16 AM

It comes from passing from wave optics to geometric optics in Landau, but I really don't like that derivation tbh

0celo7

What significance does $\mathrm{e}^{\mathrm{i}S}$ have in classical physics anyway? Note that without $\hbar$, the exponential is ill-defined because $S$ is dimensionful.

Icosahedron

@0celo7 I don't like cosmology anymore, though I'll take a look later.

0celo7

How did you decide this?

bolbteppa

Well hmmm I guess it's more complicated

0celo7

I don't understand your question.

In what context are you deriving it? You can't derive an expression.

bolbteppa

1:20 AM

Basically the idea is to motivate using the expression $e^{iS/\bar{h}}$, if you can motivate using this single expression you get a ton of quantum mechanics for free

0celo7

It's easier to motivate the SE and then derive the path integral.

user54412

@0celo7 Actually, I try to make a living doing fluid dynamics on stationary spacetimes. But everyone I talk to -- my adviser, other profs, my thesis committee, all my collaborators -- thinks coupling my method to an Einstein equation solver would be "trivial."

bolbteppa

This is like the best way to motivate Schrodinger, it's just a derivative of this baby... The method is to mimic how one goes from electromagnetic waves to geometric optics as the wavelength goes to zero, so a component of the EM field is like $e^{i \psi}$

0celo7

You could postulate that the transition amplitude is $$\int\mathcal{D}x\,G[x]$$ and then argue $G[x]=\exp\mathrm{i}S$ because of constructive/destructive phases.

The constant $\hbar$ is introduced for dimensional reasons and sets the scale at which Newtonian mechanics breaks down.

Icosahedron

@0celo7 I don't know.

Kyle Kanos

1:26 AM

@ChrisWhite Well of course it's trivial

bolbteppa

Then doing a Taylor exansion in $\psi$ in the exponent gives you the wave 4-vector, which is called the Eikonal equation in this context, and it gives you the Geometrical optics analogue of Hamilton's equations. So $\psi$ is exactly the same as the action and it is a way of taking a wave and then allowing you to interpret the motion as though it was particles.

So in QM you mimic this by setting $\psi = c S$ where the constant $c$ is $c = 1/\bar{h}$

0celo7

I'm aware of the Eikonal approx.

Have you looked into geometric quantization?

user54412

@KyleKanos Tell me the secret. Should I be using Flash? ;)

bolbteppa

You're postulating phases and stuff when you do that, I have no idea how you can accept the concept of phases tbh

0celo7

@bolbteppa Uh, the phases are simply complex numbers.

An imaginary exponential is by definition a phase.

@ChrisWhite To my eyes, the Cauchy problem in GR seems highly nontrivial.

user54412

1:32 AM

Fun fact: apparently a large part of the complexity (both mental and computational) of numerical relativity codes is in detecting event horizons.

user54412

Maybe that's not too surprising, but it's interesting to me that we still care about such non-local things.

user54412

Even though the PDEs are hyperbolic.

bolbteppa

I think I see it now it's far easier than I thought

0celo7

@bolbteppa I'm not convinced by your method. Why $\psi=cS$?

@ChrisWhite When they find a horizon, is the calculation aborted, or are they looking for them for other reasons?

user54412

@0celo7 They can't abort, otherwise they couldn't deal with black hole formation from e.g. merging neutron stars (which they do these days). They simply need to know where to excise the hole from the computational domain so calculations aren't stalled by evolving material all the way to the singularity.

bolbteppa

1:42 AM

Well it's Landau's method, what he does is consider the fact that a plane monochromatic wave has the form $f = Ae^{i\psi} = Ae^{i(\vec{k}\cdot \vec{r} - \omega t + a)}$. Because we are assuming a small wavelength $\lambda \rightarrow 0$ we can Taylor exand $\psi$ which gives us the Eikonal equation, but this shows that the Eikonal $\psi$ behaves exactly like the action when a wave behaves like a particle (due to small wavelength).

Similarly in quantum mechanics we assume that in the limit of going from quantum mechanics to classical mechanics the Eikonal will literally give us the action and Hamiltons equations, because it literally determines the path of a particle, hence $\psi = cS$ ensures the units of $S$ dissappear, hence Plancks constant is in units of action

0celo7

@bolbteppa Not totally convined.

And you still haven't explained what the heck $\exp\mathrm{i}S$ is supposed to mean.

bolbteppa

I just explained that

0celo7

I read too quickly.

bolbteppa

I cannot believe that is what he is saying

0celo7

@bolbteppa I don't understand how this shows that $\psi$ is like the action.

bolbteppa

1:46 AM

He proves it twice, the second time plugging that $f$ into the wave equation he gets the same result, he gets the four vector relation $k_i k^i = 0$ which is also the relativistic Hamilton Jacobi equation

0celo7

@ChrisWhite Do you think numerical relativity is boring? I wouldn't mind reading about the intellectual aspects, I'm not interested in the computational aspects. (Maybe when I take some computational modeling classes in my junior and senior years.)

bolbteppa

and those are derivatives of the action so it's like amazing

0celo7

I like the path integral better.

A lot better.

bolbteppa

I don't think the path integral makes any sense the way Feynman derives it out of thin air, aren't there like 5 axioms you need?

You can derive the path integral this way very directly as the green function

0celo7

He derived it as a Green's function.

bolbteppa

1:49 AM

Well Landau motivates this thing like 6 times I am not explaining the best I'd say

0celo7

I'm perfectly happy with my current understanding of the motivation.

bolbteppa

For what equation? He is supposed to derive the Schrodinger equation his way from first principles

0celo7

Perhaps something like Hall's Quantum Theory for Mathematicans would contain axiomatic motivations.

@bolbteppa The SE follows from the path integral.

bolbteppa

Yeah but you said he derived it as a Green function for some equation

It looks circular to me

0celo7

The Green function interpretation follows from the basic rules of linear algebra.

I'd explain my reasoning, but I'm writing a paper for school.

bolbteppa

1:56 AM

Okay but it is defined in terms of some differential operator so you are assuming Schrodinger already that way

0celo7

No it's defined as the evolution operator for the system.

(Plus some integrals.)

@bolbteppa i.e., write $$\psi(x,t)=\int_\mathbb{R}U(x,t;x',0)\psi(x',0)\,\mathrm{d}x'$$ and argue that $U(x,t;x',0)$ can be a path integral.

user54412

@0celo7 I find it rather interesting, but you have to appreciate PDEs to like it.

0celo7

@ChrisWhite I'll look into it when I've taken some numerics classes then.

user54412

2:12 AM

I think having exposure to numerics is pretty much the only way to appreciate PDEs. Writing them down and finding analytic solutions -- well, that never actually happens in real life. But the computational aspect is very rich, and continues to expand.

0celo7

@ChrisWhite I'll definitely get a lot of numerics in my Nuclear Engineering classes.

user54412

One of my profs likes to point out how the study of pulsar winds was stalled for a good decade or two simply because all the people working on it tried to find analytic solutions to the governing PDE. Then one day someone just found solutions numerically, and all of a sudden people were able to do science again.

0celo7

Has anyone found an analytic solution?

user54412

pretty sure one doesn't exist

0celo7

Proof by exhaustion?

user54412

2:19 AM

yes -- exhausted pencil-and-paper Russian theorists

0celo7

Lol

"no solution = gulag"

yuggib

6:55 AM

@ChrisWhite To appreciate PDEs you should prove the solution exists and perhaps is unique in a suitable (functions/distributions) space. Then the rest is pedantic machinery for which a machine is much more adapted (and something with which many mathematicians never want to deal :-D)

@0celo7 Never be happy with your current understanding...this is the worst of mistakes

ACuriousMind

10:07 AM

@Danu 1. The operator $\mathrm{d}^2$ raises the rank of a form by 2, so you must take the wedge with the Riemann tensor, how else is it gonna raise the rank? 2. The ordinary exterior derivative does not care for the metric, it doesn't know anything about it, so you must have defined the covariant exterior derivative (like you do $\partial_\mu \to \nabla_\mu$ in coordinates) as something like $\mathrm{d} + \Gamma \wedge$.

David Z

10:37 AM

1

A: Equivalent Resistance

This is niether series nor parallel circuit, because of resistor 4. We have to use Kirchoff's laws. We will assume that both ends are connected to a battery to simplify our analysis. First, reformulating the loop rule, we get that since the potential difference between the top and bottom end is ...

I don't think that quite qualifies as a complete answer to a homework-like question, since the homework problem is asking for the equivalent resistance and the answer doesn't give R_eq or a formula for it, but I could see it going either way... thoughts?

The Dark Side

An eye for an eye:

ITISNOTCLEARHOWTHISANSWERSTHEQUESTION. — ACuriousMind 1 hour ago

lol

@DavidZ I agree.

Hello bigguy (@yuggib)!

The Dark Side

11:13 AM

@DavidZ - So, did anything come out of the mod-conference?

David Z

@TheDarkSide we haven't really talked about it yet

The Dark Side

@DavidZ - OK. But I think your comment (earlier to the one linked) exposed the ambivalence, am I right?

Danu

@ACuriousMind Today, we elaborated on the definition some more and saw that one can also write it in terms of some "special wedge" (we called it $\wedge_\text{eval}$) which is really wedge composed with an evaluation map

We did not define it like that explicitly, no.

David Z

@TheDarkSide I suppose. I mean, I think it was clear from the picture that different mods have different ideas about how these things should be dealt with.

ACuriousMind

@Danu But $\mathrm{d}^2 = 0$ is one of the defining properties of the ordinary exterior derivative. What is your $\mathrm{d}$ operator supposed to be if not the covariant derivative?

yuggib

11:30 AM

@TheDarkSide hello ediS kraD ehT ;-)

without parentheses it would have been a nice palindrome

Danu

11:57 AM

@ACuriousMind A generalization of it

The exterior derivative was just the special case on sections, I think

ACuriousMind

12:13 PM

@Danu Are you purposely avoiding telling me anything of actual substance about it? ;)

0celo7

12:24 PM

Haha, @Danu just give the full definition. Calling it a "generalization" doesn't help us.

Arc676

1:28 PM

Does the equation $$\Delta t_2=\Delta t_{ proper } \sqrt { 1-\frac { r_{ s } }{ r } }$$ work for time dilation around objects that are NOT black holes?

David Z

1:41 PM

I don't think so. In general you only get the factor $\sqrt{1-r_s/r}$ for a Schwarzschild black hole, or some generalization of it for other kinds of black holes. If you have a different configuration of mass, of course it will have a Schwarzschild radius, but that value won't necessarily enter into the calculation.

ACuriousMind

I believe the Schwarzschild solution is valid outside any spherical mass distribution, no matter whether it's a black hole or not

Jim the Enchanter

I believe it's valid even if there is no mass as well.

0celo7

1:56 PM

@JimtheEnchanter Lol, with M=0 or something deeper?

@ACuriousMind Any static solution without charge, yes. This is known as Birkhoff's theorem.

Straumann proves it in chapter 4 ;)

Jim the Enchanter

@0celo7 no, just with M=0

Arc676

2:22 PM

So if I use the Earth's Schwarzschild radius and the radius of the Earth I can get the time dilation caused by the Earth's gravity right?

Jim the Enchanter

Yes

Arc676

Cool thanks

Danu

4:29 PM

@ACuriousMind Obviously

I just didn't have my notes at hand for the past few days (a friend of mine is TeXing the lecture and he uses my notes)

copy.com/s/t%3A3LVn3jhPKgvMwFiL%3Bp%3A%252Fscripts/riem.pdf

1.127 I guess

and then 1.144 is the alternative definition that I think you had in mind @ACuriousMind

Jim the Enchanter

5:41 PM

@JohnRennie You initiated a Rindler observer, but from what I understand, all Rindler observers have to be at rest wrt the inertial frame at t=0 (that's t in the inertial coordinate frame)

I'm not sure you can easily adapt that to what you have done

innisfree

In GR, is it, in general, impossible to find a global description of events on both sides of a horizon? For example, an asymptotic observer will never see something fall into a BH.

Jim the Enchanter

Well, an event horizon usually is a coordinate singularity in the metric for an asymptotic observer. So I'd say that it's often impossible. But don't quote me on that

innisfree

that sounds convincing

Hippalectryon

Does anyone here know how I can solve (numerically, no need of an exact solution) differential equations over a domain defined by conditions using Mathematica, or another (free) software? I'm solving this :

Jim the Enchanter

5:57 PM

@Hippalectryon if(r<=R) then [insert solution method to first equation] else [insert solution method to second equation] end if

Hippalectryon

@JimtheEnchanter but R is defined by the boundary conditions

(the integral one)

Jim the Enchanter

The rest should be the same as if the domain was fixed

Hippalectryon

I can't get R (can I?) without computing h too

Jim the Enchanter

looks to me like you have to compute them simultaneously

Hippalectryon

I'm kind of new to Mathematica so I don't know how to do that

Jim the Enchanter

5:59 PM

personally, I wouldn't do it in mathematica

Hippalectryon

I also have FlexPde

Jim the Enchanter

I'd write my own solving program in python probably

Hippalectryon

@JimtheEnchanter Are there any libraries that can make that easier ? I only know how to solve simple differential equations using numpy

Jim the Enchanter

@Hippalectryon Not sure. As I said, I write my own programs and use only numpy

Hippalectryon

@JimtheEnchanter Conceptually, how would you proceed to solve that kind of equation using python ?

Jim the Enchanter

6:07 PM

RK4 as a basic method most likely. I'd need to study the problem a bit more, but I'd choose a variable to integrate over (probably r) and then solve for R using the integral and h from the previous step, solving for h simultaneously using the current values for r and R.

Hippalectryon

@JimtheEnchanter thanks

0celo7

6:23 PM

@Danu Is that linked PDF the transcription of your notes?

Icosahedron

@0celo7 I can't do some of the Zee exercises, should I move on?

Also I think I should read shankar ch1, there is useful information in it.

alarge

@Hippalectryon As I see it, that's a BVP, and thus RK by itself will be useless. You can use the standard shooting method or sonething else.

Hippalectryon

@alarge BVP ? And what's the 'standard shooting method' ? (I'm just a student)

Jim the Enchanter

@alarge touche, I missed the different boundary points.

Danu

6:39 PM

@0celo7 Yes

@Icosahedron It seems crazy to try to learn GR before QM

Icosahedron

@Danu I don't see why.

Though I'm learning them at the same time.

0celo7

@Danu Either you are the best note taker in the world or your friend is an excellent writer.

Maybe both.

How detailed are your written notes?

Danu

@0celo7 Leeb lectures very well. He writes all of this out on the board.

He's nothing like the physics professors ;)

These are literally my notes, typed up and lacking a bunch of drawings (regrettably!!)

0celo7

Jesus.

How many pages per lecture?

Danu

The only thing that annoys me is that he's so fast that I often don't have time to correct his grammar :P

About 8 large pages in 90-100 minutes

...I think

0celo7

6:47 PM

You write full sentences in notes? I'm very impressed.

Danu

Leeb writes full sentences

ACuriousMind

@Danu Ah, yes, we were talking about the same thing, I forgot that the "wedges" I write for connection forms and curvatures are your $\wedge_\text{eval}$. Still, this is nothing special to Levi-Civita/Riemann, it's a general principle. (And I would still call your $\mathrm{d}$ on the $E$-valued forms covariant since they involve the connection, but that's just terminology, I guess.

Danu

I try to correct them

@0celo7 Actually, "only" about 5 pages per lecture

0celo7

@Icosahedron Which ones? I might be of assistance.

Danu

I have a really nice notebook for this :)

0celo7

6:48 PM

I've never written 5 pages by hand in one sitting.

Danu

You've never met Leeb

Not many people like his way of lecturing

but I've grown accustomed to it

Icosahedron

@0celo7 1.3 #5.

Danu

@ACuriousMind I never even mentioned anything about L-C/Riemann :P

Don't know why you keep on bringing it up

Icosahedron

Though it's because I haven't read into this stuff, I'll read the relevant chapter in arfken then try it later.

ACuriousMind

I must misremember the statement that started this then :P

Nevermind^^

Danu

6:49 PM

@Icosahedron There is such a thing as "maturity" in academics, and it matters... a lot

0celo7

@Icosahedron I think I know that one without looking at the book...does the answer involve a delta?

Icosahedron

@Danu Do you mean mathematical maturity?

@0celo7 I didn't even try it. It looks too hard.

0celo7

What is it? Zee is not nearby.

ACuriousMind

@Danu Heh, I had QM and GR at the same time.

Danu

@Icosahedron The physics equivalent

@ACuriousMind I don't think that's a good idea :P

Icosahedron

6:52 PM

@Danu I've never heard of there being a physics equivalent.

ACuriousMind

@Danu You never think anything I do is a good idea ;(

0celo7

@Danu's notetaking

Danu

@ACuriousMind Lol, you think that?

@0celo7 Thanks

I do try

@Icosahedron Well, there is and it's about knowing CM and QM very well :P

ACuriousMind

@Danu Nah, not really, hence the ; not : in the sad face ;)

Danu

@ACuriousMind ;\

0celo7

6:53 PM

I think equations work a lot better.

Icosahedron

@Danu I'm reading a beginner GR book though, and beginner QM book.

0celo7

@Danu Which of the notes on that dropbox website are complete?

Danu

@Icosahedron I would recommend getting through the beginner QM book first

@0celo7 It's not a dropbox website, and none are perfect AFAIK

0celo7

@Danu Don't forget that at his school, they do Wald and HE as juniors. I'm not sure he understands "take it easy".

Danu

I'm editing the ones on diffgeo during breaks

hopefully I'll be able to finish them this summer

0celo7

6:54 PM

@Danu I know, it's copy or something.

It's a dropbox style website.

Icosahedron

@Danu Though 0celo7 did the opposite.

Danu

@Icosahedron He knows I disagree with how he studies ;D

0celo7

@Icosahedron Danu does not approve.

@Danu You do know I've stopped reading multiple books at once, right?

Icosahedron

@0celo7 Though the people in that class have read Lee's smooth manifolds before taking it. (in advanced diff geo in junior year)

ACuriousMind

On an unrelated note, anyone else playing The Witcher 3?

Danu

6:56 PM

@0celo7 No, I did not, but good on you haha ;)

(although N\leq 3 would be acceptable too)

@ACuriousMind I would if I had free time lol

(and a good computer)

0celo7

@Danu I am breaking that rule by reading HE while having BBS and BLT open, but I really want to read it. I'll probably finish it before going back to strings.

Danu

I'm preparing a lecture on extremal black holes

0celo7

Then I'll have to re-read some stuff, but whatever.

Danu

@Icosahedron So what is junior year again? 3rd?

alarge

@Hippalectryon Boundary value problem. as for shooting method, that's probably the most elementary way of dealing with this class of problems. Google or Wikipedia it. I can maybe get back to you in more detail once I get home. Mathematica should have a BVP solver in it, as should any of the major math libraries/applications dealing with DEs.

Hippalectryon

6:58 PM

@alarge Thanks.

ACuriousMind

@Danu Yeah, my sleeping cycle has taken a major hit this week :D

Icosahedron

@Danu It's a 3rd year course though 2nd year students can take it. (but barely any do).

bolbteppa

What do you guys think of spectroscopy? It looks like some weird non-relativistic QM form of quantum electrodynamics or something?

ACuriousMind

But now it's weekend, thankfully

bolbteppa

This uk.businessinsider.com/… has convinced me why sleep really matters, no more 24-30 hour work days :p

Danu

7:00 PM

@Icosahedron Hmm yeah okay

The courses on diff geo are 4th year here @ LMU, but many 3rd year students take it

0celo7

@ACuriousMind For you...slacker

Icosahedron

@Danu What book do they use?

Danu

@Icosahedron None, since it's Leeb who's lecturing.

0celo7

7:20 PM

@Icosahedron Exercise 1.3.5 is answered in the back of the book.

Icosahedron

7:36 PM

@0celo7 I didn't know there were solutions.

Thanks.

0celo7

I guess you overlooked that when reading the preface.

If you can figure out how to do it by "just integrating" like he first says, let me know.

Also, I'm fairly sure that should be a $\sin\theta$, not $\cos\theta$.

After all, the surface element on the sphere has a $\sin\theta$.

Icosahedron

@0celo7 Do you mean can't?

0celo7

@Icosahedron I can't figure it out.

"Just integrate" is extremely vague.

(Or however he puts it is vague.)

Do you agree it should be $\sin\theta$?

Icosahedron

I still don't know what he means by Fearful

0celo7

Another thing you missed from the preface. Man, it's almost like you didn't read it!

Icosahedron

7:47 PM

I didn't.

0celo7

Zee is a delightful writer. I am forever disappointed by textbook authors who don't write as well as him.

If it's gonna be starred, might as well have correct syntax.

Got that "who" edit in with 3 seconds to spare!

Icosahedron

I didn't star that, though I don't agree.

0celo7

I didn't expect you to star it.

Icosahedron

D:

Arnold is the best physics writer.

(star if you agree)

0celo7

Are negative stars a thing?

I'm still not sure what Zee means by "average over the direction of $\vec p$". This has confused me for a year now.

Icosahedron

7:56 PM

@0celo7 Anti-stars are.

0celo7

Unless you have some insight, I think it's time I ask the German.

@ACuriousMind Please help.

I swear I've seen this problem in the context of E&M somewhere.

Icosahedron

Doesn't he just mean average integration?

0celo7

Yes, but that $\cos\theta$ is bothering me.

It's either a typo or I'm missing something.

Transcript for