The h Bar

0celo7

20:00

@Slereah Are you here?

I need a notation for antisymmetrization but w/o one of the indices in the middle

Slereah

mb

0celo7

like $\mu-\nu$ antisym of $G_{\mu\sigma\nu\rho}$

$\sigma$ should not be touched

Slereah

$A_{[b|c|d]}$ is the standard notation

0celo7

Ah, yeah. Thanks

Slereah

for antisymmetrization on $bd$

0celo7

20:03

Hmm. I need to symmetrically antisymmetrize an antisymmetrization.

$G_{(\mu|[[\sigma||\nu)|\rho]]}$?

@Slereah does that make sense?

Slereah

a bit hard to read

though when there's a lot of indices it gets hard

you know what notation is easy to read for antisymmetrization?

Penrose notation.

0celo7

Proof?

Slereah

20:16

very nice notation

0celo7

idk if serious or not

Slereah

Well for that aspect, penrose notation is usually clearer

0celo7

Ok.

Almost there.

Slereah

bad point

0celo7

10 indices

is that a PSE record?

0celo7

20:37

Condensed notation!

HDE 226868

20:49

@Secret I think I was just saying that any normal object would be hotter than this absolute-zero object, and thus there would be at least some heat flow from any other body to the object. The object could only be approximated as having constant temperature if we consider it to be an extremely large "reservoir" which could take in any amount of heat. While that's used in thermodynamics in some cases, it's not realistic in most applications, and isn't the case here.

0celo7

21:01

0

A: On the Uniqueness of the Riemann-Christoffel Tensor

Note that we're working in a normal coordinate system, so your equation for the second derivative of the metric simplifies to $$\tag{$1$}\frac{\partial^2 g_{\mu\nu}}{\partial x^\xi \partial x^\rho}=\frac{\partial\Gamma^\sigma{}_{\mu\rho}}{\partial x^\xi}g_{\sigma\nu}+\frac{\partial\Gamma^\sigma{}...

@Slereah lookie

I think I have the record for # of indices in a post.

@BalarkaSen see above for the horror of general relativity

Ben Niehoff

@0celo7 lol, I can't believe you wrote all of that out!

0celo7

@BenNiehoff It wasn't that bad. I didn't make any mistakes

Everything worked on the first try

you should read it

Ben Niehoff

well, hopefully the internet points you get should be worth it :)

0celo7

@BenNiehoff I wanted to make sure Weinberg is consistent

Btw, there's a more general proof for the uniqueness of the Einstein tensor

Ben Niehoff

more general how?

0celo7

21:15

The einstein tensor is the only tensor that is linear in the second derivatives and has zero divergence

Ben Niehoff

you mean as being the only thing built out of curvatures which is divergence-free?

ah

0celo7

yeah, basically

It only works in four dimensions. In higher dimensions you get other terms

Ben Niehoff

you mean there are other divergence-free tensors?

are they symmetric and rank 2?

0celo7

I think so. Let me get out the book to double check before I say something wrong.

Ben Niehoff

that sounds suspicious to me

0celo7

21:20

@BenNiehoff The theorem is not formulated as I remembered. I'm thinking...

Lovelock's Theorem

Lovelock's Theorem is a theory of general relativity which says that from a local gravitational action which contains only second derivatives of the four-dimensional spacetime metric, then the only possible equations of motion are the Einstein field equations. The theorem was described by British physicist David Lovelock in 1971. == Statement == The only possible second-order Euler-Lagrange expression obtainable in a four-dimensional space from a scalar density of the form L = L ...

That's the statement in 4 dimensions

Ben Niehoff

ah, yes, I do know people who have worked on so-called "Lovelock gravity"

where they add extra terms

0celo7

the proof is extremely hard btw

it uses jet bundles

@BenNiehoff apparently, yes

arxiv.org/abs/1005.2386

I might make understanding this paper a long-term goal

@BenNiehoff In any case, I have a much better description of the Riemann tensor if you want to hear it

Ben Niehoff

ok, sure

0celo7

@BenNiehoff How well versed in differential geometry are you? Do you know what a local isometry is?

Ben Niehoff

I'm pretty well versed

0celo7

21:26

Do you know the Frobenius theorem?

Ben Niehoff

vaguely...it has to do with whether first-order systems are integrable?

0celo7

Yeah.

Let me find the statement

Ben Niehoff

or whether a spray forms surfaces, etc.

what's your description of Riemann?

0celo7

@BenNiehoff The Riemann tensor being zero in an open set is a necessary and sufficient condition for the open set to be locally isometric to Euclidean/Minkowski space.

Ben Niehoff

yes, of course

it's the integrability condition to find coordinates in which the metric tensor is diag(1,1,1,1...)

0celo7

21:30

Yep

Ben Niehoff

was that all, though?

0celo7

There's an even better trick using Jacobi fields

@BenNiehoff Yeah

I figured if you were asking about Weinberg you wouldn't know any geometry, frankly.

You can use Jacobi fields to show that the exponential map is a local isometry when there's zero curvature.

Ben Niehoff

oh, I thought you were in the room when I was explaining that I'm a string theorist and I do differential geometry for a living :P

0celo7

@BenNiehoff I'm a mathematician.

Maybe I was here...I remember a string theorist being here.

Ben Niehoff

ah, maybe you didn't agree that what I do is differential geometry ;)

0celo7

21:32

:p

So is what I wrote on the main site what you're looking for?

Ben Niehoff

what I'm looking for?

I didn't ask the Weinberg question

0celo7

wait what

I need more sleep

Ben Niehoff

lol

0celo7

Ok then

Back to...hmm. I guess thermo homework.

Ben Niehoff

so, there was a question earlier about the meaning of the Einstein tensor, which got me thinking about easy ways to interpret the various curvatures

the scalar curvature is of course the deficit solid angle for geodesic balls

and the Ricci tensor measures how geodesic balls deform into ellipsoids

0celo7

21:35

Ricci controls the volume expansion of geodesic balls

Ben Niehoff

the Riemann tensor must control higher-order deformations of geodesic balls, like twisting, etc.

0celo7

Twisting? Which tensor appears in the Raychaudhuri equation?

Ben Niehoff

I don't remember, but GR gives you a deceptive idea of these things

because of the time/space split

so I want to think just in Euclidean signature

0celo7

We usually say Riemann controls geodesic spread because the Jacobi equation

Ben Niehoff

well yes

ah, I remember

0celo7

21:39

Remember?

Ben Niehoff

the Riemann tensor controls how the cross section of a geodesic ball deforms into an ellipse as you travel along a geodesic radius

which is another way of saying Jacobi fields of course

0celo7

that's what you said for Ricci curvature

My interpretation comes from $\sqrt g \sim 1-\frac{1}{6}R_{ij}x^ix^j$

Ben Niehoff

ah, yes

the Ricci tensor measures how the area of the cross section changes as you travel along a geodesic radius

whereas the Riemann tensor describes how the shape changes

0celo7

those are the kinds of things you hear, but I'm not sure anyone has written them down

Ben Niehoff

no, I don't think most people think about it often

0celo7

21:48

What do you mean by "cross section"

Ben Niehoff

take a geodesic ball centered at O. Draw a geodesic ray from O, and slice perpendicularly to the ray

0celo7

slice with what?

more geodesics?

Ben Niehoff

yeah, I know

what you really have to do is take the orthogonal subspace to your ray at O and parallel transport it

that's just the collection of Jacobi fields

I'm assuming the Levi-Civita connection, of course

so the orthogonal subspace will stay orthogonal :)

0celo7

that's the best connection

Ben Niehoff

eh, torsion is pretty neat, too

took me a long time to understand what it actually measures, since all the books just ignore it

and I actually had to use a metric-incompatible connection once :)

0celo7

21:55

I have absolutely no idea what it does

I understood it for an hour once

Then I forgot

Never seen it since

Ben Niehoff

so, take a look at the operator, built out of the Riemann tensor, that describes Jacobi fields: $R_{abcd} X^b X^d$

it's a symmetric matrix in ac

0celo7

Riemann tensor. We're working with LC?

Or general connection here

Ben Niehoff

assume LC for now

So, as a symmetric matrix, it can describe ellipsoidal deformations of the orthogonal subspace along geodesics

but it cannot describe twisting

torsion is precisely the thing that gives you twisting

0celo7

@BenNiehoff That's precisely what happens in Raychaudhuri's equation.

But with the signature, who knows if that's actually what's happening.

Ben Niehoff

Raychaudhuri's equation is on a Lorentzian manifold

so, he's only looking at a slice through the orthogonal subspace :)

0celo7

22:00

There's an $R_{abcd}X^bX^d$ term in it is what I'm saying

I've got to go

cya

Balarka Sen

22:14

@0celo7 Ugh what a horrible jumble of symbols

EternusVia

22:25

Hi everyone... anyone here understand Reynold's Transport Theorem really well?

loltospoon

Hello

Slereah

I wonder if there's physics papers for really obvious facts

Like who do you cite if you want to say "objects fall on earth"

(if they have sufficient density)

loltospoon

If I numerically solved the SEQ to get $x-y$ pairs for the wavefunction and I plot it and it looks like this, what might have gone wrong? The problem here is that my wavefunction goes below $y=0$. This is for the infinite square well, so my energies should always be above $y=0$, right?

0celo7

@BalarkaSen it's beautiful

@Slereah how much pressure is 2MPa

Slereah

22:45

About 20 atmosphere?

0celo7

@Slereah the saturation temperature is 212C

must be pretty high pressure.

Slereah

212 Coulombs isn't a temperature

0celo7

Celsius

485 K

Slereah

There is a specific K in unicode for Kelvins

fileformat.info/info/unicode/char/212a/index.htm

0celo7

...why

Slereah

22:53

KK

u can see the difference

0celo7

nope

Slereah

I'm guessing it's easier to parse for computers

0celo7

@Slereah have you been reading Munkres or nah

Slereah

not today, no

if u want me to do math every day you can hire me

I am available for a thesis

0celo7

in constructive set theory?

DanielSank

23:01

0celo7

What?

Slereah

@0celo7 as long as money is involved, sure

0celo7

will you compute scattering cross sections?

in QCD

Slereah

I've done it before

though it was effective QCD

0celo7

5th order?

Slereah

23:07

Nah, boring old first order

in the $O(N)$ sigma model

For $N = 2$ and $N = 3$

I don't even know if fifth order for the sigma model would be interesting

There's some assumptions to reduce QCD to the sigma model, i'm not sure the results will be that great at 5th order

0celo7

@Slereah should I answer physics.stackexchange.com/questions/315248/… ?

Slereah

Knock yourself out

0celo7

It would be pretty horrible

I don't know how to vary the full Riemann tensor

squared!

@Slereah The issue is that its variational derivative is a total divergence, right?

That's what we mean by "doesn't contribute to the EoM"?

Slereah

23:23

That is usually the case, yes

0celo7

Straumann says the derivative is $$(RR_{ab}-2R_{ac}R^c{}_b-R^{cd}R_{acbd}+2R_a{}^{cde}R_{bcde})-\frac{1}{4}g_{ab}‌(R_{cdef}R^{cdef}-4R_{cd}R^{cd}+R^2)$$

lol

0celo7

23:38

@dmckee What does "enthalpy out" mean?

I don't know what $h_{out}=u_{out}+(Pv)_{out}$ is supposed to mean

specifically the $(Pv)_{out}$ piece

dmckee

Enthalpy is energy that is available for use in a constant pressure environment. Enthalpy out is enthalpy removed from the system. It is useful in constant pressure contexts.

Because the system is held at constant pressure some energy added won't go to raising the temperature but (generally) to increasing the volume occupied by the system. As energy flows out some will come from the work the environment does on the system to decrease the volume occupied.

So $(Pv)_out$ is a measure of work involved in changing the volume occupied by the system, and at this point the matter of sign conventions rear's it's ugly head.

0celo7

yeah, but how do I compute the flow work?

I guess $P$ is not changing

but what about $v$?

I have water in a tank that's being emptied. I'm not sure if this is constant pressure

dmckee

@0celo7 $v$ will generally change, and you have to know enough about the system to find that.

0celo7

For all I know it explodes when the valve is opened

@dmckee then...how do I evaluate it?

dmckee

@0celo7 If it is exposed to atmosphere than it is under (at least roughly) constant pressure.

0celo7

23:45

@dmckee Must be one hell of an atmosphere. The water is at 200 celsius and is saturated

the tables tell me that means the pressure is 1555kPa

dmckee

@0celo7 You have to know enough about the system.

0celo7

@dmckee could you please be more specific?

dmckee

No. You have to figure out the volume change from the specifics of the problem you are working.

0celo7

volume change?

I don't know what this symbol means!

What is $(Pv)_{out}$?

dmckee

For that matter you have to show that the pressure is (or is not) constant from the specifics of the problem.

$P$ is the pressure.

0celo7

23:47

yes

dmckee

$v = V/N$ is the volume per particle. Or they might mean $v = V/n$ is the molar volume.

0celo7

so is $v$ the total volume that has left?

@dmckee it's volume per unit mass

dmckee

Usually it would be the current volume of the system.

@0celo7 OK. Good.

0celo7

@dmckee at the final state?

that doesn't make much sense

we're taking $h_{out}$ out of the system

shouldn't it depend on what left?

dmckee

@0celo7 Slow down. Why are they interested in $Pv$ in the first place?

Answer ... because it is related to work.

0celo7

23:49

@dmckee The first law gave me $$Q_{in}=\frac{m_1}{2}(h_{out}-u_1)$$

I need to calculate $h_{out}$ somehow

dmckee

or because it is involved in the Legendre transform between internal energy and enthalpy.

0celo7

@dmckee I suspect this is not constant specific volume

so how do I evaluate $v_{out}$?

DanielSank

@dmckee Legendre transforms is on the list of things I need to take a day to understand for real.

Transcript for