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RewCie

03:27

Morning

More Anonymous

06:07

@ACuriousMind I thought about your comments to my recent question that the term is only called "pressure" ... Doesnt that $P$ obey Naiver Stokes equations? It has to be pressure we all think about

ACuriousMind

11:08

@MoreAnonymous No, the Navier-Stokes equations are the non-relativistic equations for a viscous Newtonian fluid. But "pressure" is more general than that - we also call the spatial $T^{ii}$ components of the stress-energy tensor pressure, but this doesn't need to describe such a situation at all (in particular it's usually relativistic and the generic continuity equation is just $\nabla_\mu T^{\mu\nu} = 0$).

pressure is just "force per area"

how you compute that depends on your specific situation, your question is like complaining that "the force" is computed differently for a spring than for a charged particle in an EM field

More Anonymous

12:03

Your right I should have said the Euler equations ... So this is where Im stuck. To summarize:

1. We take a bunch stress energy tensors of discrete particles and we know they since we do not include any interaction term they cannot behave like pressure (force per unit area)
2. However when I take the divergence of the corresponding stress energy tensor and then the non-relativistic limit I get Euler's perfect fluid equation. The P in that equation is definitely the pressure we all know.

Also is it me or do physicists play a horrible misleading semantics game somtimes :/

ACuriousMind

@MoreAnonymous what you mean is that it doesn't behave like a fluid. Again, the components $T^{ii}$ are normal stress/pressure regardless of the physical situation. As an example, the dust solutions of general relativity have vanishing pressure because the "dust" particles only interact gravitationally . they don't act like you'd expect "real" particles to act, but that doesn't mean the term pressure doesn't apply.

the problem seems to be that you conflate the notion of a real-world fluid with hydrodynamic pressure with the abstract notion of pressure as momentum flux/force across surfaces

More Anonymous

@ACuriousMind But it has to behave like a fluid right if it obeys eulers equations of a perfect fluid?

ACuriousMind

the dust solutions are called "perfect fluids", I don't think that's the term you want to use there

More Anonymous

@ACuriousMind Ummm ... Ideal fluid?

ACuriousMind

probably

but I'm not sure what the question is

Euler's equations are equations for the motion of fluids

you use them if you want to describe a fluid

so it's a tautology to say that things that obey the Euler equations "behave like a fluid" - we designed these equations to models the stuff we call "fluid"!

More Anonymous

12:16

Yes ... And they have a P term which is ubiquitously means the newtonian pressure right?

ACuriousMind

your semantic problem is that you seem to think that pressure and fluids are intrinsically linked

More Anonymous

@ACuriousMind But when you derive Euler equation thats what P means

ACuriousMind

@MoreAnonymous no, again, $p$ is just momentum flux across (imaginary) surfaces

just because pressure in the context of fluids is the most common use of the term "pressure" doesn't mean you can't use it outside of that context

I can shove my fist against the wall and then I'm exerting pressure on the wall (total force divided by contact area)

there's no fluids involved there, but still the term pressure applies

More Anonymous

Sure ...

Okay how about this

If I put the stress energy tensor from Caroll's book in the Penrose box. Will all the walls after sometime feel uniform pressure?

ACuriousMind

I don't know what it means to put a stress-energy tensor in a box

the stress-energy tensor is a description of a particular physical situation - if you change the physical situation, you get a different stress-energy tensor

More Anonymous

12:22

I mean put the system with a stress energy tensor mentioned in Caroll's book

ACuriousMind

depends on what the system is

More Anonymous

@ACuriousMind Why don't you have the equations of motion? To see how the fluid will evolve?

ACuriousMind

the stress-energy tensor doesn't tell you whether the components of its system can bounce off the walls or not

and whether they can do that or not is the crucial question for what happens when you put it into a box

More Anonymous

@ACuriousMind True you can add that as a boundary condition

ACuriousMind

it doesn't even tell you whether the components of its system can bounce off each other, which is the other thing that's relevant for the Penrose box

the stress-energy tensor is not the systsem

it's like asking "What happens if I put 100 MeV into this room?"

it's a nonsensical question - are these 100 MeV heat, particles, something else?

More Anonymous

12:26

The one Caroll derives in his book assumes discrete particles which can go through each other. Are you saying I can have a collisional fluid and get the same stress energy tensor? I concede I am assuming there is some uniqueness theorem

ACuriousMind

the Carroll example is a perfect fluid, so sure, if you put that into a box, it will behave exactly like light

More Anonymous

@ACuriousMind So it will touch all the walls?

ACuriousMind

look

perfect fluids don't exist

there are no real-world particles that interact purely gravitationally

photons don't bounce of each other either, but they are also massless so they don't really interact gravitationally

More Anonymous

Yes and for photons we know they wont touch all the walls ... But photons dont obey Eulers equations. Right? (In fact I dont think there is a non relativistic limit of a photon)

ACuriousMind

if we imagine some particles that do not bounce off each other and only off the walls, then they are for the purposes of the Penrose box the same as light and won't reach everywhere, but what's the point of this exercise?

Euler equations or any other hydrodynamic equations are completely irrelevant here - they don't model the interaction with the walls of something the fluid flows through. Why would you use hydrodynamics to figure out what happens to the particles in a box when for photons it also sufficed to think about where the individual particles will bounce?

you didn't go and do relativistic dynamics for a photon gas there, either

More Anonymous

12:56

So if you are so concerned about the wall interaction. I can construct a universe where the boundary behaves like a mobius strip and any molecule that touch the wall reappears on the other side with the parity changed only (kinda similar to pacman reappearing) ...

In this case one would conclude it is impossible for the momentum flux (pressure) to be distributed (such that it is non-zero everywhere). However, I would be very surprised if simulations of an ideal fluid (obeying eulers equation) did not reach all spots.

Slereah

hello

More Anonymous

13:15

@Slereah yo!

rb3652

14:42

Hi everyone

I have a question about the Principle of Least Action

Light takes the path of least time (or action)

That's why it goes in a straight line in air, refracts in water, and reflects in equal angles.

Now, I was researching the Brachistochrone problem, which asks "What is the shortest-time path between two points in a gravitational field?"

The answer, discovered by Johann Bernoulli, is a cycloid, which he found by ingeniously exploiting the fact that light will naturally take the least-time path.

Using Conservation of Energy, we have $v=\sqrt{2gh}$

Since $\frac{\sin{\theta}}{v_1} = c$, a constant

We have $\frac{\sin{\theta}}{\sqrt{y_1}}$, which is the differential equation for a Cycloid

Now, my question is, what if we're in an arbitrary force field?

In other words, what if instead of $n\propto \frac{1}{\sqrt{y}}$, we had $n\propto \frac{1}{f(y)}$, where $f(y)$ is some general function in terms of $y$?

If anyone could suggest any ideas, that would be much appreciated.

Sorry to ping @DanielUnderwood, but any ideas?

ACuriousMind

@rb3652 please don't ping random people

rb3652

My apologies

ACuriousMind

@rb3652 I think the only answer one can give to that is "it comes arbitarily complicated"

rb3652

14:58

I'm sure an analytical solution is next to impossible, but would a computational/numerical "solution" be possible?

ACuriousMind

quick googling shows there are plenty of articles that consider various generalizations of the brachistochrone problem, see arxiv.org/abs/math-ph/0612052v2, doi.org/10.1016/S0020-7462%2897%2900026-7, sciencedirect.com/science/article/pii/S0895717711004213

rb3652

Yes, I've read many of them.

However, they are different generalizations, in which the boundary conditions are varied, and such.

Some add friction, some consider a "terrestrial" brachistochrone problem.

Slereah

I think Synge talked of weird boring problems like that

Like the ballistic suicide problem

"The usual problem of ballistics is to aim a projectile so that it hits someone else. Here we consider the ballistic suicide problem : the projectile is to hit the projector himself!
However perverse such a problem may be sociologically, it is a neat problem in relativity, because there are only two observations and both are made by the same observer. Moreover it forces us to realize that although the trajectory of a projectile fired straight upward seems sharply curved at the top, from a spacetime standpoint it is as straight as possible (geodesic). "

rb3652

Just thinking out loud here -- would it be possible to create in a program which numerically solves for the least-time path in any conservative field?

That might involve calculating the change in angle of the light ray at every interface, using Snell's Law (?)

Slereah

@rb3652 I mean sure, algorithms exist

You could just bruteforce every possibility on the mesh and that would work!

rb3652

15:02

Brute-force every possibility? Could you clarify?

ACuriousMind

@rb3652 I'm not sure what the question is, then - for a conservative force you should be able to write down an action functional and derive E-L equations, no?

Slereah

It's the first dumb idea that comes to mind as an algorithm for least action problems

ACuriousMind

once you have the E-L equations you can apply whatever numerical method for solving differential equations you want

Slereah

Just make the program generate all the paths and compute the action for each

Then sort the list

simple enough to program if long

but it shows it can exist at least

rb3652

@Slereah That's really interesting -- it gives an approximate solution, if not an exact one.

Slereah

15:03

well that is what you can expect with numerical solutions

rb3652

@ACuriousMind But will solving EL give me the path itself?

Slereah

I'm sure there are much more elaborate algorithms for it though

ACuriousMind

@rb3652 Hm? If you have $(x(t), y(t))$ you can always (locally) get $x(y)$ as "the path"

Slereah

A useful trick in computer science to check if an algorithm exists is to ask "What about bruteforcing every possible solution"

rb3652

Well, this is certainly food for thought. Thanks everyone @ACuriousMind @Slereah for the ideas. If it works out, perhaps I'll come back to report my findings.

Slereah

15:05

It's not usually the best algorithm, but it's good to show existence

bolbteppa

sites.millersville.edu/rumble/StudentProjects/Gemmer/… similar to the first link above looks good too

ACuriousMind

but note also that the difference between "an algorithm exists" and "it is feasible to implement it and it will find a solution before the end of the universe" is often annoyingly large

Slereah

@ACuriousMind I never worry

Leave the computing to the computer

I can wait $10^{500}$ years

Slereah

15:24

$10^{500}$ years of computation to work out that the line was the shortest path

Loong

15:35

Deep Thought only needed seven and a half million years of calculation.

Slereah

You can make it arbitrarily long since you can increase the mesh size of your space

More paths to check

just checking that a straight line is the shortest path up to $10^{-30}$ meters

Nihar Karve

@Slereah 1 for each string theory vacuum, nice

Slereah

Maybe there's a secret path that we missed

I think "shortest path calculations of a space that has been triangulated" is essentially graph theory

what you want is graph algorithms

Regge calculus something something

Nihar Karve

16:06

-1

Q: Anyone interested in running some high-CPU models for a genetic engineer/biotech stock trader? $ no object

Hrm....where to start....I will likely be banned for something in this post - but nothing I say herein is untrue as I currently understand it and believe. My money is also real :D NVCR from 12$, CCXI from 4$, SAVA from 9$, ANVS from 10$ <3 But first! A somewhat non-specific introduction to we! WA...

:O

John Rennie

16:59

Only thirty five more black hole mergers detected. Come on guys, stop slacking! :-)

Slereah

17:27

0

Q: Seeking physicist with access to computational quantum simulator to work on questions about acceleration of spacetime expansion and wormholes. Ill pay

I have a question about physics relating to quantum-entanglement, maximum/minimum energy levels, distance, the accelerating expansion of the universe, and the recent plans for new gravity-wave detectors to be built. These questions will require expensive computer simulations to be run - which I...

quantum-mechanics simulations biophysics wormholes low-temperature-physics

is it ethical to take a crazy man's money

ACuriousMind

can't everyone just flag these questions instead of posting them here?

Slereah

Because it is more amusing

rb3652

18:29

I'm a bit confused. I've seen various ways to get the cycloid solution $( r(\theta - sin{\theta}), r(1 - \cos {\theta}))$, but I haven't seen anyone derive it from EL (or at least, not in a way I understood)

For example, one video showed the derivation from $\int{dt}=\int{\frac{ds}{v}}$

Wait a second, I think I got it. Let me try again.

glS

18:49

@Slereah these kinds of posts really make me wonder about the person behind them.

"(tl;dr - I am a high functioning physchopathic buddhist (little b) monk and occultist with multiple degrees from good universities who wants to pay possibly up to hundreds of thousands of dollars (gradually in milestone payments) to someone to run some simulations on angular momentum, the accelerating expansion of spacetime, and quantum entanglement)" lol

Slereah

Although I suspect that if he sends me money it's gonna be all in NFTs

glS

the guy is talking about how he self-operated on his finger "cutting to the tendon", and how the dark gods later told him it was a good idea. I'm not so sure you want to try to get money from them lol

Slereah

I mean he doesn't sound any crazier than Elon Musk

glS

eh, different kind of crazy though

at least Musk doesn't go around boasting about car accidents and bad things "happen" to people that were mean to him.

Slereah

He only goes on about evil AI

rb3652

19:02

I wrote quite a simple program, hoping it would give me the equations of a cycloid, but instead, there's a bug:

x   = dynamicsymbols('x')
t   = dynamicsymbols('t')
xd  = dynamicsymbols('x', 1)

L = math.sqrt((1+(xd)**2)/t)
f = diff(diff(L, xd), 't') - diff(L, x)
pprint(f)

**TypeError: can't convert expression to float**

rb3652

19:14

Can someone explain how exactly we go from the Lagrangian to EL to a Cycloid? I think if I better understand this process, I can generalize it to any $f(y)$

bolbteppa

19:32

Chapter one of the notes I sent above explains it

rb3652

19:59

@bolbteppa OK, I'll look into it.

Bohemian relativist

22:29

is the reason why $\int pdq$ is called abbreviated action because it's only part of the action - the whole action is $\int\mathcal{L}pdq=\int (pdq-Hdt)$?

the whole action is $\int\mathcal{L}dt=\int (pdq-Hdt)$

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