The h Bar

Slereah

20:00

which means you'll get a circle

or something like that

Try to fix one of the coordinate and see which conic section time traces

bolbteppa

I'm thinking if you look at the metric in each case in some coordinates it'll explain it, but he just said it as if it's obvious from looking at those

Slereah

You can look up some (A)dS introductions for more details

0celo7

@Semiclassical sorry was in writing mode

had to compile a reference list

Semiclassical

mmkay

0celo7

@Semiclassical so the point is that you have $L^2$ functions whose distributional derivatives are also in $L^2$

Semiclassical

20:08

and that's enough to ensure that integration by parts works in the expected way

0celo7

this means that the derivatives are (i) functions (ii) don't grow too rapidly at infinity (iii) don't have deep singularities like $1/x$

@Semiclassical Yeah, the notation $$\int f'\varphi=-\int f\varphi'$$ is sensible for $f\in H^1$ and $\varphi\in C^1_c$

Semiclassical

so we've excluded stuff like the step function, which would be a function but whose derivatives wouldn't be.

0celo7

And then you can give it a norm by $||f||_{H^1}=||f||_{L^2}+||f'||_{L^2}$

@Semiclassical yeah

Slereah

You can define the step function as an integral like that

But its derivative won't be

Semiclassical

i get the idea, i suppose: if i'm doing quantum mechanics, i usually take for granted that the functions are nice enough to do integration by parts

one question: i've seen Sobolev stuff in one source, and it refers to (for instance) a "+1-index Sobolev space" @0celo7

how does that fit into this?

0celo7

20:46

@Semiclassical The "Sobolev index" could refer to the number $s$ in $H^s(\Omega)$

Semiclassical

probably

0celo7

it's basically the number of derivatives you allow

Semiclassical

hmm. so when you said

0celo7

for $s\in\Bbb N$ it's rather obvious what it means

Semiclassical

39 mins ago, by 0celo7

@Semiclassical so the point is that you have $L^2$ functions whose distributional derivatives are also in $L^2$

0celo7

20:47

for other $s$ it's more tricky

Semiclassical

the plural there was on functions, not derivatives

0celo7

@Semiclassical I meant only the first derivative

Semiclassical

kk

0celo7

But for $\Bbb R^n$ you have $n$ first derivatives

Semiclassical

right

I was seeing reference to index $s=+1-\epsilon$, so presumably that requires something more subtle.

0celo7

20:49

@Semiclassical Yeah you can define those spaces via Fourier transform, interpolation, or Riesz potentials

Semiclassical

fun.

0celo7

see arxiv.org/abs/1104.4345

Semiclassical

the source i have in mind was this one, btw: www-users.math.umn.edu/~garrett/m/v/pseudo-cuspforms.pdf

the automorphic stuff is outside my knowledge, but it tells a story about a (failed) proof of the Riemann hypothesis

0celo7

I see

the negative order Sobolev spaces are often the duals of the positive ones, btw

Semiclassical

i wondered.

0celo7

20:52

by "often" I mean "it depends on the underlying space"

For example, it's true for $\Bbb R^n$ and for compact manifolds without boundary.

Semiclassical

that source contains the following sentence: "Indeed, the Dirac delta on a two-dimensional manifold is in Sobolev spaces with index −1 − ε for all ε > 0."

0celo7

For anything with a boundary, the dual depends on boundary functionals.

Emilio Pisanty

@0celo7 that's not what that word means

(=P)

0celo7

@Semiclassical I read that which prompted my comment.

Semiclassical

ahhh

0celo7

20:53

Negative order spaces are not function spaces!

Semiclassical

lol

0celo7

They are true distribution spaces

@Slereah get link.springer.com/book/10.1007%2F978-981-287-257-9

Slereah

I already got it

0celo7

@Slereah is it interesting?

Slereah

Depends on your interest

it has some interesting tidbits on non-Hausdorff manifolds, but not that much

most of it is about non-paracompact manifolds

0celo7

20:59

yikes

no partition of unity

how is that supposed to work?

Slereah

I didn't really read those parts, so who knows

Non-paracompact manifolds aren't terribly interesting really

there isn't really a lot of interesting continuous functions on non-paracompact manifolds anyway

pZombie

hi

Slereah

Oh no it's a philosophical zombie

pZombie

Someone is observing some object placed on a far away planet (big planet with a lot of gravity). What can we say about the instantaneous reference frame that object is in?

If i was observing an object in empty space accelerating, i could think of it switching inertial reference frames continuously, given the object was infinitesimal small.

Is there any connection between those two cases?

Cows

21:18

today was a humbling one. I had my balls handed to me by an argentinian back-end developer. He dumped some code on me and i had to live code et al. Not fun. At any rate after making me feel like a superior donkey he remotely took over my pc and configured the environment. Now I have to do the rest and some mods. Sexy times. . .

Jack Of Blades

Hi folks, is this the right place to ask a question about trajectories?

Or is there antoher room specifically for that? I didn't find one.

Semiclassical

Don't ask to ask, just ask

Cows

At any rate going to starbucks for a bit to hang out and build confidence again, then will be slapping code all night like a champ

Jack Of Blades

Alright, well I have to calculate the necessary launching speed from a cannon of a projectile to hit a moving target when it passes infront. The distance to the point is known, the necessary flight time is known, the angle is known, the elevation of the cannon is known.

pZombie

cows coded any interesting recently?

anything*

Jack Of Blades

21:21

I'm not sure what to do, it seems like any formula I've googled demands a velocity from the start, without which I can't uhh.. find the velocity by "reverse-engineering it"

pZombie

Jack of Blades, does the target stay in front or does it move?

Semiclassical

that's the sort of question i really don't want to speculate without a picture

Jack Of Blades

It moves, but what's important for the ball is to hit a certain spot at a certain time. It's about hitting a windshield with a tennisball of some sort.

The cannon won't move sideways, it's just fixed at a certain angle.

Nor will it turn around.

And it points across the road.

Anonymous

You just need $\vec{r(t)}_{\text{projectile}}=\vec{r(t)}_{\text{canon}}$ at a certain time $t$. Google for "trajectory of projectile"

Anonymous

hyperphysics.phy-astr.gsu.edu/hbase/traj.html

Jack Of Blades

21:31

Yes, I saw that earlier. I had my old physics textbook up from high school, but I didn't quite understand how to combine the various formulas. I would try this one for example: x = v0 * t * cos( a ), x being the distance to the area it should hit on a flat field, and then I remade it to this: x / (t * cos( a )) = v0
But this gave me irrational results when I entered the known values in there.

pZombie

x = v0 * t * cos(a) , doesn't x here just get larger over time?

Anonymous

This is a good start: physics.ox.ac.uk/teach/FLAP/FLAP/P2.2.pdf to learn projectile motion

pZombie

that seems to be the distance the projectile would travel over time without any gravity pulling on it

hence, it would never hit the ground

Anonymous

@pZombie No

Anonymous

That's just the x-coordinate

Anonymous

21:35

It hits the ground when y coordinate becomes 0

pZombie

right

but that is not the formula to calculate when it hits the ground. Just the formula to calculate how far the object traveled in the x direction

so you need the formula now for how far the object traveled in the y direction

and then apply gravity to it, and see when y = 0

Jack Of Blades

Sorry for the silence, I'm going through the pdf

pZombie

except, now you are looking for v0... and you have y=0 or whatever height the object is at you are hitting

and you also have x i suppose

Anonymous

@JackOfBlades It's okay. It will take some time to digest that pdf

Anonymous

But after reading it you should be able to solve it

Jack Of Blades

21:42

Yes. And I should maybe mention that the distance the car needs to travel and its speed are calculated and then send a time to the cannon to say "you have to hit that particular spot when this current moment + that given time arrives".

pZombie

Jack Of Blades, the book you have from high school, does it have a formula for the y component of the trajectory?

Anonymous

@JackOfBlades Yeah, you'll need 3D coordinate geometry to deal with that.

Jack Of Blades

Yes, let me see..

Anonymous

@pZombie Lol. If it doesn't it doesn't contain that formula it doesn't even deserve to be called a physics book =P

Jack Of Blades

y = yo + vyo * t - 0.5 * g * t^2

pZombie

21:46

there you go

you have y and yo, t and g. And you are looking for v

Jack Of Blades

I'm still going through the link though, not on the clear with what I do with the two of these now. The velocities in the two aren't the same.

vyo is how fast it ascends if I understood it correctly

pZombie

y should be the height of the projectile depending on how long it flew. Hence depending on t

Anonymous

It's not trivial. You need z-coordinate too.

pZombie

yo if i understand correctly, should be the height of the cannon, which would be 0 if we assume the cannon to be at zero level.

Jack Of Blades

yes

pZombie

21:50

so just assume the cannon is at yo=0 to make it easier

then later you can work on the more complicated case :D

Jack Of Blades

I'll give it a go when I finish reading :P thanks

0celo7

@Blue I think most physics books don't have that formula

Anonymous

@0celo7 What type of physics books do you read...

Anonymous

Even the worst school book I read had it

Semiclassical

most intro level physics books will definitely have it

0celo7

21:53

@Blue General relativity and quantum mechanics, mostly.

pZombie

lol

Anonymous

@0celo7 ........

Semiclassical

i wondered if you were being that obtuse.

Anonymous

Projectile of electrons XD

Anonymous

Or projectile of black holes

Anonymous

21:54

(That's nonsense though)

pZombie

I was wondering about the photons trapped in a weightless box with ideal mirrors thought experiment.

When you push the box with the photons inside, doesn't the energy of the photons increase?

hence the box gaining more rest mass?

0celo7

@Semiclassical If you ever have to wonder if I'm being obtuse, don't.

pZombie

ok, that question didn't work either

how about warp drives then?

the mini version

Slereah

What about warp drives

pZombie

i have been told and read that the physics on which warp drives are based on are sound, except that matter with negative mass might be a problem...

but we have casimir cavities which are supposed to be negative energy volumes of space

so cannot we build mini warp drives with casimir cavities just to check if it would allow us to send information back in time?

Anonymous

22:07

Plenty of pop-sci articles on that already I suppose

pZombie

a warp drive certainly can be used as a time machine, if SR is correct

Anonymous

@pZombie Proof™?

0celo7

you're not allowed to say that

pZombie

Blue - it's in the warp drive's specs. Aren't those things supposed to be able to send matter to a space time location faster than light would travel to the same space time location?

Blue - the rest follows from SR directly if you check the math

Basically, accelerate to let's say v=0.5c relative to Alice on earth. Use your warp drive to send something at FTL towards a given space time location. For you it is just FTL, but for Alice it would be even faster... a distance done in negative time

turn around, and repeat the process into the other direction and voila, you arrived before you left

at least that is what SR tells us

now without thinking this fully through, some might be tempted to say that you do not really travel through spacetime when using a warp drive, but it does not really matter. What matters is the spacetime location you arrive at

ACuriousMind

@Blue See e.g. this answer for how FTL and causality/time machines are related

Anonymous

22:20

@ACuriousMind I get that. I was wondering about the physical plausibility. Also, I think pZombie needs GR for proof and not SR as SR doesn't allow faster than light travel (afaik)

pZombie

Blue, no, SR is enough

Blue - basically, the very short answer in pictures would be this imgur.com/a/Qs1RK

if warp drives are possible, then what you see in this image would also be possible

and that is according to SR

you arrive 8 seconds before you left in this case

ACuriousMind

@Blue GR doesn't allow faster than light travel either but you need to be more careful about what "faster than light" means in GR ;)

0celo7

$v>1$

Balarka Sen

I left 8 seconds before I arrived. How about that?

Anonymous

@ACuriousMind Yeah, I did read that "Apparent FTL is not excluded by general relativity, however, any apparent FTL physical plausibility is speculative" on Wiki

0celo7

22:22

@ACuriousMind Can you disclose if SkullPatrol is still with us?

Balarka Sen

Riddle me that general relativists

ACuriousMind

@Blue Sure it's speculative, no one has built a warp drive

pZombie

we don't know that for sure

Anonymous

@BalarkaSen You don't need GR for that. You left and you arrived again after 8 secs :P

pZombie

it seems to only be an engineering problem to build a mini version of one, if a casimir cavity counts as negative energy region

ACuriousMind

22:24

@0celo7 What's there to disclose? You can look at his user profile yourself to see whether he's suspended or not (he's not).

Balarka Sen

the skull has a lot accounts

0celo7

^

@ACuriousMind He might be using a sock

Balarka Sen

i don't know why 0celo7 wants that information tho

Semiclassical

If no one can conceive how to do it, it's not an engineering problem.

lılostafa

@ACuriousMind Hi. (Finally could find you)

Can you help with the question I sent yesterday?

pZombie

22:26

Semiclassi - I think there was a recent article about NASA planning to build a mini version of the warp drive

lılostafa

yesterday, by lılostafa

Other components of $F$ are zero.

ACuriousMind

@0celo7 He might, yes. So?

0celo7

@ACuriousMind Can you tell me if he is?

Anonymous

@0celo7 Why are you stalking skull?

ACuriousMind

@0celo7 No, I can't share information gained through moderator-exclusive tools.

Semiclassical

22:27

@pZombie [citation-needed]

0celo7

@Blue I'm not stalking anyone except for you

@ACuriousMind Fine

ACuriousMind

@lılostafa Ah. I don't really understand what's going on there. Normally, the magnetic field (which I assume is $B$) is related to the field strength as $F_{ij} = \epsilon_{ijk} B^k$ (the sign and the factor of two being normalization issues). I have no idea what the $\lvert B\rvert^3$ factor is doing there.

0celo7

@ACuriousMind How do I say Euclidean properly?

ACuriousMind

Oh dear

I think I remember a lengthy conversation also involving Danu about that at the end of which we concluded that the English standard pronounciation is probably closer to the original Greek than the German one.

lılostafa

@ACuriousMind This is a Berry phase calculation for a simple two level system (spin) in a magnetic field. The magnetic field $B$ is the parameter here. We're calculating $A$ (Berry connection) and $F$ (Berry curvature) in the space of the magnetic fields.

Here: damtp.cam.ac.uk/user/tong/qhe/qhe.pdf#page=35

ACuriousMind

22:35

Ahhh, it's not the field strength

Okay then

Semiclassical

Berry phase

how I wish I understood thee

lılostafa

It should be a conversion from spherical to cartesian coordinates but I don't know how he gets a $\frac{\mathbf B}{2|\mathbf B|^3}$ (cartesian) from $\sin \theta$ (in spherical coords)!

0celo7

@Semiclassical it's just bundle crap. Not worth it

ACuriousMind

@lılostafa I'm not even sure I understand what he means by "Cartesian coordinates" here

0celo7

@ACuriousMind Is that a sign you've become too abstract?

ACuriousMind

22:43

I mean, the space of states is just a sphere, the Bloch sphere. There are no "Cartesian coordinates" on a sphere

0celo7

You're so abstract you don't even know what Cartesian coordinates are :/

ACuriousMind

@0celo7 No, it means I don't know what "Cartesian coordinates" are supposed to mean when my space is a 2-sphere, not $\mathbb{R}^n$.

pZombie

The only reason we need GR is because of gravity or mass/energy affecting spacetime itself if you want, right? SR can deal with any acceleration just fine if we neglect the effects mass or energy have on spacetime, right?

Semiclassical

@ACuriousMind maybe it's meant in the sense of 3D as the ambient space? kinda silly, though

0celo7

@ACuriousMind take out a point and stereographically project

lılostafa

22:46

@ACuriousMind Doesn't he mean to treat $(B_x,B_y,B_z)$ as $(x,y,z)$?

and $|B|$ as $r$ in spherical coordinates

pZombie

but if one can view an accelerating spaceship, which SR can fully account for its behavior, as a spaceship being on the surface of a planet according to the equivalence principle, then why shouldn't SR be able to deal with a spaceship on a planet?

0celo7

@lılostafa does ij have to be polar coordinates?

pZombie

not that i have figured out how, but there has to be a way

Slereah

@DanielSank Is it possible to apply the path integral formalism to quantum computing

0celo7

@lılostafa I mean Cartesian

lılostafa

22:49

@0celo7 It seems Cartesian

@0celo7 Yes

Semiclassical

@Slereah it seems like it should certainly be possible; the question is whether it's useful.

(and I dunno)

Slereah

Well yes, that is what I'm asking

I am wondering if in the case of quantum computing it might reduce to a sum

at least within some limit

Semiclassical

on this note

one thing I'm trying to do in Mathematica lately is time-dependent Schrodinger stuff

all 1D, because I'd prefer not to get a headache

Slereah

aka the Schrodinger equation

Semiclassical

but whenever I try to use NDSolve for that purpose i just get junk :/

Slereah

22:52

It is pretty hard to get a good answer from Mathematica

Did you add assumptions?

pZombie

if you cannot simulate it, you either didn't understand it or it is not physics

Slereah

Usually it's best to add assumptions to Mathematica for solving differential equations

Semiclassical

well, NDSolve doesn't use assumptions per se

but I presumably should be doing something with Method

Slereah

No, but you can add them

Or can you?

I dunno

Semiclassical

Assumptions is for simplifying symbolic quantities.

I don't doubt there's something smarter I should do with NDSolve

but I think it's to do with Method and such

Anonymous

22:54

@pZombie That's too strong a statement. Mathematica does have bugs

pZombie

Blue - i wasn't really referring to the Mathematica issue, but rather made a general statement

@Blue i was expecting some know it all to come forward and tell me that i am wrong

ACuriousMind

@lılostafa Ah, yes, I managed to confuse myself. Okay, so we have $\sin(\theta) \mathrm{d}\theta \wedge \mathrm{d}\phi$ and need to convert that into Cartesian coordinates. So plugging in $\theta = \arcsin(z/r)$, $\phi = \arctan(y/x)$... (I'll post my result when I have it)

pZombie

@Blue So in general, would you agree that if one cannot simulate something in a computer, he either did not understand the Physics fully, or it is not Physical at all?

Semiclassical

depends on what one means by 'cannot simulate it'

"hasn't figured out how to simulate it" presumably doesn't count

pZombie

@Semiclassi - yes, that wouldn't count if it is just lack of programming skills

lılostafa

22:59

@ACuriousMind $\arccos (z/r)$?

Semiclassical

i also wouldn't count computational resources as a factor, e.g. "if I had X more disk space then I could compute it"

pZombie

i made a spacetime twin paradox simulation btw. You can watch/download it here youtube.com/watch?v=up6AoUShGBE

Semiclassical

and similarly time

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