7:15 AM
@Skyler Okay, so I think the code isn't actually doing a lot: It just computes the lensing of an image (source at distance src) around a point xs,ys at distance lens with a fixed coefficient 25.0. If there were any physics in there, the 25.0 in the line where shift is computed would have to come from physical parameters

7:45 AM
@Skyler To properly compute photon deflection you need to evaluate an elliptic integral. That's not as bad as it sounds because there's a fairly simple algorithm for elliptic integrals based on the Arithmetic-Geometric mean, so it converges very quickly. But for trajectories that aren't too close to the photon sphere you can get quite good results using $\theta = \frac{72}{36r - 35}$, where $r$ is the distance of closest approach, in units where the Schwarzschild radius is 1.
I guess I should update that PhysIcs.SE post with my new elliptic integral code, which is much more accurate than my old Leapfrog integrator. OTOH, the new code only works on trajectories that escape, you can't use it for trajectories that cross the photon sphere and hit the event horizon.

8:32 AM
here's something thats been bugging me
:
The relative velocity $v(u,u')$ is given by:
$\gamma(u,u') v(u,u') = | u \rangle \langle u | u' \rangle$

To find there be $2$ other 4 velocities
$u + \delta$ and $u' - k$ which have the same relative velocity $v(u,u')$ . Then

$|u \rangle \langle u| u' \rangle = |u + \delta\rangle\langle u+\delta| u' - k \rangle$
Then
$| u + \delta \rangle \langle u + \delta|k \rangle = |u \rangle \langle \delta|u' \rangle + | \delta \rangle \langle \delta| u' \rangle + | \delta \rangle\langle u |u '\rangle$
How do I simplify this further?

once again, I find your notation completely impenetrable :P
what are all the kets defined by? why is velocity equal to a ket? what's going on here?

9:29 AM
Yes velocity is a key
*ket
Gamma is the usual gamma factor
I mean I'm thinking of it as a Matrix and then mapped it to bra ket notation
@ACuriousMind would u prefer the Matrix notation?

@MoreAnonymous While I find it odd to use bra-ket notation for something that's not quantum states, the problem is more that you're being inconsistent - if $v(u,u')$ is a vector, why do you not write $\lvert v(u,u')\rangle$?
also, where does the first formula come from - it implies that $v(u,u')$ is a multiple of $u$, which strikes me as false as soon as $u$ and $u'$ are not parallel

Equation 2.2

ah, no, I'm not reading a paper to understand your problem, you'll have to find someone else :P

Does a screenshot work?

10:06 AM
Also on a side note with regards to yesterdays discussion? @ACuriousMind what are your thoughts on Chomsky?
@ACuriousMind I was reading the paper and it seems to be a spatial projection

@MoreAnonymous What do you want me to comment on? The person? His politics? Manufacturing Consent?

@ACuriousMind His politics
And your agreement or disagreement with it? I found him a superb intellect personally

I don't know a lot of specifics of his beliefs, so I can't really comment

@ACuriousMind any thoughts on universal grammar or manufacturing consent?
@ACuriousMind I'm not overly familiar either btw

10:24 AM
I think manufacturing consent is a good analysis of how the capitalist system shapes opinions via mass media without anyone actively having to be nefarious, I'm less fond of calling this "propaganda" as I think this muddles the water in comparison to actual, intentional propaganda
I never really understood the point of universal grammar. Sure, there is probably something about human genetic makeup that has allowed us to develop language where other primate species didn't, but what basis do we have for asserting this means there are certain elements of grammar that are directly encoded there, and why does it matter?

10:36 AM
@ACuriousMind Personally I saw a continuity between politics and the existence of the human condition that universal grammar assumes. It enables chomsky to come on to these topics with a confidence I enjoy.

11:15 AM
@MoreAnonymous what does universal grammar have to do with politics or the human condition?

@ACuriousMind Well if universal grammar exists I think it be saying the biases in the human condition are so apparent that they reflect in the very language we speak

that seems to me to tacitly assume a rather strong variant of Sapir-Whorf

@ACuriousMind Havent heard of that

@MoreAnonymous The idea that worldviews and language features are related is the Sapir-Whorf hypothesis

@ACuriousMind I wouldn't say they related it's not like cause I speak hindi I welcome diversity (i doubt you were implying that)
But more like there are human biases
which philosophers don't always acknowledge

11:19 AM
I'm confused what this has to do with grammar
how exactly does e.g. a supposedly universal distinction between verbs and nouns reflect a human bias?

@ACuriousMind Consider the way you structure a sentence. Subject before object, etc

@MoreAnonymous but that's precisely not universal
word order varies a lot between languages

@ACuriousMind Yes but if I look at the distribution

with some languages having essentially free word order because of strong inflection

Some choices are very infrequently made
which seems to suggest a bias

11:21 AM
@MoreAnonymous but isn't universal grammar precisely about the claim that some language features are genetic, i.e. truly universal and unavoidable?
you're now arguing for a weak version of Sapir-Whorf, but I don't see what this has to do with the claim that there is grammar that is universal

@ACuriousMind yes. I don't think CHomsky's orginal formulation have held the test of time
But I think Chomsky believed it in this kind of fashion
@ACuriousMind Also if true because it was genetic an unavoidable that also prove the point of the human condition

@MoreAnonymous I think there are two claims here: 1. Language features can reflect/shape/influence thought (Sapir-Whorf) 2. There is a genetic part of human language faculty that encodes certain grammatical features.
The second claim is what I understand to be the point of "universal grammar"
the first claim is, in various versions, just something a lot of linguists discuss

So the way I think of it.

1. There exists a human condition (/bias)
2. These biases reflect in language
3. Universal Grammar is one model that validate 1 and 2

but what is the model?
what exactly is the additional claim beyond 1 and 2 that universal grammar makes in your understanding?

Well the idea that certain features of grammar have to exist (as u said urself previously)

11:27 AM
what does "have to exist" mean
isn't the idea here that some features of grammar are "hardcoded" in whatever our brains use to process language?

@ACuriousMind yes

that's the part I don't get - we could just as well assume "the human condition" shapes language without assuming this means grammatical features are directly encoded
what do we gain by assuming that certain grammar features are really intrinsic, and what sort of evidence/counterevidence could we possibly collect for that?

@ACuriousMind Ah which is why I said is one model. My point is universal grammar is verifiable.

it is?

@ACuriousMind Well thats why most linguists don't follow it anymore. Cause the evidence (in its original formulation) is not in its favor

11:31 AM
I mean, even if every language on earth clearly and obviously shared some grammatical feature, the explanation that they all come from a common ancestral language would work just as well as assuming that that feature must be genetically encoded, wouldn't it?

@ACuriousMind Umm ... yes but not all philosophers think that way

not sure what that means - if you agree with me there, then in what sense is universal grammar "verifiable"?

@ACuriousMind I think there were some languages found in the pre 2002 revision (I'm literally at the edge of my knowledge here) which did not agree with Chomsky's orginal formulation

that's not my point, really

@ACuriousMind It got disproved then no?
@ACuriousMind Ah wait just re-read this

11:35 AM
sure, it's obviously false if languages don't actually share universal features
but I'm saying that even if all languages shared universal features, this does in no way imply those features are due to genetics or anything else in our brains
you'd have to show me the "verb/noun distinguisher lobe" in the brain for me to believe that :P

@ACuriousMind If I just think of this as some kind of system. I think the claim is the phase space of all possible evolutions is governed by the phase space of universal grammar
given the initial language

now that's a bit of a tautological claim

possible evolutions = new languages

of course the space of all possible languages is the space of all languages humans can possibly speak :P

@ACuriousMind But they claim is the theory behind it which I'm by no means an expert in

11:39 AM
though, consider: Are programming languages languages in this context? Do humans "speak" C?

I was working on a joke like "are you guys talking about math?" and then we came to spaces of languages

@ACuriousMind ofcourse humans don't speak C

@MoreAnonymous but someone else can type a bunch of code and I can (sometimes :P) read it and understand what they meant and aimed to accomplish. We're certainly communicating via code

In the context of programming languages. Lets say I just type a bunch of words together in C and ask what are the odds this program will run? Lets say its y for 10 words. I would bet if we randomly created a language it would be say x. Now, I would be y < x because we structure our programming language on the basis useability
Interesting digression: en.wikipedia.org/wiki/….

that's just a difference in the strictness of grammar of natural and programming languages
I don't see it's an argument either for or against counting them as languages humans "speak"

11:43 AM
Now, in the context of programming we know this came from intelligent design. For language I suspect evolution?

both "intelligent" and "designed" are very generous words for some programming languages

@ACuriousMind lol XD

and they "evolve", too - not in any way similar to natural selection, but just look at how e.g. C++ has changed over the years
and many subsequent decisions are constrained by prior ones rather than being good intentional design

true but I don't see this as going against the point I made

I'm not sure what point you made - you're correct that it is probably harder to write a correct C program than say a correct English sentence
what does that have to do with whether programming languages are languages humans speak for the purpose of this discussion?
@Feynman_00 math is also a good candidate for another category of "language"
the way mathematicians use words to communicate ideas is often rather idiosyncratic :P

11:48 AM
my point was that if I randomly created a language lets call it B and typed 10 words in B then C++ is more likely to give me a non crashing output since C++ is structured to be helpful.

In the same sense is our language structured to be helpful? If so was it language evolution or genetics?

B is actually a precursor to C

lol
didnt know

@ACuriousMind The most universal we have I would argue, just after the italian gesture for "what"

@MoreAnonymous What does it mean to "randomly create a language"?
what's the sample space here?

@ACuriousMind oh no ... Sometimes I think your a mathematician in disguise
:P
@ACuriousMind I wanna all possible languages but I think we're gonna get into measure theory?

11:51 AM
people saying "random" without having an actual random selection process in mind is somewhat of a pet peeve of mine :P
@MoreAnonymous what are "all possible languages"? What defines a language?
(you can't say something like "a formal grammar" because natural languages aren't described by formal grammars)

hmm ... for spoken language its probably the finite subspace of audible words
?
Something like tha
that ... *inserts physics hand waving here
@ACuriousMind you should hang out with this guy :P

@MoreAnonymous "Languages are just jiggling words"

@Feynman_00 waves hand even more :P
I mean Im pretty sure its definable cause they manage to do it for Chaitains constant
This
In the computer science subfield of algorithmic information theory, a Chaitin constant (Chaitin omega number) or halting probability is a real number that, informally speaking, represents the probability that a randomly constructed program will halt. These numbers are formed from a construction due to Gregory Chaitin. Although there are infinitely many halting probabilities, one for each method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one. Because Ω depends on the program encoding used, it is sometimes called Chaitin's construction when not...

I made a GIF of RFP handwaving for these occasions

@MoreAnonymous "Although there are infinitely many halting probabilities, one for each method of encoding programs, it is common to use the letter Ω to refer to them as if there were only one. Because Ω depends on the program encoding used, it is sometimes called Chaitin's construction when not referring to any specific encoding."

11:58 AM
@ACuriousMind Claim all $\Omega <$ C++ version
*conjecture

the Wiki article makes exactly the point I make - there is no universal notion of what a language or an admissible program is, so there is no universal notion of how likely it is to produce valid programs

@ACuriousMind Dang :/
And here I thought I had some definite proof of the human condition

(also this is again about programming languages and so doesn't really have anything to say about human languages)

I mean if there was a human condition I'm sure we'd have a probability
@ACuriousMind I feel by now you're just beating a dead horse
:P

and thirdly, the inputs to this function are actually all already assumed to be valid, executable programs - the question is just whether they halt, not whether they are valid programs

12:00 PM
any1 here?

the ongoing discussion would indicate that that's the case :P

ok here it is. why we differentiate between static friction and friction while moving?

@ACuriousMind Can you prove that formally?

@Jam because it is an experimental fact that these two things are different?

so an object which is not moving needs an initial force greater than the static friction to start moving which makes sense but after that i need less force to keep it moving
am i right?

12:02 PM
yes, that's generally how it works

@ACuriousMind I think you might enjoy this proof btw

@Feynman_00 no, that's why I said "indicate" :P

though the title is a bit hyped

You have a point

physics are fun everytime i have a question the answer is "thats what is observed experimentally"
isnt there a reason? like maybe when you are on ther move there is less "contact"

12:05 PM
Friction is a phenomenological macroscopic description of microscopic interactions, there is nothing fundamental about friction force

@Jam I guess one could reason it like that. Like if surface A and B are in contact. Then when A is motion it hits more molecules of B and interacts more?

also another question. There is a classic example of an airplane with a pilot doing a circular motion and at the top of the circle since the pilot is upside now. We sum the force of gravity along with the Normal force .BUt my question is since gravity points down how does there exist a normal force . Opposite to what force??
@MoreAnonymous yeah i was thinking something like that

@Jam of course there's some argument of why this is so: Friction occurs because surfaces are rough on a microscopic level, and their protrusions interlock with each other. When you drag on something when it rests, you have to pull all these protrusions out of each other - but when it's moving, it doesn't have time for all of them to settle back down and interlock, so there's fewer obstructions to your drag

@Jam seatbelt?
Sounds like a riddle :P

@ACuriousMind of thank you!!!

12:09 PM
@Jam in the frame of the pilot, there's a centrifugal force pressing them into the seat at the top of the loop if they're fast enough

@ACuriousMind Oh yea I remember how horribly that one was taught. It doesn't surprise me it confuses students

ohhhh
so the seat is kinda the rope that keeps him form going to infinity

@Jam what? I was joking

in the frame of an outside observer, the plane accelerates downward at the top of the loop (acceleration in circular motion always points inward), and that's the centripetal force pressing the seat into the pilot, if you will

12:11 PM
Think of it like this:

$\vec F = m \vec a$

@Jam yes, exactly, this is just the same as what happens to a ball on a string

we know $\vec r = r \cos( \omega t) \hat i + r \sin( \omega t) \hat j$
Differentiate $\vec r$ twice and take the modulus

i think i got it
so the seat pulls to the centre haha
well actually is not the seat it is the engine of the airplane that gives all this force to have the circular motion but it comes down to that
ok next question would be then how do machines form a force that pulls to the centre in order to change direction. :P
i imagine friction would do the work on cars
and air resistance on airplanes?
ok ok thanks everything is in place

3 hours later…
3:43 PM
a mass tied to a strting in a table at to is given an initial velocity uo when the string is at full length so it starts doing circular motion.The table has friction$=-mbu^2$ how to calculate the tension
i use newton somehow and integrate and find the tension in temrs of time
its doesnt do a uniform accelaration
so in the direction of the centre will i have 2 forces the tension and friction? $F_N+F_r=mu^2/R$ ??

4:00 PM
"In GR there is no external time parameter. Coordinate time is a gauge variable which is not observable, and the physical variable measured by a clock is a nontrivial function of the gravitational field."
is the proper time the physical variable measured by a clock?

1 hour later…
5:24 PM
@Bohemianrelativist Isn't proper time just the line element (upto a factor of c)?

Just a weird fun fact: I logged in to SE and went to PhysSE homepage. 4 of the top 5 posts are modified by Qmechanic! Though the frequency is less, you'll still find noticable amount of edit by him/her for the rest of the questions too xD

@MoreAnonymous I think proper time is just a function of metric, which is characteristic of the gravitational field in GR.

@Earman Qmechanic is the most prolific editor by an order of magnitude
2
so that's not really a surprise

@Bohemianrelativist I thought if I put all spatial derivatives to 0 then upto a coordinate transformation I have $ds^2 = - \lambda dt^2$ where $\lambda$ is an arbitrary function. My point is in some sense proper time corresponds to the time experienced when one is in the his own frame

@Bohemianrelativist proper time is a quantity along a specific worldline
of course, along that worldline, it's a function of the metric
but it's not a function on spacetime

5:37 PM
Speaking of QMechanic, could someone more enlightened than myself help me understand their answer to this: physics.stackexchange.com/questions/645149/… (which is one of my questions from a long time ago)

I think I understand now, thank you! — Charlie Jun 12, 2021 at 22:01

I was misguided
A younger charlie
Understanding the time ordering operator has been a long-standing and upsetting topic to me. I am still hung up on the fact that even though we say that it takes symbols to operators, QMechanic has equated this to the expression involving the Heaviside functions

so what's the problem with that? $\mathcal{T}$ is a map from symbols to operators. The stuff inside $\mathcal{T}\{\dots\}$ is symbols, the stuff with the Heavisides is operators

I could imagine arriving at this conclusion: the time ordering operator is entirely fictitious and nothing written within it's brackets is expected to hold, because we are simply ignoring non-commuting objects, hence the "these are symbols not operators" statement
Ok I totally 100% get rewriting something like $T(\phi(x)\phi(y))=\Theta(x-y)\phi(x)\phi(y)+\Theta(y-x)\phi(y)\phi(x)$
This is fine - clearly, we must have one of these terms vanishing
What I find troubling, and the point that I was alluding to in my question which QMechanic answered, is that we appear to take advantage of the fact that everything commutes within the $\mathcal T\{...\}$ and manipulate expressions to get exponentials which we then Taylor expand
If I look at the example QMechanic gives, \begin{align}{\cal T}[e^{A(t_1)+B(t_2)}]~=~&{\cal T}[e^{A(t_1)}e^{B(t_2)}]\cr ~=~&\theta(t_1\!-\!t_2)e^{\hat{A}(t_1)}e^{\hat{B}(t_2)}\cr ~+~&\theta(t_2\!-\!t_1)e^{\hat{B}(t_2)}e^{\hat{A}(t_1)}.\end{align}
The BCH formula is necessary because $A$ and $B$ do not commute, and thus the exponentials can't just be pulled apart because otherwise terms involving the (non-zero) commutator are needed
However lets say $t_1<t_2$, so the first term vanishes, we arrive at:

$$\mathcal T[e^{A(t_1)+B(t_2)}]=e^{B(t_2)}e^{A(t_1)}$$
My problem is that we are equating an expression which is inside $\mathcal T\{\}$, where we ignore commutators, and thus nothing actually has to make mathematical sense, with a real expression

okay, I'm writing an answer :P
I'm not sure you're focusing on the correct part but you're right that "applying the time-ordering operator" as P&S do it doesn't make a lot of sense

5:56 PM
If I had to hazard a guess, I imagine what's causing my objection is fairly simple and a conceptual error rather than something mathematical I am missing
Hmm
@ACM If I may just ensure I am headed in the right direction on the answer you've given. Is (part of) the point you are making that "applying $\mathcal T$" would be better said "let's instead compute the original expression but involve $\mathcal T$"
So rather than introducing it ad-hoc in the middle of a derivation, we instead opt to compute (what is essentially) a different quantity

I'm saying the argument is often presented in a confusing way: We want to compute $\langle T\phi(x)\phi(y)\rangle$ to begin with. We do so by applying the definition, i.e. $\langle T\phi(x)\phi(y)\rangle = \theta(x-y)\langle \phi(x)\phi(y)\rangle + \theta(y-x)\langle \phi(y)\phi(x)\rangle$ and then putting the pieces back together
at no point is anyone "applying $T$" to an equation

6:13 PM
Would I be correct to say that if I were to write out the full expression for $\langle \Omega \vert \mathcal{T}[\phi(x)\phi(y)]\vert \Omega\rangle = \theta(x^0 - y^0)\text{(i)} + \theta(y^0-x^0)\text{(ii)}$, including the full (i) and (ii) expressions, I could derive the RHS of 4.31? Without choosing which time is earlier or later?

Inside the time-ordering symbol, everything (at different times) commutes

@Charlie I'm not sure what you mean
by virtue of the theta functions, the order of the times is fixed in each of the subexpressions (i) and (ii)

$T[e^{A(t_1) + B(t_2)}] = T[I + (A(t_1) + B(t_2)) + \frac{1}{2!}(A(t_1)^2 + A(t_1) B(t_2) + B(t_2) A(t_1) + B(t_2)^2) + ...]$
and then you can commute the individual terms because of the $T$, so it reduces to the usual proof of $e^{x+y} = e^x e^y$

Is the RHS of the your expression involving (i) and (ii) actually equal to the RHS of 4.31? As in, I could in principle derive one from the other without magically commuting anything past anything else as if they commute

you're not magically commuting anything
it's just a fact that if you have an operator expression $A(t_1)B(t_2)C(t_3)$ where the times are in order this is equal to $T[ABC] = T[BCA] = T[CAB] = \dots$

6:18 PM
So they are equal in the normal sense of the word

If you agree with $T[A(t_1) B(t_2)] = T[B(t_2)A(t_1)]$ then the rest follows from this

there's no magic there, that's just the definition of time-ordering
given the questions you ask now I think my answer actually misses the point and Qmechanic's hit the spot :P

ok this has given me a lot to think about, I am going to take a quick shower and have a think about it
I am determined to get it now
ACM your answer here, that normal ordering is only defined on a single product of operators, is this also true for time ordering?

@Charlie don't read that, it's just confusing :P

>:(
Very well, I will have my think shower now

6:31 PM
as the comments say, I'm imagining a version of normal ordering there that could act on (a subset of) operators

Ok, I will disregard it in my shower

the better approach is to just think of normal ordering as a function on symbols, and then it's defined on all symbols

I'm going to apply normal ordering twice, just watch me

actually, let me delete that, it's really just confusing

$::a^{\dagger} a:: = :a^{\dagger} a: = a^{\dagger} a$ :o

7:30 PM
Differential geometry is now my favorite branch of mathematics

1 hour later…
8:58 PM
@ACuriousMind imo the term capitalism itself is kind of an interesting piece of manufactured consent. It was never used by Adam Smith, and was actually a term made by socialists (Louis Blanc). Before that "commercial economics" applied in an important but limited sense to describe a large family of market dynamics that occurred in what i call a communitarian producer society, but there wasnt this "everything for sale" culture that "capitalism" takes on. The word itself is a social trap
@ACuriousMind the coefficient here was the confusing part for me btw, it causes some major localized warping but you're saying its just a fudge factor for physical effects right.

@Skyler yeah, in a real scenario that would be the part that's dependent on the lensing object's mass

@ACuriousMind so is this more of a generalized lensing equation, in the sense that it also describes traditional optical phenomena, or is it still just structured kind of mathematically like GR lensing sans the physics

I don't think there's any other way except gravity to get this kind of lensing effect
the "special" thing is that this "lens" just pulls all rays towards it, while normal optical lenses have a focal point
i.e. this isn't actually a "lens" in the normal sense of optics

yea, its a mechanism to look around something I suppose
Which actually is why I initially started looking at this, I wanted a "lensing" mechanism that allows for you to look around an object/shape, but the GR aspect of it left me having a hard time understand how to control the manipulation.
wish i got to take that class in undergrad...
anyways what would be the physical terms that this guy simplifies down to a 25 there. I guess that would let me kind of reverse engineer some intuition about the algorithm

note that the 25.0 would not only depend on the mass of the hole but also the distance of the observer
this function has the distance of the source as an input, but I don't see anything for the distance of the camera
except for mass of the hole and distance of the camera I don't think anything else enters
@Skyler while the history of the term is interesting, I don't think it's an example of manufactured consent: What would be the incentive for mass media to propagate that term? What dissenting views are being left out by saying "capitalism" as opposed to something else?

9:24 PM
@ACuriousMind this is a very effective way of introducing ideological shift, basically the balancing game of mercantilism greed with financial longevity and growth was only a small part of how society worked. By getting a "capitalist" culture accepted you created the consent to financialize all of society. This creates a bunch of new pathologies that you can create consent for under the guise of profits/returns

ah, I see what you mean

Mutual aid societies collapsed into juggernaut societies, broad public markets detached from anything except self proliferation of a corporation, and you could eventually get people to even buy their houses like how a asset ownership business would want to financially structure it
The whole incentive and punishment structure of mortgages for example makes no sense for individuals, if America wasn't leveraging it's future monetary supply to create mortgage funds it wouldnt be sustainable in any capacity. Mortgages make sense for a business which can run a portfolio of properties to weather dry spells
juggernaut insurance companies*

1 hour later…
10:51 PM
@ACuriousMind how would distance from camera be figured into that term though i guess is my question

@Skyler here's a good answer for distant sources, I'd say