@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
@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.
@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
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?
@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.
@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
@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
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?
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 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
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
@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
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?
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
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
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
@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
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
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? $F_N+F_r=mu^2/R$ ??
"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?
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
@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
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)
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:
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
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
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?
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
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 $T[ABC] = T[BCA] = T[CAB] = \dots$
@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.
@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
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?
@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
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