@Slereah I think it's not given a name because it's not actually that useful? :P
Whitney's embedding theorem is great and useful because it shows the equivalence between the intrinsic and extrinsic manifold definitions, but since the choice of smooth structure in the "smoothing theorem" isn't unique, what are you actually gonna use it for?
I mean strictly speaking that's not really a good argument? You can have different smoothings of the same $C^k$ manifold (something like exotic spheres exists in principle for any $C^k$ to $C^\infty$ smoothing)
i.e. two manifolds can be isomorphic as $C^k$ manifolds but not as $C^\infty$ manifolds
My biggest problem with Mach is that he kept saying that Hertz completed his mechanics textbook that totally eliminated the concept of forces, but actually Hertz's text is very rudimentary and did not do much. I did enjoy that it really removed the concept of forces, but it would not have been a good textbook to learn from.
Mach was asking the wrong question. Back then, people didnt think of spacetime as its own entity. this is what led to weirdo ideas like "all motion is relative"
but the problem is that the definition of "self aware beings" is vague. according to Descartes, animals arent conscious
and according to solipsists, only u r conscious
solipsism is the only well defined way to base Mach's philosophy
he wud have to believe that only his sensations existed
if he believed in other peoples' sensations as well, then he has no reason to not include animals. and then the lines get blurry about who isnt conscious
@Slereah Careful: Mach's empiricist approach is today usually called phenomenalism, not idealism: Idealism is the ontological claim that only the contents of the mind exist, phenomenalism is the epistemological claim that the only "things" that knowledge/science deals with are the perceptions/contents of our minds.
Mr. Poincaré was writing in 1902 that if we found out that astronomical observation told us that light did not seem to obey Euclidian geometry, we would have to either abandon Euclidian geometry or go with the idea that light does not move in a straight line, and that we would probably do the latter
@RyderRude you're doing that thing again where you project back our modern understanding of "spacetime", "metric" etc. onto people who did not yet understand it that way
this is not useful to understand what they actually thought
even tho the first law is not really violated. it just gets modified a bit
@ACuriousMind then on what basis was Poincaire predicting particle behavior based on metric behavior if people had no idea of non euclidean dynamics
i think people tried to generalise the dynamics into curved space, and got their predictions
people had to have made a connection between geodesics in curved space and newton's first law
i think this is what Poincaire's claim was based on. he thought we ought to observer violations of first law if we were living in curved space
but yeah, this reasoning doesnt require understanding the role of the metric in the Lagrangian
in the quantum eraser experiment, how can a single particle perform a measurement
the particle causes an effect akin to decoherence
it seems like a measurement device in quantum mechanics is anything which lets out the "which path" information
decoherence is not even necessary. a single particle can measure stuff
even tho it was told to me that a quantum particle cannot measure another quantum particle
they shud just get entangled
but again, this observation is a conflict between the state reduction postulate and the unitary evolution postulate
it seems like nature employs a concept of "system" and "outside the system". whenever the "which path information" of a system leaks out to the outsider, the evolution isnt unitary, but state reduction starts applying
but there doesnt seem to be an inherent division between "system" and "outside the system". it seems arbitrary what you decide to be your system
but probably, nature holds some undiscovered rule which determines "system" and "outside the system" and "information leaks to the outside"
without this rule, u get either MWI or consciousness causes collapse
@RyderRude This is not at all what I said. Who claimed that Poincaré predicted behavior "based on metric behavior"? Who claimed "people had no idea of non-euclidean dynamics"?
again, all I said is that you shouldn't project our modern understanding of these topics onto people in the past. That does not mean those people had no understanding of these topics, it means they had a different understanding
@Slereah What do you mean by this? Why is matter suddenly involved in this? Did we not make those measurements in the limit of low matter density where the light's trajectory passes though?
@naturallyInconsistent to observe light you need some detector made of matter ;)
you can't actually observe the path of a light ray, all you have is start and end points and then we believe that the ray took a particular path because it is consistent with our observations
@Slereah you mean that either we assume the geometry by convention and then have to figure out how stuff moves or we assume motion in straight lines by convention and then have to figure out what the geometry is?
We can either say that light moves in a straight line and then figure the geometry from there, or we assume that geometry is Euclidian and if light doesn't behave nicely we say it doesn't move in a straight line
and the trick of conventionality is that we can't really decide if any choice makes more sense than the other
and then we claim that the latter term in the split is a force field
this can be done mathematically. but "gravity is not a force" will always be the preferred convention because the split metric is not an observable. for e.g. in quantum mechanics, the parameter of unitary evolution must be given by the full metric
@Slereah I think things are much more complicated than that. We know from classical physics, those before-university experiments with light boxes, that light moves in "straight lines", a concept that later gets generalised to geodesics. We know from quantum considerations that paths that are not near a stationary point of the action contribute very little. The experiment that propelled Einstein to fame measured a deviation due to the Sun moving near the old predicted trajectory of light.
i.e. the trajectory could be verified before and after the experiment for some geometry, I don't think there is much to that, but deviation is kind of difficult to assert to be that we are assuming some prior geometry, other than that we are assuming things to stay the same.
@Slereah Pins on paper. With light boxes the entire trajectory shows up on the paper, so it is visually there before the student. Of course, there will still be some theoretical issues, like we have to assume that what appears straight for us is actually straight in the Euclidean sense, but it becomes too pedantic for students too young for the pedantry.
if one observed light moving in a curved path, one could also assert that it's the gravitational force acting on the light and that the geometry is euclidean
this observation does not in itself prove that geometry must be non euclidean
in fact, nothing can prove that. both views are experimentally identical
it's just that the underlying Minkowski spacetime is irrelevant to any observations. it's an unnecessary extra strcuture that's shaved off by Occam's razor
but what we objectively cant do is treat gravity as a force in quantisation, because the parameter of unitary evolution is not irrelevant. it's an observable
Also, I was not saying that the light is moving in a straight line either. I am saying that whatever geodesic it was already observed to be moving on, should still be there before and after the mass moves near and past whatever that geodesic/trajectory was there, and only during this "moves near", we observe a deviation, and that should be a informative experiment
@Slereah Oh, I am perfectly happy with that. As long as it is that it isn't one same curve, we can say something extraordinary (at least to Kant, muahahahaha)
@Slereah I'm sorry, why are we fixating on straightness of lines again?
I know vaguely that you were talking about how we could talk about straight light + curved space or straight space + curving light, but I had a hard time seeing how either is to map to the actual situation in GR
In terms of GR this corresponds usually to the perturbative case where we pick a fixed geometry $g'$ and a perturbation $h$, and our metric tensor is then $g=g'+h$
At least that is one of the kind of process you can use to illustrate this, since that is a general process
> The book was written as a reaction and criticism to the three-volume work Empiriomonism (1904–1906) by Alexander Bogdanov, his political opponent within the Party. In June 1909, Bogdanov was defeated at a Bolshevik mini-conference in Paris and expelled from the Central Committee, but he still retained a relevant role in the Party's left wing. He participated in the Russian Revolution and after 1917, he was appointed director of the Socialist Academy of Social Sciences.[citation needed]
@ACuriousMind yes, the story is pretty interesting, they even met in a villa in Capri to try to reconcile, there's a nice story/picture of them playing chess
They=Lenin vs Bogdanov
I don't remember if Rovelli's book has good citations but I read similar accounts elsewhere (Rovelli is the only one I can think of on English)
@Slereah I mean with Posadas I can see how believing in aliens fits into the ideology - if you believe that the kind of scientific progress needed for interstellar flight is only possible after overcoming capitalism, then believing in UFOs means believing that overcoming capitalism is actually possible
@ACuriousMind yeah not right away, but a few years down the line he was in charge of the institute of haematology and died after some transfusion experiments
Proletkult by Wu Ming Is also a great read about this stuff (not sure I'm which languages it's been translated) although it's a novel so not everything is historically accurate (eg I suspect the part where communist aliens kidnap one of the bolsheviks might be made up)
@Slereah lets say we write down our laws of motion first, and then we hypothesize that they hold for some constants $m1, m2...m_n$. From the laws, we can deduce the experiments that can determine these values upto multiplication by a constant
this only tries to make sense of things in hindsight. it wud b hard to come up with the laws
and if the our hypothesis is successful, then we've got a predictive theory
laws of motion are just recipes to get trajectories. and trajectories are directly measurable
@Slereah when we write down the laws of motion, we make no mention of forces and stuff. we just write down a lagrangian or a differential equation
@Slereah if JC did not contribute he should not get single author credit (that’s unethical). He could conceivably be co-author to very many papers though.
Poincaré is saying that Lagrangian mechanics is striving for least action
Take that extremal action people
he is also saying that using energy instead of Newtonian mechanics allows us to not use atoms, which I can maybe see why but it is a weird thing to drop like that
And he is calling the kinetic and potential energy T + U, I guess the filthy Germans already got to him even in 1902
@ACuriousMind can it just be that if V is taken, say, because we may need a symbol for each of potential energy (vis viva) and potential, then the other is chosen to be looking very similar to V?
@ACuriousMind There is a tonne of such notational cargo-culting going on. Kamiltonian is one, apparently the duplicity in the form of $\epsilon_0$ v.s. $\varepsilon_0$ too, and countless others.
@RyderRude But part of the frustration currently is that you don't even know what is the limit of your knowledge. If we knew, we would at least know when to stop struggling with nature
When you take a time derivative of some function $f = f(p,q,t)$ like energy $E = E(t,q,p)$ and then invoke Hamilton's equations for $\dot{q}$ and $\dot{p}$, you immediately find that $\frac{d}{dt}f$ can be re-written as $\frac{d}{dt}f = (\frac{\partial}{\partial t} + \{H,...\})f$
> Another version involves a philosophy instructor who, for a final examination, places a chair at the front of the room and challenges the class to prove it exists. One student receives an 'A' for writing "What chair?"
@RyderRude What's wrong with bringing it back to Hamilton's equations? Maybe you want something like this $\oint_{\partial S} P dQ = \iint_S dP \wedge dQ = \iint_S [P,Q]_{p,q} dp dq = \iint_S dp dq = \oint_{\partial S} p dq$ i.e. thinking in terms of integral invariants as a motivation for it, but this way you're still basically invoking Hamilton's equations too
@bolbteppa the evolution must be volume preserving. but this doesnt automatically imply Hamilton's equations, I think. so Hamilton's equations are taken for granted here
How can I know which is the oldest tag of Physics Stack Exchange?
