@naturallyInconsistent Which books are you talking about? All book I know about just establishes a derivation of the wave equation from taking the continuum limit of an infinite chain of harmonic oscillators. I was specifically asking for the mass term. I already understood all the sentences you wrote apart from the last one.
@naturallyInconsistent I need some details on the last sentence. So do you mean to say that we add a $\frac{1}{2} m_i^2 x_j^2$ to the Lagrangian in addition to the $\frac{1}{2} k (\x_j -x_j+1)^2$ like terms or something?
@Slereah No, but the two definitions (preserving limits and colimits or preserving exact sequences) are equivalent in abelian categories, se ee.g. nLab
hi i want to understand some sections of this answer https://physics.stackexchange.com/a/14944 the sections : "what finitary means", "ordinal religion", "ordinal analysis" and "other interpretations"
> Given a sequence of ordinals which approaches the Church-Kleene ordinal, the theories corresponding to this ordinal will prove every theorem of Arithmetic, including the consistency of arbitrarily strong consistent theories.
later on :
> To go further requires an advance in our systems of ordinal notation, but there is no limitation of principle to establishing the consistency of set theories as strong as ZF by computable ordinals which can be comprehended.
Doing so would complete Hilbert's program--- it would removes any need for an ontology of infinite sets in doing mathematics. You can disbelieve in the set of all real numbers, and still accept the consistency of ZF, or of inaccessible cardinals (using a bigger ordinal), and so on up the chain of theories.
what does this stuff mean? I know that the consistency of a theory can be proved by a bigger theory, but then the consistency of the bigger theory is in question is this stuff expressing the same idea
@RyderRude Ron has deeply idiosyncratic opinions about history and philosophy of science and mathematics, and in particular about the foundations of mathematics (see also the MO thread linked in the comment), most of this is not a presentation of mainstream thought. I would not suggest taking it at face value.
@ACuriousMind yes. Thats why i was asking. I found this other post discussing it mathoverflow.net/q/432470
apprently, this is not the "trivial" idea i was talking about.
It is a somewhat deeper theorem
But Ron somewhat overstates its importance. It is debatable if this idea makes things like consistency of Peano provable or more believable than it already was
Ron has an obscurantist writing style - in particular his reluctance to provide specific references for any claims - that makes actually assessing the content of his longer answers on these soft topics at lot of work; I've usually not found it worth the effort since he mixes facts and his own opinion so liberally you have to double-check every sentence against other sources.
You also have to be careful with Wiki links he provides, since he was also an active Wikipedia editor and these can be his own work, i.e. not actually a reference
@Amit The "completely unrelated Latin text" you flagged was the source text of Lorem Ipsum, by the way, i.e. it's not random Latin but the one Latin text you post when you just want nonsense content.
its relation to Godel's theorem is discussed in the article
> Kleene (2009, p. 479) made the following comment in 1952 on the significance of Gentzen's result, particularly in the context of the formalist program which was initiated by Hilbert.
The original proposals of the formalists to make classical mathematics secure by a consistency proof did not contemplate that such a method as transfinite induction up to ε0 would have to be used. To what extent the Gentzen proof can be accepted as securing classical number theory in the sense of that problem formulation is in the present state of affairs a matter for individual judgement, depending on how r…
this technique proves consistency of PA. But it uses transfinite induction
@ACuriousMind Right, I actually did find that out but only after raising the flag :) because a bit later I noticed the title was also changed to something containing "Lorem Ipsum" and that phrase rung a bell so I googled it... :)
For some reason I still find it surprising every time I see someone changing a question's title. My intuition is always that titles are uneditable, perhaps because in many of the older forums it was true
I think it's just the math that developed first. Like Euclidean geometry, which makes the more "common sense" assumptions. Later we began to study all kinds of crazy structures, and today mathematicians generally speaking don't really care about empiricism do they?
It is an interesting question, just how much of logic is forced once you at least accept that A and ~A is impossible. Then in the rule I suggested above, A=B can't work and you can't do it.
Then again, even saying that is relying on the existing logic axioms... is there a system where A and ~A = F is not equivalent to A or ~A = T? Dunno...
“Life is like riding a bicycle. To keep your balance, you must keep moving.” Albert Einstein's life advice in a letter to his son Eduard on 5 February 1930.
@RyderRude answered your questions about what consistency refers to, meanings of finitary and ordinal analysis in Mathematics. The rest I don't know for I am not a logician/set theorist
@RyderRude One often finds intersections between topology and set theory and I find those very interesting, for example the existence of $P$-points in $\beta\mathbb{N}\setminus\mathbb{N}$ is equivalent to continuum hypothesis. About dimension theory, I like it because of the concept of dimension and its different interpretations and how the dimension functions interact with each other.
this was the msg. It is indeed about relation of continuum hypothesis and topology
@RyderRude Yes but CH isn't the only thing I am interested in. For another example, the existence of countably compact $T_6$ spaces which aren't compact.
It's more that I am interested in when existence of some space with particular properties can exist, and it's not that I particularly care about which set theory axioms do I need to assume for this to be true (or false). Of course they are important, but secondary to me
I am interested in dimension theory as I said before, and this means I am also interested somewhat into subsets of $\mathbb{R}^n$.
Okay - theoretically those can be much uglier than manifolds, but still. Manifolds not excluded
In particular in the problem of compactifying an $n$-dimensional space so that the dimension is preserved and embeddability into Euclidean spaces gets preserved
Hi people. I have a question. I was reading the following statement: "The point of the thought experiment is to illustrate that any freely falling frame is locally inertial, meaning free from gravity and acceleration, if you look at it over a sufficiently small scale. "
But consider for example Niemytzki plane. This is just the normal half-plane but with different topology. So how do I visualize it? By visualizing how would sequences approach a point on the x-axis.
@RyderRude If you find their current username inappropriate, please say so - while users can in principle name themselves however they want, mods can reset usernames that are offensive or otherwise inappropriate.
@imbAF e.g. if gravitational field were uniform, then freely falling frame would be globally inertial. But in general, it can be thought of as uniform over a small region, so it is locally inertial
@imbAF When you're freely falling, you don't feel acceleration/gravity where you are, but stuff that's further away from you may still fall/accelerate relative to you
But, if I say, that the object that is freely falling is considered in 5 different instances of different locations in space, in the presence of some gravitatinal field
In every location, if we assume that the object is small enough
the field, locally is uniform
So, in 5 different locations, the field is uniform, albeit different in values, in each case.
If the object in position r_1 and small enough to assume uniformity of the field, then the object represents a locally intertial reference frame
If the object in position r_2 in another scenario and small enough to assume uniformity of the field, then the object represents a locally inertial reference frame
so no matter, where the object is, because of its small size, will always represent an inertial frame of reference
the fact that is inertial, from what I understand, is related, primarily with the size of the object
for instance, close to a black hole most objects will eventually undergo spaghettification as they fall in, decidedly something that does not happen in "free fall", even if those objects are fine freely falling in most weaker gravitational fields
I often hear the story of how Einstein came up to the conclusion that time would slow down the faster you move, because the speed of light has to remain the same.
My question is, how did Einstein know that measuring the speed of light wouldn't be affected by the speed at which you are moving. W...
And one of the replies, says: How does maxwell equation tell you speed of light has to be constant? You would have to know the transformatiln properties in different frames of E and B. Its definitely not that simple.
And another one: Your three points are right, but the derivation from Maxwell eqns sort of misses the point. The maths is correct, but the whole issue is the nature of "vacuum". Saying "and there you are" at the end assumes the hindsight of knowing how to interpret the role of what we now call vacuum but was thought to be aether.
While I understood the answer to the question.
I didn't think there was an issue with the proof that light speed is constant
But these two replies do say that
And I am trying to understand what are the issues that they raise
Could you help me
Is one of the problems, the fact that the person who answers, is not considering the calculation of the speed of light in different frames of reference
why are you focusing on the short comments when there's a comment by dmckee that actually explains the issue? :P
This doesn't really address the question because classical physics is full of waves whose wave equation you can derive to find a speed that depends only on constants (albeit constant properties of matter). And the speed of those waves did depend on the motion of the observer because they were only correct relative the medium. This is the reason that many 19th century physicist pre-supposed the existence of a medium for light. — dmckee --- ex-moderator kittenJul 2, 2019 at 20:31
@imbAF He's saying that if this proved that the speed of light was the same in all frames (rather than perhaps just suggesting it), then the wave equation for waves in water would also prove that the speed of a water wave is the same in all frames, since the speed you can compute from that equation is just a bunch of constants, too
@ACuriousMind Ok i understand that solving the equation doesn't really prove that the sspeed of light is constant, just because it can be expressed as the result of constants
this is the reason why you say this: "Just looking at the equation doesn't prove the result holds in all frames" ?
@ACuriousMind To my understanding, while the 3 bullet points are a good starting point for the idea that the speed of light might be the same in all inertial frames of reference, ultimately, Einstein, simply, assumes that
If you have to assume instances in science, isn't that somehow showing lack of logic in the way we conduct it? And I am not saying Einstein lacks logic :D
Your SR big rule is that your spacetime interval is constant under Lorentz transform, so $$t'^2 - x'^2 = \gamma^2 t^2 + \beta^2 \gamma^2 x^2 - 2 \gamma^2 \beta tx - \gamma^2 x^2 - \gamma^2 \beta^2 t^2 + 2 \gamma^2 \beta tx $$
@imbAF I don't know what you mean - how would we ever get anywhere without first making some assumptions (based preferably on what we already know) and then testing them against reality?
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the thought experiments are just to figure out what "the speed of light is constant" as an assumption would imply, so you know what to test in the real experiments
Meaning that using that convention, you will have $\gamma^2 = \cosh(u)$
Einstein's theory comes within the context of science methodology of the era where it was in vogue to remove hypothesis that were seen as not indispensible to the theory
Given the lack of results of the Michelson Morley experiment, he tried to figure out if we just couldn't do away with the speed of light having to change at all
You can also show the hyperbolic nature of SR in a more visual way too, by looking at the set of points at the same spacetime interval within a lightcone
@ACuriousMind If you remember, in a previews discussion of ours, I made a statement about the significance of the LT and the fact that two observers in two different frames of reference would use them, to experience/witness what the other experiences. And I gave the the example with the rod being at rest in the frame S, while S' moves relative to it. I said, that if the observer at S performs the LT, he will
find out what value for the length of the rod, the observer in S' measures, and he will be aware of length contraction. You agreed with my statement. But there is one thing that I missed/didn't notice. Now, if the observer in S' would perform his LT, according to what I said above, he would experience what the observer in S experiences, which means, the observer at S' would measure the proper length of the rod? Is that possiblle?
@imbAF Normally when we want to say that one observer changes inertial reference frame we say he performed a boost from frame S to frame S' for example. So if physically one observer moves to another inertial reference frame, of course he will measure the same length as another observer in this same frame. They are both now in S'.
It becomes more a matter of semantics whether you're saying he performed the LT to find out how an observer in S' measures things or alternatively he moves to S' and this change is described by a Lorentz boost (LB - I suppose? :)) and then find out how things are measured in S' in that manner
Yes but I am considering the case that you have two observers, one in S and one in S'. And both of them measure the length of the rod in their own frame
So each of them will calculate a location for each end of the rod and then find the length
Then, the observer in S, performs LT in order to find out what the observer in S' measures for the location of the two ends of the rod. And he find the length
What he will measure will be a length shorter than what he measures in his own frame of reference
which is the rest frame of the rod at the same time (I take S to be the rest frame of the rod)
Now, let's do the opposite, the observer in S' performs LT.
If we are consistent
with the above statement, that you agreed with
The observer in S', when performing the LT, must measure for the two ends of the rod, values, such that, when calculating the rod's length he must calculate a larger value than what he measures
If the observer in S' tries to find out how the observer in S measures the length of the rod, yes, he finds out what you might call length extension. However note that this isn't a useful phrase because this is disconnected from what he phenomenologically, actually measures. You either measure the proper length when at rest wrt to the object, or you measure a contracted length when it is moving...
You can define a length measurement in terms of the spacetime interval, if that's what you're looking for. I suppose something like $L=|x_2(t_0)-x_1(t_0)|$ with the key point that both measurements are done at $t=t_0$. The squares really come in when you have more than one dimension :)
It's a very specific no yeah... when you do the LT the point is that $x_2(t_0)$ and $x_1(t_0)$ will not be simultaneous, so if you use my formula in one frame it doesn't keep the same form in the other after LT
yes, I suspected that's what you mean. If that's the case, I just wanted to note that the formula I wrote for $L$ will look different in S'. It will depend also on $t'$ and not only on $x'$.
Can I think of this as a differential form $dx_1 \wedge dx_2 \wedge dx_3$? If yes, then suppose I change coordinates $x_i \mapsto y_i(x_1,x_2,x_3)$. To get the new "volume element" am I trying to solve $\wedge_i dx_i = [\text{something}]\wedge_i dy_i$?
As a workaround while this request is pending, there exist several client-side workarounds that can be used to enable LaTeX rendering in chat, including:
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