My intuition about the Lie algebra is that it tries to capture how infinitestimal group generators fails to commute. This means ##[a, a] = 0## makes sense naturally. However the Jacobi identity ##[a,[b,c]]+[b,[c,a]]+[c,[a,b]] = 0## makes less sense. After some search, I found this article...
Sorry, the question may sound a bit weird. I will try to give a more detailed explanation on what structure I have in mind before rasing the question.
So first, we have a 4d spacetime, there are n worldlines in the spacetime representing n particles forming a system with certain interation...
We can formulate the spacetime in an observer/coordinate independent way, i.e. a particle becomes a worldline in the 4d space. Then relative to each observer, the worldline can be casted to a function in R^3. However, I haven't found any reference on formulating configuration space in a...
I think I started to understand why it's an unnatural thing to think about Galilean transformation on non-inertial observers, because the Galilean group naturally applies to the set of all inertial observers and is in one-to-one relationship to it (not sure what's the right math word here?).
I...
What I originally had in mind is below:
So we have an affine spacetime ##N^4##, the associated vector space ##V^4##, with the simultaneity subspace ##V^3##. An oberserver in my mind is the combination of a smooth worldline ##\gamma: \mathbb{R} \to N^4## and a smooth orthonormal basis assignment...
Hmmmm...there must be something I understood wrongly on this (why Galilean transformation is not applicable to non-inertial frame). I will try to be a bit more explicit:
Let's say the world is the 4 dimensional space with Galilean structure defined in the Arnold's. A Galilean transformation is...
If we take the definition from Arnold's
The galilean group is the group of all transformations of a galilean space which preserve its structure. The elements of this group are called galilean transformations. Thus, galilean transformations are affine transformations of A^4...
It's frequently discussed Galilean transformation brings one inertial frame to another inertial frame, and such a transformation leaves Newton's second law invariant (of the same form). I wonder what happens for non-inertial frame? If we start with a non-inertial frame, and Galilean transform...
That's a really good recommendation! I wonder if there're classical mechanics books that follows a similar philosophy? E.g. starting from a mathematical formulism in coordinate free langauge.
Thanks! I watched Susskind's lectures before but I'm looking for something with a bit more mathematical...
I found some new reference on this. In book "Special Relativity in General Frame", the concept of "local frame" is defined, similarly the concept of observer, in section 3.4.
I wonder if this is a common definition, and does it generalise to GR well?
Some introduction books on Lagrangian and Hamiltonian mechanics use classical mechanics as the theoretical framework, and when it come to special relativity it goes back to the basics and force language again. I would like to ask for some recommendations on good books that introduces Lagrangian...
I'd like to get some help on checking my understanding of special relativity, specifically I'm trying to clarify the idea of coordinates. Any comment is really appreciated!
The spacetime is an affine space ##M^4##, which is associated with a 4 dimensional real vector space ##\mathbb{R}^4##...
About this particular benefit, can't we say that Newton's second law is on the same footing here? Because we can also transform the coordinates to get Newton's equation of motion into a non-inertial frame? Similar to transforming coordinates for the Lagrangian?
Thanks for the info! Looks like some more advanced stuff. I guess I will need to get the basic GR firmly understood before getting into this...
Good tip, thank you!
Ok that's fair, no need to get deep into what exactly the author had in mind ;)
Thinking about this again, I think an observer is more than just a world line. For example, we can think about an observer moving with uniform velocity, but rotating all the time. Then its world line appears to be...
I have sometimes seen the claim that one advantage of Lagrangian mechanics is that it works in any frame of reference, instead of like Newtonian mechanics which will hold only in the inertial frame of reference. However isn't this gain only at the sacrifice that the Lagrangian will need to take...
Ok this is helpful. I was confused between inertial observer (world line) vs inertial coordinates. And it caused me to think that there's a different coordinates at each point on the world line.
Thanks for the confirmation, now I think I don't need to consider this "tetrad field" concept at all...
It sounds from your explanation that tetrads don't have any physical meaning? I thought a tetrad is something that characterises your motion frame, i.e. inertial or non-inertial. But it seems that they're only as real as coordinate charts? So what's the point of having this mathematical tool...
Maybe being more specific would be useful here. Let's say we consider a non-inertial observer in SR. Surely it has a world line. At each event of the world line, will this observer have a tetrads (frame)? And will these tetrads be different? (Assuming we can compare vectors at different points...
If it's not available in classical mechanics, SR would also be good. Basically I'm used to thinking about a single frame of reference (usually inertial) that can be transformed Lorentz transformation, so that we can give coordinates numbers to events in spacetime. But I don't know how do we...
I'm trying to bridge my understanding from classical mechanics to relativity by thinking about these problems more carefully. Is frame field also a concept in classical mechanics? Is there any resources explaining things from this perspective? I feel I might ask better questions once I read a...
Oops, you caught me again ;)
Ok I think for my purpose, we don't need to consider (3), so the difference I'm considering is "frame field" vs "coordinate chart". For the moment let's consider flat spacetime only (classical or SR). Based on your definition, "frame field" is a mapping from...
Just want to clarify some concepts.
There seems to be difference between reference frame and coordinate system. See https://en.wikipedia.org/wiki/Frame_of_reference#Definition . A reference frame is something has physical meaning and is related to physical laws, whereas coordinate system...