@punkerplunk So, I think I get what you mean. If S1 is an inertial spaceship, S2 is one triplet doing the accelerating weirdness, and S3 is another one. Your question is, why is "S2 in S1's frame" valid to look at the twin paradox, but not "S3 in S2's frame"?
There really is nothing at all less philosophically pleasing about special relativity than there is about Newtonian mechanics/galilean relativity.
@NeuroFuzzy sort of yes. More specifically, though, I'm wondering if we can just consider the earth as another spaceship, and say it exists in another 'inertial' frame.
@punkerplunk well but it's not inertial in quotes. It's a real physical assumption for the frame to be inertial. So if you consider the earth moving slowly (much less than the speed of light) then yeah, you can just consider the earth as another spaceship, and say that its reference frame is inertial (ie, you neglect the Foucault pendulum and tides and blah blah noninertial frame stuff) .
@0celo7 It means you take the free algebra on the space (i.e. just take the basis vectors $e_i$ and define additional objects $e_i\cdot e_j$,$e_i\cdot e_j\cdot e_k$ and so on, and then you impose the constraint $v^2 - g(v,v) = 0$.
@punkerplunk okay, I'm out, we're going in circles. The bottom line is inertial frames are special. They're special in special relativity, they're special in newtonian mechanics, they're absolute. There's a big difference between inertial and noninertial frames, and the laws of physics in inertial frames are different than in noninertial ones. You cannot apply the same laws in noninertial frames as you can in inertial ones.
In this case, the ideal generated by the relation $v\otimes v = 0$ is the set of all tensors such that they can be written as $v\otimes v \otimes \dots$ for some $v$.
(We must take the ideal because the only things you can divide out of rings such that the quotient is still a ring are ideals)
You people need to take some sweet abstract algebra ;)
@0celo7 for example, in introductory courses you say the laws are newton's laws. Well, the form of Newton's law F=ma changes. So the law changes from frame to frame. Of course that's not true in the lagrangian formulation though, because that transforms with the coordinate system. That's the sense in which I mean it, and it doesn't confuse things to say that.
It elucidates that in newtonian mech/spec rel, inertial frames are real/physical/tangible/important. Different definitions give different theorems.
Random math question here. When solving Laplace's equation to find spherical harmonics, the expression$$\frac{1}{\Phi}\frac{\mathrm{d}^2\Phi}{\mathrm{d}\phi^2}=-m^2$$is always used. Why $\Phi$, and not $\Theta$?
@HDE226868 Uh...you did assign the meanings of the angle variables $\phi$ and $\theta$ somewhere prior, I hope, and the convention is to measure the latitude as $\phi$, no?
Well, the naming in the standard solution is kinda pre-cognitive because the author knows that $\Phi$ will be a function of $\phi$ and $\Theta$ a function of $\theta$.
I consider it good style to make a comment when one does such naming in anticipation of the result, because people are often left wondering how one could have known that at the step of naming it, when the simple fact is one couldn't
