Relativity 105c: Acceleration - The Jacobian (changing basis in curvilinear Rindler coordinates)

Previous videos on the Jacobian from my Tensor Calculus series:

0:00 Intro to Jacobian
2:45 Multivariable Chain Rule
7:20 Basis vectors as Partial Derivatives
14:23 Change to Rindler Basis (Jacobian)
18:42 Change back to Inertial Basis (Inverse Jacobian)
22:28 All change-of-basis matrices are Jacobians
25:05 Change-of-components to Rindler coordinates
31:21 Change-of-components to Inertial coordinates
33:23 Summary

[eigenchris]

Error at 29:27: I've mixed up a lot of the terms in the multi-variable chain rule expressions for ct~ and x~. Sorry about that.

[HighWycombe]

I've been carefully transcribing all your Relativity video narration, and hand drawing all your diagrams to help my understanding and for reference. At the moment, each half-hour video is coming out at something like 20-24 pages of notes with much stopping and starting. (My notes for Relativity 101a only stretched to one page :-) ) I'm 68 years old and retired from a job in IT. Enjoying your videos very much.

[thomaskokoska]

Thank you, Chris for a wonderful series so far. I look forward to future chapters. To a retired engineer revisiting the topic of relativity after 40 years, I find the online video mode of learning to be fascinating and your use of graphics is powerful. Keep up the good work!

[ericbischoff]

"Because the Internet told me so" the first time I see this in a math demonstration :-)

[simatwokirwa]

Interesting intuition - from y = mx (where m is slope which is dF/dX .. thus dF is dF/dX times dX) .. good job Eigen-Chris ... building what seemed at first an enigma to many, in a simply easy to comprehend approach: - from Algebra->Vectors->Tensors---> to the real deal RELATIVITY (Newtonian/Galilean, Special and ultimately General).

[SaberToothPortilla]

Great video as always!

While the explaination at 8:00 is great for intuition purposes, it might be worth making the caveat that the positions that the curves describe aren't literally the same as the displacement vectors from the origin. I see students confuse that a lot.

I only say this because once you apply functions to the field, the distinction becomes very relevant. A function from a set of points to scalars is very different than one from a set of vectors to scalars.

A quick solution is just to say that all points add with vectors to make another point, but points *don't* add with points, and vectors have the typical addition. Then f(P) can be expressed as f(P) = g(v) + f_0, where P is any member of the domain, f_0 is the value of f at the origin, and g is a proper function from vectors to the same codomain as f, and it's g that the vector calculus happens on, with displacement vectors v.

[mot]

Thanks for you lucid explanations with great graphics. I have to review it again and again because I am not a math or physics major but I am understanding this. Great job.

[BlackEyedGhost]

19:23 Lol, nice. For reference:
artanh(z) = ½ln( (1+z)/(1-z) )
A formula which is a lot of fun to derive.

[DmAlmazov]

Will you discuss rotating non-inertial frames and Ehrenfest paradox in this siries?

[MasterHigure]

19:25 The inverse hyperbolic functions are most correctly given the prefix ar- rather than arc-. There is a reasoning behind it, stemming from the fact that the hyperbolic trig functions relate areas to Cartesian coordinates, rather than arc lengths to Cartesian coordinates.

That being said, the arc- misnomer is so common it's barely wrong.

[patriciacosson]

Toujours aussi remarquable félicitations!

[nickst]

Thank you very much for your excellent series! Do you plan to cover General Relativity as well? That is the hardest thing to cover. But if you approach it like you do so far, you can help us a ton! Thanks!

[klgamit]

Great video but I think the way you have defined the vector S that you name as tangent to the coordinate curves is a bit lacking, because first you introduce it as a general tangent vector for a curve. But then, for example in 10:03 you take S(x) to necessarily mean it moves only in the positive x direction, implying it doesn't move in the ct direction, but that's not necessarily true for all curves. Similarly S(ct) won't necessarily move along the positive direction of time only. All the later usage of S implies that deriving it with respect to each direction you basically get the Kronecker delta. So I assume S = (ct, x) in the non tilde basis. If I understood it correctly and that's the case it could be good to define it explicitly... but I'll be glad to know if I am just missing something obvious.

[nityadas]

Dear Chris, thanks a lot for one more excellent video.

[derickd]

Could you make a list of books where we could read further about and apply everything you've been teaching us?

[nellvincervantes]

Question sir. Not directly related to this video. It is about how to derive Unruh Temperature.

Wick rotation, 1/kT = i/h_bar*w

According to acceleration observer, the speed of light (light - the vacuum particle):
x = x_o e^(at/c)
dx/dt = (a/c)(x_o)e^(at/c)
dx/dt = ax/c

If 2pi*w = a/c
dx/dt = 2pi*w*x
r = 2pi*x
dx/dt = v = w*r (where r = radius of curvature).

Youll get T = h_bar*a/2pi*k*c (as the unruh temperature). Maybe im wrong

[CosmicLog]

Hi chris, I´m starting my thesis and your videos are being really helpfull, I just wish I could find these topics in some book, I´ve really been struggling to find non-inertial reference frames, books just jump straight to General Relativity instead.

[educationtarunramkanuri]

Proof by internet told me so.

Seems pretty legit to me. I'll take it.

[pythoncure]

Will we get a topology series you make math interesting....

[redbel]

I have not started with the general relativity series yet, but I watched your series on all tensors, and I don’t feel that I understood tensors. Can I understand general relativity even though your videos look very simple?