Friday, 20 September 2019

Exercise 3.04 Paraboloidal coordinates

Question

In Euclidean three-space we can define paraboloidal coordinates ##\left(u,v,\phi\right)## via$$
x=uv\cos{\phi}\ \ y=uv\sin{\phi}\ \ z=\frac{1}{2}\left(u^2-v^2\right)
$$(a) Find the coordinate transformation matrix between paraboloidal and Cartesian coordinates ##{\partial x^\alpha}/{\partial x^{\beta^\prime}}## and the inverse transformation. Are there any singular points in the map?

(b) Find the basis vectors and the basis one-forms in terms of Cartesian basis vectors and forms.

(c) Find the metric and inverse metric in paraboloidal coordinates.

(d) Calculate the Christoffel symbols.

(e) Calculate the divergence ##\nabla_\mu V^\mu## and Laplacian ##\nabla_\mu\nabla^\mu f##.

Overview

I completely screwed up calculation of the metric, getting one that was very non-diagonal and would have been hard work and I had to go back to spherical polar coordinates to see where I had gone wrong, then it all worked except for a sign error in the inverse transformation matrix. It was fairly obvious which component contained the error.

The way that a covariant tensor ends up in the wrong coordinates after using the general tensor transformation law (e.g. at (81)) is very odd and needs further investigation. I think it's just inevitable and would often need transformation back into the correct coordinate system.

I learned a bit more about basis vectors, but this exercise mainly seemed to be teacher-torture with plenty of exercise on differentiation. The Laplacian ##\nabla_\mu\nabla^\mu f## is not in the index of this book. I hope it is discussed somewhere. I spent about half the time developing drawings of the basis vectors and one-forms but sadly the animation broke down. It was a surprise that the coordinate transformation matrix and its inverse not only take tensors back and forth between coordinate systems but are also inverses in the matrix sense. I suppose I should have known.

When calculating the Christoffel symbols, my first shot was 77% correct. I suppose if this was an exam question that would be quite good, except that I have taken too much time!

Answer

See: Ex 3.04 Paraboloidal coordinates.pdf (15 pages, 147 equations)

6 comments:

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    1. Hi,George. When i calculated the divergence of vevtor V,i found that i use two different formulas (3.32)and(3.34) in carroll's to get different results, which confusing me. Then i found that when u calculated the connection, u first fixed the up index of Γ and next u fixed the two down index together of g,like (94) (104) and (114).However, when we choose different μυρ of g, it turns out that the index of g are not fixed, some of them are going to be vanish.Excuse my poor English!Thx u!

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    2. Glad you are still at it Chun! Is there a question here? Or have I made another mistake?

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    3. One mistake:I find that you fixed the down index of g_μυ(like g_11 g_22 g_33) at the beginning when you calculated connection Γ.But after you fixed them, some ∂_σg_μυ which should be vanish do not be vanish.This resulted in a miscalculation of the connection Γ.Am I making myself clear? If not, I can show you my manuscript.

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    4. Not clear! I suppose you are at equations (94) and onwards. I checked a few they seem OK. Do send me your manuscript: george@general-relativity.net. I will check again later today.

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  2. I have sent my handscript to your email.I hope it won't be classified as spam:(

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