Godel's metric in cylindrical coordinates

[Ramon03]

Hello,

In Godel's paper: an example of a new type of cosmological solutions of einstein's field equations of gravitation, he passes from his original metric to cylindrical coordinates by giving some transformation formulas. Can someone tell me how is this transformation obtained, or at least what does it mean geometrically?

Thanks

 

[Mentz114]

Hello,

In Godel's paper: an example of a new type of cosmological solutions of einstein's field equations of gravitation, he passes from his original metric to cylindrical coordinates by giving some transformation formulas. Can someone tell me how is this transformation obtained, or at least what does it mean geometrically?

Thanks

What were the original coordinates ? Have a look at this where the transformations from Cartesian to cylindrical and spherical polar to cylindrical coords are written out.

http://en.wikipedia.org/wiki/Cylindrical_coordinate_system

 

[Ramon03]

Ok, I should have explained myself better. The original metric is written in cartesian coordinates x_0, x_1, x_2 and x_3 (metric signature (+,-,-,-)).

He wants to show his metric has rotational symmetry so he goes to cylindrical coordinates. However, as we have local rotation around the x_3 axis, the transformation is done in a x_3 = constant hypersurface, so the cylindrical coordinates involve, space and time.

The transformation laws are given at the beginning of page 3 http://rmp.aps.org/pdf/RMP/v21/i3/p447_1

 

[pervect]

Hello,

In Godel's paper: an example of a new type of cosmological solutions of einstein's field equations of gravitation, he passes from his original metric to cylindrical coordinates by giving some transformation formulas. Can someone tell me how is this transformation obtained, or at least what does it mean geometrically?

Thanks

The metric is a tensor, and it transforms according to the standard tensor transformation rules when one changes coordinate systems.

Geometrically, coordinates have no significance at all. You can assign coordinates in any way you like to make the problem simpler. The way this works is that tensors are geometric objects, and the coordinate system used is just a representation, or view, of those fundamental objects. Changing the "view" isn't considered to actually change the object, and the rules of using tensors are such that if a tensor equation is valid in one coordinate system, it's valid in all.

I have the feeling that going into more mathematical detail wouldn't necessarily be productive, but if you want or need to know more, please do ask.

 

[Ramon03]

Yes pervect, I get your point, but still is quite annoying for me to think something came out from nowhere. I suppose the transformation law given (http://rmp.aps.org/pdf/RMP/v21/i3/p447_1 see beginning of page 3) should have some meaning and I would like to have an idea how to get it. If you have some more ideas I would like to hear them. Thanks

 

[pervect]

Yes pervect, I get your point, but still is quite annoying for me to think something came out from nowhere. I suppose the transformation law given (http://rmp.aps.org/pdf/RMP/v21/i3/p447_1 see beginning of page 3) should have some meaning and I would like to have an idea how to get it. If you have some more ideas I would like to hear them. Thanks

The good news is that it's just linear algebra - because that's what tensors are. The bad news is that it's something that all the gory details are messy enough that you'll need to read a textbook about it, I'm not going to try to explain it fully in a post. Many textbooks will simply DEFINE tensors by their transformation properties.

I can give a pretty simple motivational example, though.

Suppose you have some coordinate system , and a metric on it with some component, so that the line element is g_00 dt^2 + (some space terms)

And you assert that said line element has some physical significance independents of the coordinates used.

Now, suppose you change coordinates, so that t' = \alpha \,t. And you don't change the space terms at all. You consider the same "physical" volume under a passive change of coordinates, which re-label the volume. If the quantity g_00 dt^2 has some physical significance, it can't change when you change the labels. Then

then dt' =\alpha \,dt , so that g_00 dt^2 = g_00 (dt' / \alpha)^2. You can conclude then that g_00 must transform so that in the primed system

g_0'0' dt'^2 = g_00 (dt/dt') (dt/dt') dt^2= g_00 \alpha^2

This is part of the general transformation law, which you'll see written out in lots of textbooks, for instance http://www.scribd.com/doc/56304177/22/The-tensor-transformation-laws

In general, things will transform either proportionally to (dt/dt'), or (dt'/dt), which corresponds to covariantly or contravariantly.

In general, you'll have a lot more partial derivatives than the simple example I got here, the formulae will have a lot more terms, and you'll need to use tensor notation to keep tract of them. Though that won't necessarily explain correctly all the transformation rules, it's a start.

I suppose the really short version is that if you write your metric out in terms of the line element, you can use algebra to perform your tensor transformations, like I did in the motivational example.

To get the full details, you'll need to study tensors. To get all the details right and fuly understand them, you'll need to take a course, not read a post on the net.

ps- reading over this, I can see I've been a bit sloppy on the notation. Sorry about that, but it's another reason to go to a textbook on the topic, now that you know what sort of textbook you need.