New Bee question on Gauss Coordinates systems

[eon_rider]

Hello, :)

The below link has great examples and simple explanations of
4D space-time light cone diagrams and how they are used in Special Relativity.
http://www.phy.syr.edu/courses/modules/LIGHTCONE/LightClock/default.html
The university site above was very helpful for visualising how special relativity is graphed.

It’s all good,

But does any expert or enthusiast of General Relativity know of
any sites that show decent visual examples and straight forward explanations
of a Gauss or Gaussian co-ordinate systems? The short book I’m reading on G.R. refers to Gauss co-ordinates but does not show any good examples of what the Gauss co-ordinate graphics look like in full.

I did do a search for images of gauss co-ordinates but
came up with very little. I must not be using the best key words.

Thanks in advance for any helpful links.

Very best

Eon.

 

[eon_rider]

I found the short answer may be that gaussian coordinates are just Cartesian coordinates wrapped over a curvature.
or
a Gaussian coordinate system is just a curved Cartesian coordinate system

 

[pervect]

Imagine shooting 3 bullets all at right angles from each other from a point. The trajectory of the bullets will trace out the axes of a Gaussian coordinate system. You can use the "proper time" of each bullet (imagine that each bullet contains a clock, too) to mark the "tic marcs" on the coordinate axes.

When you imagine doing this in an inertial coordinate system, you get a Cartesian coordinate system. In a general curved space-time, this coordinate system comes as close as possible to a Cartesian coordinate system near the origin (the point from which the bullets were fired).

You should imagine that the bullets are flying a 'free-fall' trajectory and not hitting anything, BTW.

 

[eon_rider]

Imagine shooting 3 bullets all at right angles from each other from a point. The trajectory of the bullets will trace out the axes of a Gaussian coordinate system. You can use the "proper time" of each bullet (imagine that each bullet contains a clock, too) to mark the "tic marcs" on the coordinate axes.
When you imagine doing this in an inertial coordinate system, you get a Cartesian coordinate system. In a general curved space-time, this coordinate system comes as close as possible to a Cartesian coordinate system near the origin (the point from which the bullets were fired).
You should imagine that the bullets are flying a 'free-fall' trajectory and not hitting anything, BTW.

That makes sense. Thanks.
I'm reading "Albert Einstein's Relativity: The Special and General Theory"

A.E. writes "Under these conditions, the u-curves and v-curves are straight lines in the sense of Euclidean geometry, and they are perpendicular to each other. Here the Gaussian coordinates are simply Cartesian ones. It is clear that Gauss co-ordinates are nothing more than an association of two sets of numbers with the points of the surface considered, of such a nature that numerical values differing very slightly from each other are associated with neighbouring points " in space."

So according to the link below, the bullet's trajectory is like the "U" curve. Each "tic marc" is a part of the "V" curve and the intersections are P1, P2, Pn...etc.
Here's a link to the small chart he draws of a gassian system. http://www.bartleby.com/173/25.html
Thanks again.
eon

 

[pervect]

I have to apologize -the coordinates I were describing were actually Riemann normal coordinates, not Gaussian coordinates :-(.

It's clear from the text that Gaussian coordinates are perfectly general, the u and v curves, for instance, are not constrained to cross each other at right angles. There is also no requirement for the "tic marks" to be uniformly spaced.

 

[eon_rider]

the coordinates I were describing were actually Riemann normal coordinates, not Gaussian coordinates :-(.

No worries and no need to apologize.
I didn't see any issue with your description.

Honestly, the way you described Riemann normal co-ordinates was
close enough for a non-expert like myself, that your description appeared to work for the gaussian coordinates.

You didn't mention right angles. And your feedback did spark my own further research into some of the geometry.

Thanks for the clarification.

best,

Eon.

EDIT: oops...lol...I re-read your description.
You did mention right angles...NO Worries. Still I got it. Thanks. :)