m4r35n357 said:
In my, er, studies I've encountered descriptions of what I understand to be various ways to go from global to local coordinates. These are: tetrads, Riemann Normal Coordinates and Fermi Normal coordinates. Until now I haven't investigated much further than that, mostly because I've not been able to really see where I would want to chose anyone over the others.
I am reasonably OK with getting orthonormal tetrads from a metric, but even then I'm not really sure where to go next. I only understand RNC as some kind of local polar-ish coordinate sytem, and I have only really been alerted to FNC via a recent comment by pervect on a thread here.
Anyway, the question is: Is there a reasonably snappy way of describing what sort of problems or applications each technique is most suited to?
Let's start by imagining the curved 2d surface of some 3d object, which we will call M (think Manifold) and another 2d surface, P which is a flat plane, tangent to M at some particular point. I'll borrow a wiki image that's close:
"Tetrads", which only have two vectors, because it's a 2d surface, span the tangent plane P. Any unit vector v in the space of P will have some counterpart geodesic curve, ##\gamma## that exists in M rather than P, that starts out "in the same direction" as v.
You need some idea of a geodesic to appreciate this, in GR the "shortest distance between two points" will do. Otherwise you have the problem of a lot of curves being associated with every vector v, rather than just one.
You can mark lengths along the curve ##\gamma##, and via this mechanism associate vectors that have both a magnitude and a direction with points on M, by using the direction of the vector v to pick out the curve ##\gamma## that "goes in the same direction", and the length of the vector to pick out some point that's the specified distance along the curve away from the origin. (The origin is where P touches M).
Up until the point where your geodesic curves ##\gamma## cross, you can take a point on M, convert it back to a point on P via the above mapping technique, and express the coordinates of the point in M by the coordinates on the flat plane P which you already know how to do in terms of "tetrads".
Your coordinate system, which is the Reimann normal coordinate system, only handles points in the region before geodesics cross.
Fermi normal coordinates are a bit harder to visualize, but they handle time and space in a familiar manner. You imagine some timelike worldline, traced out by some "observer" carrying a clock. You identify points at regular times along the worldline (as measured by the observer's clock), and construct sets of point that are "simultaneous" with point via the mechanism of drawing spacelike geodesics that are orthogonal to the observers wordline.
http://relativity.livingreviews.org/Articles/lrr-2004-6/fig_6.html
http://relativity.livingreviews.org/Articles/lrr-2004-6/figure06.png
To appreciate this, you need to appreciate both "geodesics" and "orthogonal geodesics" Recall that there is a 1:1 correspondence between vectors in the tangent plane, and geodesics, I think of this as the direction of the vector determining the direction of the geodesic.
Bascially, if you take two vectors, ##\mu## and ##\nu## that exist in the tangent plane P that by the above 1:1 correspondence determine geodesics ##\gamma_\mu## and ##\gamma_\nu##, the geodesics are orthgonal if the vectors are orthogonal. The vectors are orthogonal if their dot product is zero. You use the space-time metric to determine orthogonality, if you have vectors (t,x,y,z) and (t', x', y', z'), in a Minkowskii space with a diagonal metric (-1, 1, 1, 1) their dot product will be
##x \, x' + y \, y' + z \, z' - t \, t'##
The minus sign makes the product rule different than it is for the Euclidean vectors you're hopefully more familiar with.
You can see, hopefully, that a space-like vector with no time component (the t component is zero) is orthogonal to a time-like vector with no space component (x,y,z are all zero).
So, in Fermi normal coordinates, you define the time coordinate of every event by a clock carried along your worldline and using the orthogonal space-like geodesics to your worldline you define spacelike "surfaces of simultaneity". All events in this surface of simultatneity get the same time coordinate t, the time that is read on the clock that you imagine being carried along the worldline.
You basically use Riemann normal coordinates to get the spatial part of the Fermi Normal coordinates. You've already imagined a set of spatial geodesics going through your worldline at some particular point in time. The "starting direction" and "distance" along said spatial geodesics determines the spatial coordinates in the way we did before for Riemann normal coordinates.