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Which local coordinates?

  1. Nov 8, 2014 #1
    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 any one 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?
     
    Last edited: Nov 8, 2014
  2. jcsd
  3. Nov 8, 2014 #2

    aleazk

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    Tetrads are not usually associated to coordinate systems, they are non-coordinate orthonormal bases on the tangent spaces. If they were associated to a coordinate system, then the curvature would be zero. In general metrics, some of the elements of the tetrads don't commute, and therefore can't be associated to a coordinate system. The non-zero commutators can be used to calculate the components of the curvature tensor in this basis. This is called the 'tetrad method for calculating curvature', 'Ricci rotation coefficients method', etc. It can simplify the computations when compared to the usual coordinate approach.

    Since tetrads are orthonormal, they are also often used to study the physical interpretations of certain quantities, or to simplify certain computations.

    The Riemann Normal Coordinates are central in abstract causality theory, since one of the main theorems there states that the causality properties in any normal neighborhood of a point p are the intuitive ones we have from flat spacetime (this theorem is naturally proved by using RNC). A lot of other more global causal results can be obtained by applying this. Usually, one takes a segment of a causal curve and a normal neighborhood at each point of the segment; the collection of these neighborhoods form an open cover of the segment; since the segment is compact, you can take then a finite subcover. You then try to prove 'global' properties about the segment by 'joining' the results obtained in each of these neighborhoods. This can be done without further complications since the process is finite.
     
    Last edited: Nov 8, 2014
  4. Nov 8, 2014 #3

    pervect

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    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:

    640px-Tangentialvektor.svg.png

    "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 [Broken]

    http://relativity.livingreviews.org/Articles/lrr-2004-6/figure06.png [Broken]

    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.
     
    Last edited by a moderator: May 7, 2017
  5. Nov 9, 2014 #4
    Phew, thanks for all that. I've got some reading to do today, but it seems like you are saying that they are all part of an integrated picture rather than and either-or. That would be a good thing for me.
     
  6. Nov 9, 2014 #5

    pervect

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    The concepts are all related, though I'm not sure if this will be stressed or not in whatever formal treatment you find. The way I'd describe it is that Fermi Normal coordinates best match the idea of an "observer", as you can imagine an abstract worldline parameterized by proper time as representing an observer, flying through space-time, carrying around a clock. It has a mechanism for generating local surfaces of simultaneity, which I described rather abstractly, but you could instead imagine that locally the simultaneity was determined via radar methods for the observer in regions that are so small that curvature effects aren't important. Once you have this "local" defintion of simultaneity, you still need the uniqueness of geodesics to propagate the local notion of simultaneity over a larger region. So you start your spacelike geodesics in the "direction" of some nearby point that you've determined to be simultaneous by radar measurements, then you propagate the spacelike geodesic along that direction to determine the full surface of simultaneity.

    Riemann normal coordinates all work around a single point, or event. There is nothing that's directly analogous to an observer, and nothing that distinguishes space from time, so there's no real concept of time or simultaneity.

    As far as references go, MTW's "Gravitation" has some good discussions. I'm not sure who else does.
     
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