smoothoperator said:
So, when we define a metric in curved space-time, what have we defined in general? These coordinate systems that are so oftenly mentioned confuse me very much.
The paper I like best on this issue is "Precis of General Relativity",
http://arxiv.org/abs/gr-qc/9508043. It's a rather abstract read. I'll attempt to simplify it some, the simplifications will hopefully be more helpful in understanding than they will be hurtful in the lost of preciseness.
We have three things we want to define - not necessarily a rigorous definition, but one of enough information that we can communicate sensibly about them. These three things are: metrics, coordinates, and physical measurements.
A metric is fundamentally a map of space-time. Coordinates are arbitrary labels on the map (like grid markers, B7, on a hopefully familiar 2d paper map of space - except that the coordinates in GR are all numeric rather than a letter-number combination like I used above. Those are 2 of the 3 things we want to define, the last thing we need to define are physical measurements. On the paper map, we can take (for our purposes) physical measurements as being the bearings of landmarks. Then by triangulation, with enough physical measurements (bearings), you can find your location (and the coordinates you labelled that location with on the map).
In GR, we can idealize the physical measurements as consisting of the readings of physical clocks, and the transmission and reception of radar signals (which can carry timestamp information from the clocks). We will assume for simplicity the signals propagate in a vacuum. If the signals don't propagate in a vacuum, nothing fundamental changes except that everything gets much more complicated and hard to explain. The theoretical model needs not only a map of space-time, but a map of the characteristics (velocity, density, composition) of the matter that the space-time contains, and details of how the presence of this matter affects the propagation of the specific signals you used. We really regard the underlying properties of the space-time itself as being fundamental, the presence of the matter is an experimental distraction that we need to compensate for.
How might we use these very limited tools to measure distance? Well if we send out a radar signal and the clock reads 0, and we receive a reflection of the radar signal and the clock reads 1 second, we know that when the clock read .5seconds, the object in question was half a light second away.
How do we use these very limited tools to measure coordinates? GPS, which the paper is about, serves as a good example. We send out signals from at least four reference satellites. If the reference satellites were fixed, we'd have an easier job, but it doesn't matter if they move as long as we know how they move. Then if the signals encode their transmission time, all we need to know is the reception time of the 4 signals to triangulate our position in space-time.
Now, one interesting point the paper makes is that specifying a metric operationally specifies the coordinates. While I suspect that the reaction of the average lay-person to being given a metric is one of confusion, rather than one of saying "Oh, this defines our coordinates", a metric does operationally specify coordinates.
The way the metric specifies the coordinates is rather similar to the way having a good 2d paper map of a tract of land specifies coordinates. You make measurements (reception and transmission times of radar signals in the space-time case, triangulation of landmarks in the paper map case) and you can operationally find where you are on the map, then you use the map to read out your coordinates. Recall that the coordinates are just the labels you've put on the map. So, once you've put the labels on the map, then when you make physical measurements, you can determine where you are on the map, and then you can communicate this information concisely by the labels you've put on your map, the coordinates.
There is one thing I've skipped over here, which it is assumed by the author of the above paper that you already knew, and which might not be obvious. This is that the space-time map specifies not distances, not times, but the observer independent "Lorentz Interval" that is the square root of distance^2 - time^2, the only observer-independent interval that special relativity has.
Now suppose we ask - how do we survey space-time? We often survey land , to draw an accurate map, what are some of the ideas we use to survey space-time? All we need to do is to have a large number of observers, who constantly exchange radar signals, and who compute the Lorentz interval between any pair of events. The metric can be viewed as a mathematical model that gives the Lorentz interval in terms of coordinate displacements, as well as being viewed as a map. This is how the map works, saying that the space-time map gives us the Lorentz intervals between all nearby points is similar to saying that a paper map gives the distances between all nearby points. The point is that the distances on the paper map are (in this non-relativistic context) observer -independent, and the Lorentz intervals are the equivalent observer-independent concept in the space-time case..
The abstraction here is all this focus on the Lorentz interval, which will become a familiar and friendly quantity if you seriously study relativity, but may not start out that way. If you had just the Lorentz interval between all pairs of possible points, how do you turn this into (for example), distances and times?
What you need to interpret the information about the Lorentz interval into space and time displacements are coordinates. These used to be just the labels on the map (in our previous discussion), but now we are applying them in a semi-physical way. Given coordinates, you can break the observer independent Lorentz interval into two parts, one part that is spatial only, the length of a path that has a constant time coordinate, and another path that has constant spatial coordinates, and progresses only through time. So by using your coordinates, you can split space-time into space, and time.
I think it might be less circular to say that you need information about the simultaneity, rather than a coordinate choice, to break space-time into space and time in a familiar way. Unfortunately, this got difficult to explain, so I went with the former view as being easier, and just about as good.
