Physical Interpretation of Coordinates in GR

[MrBillyShears]

What is the relationship between the differentiable manifold that is space-time and the physical space around us? How does one relate the three seemingly Cartesian coordinates around us, those which we can measure out with a ruler, to the coordinates of the Lorentzian manifold? If i say, measure out a length with a ruler, how would that relate to the three spatial coordinates of space-time? I'm just getting all confused thinking about this. Maybe this question doesn't make much sense, but I just want to see if anyone can help me with this.

 

[WannabeNewton]

What is the relationship between the differentiable manifold that is space-time and the physical space around us?

A local observer that makes measurements using local rods, clocks, gyroscopes etc. can represent these measurements in the form $T = T^{abc...}e_{a}e_{b}e_{c}...$ where $\{e_a \}$ is an orthonormal basis along the worldline of the observer (that is, a local Lorentz frame) and $T$ is a tensor that represents the measurements of the quantity in question. The components $T^{abc...}$ relative to the local Lorentz frame represent the measurements of the components of this quantity along the time and spatial axes carried by the observer. However these measurements can only be made at the location of the observer e.g. it could be the relative velocity of another observer just when passing by the first. But what about measurements that aren't made right on the worldline?

Taking this basis $\{e_a \}$ the observer can use parallel transport $\nabla_{\xi}e_a = 0$ along space-like geodesics with tangent $\xi$ emanating orthogonally from the worldline to define a local coordinate system, called a Fermi-normal coordinate system, in a sufficiently small neighborhood around the worldline. Imagine the basis as a set of three mutually orthogonal meter sticks, oriented say along gyroscopes, and a clock. Then the Ferm-normal coordinate system represents a very small laboratory comoving with the observer that contains a lattice of these rods and clocks, the latter all being synchronized assuming the lab is small enough.

The observer can now make measurements anywhere within this laboratory, e.g. the spatial distance between two flashes of light going off simultaneously relative to this observer in the Fermi-normal coordinate system. Coming back full circle to your question, values of quantities in an arbitrary coordinate system do not represent the physical values as measured by local observers. Coordinate values are in general physically meaningless. You must convert these values into the local Lorentz frame (or Fermi-normal coordinate system), as in the first paragraph above, in order to get the observables relative to a given observer.

So for example in Schwarzschild space-time the coordinate velocity $\frac{dr}{dt}$ of an observer in a radial free fall is not physically meaningful as is obvious from the fact that $\frac{dr}{dt}\rightarrow 1$ as one approaches the event horizon. If we convert the quantities $dr$ and $dt$ to the proper distance and proper time measurements made by local observers then we get physical measurements.

Now let $\{\partial_{\mu} \}$ be a coordinate-basis for the space-time. If we have a local Lorentz frame $\{e_a \}$ for a specific observer then we know $e_a = e^{\mu}_a \partial_{\mu}$. The matrices $e^{\mu}_a$ tell us how to go from the values of quantities in the space-time coordinates to the physical measurements made by the observer.

C.f. chapter 6 of MTW and http://en.wikipedia.org/wiki/Frame_fields_in_general_relativity

 

[cosmik debris]

A differentiable manifold is taken to be locally flat, just as on Earth we measure things close to us in kilometres and assume angles in triangles add to 180 degrees even though we know we live on a curved surface. The units we measure here are the same as the spatial co-ordinates on a flat Lorentzian manifold, time is an extra dimension of the manifold instead of a parameter in 3D.

 

[pervect]

What is the relationship between the differentiable manifold that is space-time and the physical space around us? How does one relate the three seemingly Cartesian coordinates around us, those which we can measure out with a ruler, to the coordinates of the Lorentzian manifold? If i say, measure out a length with a ruler, how would that relate to the three spatial coordinates of space-time? I'm just getting all confused thinking about this. Maybe this question doesn't make much sense, but I just want to see if anyone can help me with this.

[add]
The very short answer is that coordinates are just labels, so they don't in and of themselves have much physical significance. But this isn't too helpful, really, so instead consider the following analogy, which hopefuly will be helpful.

Consider the surface of a sphere. It's a 2 dimensional. It's also a manifold, but it's not a plane.

The surface of the sphere is embedded in a higher dimensional space, but the surface itself is two dimensional.

You can apply coordinate systems to the surface of the sphere, lattitude and longitude for instance, but they aren't cartesian coordinates, at least not globally. The coordinates are really just labels anyway.

Any small part of the sphere looks flat. You can generate coordinates that are nearly cartesian for a small part of the sphere, but you can't cover the whole sphere with them. Similar remarks apply to the curved manifolds in GR.

What you need to get an actual description of the geometry is distances. You have available the coordinates, you now need a tool that takes in the coordinates (and coordinate changes) and outputs the distances. The mathematical tool you need to do this is called a "metric". This same mathematical tool is what's used in GR. I'm not sure how much detail you want or need, so won't go into the specifics of how the metric works unless I get a further question, except to say that it coverts coordinate changes into distances.

The mathematical approach of the metric doesn't rely or need the concept of embedding a lower-dimensional manifold in a higher dimensional space, as we did with the surface of the sphere (2d) in a higher dimensional (3d) space. The embedding is helpful for purposes of visualization, but it's not unique or required to do the math.

As far as your questions about rulers go, note that if you have small rulers, then you can construct a coordinate system that will directly measure distances in the way you are used to - a distance will be the same as a change in coordinates, at least to a high degree of precision. So there isn't any real mystery about how to handle small distances.

If you've got larger rulers, you need to do things like replace "straight lines" with geodesics. For instance in our example of the surface of a sphere, the geodesics are "great circles". There are some details on defining geodesics that I will skip over, but for the purpose of GR, and the limitation of "short" distances, you can think of a geodesic as the shortest curve on the manifold between two "close" points. You can find the length of any curve via the metric and integration, geodesic or not.