Physically measure the covariant and contravariant components of a vector? I'm just wondering if there is a way to physically measure the covariant and contravariant components of a vector without prior knowledge of the metric.
Suppose I have a speedometer of some sort to measure the speed relative to me, of some faraway slowly moving and non interacting dust cloud in some area with dark matter. Said speedometer would give me measurements of the vector field in contravariant form (or covariant, doesn't matter).
However as one can't yet observe anything about dark matter save it's gravitational effects, I am unable to write out the energy momentum tensor to derive the metric.
Simply by knowing the covariant and contravariant components of the vector field of the cloud, I could probably derive the metric and thus solve for the EM tensor and therefore derive the matter distribution and movements of said dark matter (I think?).
But from the measurements of my speedometer, I would only get covariant components of the vector field. Since I do not have knowledge of the metric itself, and as far as I am concerned the coordinates I am using may as well be Cartesian,or at least Minkowskian since it 'looks' flat, without the knowledge of the matter and momentum distribution of dark matter, how do I find the covariant components and therefore be able to derive a metric?
I'm sure there is a way to do so, else how do astronomers calculate the masses of black holes or the amount and distribution of the missing mass that is dark matter?
 A: As a consequence of the equivalence principle, the metric is not observable at a point in general relativity. Only scalars formed from its second and higher derivatives are observable at a point. This means that there is no meaningful sense in which we could measure, or need to measure, both a contravariant vector and its covariant version. If we could measure them both, then we would be able to infer the metric from them.
There is not really any notion of finding an unknown metric in general relativity, although we often say it that was as shorthand for something that is mutually understood. What we're really finding is a metric over some region of space, up to an equivalence class based on an arbitrary smooth change of coordinates. The metric at one point doesn't need to be found. We can make it anything we like, as long as it has the correct signature.
The question seems to be written as if the Einstein field equations need to be expressed in a form where the stress-energy tensor is in covariant form. You can freely raise and lower indices on both sides of the EFE.
