Contravariant components to covariant

[cavus]

Hi, everyone

I was playing with the coordinate transformations and metric tensors to get a feeling of how it all behaves, and got stuck with some basic problem I am hoping you can help me with.

So, I have defined a coordinate system (s,t), with the s axis going along the x-axis in the cartesian coordinates, and t axis going along the y=x line:
s = x-y
t = y*sqrt(2)

with inverse transformation:

x = s + t/sqrt(2)
y = t/sqrt(2)


If I am differentiating correctly, the metric tensor in these coordinates looks like:
1 (2+sqrt(2))/2
(2+sqrt(2))/2 1

g₁₁ = g₂₂ = 1,
g₂₁=g₁₂ = (2+sqrt(2))/2

Now, I pick a point (3,1) in cartesian coordinates, and transform it to my new frame, and get the contravariant coordinates as (2, sqrt(2)).
So far so good. What I am trying to do is find out what its covariant coordinates are going to be. I think, that covariant coordinates are supposed to be the lengths of orthogonal projections of the vector on the respective axes. From basic geometry, I get (3, 2*sqrt(2)).

The problem is that when I try to multiply my metric tensor by the contravariant vector, I get a different answer - (3+sqrt(2), 2+2*sqrt(2))
Clearly, there is something I am doing wrong here, but I can't figure out what it is :( Can somebody please help me spot the problem?

Thanks a lot for your help!

 

[weaselman]

Your metric is wrong. The mixed components (g₁₂ and g₂₁) should be 1/sqrt(2).
Because you did not show how you got your metric tensor, I can't say where you went wrong, but if your check your index dropping with the correct metric, you'll see that it fits.

Hope, it helps ...