Hi, everyone(adsbygoogle = window.adsbygoogle || []).push({});

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

g_{11}= g_{22}= 1,

g_{21}=g_{12}= (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!

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# Homework Help: Contravariant components to covariant

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