From Orthogonal to Non-Orthogonal Systems

[Anamitra Palit]

Let's consider a 2D orthogonal [x-y] system having origin at O in the
flat space context..A is a point on the x-axis and B is another on
the -axis.ABC is a right angled triangle with

AB^2=OA^2+OB^2

We now transform to a non-orthogonal system in the same manifold[flat
space]We make the angle between the axes x' and y' =theta not equal to
a right angle.The axes are maintained as straight lines.The value of
the line element is preserved .

A----->A'
B----->B'
O---->O'

Since ds^2 is preserved

OA=O'A'
OB=O'B'
AB=A'B'

AB is the shortest distance between A and B in the original distance.
If you try to figure the situation on a piece of flat paper[after the
transformation], the distance A'B' along a straight line path in the
new frame will not be equal to AB

Actually the straight line AB is becoming a curved line. A'B' is a
curved line. What about the straight line distance between A' and B'?
Should we allow such straight lines in the transformed situation.



We indeed have the relation:

A'B'=O'A'^2+O'B'^2
In the new frame.We have Pythagorean for a triangle for a triangle
without a right angle--of course on a curved surface. The problem
consists in the fact that we may A'B' as a straight line. We may
consider the x'-y' surface to become a curved one after the
transformation from the x-y system to remove the problem. We simply do
not have straight line path between A' and B' in the new frame!

We have passed into a new manifold with the preservation of ds^2 but
with the non-preservation of angles.We may do some sort of re-
labeling of coordinates to make the new system orthogonal.

The significance of these Transformations:

Tensors are usually defined by relations like T'^[mu,nu]=[del
x'^[mu]]/]del x^[alpha]] [del x'^[mu]]/]del x^[alpha]] T^[alpha,beta]

Such definitions assume relations like:
x'^[mu]=f1(x^[alpha],x^[beta]
ds^2 does not change in these relations. But angles between curves
might change.
The salient feature or the point of advantage is that we may pass from
one manifold to another, giving a mathematical foundation to the
universality of the physical laws--that they have the same tensor
form[covariant form] in all distinct manifolds.

Any failure to pass between different manifolds[by suitable
transformations] would restrict the tensor object to a particular type
of manifold.Such a situation would be a hindrance towards the claim
of the universality of the physical laws [in covariant form].