Tensors and Relativity: Chapter 6

Tensors in polar coordinates

next up previous index
Next: Parallel transport and geodesics Up: Title page Previous: Calculating from the metric

The covariant derivative differs from partial derivatives even in flat spacetime if one uses non- Cartesian coordinates. This corresponds to going to a non inertial frame. To illustrate this we will focus on two dimensional Euclidian space  with Cartesian coordinates ( tex2html_wrap_inline3575 , tex2html_wrap_inline3577 ) and polar coordinates ( tex2html_wrap_inline3579 , tex2html_wrap_inline3581 ). The coordinates are related by

equation2908

For neighboring points we have

equation2910

and

equation2912

We can represent this by a transformation matrix tex2html_wrap_inline3420 :

equation2915

where

eqnarray2919

Any vector components must transform in the same way.

For any scalar field tex2html_wrap_inline3392 , we can define a one- form:

equation2922

We have

eqnarray2924

and

eqnarray2926

This transformation can be represented by another matrix tex2html_wrap_inline3587 :

equation2929

where

eqnarray2933

Any one- form components must transform in the same way.

The matrices tex2html_wrap_inline3420 and tex2html_wrap_inline3587 are different but related:

eqnarray2938

This is just what you would expect since in general tex2html_wrap_inline3593 .

The basis vectors  and basis one- forms  are

eqnarray2946

and

eqnarray2948

Note that the basis vectors change from point to point in polar coordinates and need not have unit length so they do not form an orthonormal basis :

equation2950

The inverse metric tensor is:

eqnarray2952

so the components of the vector gradient tex2html_wrap_inline3595 of a scalar field tex2html_wrap_inline3392 are :

eqnarray2957

This is exactly what we would expect from our understanding of normal vector calculus.

We also have:

equation2959

and

equation2961

Since

equation2963

we can work out all the components of the Christoffel symbols :

equation2970

and all other components are zero.

Alternatively, we can work out these components from the metric:

equation2972

This is the best way of working out the components of tex2html_wrap_inline3599 , and it is the way we will adopt in General Relativity.

Finally we can check that all the components of tex2html_wrap_inline3601 as required. For example

eqnarray2987


next up previous index
Next: Parallel transport and geodesics Up: Title page Previous: Calculating from the metric