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Tensors and Relativity: Chapter 6

# Tensors in polar coordinates

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 ( , ) and polar coordinates ( , ). The coordinates are related by

For neighboring points we have

and

We can represent this by a transformation matrix :

where

Any vector components must transform in the same way.

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

We have

and

This transformation can be represented by another matrix :

where

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

The matrices and are different but related:

This is just what you would expect since in general .

The basis vectors  and basis one- forms  are

and

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 :

The inverse metric tensor is:

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

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

We also have:

and

Since

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

and all other components are zero.

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

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

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