Fun With Tensors

Fun with Tensors

George E. Hrabovsky

James Firmiss

Example 2-3 from David McMahon’s book Relativity Demystified.

Given the line element

we seek to find the metric tensors for raising and lowering indices for vectors and one-forms. We begin by noting that the metric tensor is symmetrical,

We then identify the terms of the metric tensor. We begin with the first term,

We also have the last term, 

Then we have the middle term, 2 dx dy. You might jump at this and say it is 2, but that would neglect that product of the differentials—each of which contributes half, so we have,

We can write the metric tensor,

We note that the product of the metric tensors is,

so,

We expand this term-by-term,

(1)

(2)

(3)

(4)

By (3) we get,

Rewriting (1)

or,

so,

Rewriting (2)

so,

or,

This gives us all of the components of the raised index metric tensor,

Given the vector



we can raise the index by this formula,

Expanding in coordinates,

This gives us

.

Given



we can lower the index by,

Expanding this in components,

This gives us the answer



Note that V=W.