# Tensor Analysis

1. Feb 25, 2010

### iamalexalright

1. The problem statement, all variables and given/known data
For ease of writing, a covariant tensor $$\bf G..$$ will be written as $$\bf G$$ and a,b,c,d are vectors.

Let $$\bf S$$ and $$\bf G$$ be two non-zero symmetric covariant tensors in a four-dimensional vector space. Furthermore, let S and G satisfy the identity:

$$[\bf G \otimes \bf S](\vec a, \vec b, \vec c, \vec d) - [\bf G \otimes \bf S](\vec a, \vec d, \vec b, \vec c) + [\bf G \otimes \bf S](\vec b, \vec c, \vec a, \vec d) - [\bf G \otimes \bf S](\vec c, \vec d, \vec a, \vec b) \equiv 0$$

for all a,b,c,d in V. Prove that there must exist a scalar $$\lambda \neq 0$$ such that

$$\bf G = \lambda \bf S$$

2. The attempt at a solution
First we write it as:
$$\bf G(\vec a, \vec b)\bf S(\vec c, \vec d) - \bf G(\vec a, \vec d)\bf S(\vec b, \vec c) + \bf G(\vec b, \vec c)\bf S(\vec a, \vec d) - \bf G(\vec c, \vec d)\bf S(\vec a, \vec b)$$

My first thought it to set $$\alpha = \bf G(\vec a, \vec b) \ and \ \beta = \bf G(\vec a, \vec d) \ and \ \gamma = \bf G(\vec b, \vec c) \ and \ \delta = \bf G(\vec c, \vec d)$$ since they are just scalars. After utilizing the symmetric properties of the tensors we get:

$$\alpha \bf S(\vec c, \vec d) - \beta \bf S(\vec c, \vec b) + \gamma \bf S(\vec a, \vec d) - \delta \bf S(\vec a, \vec b)$$

Simplifying:
$$\bf S(\vec c, \alpha \vec d - \beta \vec b) + \bf S(\vec a, \gamma \vec d - \delta \vec b)$$

Which doesn't seem to get us anywhere.

I next tried to use a different substitution(same alpha and beta but this time gamma and delta I set to be S(...) and I get:

$$\bf S(\vec c, \alpha \vec d - \beta \vec b) = \bf G(\vec c, \delta \vec d - \gamma \vec b)$$

This looks slightly more promising as far as I can tell but I don't know where to go.
I tried to do this:
$$\alpha \vec d - \beta \vec b = \bf G(\vec a, \vec b)\vec d - \bf G(\vec a, \vec d) \vec b =$$
$$= \bf G(\vec a, \vec b)d_{i} - \bf G(\vec a, \vec d)b_{i}$$

But that doesn't seem to pan out (I took a few more steps in this direction but it doesn't seem to go anywhere useful).

Any suggestions?