# Tensor contraction proof

1. Feb 3, 2010

### quasar_4

1. The problem statement, all variables and given/known data

Show that the definition [of tensor contraction]

$$A^{ae}{}_{cde} = u^a \nu^e \sigma_c \tau_d \omega_e + w^a x^e \zeta_c \eta_d \xi_e + ...$$

implies

$$A^{im}{}_{klm} \equiv \sum_{m=0}^{n-1} A^{im}{}_{klm}$$

first by looking at tensors of the form $$u^a \sigma_b$$, then of the form $$u^a...\nu^b \sigma_c... \tau_d$$, and finally of linear combinations of these.

2. Relevant equations

We're working in Minkowski space, so we have n basis vectors.

3. The attempt at a solution
I guess I've been thinking I'll just expand into a basis. Does that make sense for this? Here's what happens (if I'm getting notation correctly):

$$A^a_b = u^a \sigma_b = a^a_i e^i a_b^j e_j$$

so if a = b = m, then

$$A^m_m = a^m_i e^i a^j_m e_j = a^m_i a^j_m \delta^i_j$$

and then (this is the part I'm not sure about...)

$$A^m_m \equiv \sum_m a^m_i a^j_m \delta^i_j = \sum_m A^m_m$$

If that is correct I have no problems generalizing to higher rank tensors, just wasn't sure if the sum was introduced correctly. I guess I'm thinking that this is only non-zero for i = j, and then we can sum over m to get all the components... is that right?