Trace and index manipulation

Question

Imagine that I have a quantity $F_{ab}$ multiplying the stress tensor $T^{ab}$:

\begin{equation} F_{ab} T^{ab}. \end{equation} There is also a metric, say $h_{ab}$. If I want to write the above equation in terms of the trace, $T^a_a = T^{ab}h_{ab}$ (I think this is the right definition). Then can I write \begin{equation} F_{ab} T^{ab} h_{ab}h^{ab} = F_{ab}T^c_c h^{ab} \end{equation} or is this not an allowed computation? Since the summation convention is implicit, I changed notation from $T^a_a \to T^c_c$.

Edit: so it seems like the problem is that the indices appears too many times. But then the question is if a trace could be introduced on the RHS.

Answer 1

You cannot write $F_{ab}T^{ab}$ only in terms of the trace $T^a_a$. A simple argument is that $$F_{ab}T^{ab}=F_{00}T^{00}+F_{01}T^{01}+\ldots + F_{10}T^{10}+\ldots + F_{dd}T^{dd}$$ involves all the matrix elements $T_{ab}$. In the special case of a diagonal metric, the trace involves only the diagonal elements $T_{aa}$. Therefore, $F_{ab}T^{ab}$ cannot be expressed only in terms of the trace.

In your calculation, you cannot use more than twice the same index: $$F_{ab}T^{ab}=F_{ab}T^{ab}=F_{ab}T^{a}_{\ \ c}h^{cb}$$

Answer 2

Simply to find out the trace $T^{ab}h_{ab}$ works and gives a scalar (easy to visualize when the metric is diagonal).

To get the RHS, try $F_{ab}T^{ab}h_{bc}h^{cd}\delta_d^b$=$F_{ab}T^a_ch^{bc}$.

Now do not rewrite the dummy $a$ to $c$ since new $c$ will be different than the esisting one, it becomes $F_{c'b}T^{c'}_ch^{bc}$

(Considered only symmetric tensors)