Trace and index manipulation

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

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

• Indices ought to appear two times only. Jan 23 at 9:41
• Could you provide a bit more detail on how you got to this calculation? While this might initially seem superfluous, knowing a bit more about the context might help us understand what you are trying to do and what is going wrong. Jan 23 at 10:24

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: $$F_{ab}T^{ab}=F_{ab}T^{ab}=F_{ab}T^{a}_{\ \ c}h^{cb}$$

• Is it possible to introduce the trace on the RHS? Jan 23 at 10:16
• I don't think so. Jan 23 at 10:48

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)