# Contraction over two indices on a symmetric tensor

I'm reading a book on GR, and it is going over tensors. They say,

When contracting over a pair of upper indices that are symmetric on one tensor, only the symmetric part of the lower indices will contribute; thus, $$X^{(\mu\nu)}Y_{\mu\nu} = X^{(\mu\nu)}Y_{(\mu\nu)}$$

Here, $$X^{(\mu\nu)} = \frac{1}{2}\left(X^{\mu\nu} + X^{\nu\mu}\right)$$ is the symmetrization of the tensor. I don't follow this step - why do the asymmetric parts of the lower indices not contribute, and if they don't, even then what then allows us to write that equation?

• Have you tried writing it out and relabelling some dummy indices? Mar 27, 2021 at 17:14
• Ah, yeah, it turns out I just needed to do some algebra, which I haven't thought of. I am pretty new to tensors, so I haven't built up a good intuition of what to do when working with them yet. I suppose I'll answer the question and leave it up. Mar 27, 2021 at 17:19
• You could also use that $Y_{\mu \nu} = Y_{(\mu \nu)} + Y_{[\mu \nu]}$, then the above relationship is more intuitive since the product of the symmetric part and the antisymmetric part is zero due to antisymmetry. Mar 27, 2021 at 19:31

\begin{align*} X^{(\mu\nu)}Y_{(\mu\nu)} &= \frac{1}{4}\left(X^{\mu\nu}Y_{\mu\nu} + X^{\mu\nu}Y_{\nu\mu} + X^{\nu\mu}Y_{\mu\nu} + X^{\nu\mu}Y_{\nu\mu}\right) \\ &= \frac{1}{4}\left(2X^{\mu\nu}Y_{\mu\nu} + 2X^{\nu\mu}Y_{\mu\nu}\right) & \text{Renaming dummy variables} \\ &= \frac{1}{2}\left(X^{\mu\nu} + X^{\nu\mu}\right)Y_{\mu\nu} \\ &= X^{(\mu\nu)}Y_{\mu\nu} \end{align*}