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Suppose I have a symmetric tensor field $h_{\mu\nu}$ I want to implement somehow the following gauge variation of this tensor field as follows

$\delta h_{\mu\nu} = \nabla_{\mu}\epsilon_{\nu} + \nabla_{\nu}\epsilon_{\mu}$

and with this operator compute various gauge variations of different objects.

For example, suppose I have the following tensor in a flat background(which is gauge-invariant by definition)

$k_{\mu\nu} = \Box h_{\mu\nu} - \nabla_\mu \nabla^{\lambda} h_{\nu\lambda} - \nabla_\nu \nabla^{\lambda} h_{\mu\lambda} + \nabla_{\mu\nu}h_{\lambda}^{\lambda}$

where $\Box = \nabla_{a}\nabla^{a}$

The gauge variation of the $k_{\mu\nu}$ tensor should be zero

${\delta k_{\mu\nu}} = 0$

How can I implement the variation operator in Wolfram Mathematica using xAct package? Is it possilbe?

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