# Why does the Minkowski metric in special relativity not change under a Lorentz transformation?

If the metric is a (0,2)-tensor then it transforms twice covariently under a basis transformation, but in special relativity the metric seems to be defined globally as:

$$\eta_{\mu\nu}=\begin{pmatrix}-1 && 0 && 0 && 0 \\ 0 && 1 && 0 && 0 \\ 0 && 0 &&1&&0\\0&&0&&0&&1\end{pmatrix}$$

in most texts I've seen introducing tensors when we change between basis we also have to change the metric, so why does the Minkowski metric not have to be transformed in the same way when we change our basis in Lorentz transformations:

$$\eta_{\mu\nu}'=\Lambda_{\mu}^{\nu}\Lambda_{\mu}^{\nu}\eta_{\mu\nu}$$

• That makes sense now thank you, however I've now written it out explicitly and the only way I can get the metric to be unchanged under the transformation is with the order $\eta'_{\mu\nu}=\Lambda_{\mu}^{\nu}\eta_{\mu\nu}\Lambda_{\mu}^{\nu}$, but in general with a (0,2)-tensor I see the transformation written as $\eta'_{\mu\nu}=\Lambda_{\mu}^{\nu}\Lambda_{\mu}^{\nu}\eta_{\mu\nu}$ when changing basis, why are the transformations on either side of the metric? From what I've seen this is what you would do for a (1,1)-tensor. Dec 7, 2019 at 18:48