Appendix

Abstract

Consider an observer with 4-velocity $\hat{u}$ performing measurements in an infinitesimal neighborhood $\hat{\mathcal{I}}$ of her position $\hat{P}$. She uses a local coordinate system $x^{\mu }$ to parametrize $\hat{\mathcal{I}}$ and therefore the basis $\partial _{\mu }$ to decompose tensors at $\hat{P}$. In particular, she parametrizes evolution with her proper time, meaning that the norm of the time-coordinate intervals $\mathrm{d}t \equiv \mathrm{d}x^0$ is unity.