❓Is the gravitational tidal tensor symmetric?

Yes, the gravitational tidal tensor $T_{i j}$ (or $T_j^i$ with appropriate lowering of the index) is symmetric.

$$T_{i j}=-\frac{\partial^2 \phi}{\partial x^i \partial x^j}$$

Since the gravitational potential $\phi$ is a continuous, smooth function in regions of space free of mass, the order of differentiation does not matter (Clairaut's Theorem or Schwarz's Theorem):

$$\frac{\partial^2 \phi}{\partial x^i \partial x^j}=\frac{\partial^2 \phi}{\partial x^j \partial x^i}$$

Thus, $T_{i j}=T_{j i}$.

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Is the gravitational tidal tensor symmetric? | FAQsviadean on Notion