🧄Tensor Form of the Centrifugal Force in Rotating Frames

The centrifugal force in a rotating frame can be elegantly expressed in tensor form as $F_c^i=T^{i j}{ }_x{ }^j$, where the tensor $T^{i j}=m\left(\omega^i \omega^j-\omega^2 \delta^{i j}\right)$ captures the dependence on angular velocity and mass. This formulation reveals that the force is symmetric and linear in position, acting radially outward from the axis of rotation. It vanishes precisely when the particle lies along the rotation axis, where $\vec{\omega} \times \vec{x}=0$, highlighting the geometric nature of fictitious forces in non-inertial frames.

Tensor Form of the Centrifugal Force in Rotating Frames | Proof and Derivationviadean on Notion