General Relativity: An Introduction for Physicists · Problem 5.9

When a particle's ordinary acceleration is perpendicular to its velocity, you're in a special geometric situation—the speed isn't changing, only direction. The key is to use the constraint that the 4-velocity has constant magnitude ($u^\mu u_\mu = -c^2$) to relate the 4-acceleration components to the 3-acceleration, then exploit the orthogonality $\vec{a} \perp \vec{u}$ to show the time component vanishes. This perpendicularity is the crucial simplification that makes the 4-acceleration take this clean form.

When a particle's ordinary acceleration is perpendicular to its velocity, you're in a special geometric situation—the speed isn't changing, only direction. The key is to use the constraint that the 4-velocity has constant magnitude ($u^\mu u_\mu = -c^2$) to relate the 4-acceleration components to the 3-acceleration, then exploit the orthogonality $\vec{a} \perp \vec{u}$ to show the time component vanishes. This perpendicularity is the crucial simplification that makes the 4-acceleration take this clean form.