A geometric framework for spin relaxation.

Spin relaxation is conventionally described by two independent phenomenological rates - longitudinal (R₁) and transverse (R₂) - whose separation obscures a deeper structural unity. Here we develop a geometric framework in which dissipation is represented by a single covariant relaxation tensor acting in Liouville space, from which R₁ and R₂ emerge as complementary projections. This tensor structure is not merely formal but is experimentally accessible through pulse sequences that probe noncommuting directions in spin space. Using hyperpolarized 13C spins in diamond with nitrogen-vacancy centers, we show that commuting pulse trains yield effective relaxation matrices that are approximately diagonal, while noncommuting sequences produce off-diagonal components that vary with transmitter frequency offset and pulse ordering, providing evidence that relaxation is a directional process governed by a tensor rather than a pair of scalar rates. Complementary measurements of geometric phase demonstrate that noncommuting dynamics introduce ordering-dependent effects that are separable from dissipation, consistent with the interpretation of relaxation as geometric transport on the state manifold. This framework unifies Bloch, Redfield, and Lindblad descriptions within a coordinate-independent formulation and provides a natural language for relaxation in driven, anisotropic, and non-equilibrium spin systems.