  An exact kinematic law for the motion of disclination lines in nematic liquid
crystals as a function of the tensor order parameter $\mathbf{Q}$ is derived.
Unlike other order parameter fields that become singular at their respective
defect cores, the tensor order parameter remains regular. Following earlier
experimental and theoretical work, the disclination core is defined to be the
line where the uniaxial and biaxial order parameters are equal, or
equivalently, where the two largest eigenvalues of $\mathbf{Q}$ cross. This
allows an exact expression relating the velocity of the line to spatial and
temporal derivatives of $\mathbf{Q}$ on the line, to be specified by a
dynamical model for the evolution of the nematic. By introducing a linear core
approximation for $\mathbf{Q}$, analytical results are given for several
prototypical configurations, including line interactions and motion, loop
annihilation, and the response to external fields and shear flows. Behaviour
that follows from topological constraints or defect geometry is highlighted.
The analytic results are shown to be in agreement with three dimensional
numerical calculations based on a singular Maier-Saupe free energy that allows
for anisotropic elasticity.
