Motion of Space Curves in Three-dimensional Minkowski Space $R_1^3$, SO(2,1) Spin Equation and Defocusing Nonlinear Schr\"odinger Equation

We consider the dynamics of moving curves in three-dimensional Minkowski space $R_1^3$ and deduce the evolution equations for the curvature and torsion of the curve. Next by mapping a continuous SO(2,1) Heisenberg spin chain on the space curve in $R_1^3$, we show that the defocusing nonlinear Schr"odinger equation(NLSE) can be identified with the spin chain, thereby giving a geometrical interpretation of it. The associated linear eigenvalue problem is also obtained in a geometrical way.