  For a compact spacelike constant mean curvature surface with nonempty
boundary in the three-dimensional Lorentz-Minkowski space, we introduce a
rotation index of the lines of curvature at the boundary umbilic point, which
was developed by Choe \cite{Choe}. Using the concept of the rotation index at
the interior and boundary umbilic points and applying the Poincar\'{e}-Hopf
index formula, we prove that a compact immersed spacelike disk type capillary
surface with less than $4$ vertices in a domain of $\Bbb L^3$ bounded by
(spacelike or timelike) totally umbilic surfaces is part of a (spacelike) plane
or a hyperbolic plane. Moreover we prove that the only immersed spacelike disk
type capillary surface inside de Sitter surface in $\Bbb L^3$ is part of
(spacelike) plane or a hyperbolic plane.
