  Penrose's spinor calculus of 4-dimensional Lorentzian geometry is extended to
the case of 5-dimensional Lorentzian geometry. Such fruitful ideas in Penrose's
spinor calculus as the spin covariant derivative, the curvature spinors or the
definition of the spin coefficients on a spin frame can be carried over to the
spinor calculus in 5-dimensional Lorentzian geometry. The algebraic and
differential properties of the curvature spinors are studied in detail and as
an application we extend the well-known 4-dimensional Newman-Penrose formalism
to a 5-dimensional spacetime.
