  We develop a new approach on the (1+3) threading of spacetime $(M, g)$ with
respect to a congruence of curves defined by an arbitrary timelike vector
field. The study is based on spatial tensor fields and on the Riemannian
spatial connection $\nabla^{\star}$, which behave as $3D$ geometric objects. We
obtain new formulas for local components of the Ricci tensor field of $(M, g)$
with respect to the threading frame field, in terms of the Ricci tensor field
of $\nabla^{\star}$ and of kinematic quantities. Also, new expressions for time
covariant derivatives of kinematic quantities are stated. In particular, a new
form of Raychaudhuri's equation enables us to prove Lemma 6.2, which completes
a well known lemma used in the proof of Penrose-Hawking singularity
theorems.Finally, we apply the new $(1+3)$ formalism to the study of the
dynamics of a Kerr-Newman black hole.
