https://doi.org/10.1140/epjc/s10052-015-3390-0
Regular Article - Theoretical Physics
On the (
) threading of spacetime with respect to an arbitrary timelike vector field
1
Department of Mathematics, Kuwait University, P.O.Box 5969, 13060, Safat, Kuwait
2
Department of Mathematics, Technical University “Gh.Asachi”, B-dul Mangeron no. 67, 700050, Iasi, Romania
* e-mail: aurel.bejancu@ku.edu.kw
** e-mail: bejancu@sci.kuniv.edu.kw
Received:
28
November
2014
Accepted:
31
March
2015
Published online:
23
April
2015
We develop a new approach on the () threading of spacetime 
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 
, which behave as 3D geometric objects. We obtain new formulas for local components of the Ricci tensor field of with respect to the threading frame field, in terms of the Ricci tensor field of 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 6.3, which completes a well-known lemma used in the proof of the Penrose–Hawking singularity theorems. Finally, we apply the new 
formalism to the study of the dynamics of a Kerr–Newman black hole.
© SIF and Springer-Verlag Berlin Heidelberg, 2015
