On the ( $$1+3$$ 1 + 3 ) threading of spacetime with respect to an arbitrary timelike vector field

Abstract

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 ∇ , 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 ∇ 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 (1 + 3) formalism to the study of the dynamics of a Kerr-Newman black hole.

Equations of motion with respect to the $$(1+1+3)$$ ( 1 + 1 + 3 ) threading of a 5D universe

“…The present paper has its roots in [1,8], wherein we developed new approaches on the (1 + 1 + 3) threading of a 5D universe and on the (1 + 3) threading of a spacetime, respectively. The main geometric objects used in the paper are: the adapted frame and coframe fields, the kinematic tensor fields, and the Riemannian spatial connection.…”

Kinematic quantities and Raychaudhuri equations in a 5D universe

“…This can be considered as an extension of the (1 + 3) threading of spacetime with respect to an arbitrary timelike vector field that has been developed in [18].The main tools in our approach are the spatial tensor fields and the Riemannian connection defined on the spatial distribution. It is worth mentioning that all the kinematic quantities (acceleration, expansion, shear and vorticity) are defined as spatial tensor fields, and therefore they should be considered as 3D geometric objects in a 5D universe.…”

On the $$(1+3)$$ ( 1 + 3 ) Threading of Spacetime

“…[1,2]). In this general setting we obtained in a covariant form, the fully general 3D equations of motion and a 3D identity satisfied by the geodesics of a spacetime.…”

“…The study is based on both the Riemannian spatial connection and the spatial tensor fields defined in [2]. It is worth mentioning that each group of the EFE given by (7.3) is invariant with respect to the transformations of coordinates on the spacetime.…”

“…2 we introduce the kinematic quantities determined by a non-normalized timelike vector field ξ . Note that in [2] we put on Φ given by (2.2a) the condition that it is independent of time. This condition is satisfied by all stationary black holes (cf.…”