  I present a covariant approach to developing 1+3 formalism without an
introduction of any basis or coordinates. In the formalism, a spacetime which
has a timelike congruence is assumed. Then, tensors are split into temporal and
spatial parts according to the tangent direction to the congruence. I make use
of the natural derivatives to define the kinematical quantities and to
investigate their properties. They are utilized in the splitting of covariant
derivatives. In this way, the Riemann curvature is split into the temporal and
spatial part, i.e. Gauss, Codazzi, and Ricci relation. Finally, the splitting
of the Einstein equation is achieved by contraction. Choosing congruence as
normal to a spacelike hypersurface, the formalism reduces to 3+1 formalism.
This approach deepens our understanding of 3+1 formalism. All these processes
are performed in a covariant manner without the complexities caused by the
introduction of a coordinate system or basis.
