Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: On the 1+3 Formalism in General Relativity   We present in this paper the formalism for the splitting of a
four-dimensional Lorentzian manifold by a set of time-like integral curves.
Introducing the geometrical tensors characterizing the local spatial frames
induced by the congruence (namely, the spatial metric tensor, the extrinsic
curvature tensor and the Riemann curvature tensor), we derive the Gauss,
Codazzi and Ricci equations, along with the evolution equation for the spatial
metric. In the present framework, the spatial frames do not form any
hypersurfaces as we allow the congruence to exhibit vorticity. The splitting
procedure is then applied to the Einstein field equation and it results in an
equivalent set of constraint and evolution equations. We discuss the resulting
systems and compare them with the ones obtained from the 3+1 formalism, where
the manifold is foliated by means of a family of three-dimensional space-like
surfaces.

19
The categories of this paper are: eess
30
The categories of this paper are: q-alg
23
The categories of this paper are: solv-int
35
The categories of this paper are: mtrl-th
31
The categories of this paper are: ao-sci
9
The categories of this paper are: nlin
6
The categories of this paper are: astro-ph
27
The categories of this paper are: comp-gas
17
The categories of this paper are: q-fin
26
The categories of this paper are: bayes-an
33
The categories of this paper are: plasm-ph
20
The categories of this paper are: acc-phys
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
8
The categories of this paper are: hep-ex
36
The categories of this paper are: cmp-lg