The generic model of General Relativity

TL;DR

This paper introduces a comprehensive spacetime model in General Relativity that incorporates geometric assumptions, observer-based decompositions, and symmetry constraints to unify and extend gravitational modeling.

Contribution

It develops a generic, flexible framework using 1+3 and 1+1+2 decompositions, including a double congruence, to describe gravitational and matter dynamics in spacetime.

Findings

Unified framework for gravitational models

Decomposition of spacetime variables into kinematic and dynamic sets

Extension to double congruence with 1+1+2 decomposition

Abstract

We develop a generic spacetime model in General Relativity which can be used to build any gravitational model within General Relativity. The generic model uses two types of assumptions: (a) Geometric assumptions additional to the inherent geometric identities of the Riemannian geometry of spacetime and (b) Assumptions defining a class of observers by means of their 4-velocity $u^{a}$ which is a unit timelike vector field. The geometric assumptions as a rule concern symmetry assumptions (the so called collineations). The latter introduces the 1+3 decomposition of tensor fields in spacetime. The 1+3 decomposition results in two major results. The 1+3 decomposition of $u_{a;b}$ defines the kinematic variables of the model (expansion, rotation, shear and 4-acceleration) and defines the kinematics of the gravitational model. The 1+3 decomposition of the energy momentum tensor representing all gravitating matter introduces the dynamic variables of the model (energy density, the isotropic pressure, the momentum transfer or heat flux vector and the traceless tensor of the anisotropic pressure) as measured by the defined observers and define the dynamics of he model. The symmetries assumed by the model act as constraints on both the kinematical and the dynamical variables of the model. As a second further development of the generic model we assume that in addition to the 4-velocity of the observers $u_{a}$ there exists a second universal vector field $n_{a}$ in spacetime so that one has a so called double congruence $(u_{a},n_{a})$ which can be used to define the 1+1+2 decomposition of tensor fields. The 1+1+2 decomposition leads to an extended kinematics concerning both fields building the double congruence and to a finer dynamics involving more physical variables.