Editorial note to: On the Newtonian limit of Einstein’s theory of gravitation (by Jürgen Ehlers)

Abstract

We give an overview of literature related to Jürgen Ehlers' pioneering 1981 paper on Frame Theory-a theoretical framework for the unification of General Relativity and the equations of classical Newtonian gravitation. This unification encompasses the convergence of one-parametric families of four-dimensional solutions of Einstein's equations of General Relativity to a solution of equations of a Newtonian theory if the inverse of a causality constant goes to zero. As such the corresponding light cones open up and become space-like hypersurfaces of constant absolute time on which Newtonian solutions are found as a limit of the Einsteinian ones. It is explained what it means to not consider the 'standardtextbook' Newtonian Theory of gravitation as a complete theory unlike Einstein's theory of gravitation. In fact, Ehlers' Frame Theory brings to light a modern viewpoint in which the 'standard' equations of a self-gravitating Newtonian fluid are Maxwell-type equations. The consequences of Frame Theory are presented for Newtonian cosmological dust matter expressed via the spatially projected electric part of the Weyl tensor, and for the formulation of characteristic quasi-Newtonian initial data on the light cone of a Bondi-Sachs metric. 4 Rather, we may call it an "incomplete theory" for the reasons listed below. Jürgen thought of the hyperbolic character of a theory that completely determines the system from initial data only. We may add the property of Einstein's theory being background-free, in contrast to Newtonian gravitation. (As the constraint equations of Einstein's theory require the specification of boundary conditions (in addition to the initial conditions) the global topology is to be specified too, see the discussion in [30]. Einstein's theory in its classical formulation determines the global topology for all times by the topology of the initial Cauchy hypersurface; generally, however, one does not consider the possibility of a dynamical topology change.)

Einstein’s Universe: Cosmological structure formation in numerical relativity

“…Equations (14) contain both a growing and decaying mode for the density and velocity perturbations. We choose g = 0 to extract only the growing mode of the density perturbation, and our solutions become…”

“…We use (14) along with the Code for Anisotropies in the Microwave Background (CAMB) [53] to generate the matter power spectrum at z = 1100, with parameters consistent with Planck Collaboration et al [8] as input. Figure 1 shows the matter power spectrum from CAMB (grey curve), as a function of wavenumber |k| = k 2 x + k 2 y + k 2 z .…”

“…The smallest k component sampled represents the largest wavelength of perturbations -approximately the length of the box, L -and the largest k component sampled represents the smallest wavelength of perturbations -two grid points. To relate the initial density perturbation to the corresponding velocity and metric perturbations, we transform (14) into Fourier space. Initially, ξ = 1 which gives a density perturbation of the form…”

“…The assumptions underlying the standard cosmological model are based on observations that our Universe is, on average, homogeneous and isotropic. However, the averaged evolution of an inhomogeneous universe does not coincide with the evolution of a homogeneous universe [14,15]. Additional "backreaction" terms exist, but their significance has been debated [e.g.…”

A new method to probe the mass density and the cosmological constant using configuration entropy

“…Recently Das & Pandey (2019) used the evolution of the configuration entropy to distinguish various dynamical dark energy parameterizations. The importance and some interesting implications of inhomogeneities in cosmology has been highlighted earlier in Buchert & Ehlers (1997) and Buchert (2000).…”