Averaging spherically symmetric spacetimes in general relativity

Abstract

We discuss the averaging problem in general relativity, using the form of the macroscopic gravity equations in the case of spherical symmetry in volume preserving coordinates. In particular, we calculate the form of the correlation tensor under some reasonable assumptions on the form for the inhomogeneous gravitational field and matter distribution. On cosmological scales, the correlation tensor in a Friedmann-Lemaître-Robertson-Walker (FLRW) background is found to be of the form of a spatial curvature. On astrophysical scales the correlation tensor can be interpreted as the sum of a spatial curvature and an anisotropic fluid. We briefly discuss the physical implications of these results.[PACES: 98.80.Jk,04.50.+h] The gravitational field equations on large scales are obtained by averaging the Einstein equations of general relativity (GR). The Universe is not isotropic or spatially homogeneous on local scales. An averaging of inhomogeneous spacetimes on large scales can lead to important effects. For example, on cosmological scales the dynamical behavior can differ from that in the spatially homogeneous and isotropic Friedmann-Lemaître-Robertson-Walker (FLRW) model [1]; in particular, the expansion rate may be significantly affected. Consequently, a solution of the averaging problem is of considerable importance for the correct interpretation of cosmological data. It is also of importance for physical phenomena on astrophysical (galactic) scales.There are a number of theoretical approaches to the averaging problem [2][3][4]. In the approach of Buchert [4] a 3+1 cosmological space-time splitting is employed and only scalar quantities are averaged. The perturbative approach [2] involves averaging the perturbed Einstein equations; however, a perturbation analysis cannot provide any information about an averaged geometry. On the other hand, the macroscopic gravity (MG) approach to the averaging problem in GR [3] gives a prescription for the correlation functions which emerge in an averaging of field equations. The MG approach is a fully covariant, gauge independent and exact method. We shall adopt the MG averaging approach.Averaging of the structure equations for the geometry of GR then leads to the structure equations for the averaged 1 (macroscopic) geometry and the definitions and the properties of the correlation tensor. The averaged Einstein equations can always be written in the form of the Einstein equations for the macroscopic metric tensor when the correlation terms are moved to the right-hand side of the averaged Einstein equations [3].Spherical symmetry is of particular physical interest, and it is especially important to study the averaging problem within the class of spherically symmetric cosmological models. In [5] the microscopic field equations were taken and the averaging procedure was effected to determine the precise form of the correlation tensor in this case. In volume preserving coordinates (VPC), the spherically symmetric line element is given bywhere the functions A and B depend on t and r....

Averaged Lemaître–Tolman–Bondi dynamics

“…The emergence of a spatial curvature term as the result of the averaging procedure is in accord with a corresponding result from Macroscopic Gravity [8][9][10]. For an inhomogeneous bang time the kinematic backreaction is generally different from zero.…”

“…Various combinations of the parameters t B0 , r c and r E were checked, but even though one of them reproduces roughly the same value as the H D (z D ) analysis, the physical significance of this term remains doubtful. curvature term is also known from Macroscopic Gravity [8][9][10]. At first order in E one has R D ∝ a −2 D for the averaged curvature which is similar to the FLRW case.…”

Explicit cosmological coarse graining via spatial averaging

“…Some examples of averaging techniques employed in General Relativity (and in Cosmology in particular) can be found in Ref. [50][51][52][53][54][55][56][57][58]. Here we restrict ourselves to the formalism developed by Buchert [18,19] for volume averages of scalar quantities in a given choice of time-slicing.…”

Expansion and growth of structure observables in a macroscopic gravity averaged universe

“…The exact solutions for Macroscopic Gravity are known only for the flat homogeneous and isotropic and the static spherically symmetric cases [85][86][87]122]. The non static spherically symmetric solution has been found [123] using "volume preserving coordinates" and an approximation rather than by solving equations (11)- (19) directly. In this section we will consider the solution for a macroscopically homogeneous, anisotropic and spatially flat metric (i.e a macroscopically Bianchi type I metric) of the form…”