Spherical and plane symmetric solutions in macroscopic gravity

The theory of macroscopic gravity provides a formalism to average the Einstein field equations from small scales to largest scales in space-time. It is well known that averaging is an operation that does not commute with calculating the Einstein tensor and this leads to a correction term in the field equations known as backreaction. In this work, we derive exact solutions to the macroscopic gravity field equations assuming that the averaged geometry is plane or spherically symmetric, and the source is taken as vacuum, dust, or perfect fluid. We then focus on the specific cases of spherical symmetry and derive solutions that are analogous to the Schwarzschild, Tolman VII, and Lemaitre-Tolman-Bondi solutions. The geodesic equations and curvature structure are contrasted with the general relativistic counterparts for the Schwarzschild and Lemaitre-Tolman-Bondi solutions.

Automated summary

The theory of macroscopic gravity provides a formalism to average the Einstein field equations from small scales to largest scales, and this work derives exact solutions under the assumptions of averaged geometry being plane or spherically symmetric, and the source being vacuum, dust, or perfect fluid. The specific cases of spherical symmetry are studied and solutions analogous to the Schwarzschild, Tolman VII, and Lemaitre-Tolman-Bondi solutions are derived, with comparison to the geodesic equations and curvature structure in general relativity.