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 back-reaction. 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 Lema\^{\i}tre-Tolman-Bondi solutions. The geodesic equations and curvature structure are contrasted with the general relativistic counterparts for the Schwarzschild and Lema\^{\i}tre-Tolman-Bondi solutions.