The problem of backreaction in GR concerns the question how the field equations of the modern theory of gravity, i.e. general relativity (GR), behave under coarse-graining. These nonlinear partial differential equations, called Einstein's equations, relate the curvature of the spacetime metric tensor, composed of the second derivatives of the metric, with the stress-energy energy tensor representing the matter content of the spacetime. By coarse-graining I mean averaging out the complicated, fine details of the solution while keeping its large-scale structure. It is well known that linear field equations like the Maxwell's equations of
electromagnetism preserve their form exactly under coarse-graining: if we impose them on the local electric and magnetic fields as well as charges and currents then exactly the same Maxwell equations hold for their averages over finite-size domains. For nonlinear PDE's, this is not true in general, but in GR we may demand any additional terms appearing in the averaged equations to be of the form of a correction to the averaged stress-energy energy tensor, called the backreaction. Coarse-graining is a necessary step in application of GR to any astrophysical situation, because it is in general impossible to solve the Einstein's equations exactly for physical objects while taking into account every fine detail
of their matter distribution. However, the problem of applicability and the precise definition of coarse-graining in GR has been poorly researched in the literature.
Black hole lattices, i.e. regular arrangements of black holes, offer an excellent example to study the backreaction effects. Instead of a uniform matter distribution we have localized, Schwarzschild-like sources with vacuum metric outside, arranged in a lattice with a discrete symmetry group. I will discuss recent results about solutions of this kind, including my own results concerning the continuum limit in the black-hole lattices.