Slides of 5 lectures at XV Brazilian School on Cosmology and ...

II. Averaging and backreactionIn general 〈G µ ν(g αβ )〉 ≠ G µ ν(〈g αβ 〉)〈G µ ν〉 need not be Einstein tensor for an exact geometry(1)〈G µ ν〉 = 〈g µλ R λν 〉 − 1 2 δµ ν〈g λρ R λρ 〉 = 8πGc 4 〈T µ ν〉E.g., Zalaletdinov (1992,1993) works with the averageinverse metric 〈g µν 〉 and the average Ricci tensor 〈R µν 〉,and writes(2)〈g µλ 〉〈R λν 〉 − 1 2 δµ ν〈g λρ 〉〈R λρ 〉 + C µ ν = 8πGc 4 〈T µ ν〉,Correl<strong>at</strong>ion functions C µ ν defined by difference <strong>of</strong> thel.h.s. <strong>of</strong> (1) and (2): these are backreaction terms15th <strong>Brazilian</strong> <strong>School</strong> on Cosmology and Gravit<strong>at</strong>ion, August 2012 – p.24/143